1 p .-r ^ '*,.^CMJr 4? '% BARTLETT'S MECHANICS. ELEMENTS NATURAL PHILOSOPHY. BY W. H. C. BARTLETT, LL.D., . >*jaaaBji PEOFESSOE OF NATUEAL AND EXPERIMENTAL PHILOSOPHT IN THE UNITED STATES MILITAEY ACADEMY AT WEST POINT. SECTION I. "" MECHANICS. NEW YORK: PUBLISHED BY A. S. BARNES AND COMPANY, NO. 51 JOHN-STREET. CINCINNATI: H. W. DERBY AND COMPANY. 1850 QC I Entered, according to Act of Congress, in the year One Thousand Eight Hundred and Fifty, By W. H. C. BARTLETT, In the Clerk's Office of the District Court of the United States for the Southern District of New York. STEREOTYPED BY BILLIN & BROTHERS, No. 10 North William Strbbt, Niw York. P. C. GUTIERREZ, Printer, No. 51 John Street, oor. ov Dutch PREFACE, The present volume is tbe first of tbree in wbicb its autbor desires ffer, to academies d colleges, a course of Natural Pbilosopby, /- ding Astronomy It embraces tbe subject of Mecbanics—tbe ground-work of tbe wbole. It is intended to be complete witbin ' itself, andyto bave no necessary dependence, for tbe fall compre- hension of- its contents, upon tbose wbicb are to follow. In its preparation, constant reference was made to tbe admirable labors of M. Poncellet, and mucb valuable assistance was derived from tbe work of M. Peschel. Large type, marginal notes, tables of reference, and numerous diagrams, often repeated, bave swollen tbe volume beyond tbe bmits originally intended; but wbatever of inconvenience may tbence arise, wjU, it,is boped, be more tban compensated by tbe facilities wbicb ' these sources of increased size cannot fail to bring to tbe aid botb of tbe teacber and student. CONTENTS. INTRODUCTION. Page Objects of Physical Science............................................. 9 Physics of Ponderable Bodies........................................... 12 Primary Properties of Bodies........................................... 13 Secondary Properties.................................................. 14 Force................................................................ 19 PART FIRST. MECHANICS OF SOLIDS. I. Space, Time, Motion, and Force......................................... 29 n. Action of Forces, Equilibrium, Work..................................... 42 m. Varied Motion........................................................ 6<1 rv. Forces whose Directions meet in a Point.................................. 102 V. Forces whose Directions are Parallel..................................... 123 6 CONTENTS. Page VI. Centre of Gravity of Bodies............................................ 134 VII. Motion of Translation of a Body or System of Bodies...................... 150 vin. Equilibrium of a System of Heavy Bodies................................. 162 IX. Equilibrium of Several Forces, Virtual Velocities, and Motion of a Solid Body.. 167 X. Motion and Equilibrium of a Body about an Axis.......................... 180 XI. Central Forces........................................................ 200 XII. Motion of the Heavenly Bodies......................................... 230 XIII. Pendulum............................................................ 241 Balistic Pendulum..................................................... 266 xrv. Funicular Machine.................................................... 271 XV. Action of Bodies resting upon each other, and upon Inclined PlaneB........... 299 XVI. Friction and Adhesion................................................. 313 Tables of Friction without Unguents..................................... 319 Tables of Friction of Unctuous Surfaces.................................. 322 Tables of Friction with Unguents........................................ 324 Table of Friction on Trunnions in their Boxes............................. 365 CONTENTS. 7 XVII. Page The Wedge.......................................................... 345 XVIII. Stiffness of Cordage................................................... 379 Table of Rigidity of Cordage........................................... 382 XIX. Wheel and Pulley..................................................... 387 Wheel and Axle...................................................... 406 XX. Screw............................................................... 428 XXI. Lever............................................................... 440 xxn. Atwood's Machine..................................................... 444 xxni. Impact of Bodies...................................................... 458 PART SECOND. MECHANICS OF FLUIDS. I. Introductory Remarks................................................. 476 n. Mechanical Properties of Fluids......................................... 480 III. Work of the Power and of the Resistance................................. 488 IV. Pressure of Heavy Fluids............................................... 493 Table of Tenacities.................................................... 508 8 CONTENTS. V. Page Equilibrium of Floating Bodies.......................................... 510 VI. Specific Gravity....................................................... 517 Table of the Densities of Water at different Temperatures................... 522 Table of Specific Gravities.............................................. 531 VII. Compressible Fluids................................................... 536 VIII. Air-Pump.................................................'........... 540 IX. Weight and Pressure of the Atmosphere.................................. 550 X. Mariotte's Law........................................................ 556 XI. Law of the Pressure, Density, and Temperature............................ 559 XII. Barometer........................................................... 556 Table for Finding Height by the Barometer............................... 578 XIII. Water-Pumps.................,...................................... 582 XIV. Siphon.............................................................. 595 XV. Motion of Fluids...................................................... 598 XVI. Motion of Gases and Vapors............................................ 606 Table of Aperture Coefficients........................................... 618 xvn. Discharge of Fluid through Pipes........................................ 622 ELEMENTS NATURAL PHILOSOPHY. INTRODUCTION. The term nature is employed to signify the assemblage Nature. of all tbe bodies of the universe; it includes whatever exists and is the subject of change. Of the existence of these bodies we are rendered conscious by the impressions Bodies. they make on our senses. Their condition is subject to a variety of changes, whence we infer that external causes are in operation to produce them; and to investigate Physical science. nature with reference to these changes and their causes, is the object of Physical Science. All bodies may be distributed into three classes, viz.: classification of unorganized or inanimate, organized or animated, and the heavenly bodies or primary organizations. The unorganized or inanimate bodies, as minerals, inanimate water, air, form the lowest class, and are, so to speak, J,,^ p°eriod) the substratum for the others. These bodies are acted no life. on solely by causes external to themselves; they have no definite or periodical duration; nothing that can properly be termed life. The organized or animated bodies, are more or less Animated bodies, perfect individuals, possessing organs adapted to the per- or formance of certain appropriate functions. In consequence of an innate principle peculiar to them, know as vitality, bodies of this class are constantly appropriating to them- selves unorganized matter, changing its properties, and 10 NATURAL PHILOSOPHY. deriving, by means of this process, an increase of bulk. Reproduction, They also possess tbe faculty of reproduction. They 7m*^Td retain only for a limited time the vital principle, and, when life is extinct, they sink into the class of inanimate Animal and bodies. The animal and vegetable kingdoms include all vegetable fch ecies 0f t^[s c]ass 0n our earth. kingdoms. * . ■■ celestial bodies; The celestial bodies, as tbe fixed stars, the sun, the comets, planets and their secondaries, are the gigantic individuals of tbe universe, endowed with an organization organs- on the. grandest scale. Their constituent parts may be continents, ocean, pared to the organs possessed by bodies of the second atmosphere. r ° A class; those of our earth are its continents, its ocean, its atmosphere, which are constantly exerting a vigorous action on each other, and bringing about changes the most important. Earth existed The earth supports and nourishes both the vegetable long before plants and animai WOrld, and the researches of Geology have and animals. ' demonstrated, that there was once a time when neither plants nor animals existed on its surface, and that prior to the creation of either of these orders, great changes must have taken place in its constitution. As the earth existed thus anterior to the organized beings upon it, we may Heavenly bodies infer that the other heavenly bodies, in like manner, were a^majs3rt°f called into bein§ before any of tne organized bodies which probably exist upon them. Reasoning, then, by analogy from our earth, we may venture to regard the heavenly bodies as the primary organized forms, on whose surface both animals and vegetables find a place and support. Natural Natural Philosophy, or Physics, treats of tbe general externaTchan<*e9 properties of unorganized bodies, of the influences which act upon them, the laws they obey, and of the external changes which these bodies undergo without affecting their internal constitution. chemistry; Chemistry, on the contrary, treats of the individual properties of bodies, by which, as regards their constitu- INTRODUCTION. 11 tion, they may be distinguished one from another ; it also internal changes. investigates the transformations which take place in the interior of a body—transformations by which the sub- stance of the body is altered and remodelled; and lastly, it detects and classifies tbe laws by which chemical changes are regulated. Natural History, is that branch of physical science Natm-ai mstory- which treats of organized bodies; it comprises three ^e^try, divisions, the one mechanical—the anatomy and dissec- physiology. tion of plants and animals; the second, chemical—animal and vegetable chemistry; and the third, explanatory— physiology. Astronomy teaches the knowledge of the celestial Astronomy, bodies. It is divided into Spherical and Physical astron- p^™^ an omy. The former treats of the appearances, magnitudes, distances, arrangements, and motions of the heavenly bodies ; the latter, of their constitution and physical con- dition, their mutual influences and actions on each other, and generally, seeks to explain the causes of the celestial phenomena. Again, one most important use of natural science, is of the application of its laws either to technical purposes— Application rr oil physical laws. mechanics, technical chemistry, pharmacy, &c.; to the phe- nomena of the heavenly bodies—physical astronomy ; or to the various objects which present themselves to our notice at or near the surface of the earth—physical geography, meteorology—and we may add geology also, a science which has for its object to unfold the history of our planet from its formation to the present time. Natural philosophy is a science of observation and ex- Natural periment, for by these two modes we deduce the varied gCien^° of' information we have acquired about bodies; by theobservationand • . .. experiment. former we notice any changes that transpire in the condi- tion or relations of any body as they spontaneously arise 12 NATURAL PHILOSOPHY. Apparatus; experimental physics. Observation, experiment. Laws of nature. Hypotheses and probability of their truth. / I without interference on our part; whereas, in the per- formance of an experiment, we purposely alter the natural arrangement of things to bring about some particular con- dition that we desire. To accomplish this, we make use of appliances called philosophical or chemical apparatus, the proper use and application of which, it is the office of Ex- perimental Physics to teach. If we notice that in winter water becomes converted into ice, we are said to make an observation: if, by means of freezing mixtures or evaporation, we cause water to freeze, we are then said to perform an experiment. These experiments are next subjected to calculation, by which are deduced what are sometimes called the laws of nature, or the rules that like causes will invariably pro- duce like results. To express these laws with the greatest possible brevity mathematical symbols are used. When it is not practicable to represent them with mathematical precision, we must be contented with inferences and assumptions based on analogies, or with probable ex- planations or hypotheses. A hypothesis gains in probability the more nearly it accords with the ordinary course of nature, the more numerous the experiments on which it is founded, and the more simple the explanation it offers of the phenomena for which it is intended to account. PHYSICS OF PONDERABLE BODIES. physical § 1.—The physical properties of bodies are those ex- ^iTf1635 the ternal signs by which their existence is made evident to sense s# *^ our minds; the senses constitute the medium through which this knowledge is communicated. All our senses, however, are not equally made use of for this purpose ; we are generally guided in our decisions by the evidence of sight and touch. Still sight alone is All the senses not equally employed. INTRODUCTION. 13 frequently incompetent, as there are bodies which cannot be perceived by that sense, as, for example, all colorless gases; again, some of the objects of sight are not sub- stantial, as, the shadow, the image in a mirror, spectra formed by the refraction of the rays of light, &c. Touch, Touch. on the contrary, decides indubitably as to the existence of any body. The properties of bodies may be divided into primary Primary and or principal, and secondary or accessory. The former, are ^ert^ f such as we find common to all bodies, and without which bodies. we cannot conceive of their existing; the latter, are not absolutely necessary to our conception of a body's ex- istence, but become known to us by investigation and experience. PRIMARY PROPERTIES. § 2.—The primary properties of all bodies are extension and impenetrability. Extension is that property in consequence of which Extension; every body occupies a certain limited space. It is the ^d th'ictoessh' condition of the mathematical idea of a body; by it, the volume or size of the occupied space, as well as its boun- dary, or figure, is determined. The extension of bodies is expressed by three dimensions, length, breadth, and thick-. ness. The computations from these data, follow geometri- cal rules. Impenetrability is evinced in the fact, that one body impenetrability. cannot enter into the space occupied by another, without previously thrusting the latter from its place. A body then, is whatever occupies space, and possesses extension and impenetrability. One might be led to im- Body defined. agine that the property of impenetrability belonged only to solids, since we see them penetrating both air and Air and water water; but on closer observation it will be apparent that iraPenetiable- this property is common to all bodies of whatever nature. 14 NATURAL PHILOSOPHY. If a hollow cylinder into which a piston fits accurately, be filled with water, the piston cannot be thrust into the Experiment. water, thus showing it to be impenetrable. Invert a glass tumbler in any liquid, the air, unable to escape, will pre- vent the liquid from occupying its place, thus proving the impenetrability of air. Tbe diving-bell affords a familiar illustration of this property. The difficulty of pouring liquor into a vessel having only one small hole, arises from the impenetrability of the air, as the liquid can run into the vessel only as the air makes its escape. The following experiment will illus- trate this fact: In one mouth of a two- necked bottle insert a funnel a, and in the other a siphon b, the longer leg of which is im- mersed in a glass of water. Experiment. Now let water be poured into the funnel a, and it will be seen that in proportion as this water descends into the vessel F, the air makes its escape through the tube b, as is proved by the ascent of the bubbles in the water in the tumbler. Fig. l. SECONDARY PROPERTIES. Secondary properties. The secondary properties of bodies are compressibil expansibility, porosity, divisibility, and elasticity. Compressibility, expansibility. § 3.—Compressibility is that property of bodies by virtue of which they may be made to occupy a smaller space; and expansibility is that in consequence of which they may be made to fill a larger, without in either case altering the quantity of matter they contain. INTRODUCTION. 15 Both changes are produced in all bodies, as we shall change of presently see, by change of temperature; many bodies temPerature' may also be reduced in bulk by pressure, percussion, &$. percussion. § 4.—Since all bodies admit of compression and ex- pansion, it follows of necessity, that there must be interstices between their minutest particles; and that property of a body by which its constituent elements do not completely fill the space within its exterior boun- dary, but leaves holes or pores between them, is called Porosity. porosity. The pores of one body are often filled with Pores ailed with some other body, and the pores of this with a third, as in other bodie8, the case of a sponge containing water, and the water in its turn, containing air, and so on till we come to the most subtle of substances, ether, which is supposed to pervade Ether pervades all bodies and all space. a11 bodies andaU r space. In many cases the pores are visible to the naked eye ; visible and in others they are only seen by the aid of the microscope, in and when so minute as to elude the power of this instru- ment, their existence may be inferred from experiment. Sponge, cork, wood, bread, &c, are bodies whose pores are noticed by the naked eye. The human skin appears full of them, when viewed with the magnifying glass ; the porosity of water is shown by the ascent of air bubbles when the temperature is raised. § 5.—The divisibility of bodies is that property in Divisibility. consequence of which, by various mechanical means, such as beating, pounding, grinding, &c, we can reduce them to particles homogeneous to each other, and to the entire mass; and these again to smaller, and so on. By the aid of mathematical processes, the mind may infinite be led to admit the infinite divisibility of bodies, though dmsIblhty; their practical division, by mechanical means, is subject practical to limitation. Many examples, however, prove that ithmitatlon- may be carried to an incredible extent. We are fur- smaiinessofsome nished with numerous instances among natural objects, na m jec 8" 16 NATURAL PHILOSOPHY. Mechanical subdivisions in the arts Divisibility of gold. Divisibility of dyes. whose existence can only be detected by means of the most acute senses, assisted by the most powerful arti- ficial aids; the size of such objects can only be calculated approximately. Mechanical subdivisions for purposes connected with the arts are exemplified in the grinding of corn, the pulverizing of sulphur, charcoal, and saltpetre, for the manufacture of gunpowder; and Homoeopathy affords a remarkable instance of the extended application of this property of bodies. Some metals, particularly gold and silver, are suscep- tible of a very great divisibility. In the common gold lace, the silver thread of which it is composed is covered with gold so attenuated, that the quantity contained in a foot of the thread weighs less than g^Vo of a grain. An inch of such thread will therefore contain -7-2o0o of a grain of gold; and if the inch be divided into 100 equal parts, each of which would be distinctly visible to the eye, the quantity of the precious metal in each of such pieces would be -7200000 of a grain. One of these particles ex- amined through a miscroscope of 500 times magnifying power will appear 500 times as long, and the gold covering it will be visible, having been divided into 3,600,000,000 parts, each of which exhibits all the characteristics of this metal, its color, density, &c. Dyes are likewise susceptible of an incredible divisi- bility. With 1 grain of blue carmine, 10 lbs. of water may be tinged blue. These 10 lbs. of water contain about 617,000 drops. Supposing now, that 100 particles of car- mine are required in each drop to produce a uniform tint, it follows that this one grain of carmine has been sub- divided 62 millions of times. According to Biot, the thread by which a spider lets down is composed of more tban 5000 single In the spider's thread, thread of flgygelf the silkworm. threads. The single threads of the silkworm are also of In blood. an extreme fineness. Our blood which appears bke a uniform red mass, con- INTRODUCTION. 17 sists of small red globules swimming in a transparent fluid called serum. Tbe diameter of one of these globules does not exceed tbe 4000th part of an inch : whence it follows that one drop of blood, such as would hang from the point of a needle, contains at least one million of these globules. But more surprising tban all, is the microcosm of organ- in the infusoria. ized nature in the Infusoria, for more exact acquaintance with which we are indebted to the unwearied researches of Ehrenberg. Of these creatures, which for the most part we can see only by the aid of the microscope, there exist many species so small that millions piled on each other would not equal a single grain of sand, and thou- sands might swim at once through the eye of the finest needle. The coats-of-mail and shells of these animalcules exist in such prodigious quantities on our earth that, according to Ehrenberg's investigations, pretty extensive Ehrenberg's strata of rocks, as, for instance, tbe smooth slate near Bilin, investlsatI0ns- in Bohemia, consist almost entirely of them. By micro- scopic measurements 1 cubic line of this slate contains Microscopic about 23 millions, and 1 cubic inch about 41,000 millions measuiement' of these animals. As a cubic inch of this slate weighs 220 grains, 187 millions of these shells must go to a grain, weight. each of which would consequently weigh about the xhr millionth part of a grain. Conceive further that each of these animalcules, as microscopic investigations have proved, has his limbs, entrails, &c, tbe possibility vanishes of our forming the most remote conception of the dimen- sions of these organic forms. In cases where our finest instruments are unaoie to Divisibility render us the least aid in estimating the minuteness of dete°tedby ° smelling. bodies, or the degree of subdivision attained; in other words, when bodies evade tbe perception of our sight and touch, our olfactory nerves frequently detect the presence of matter in the atmosphere, of which no chemical analysis could afford us the slightest intimation. Thus for instance, a single grain of musk diffuses in a instance of musk. 18 NATURAL PHILOSOPHY. large and airy room a powerful scent that frequently lasts for years; and papers laid near musk will make a voyage to the East Indies and back without losing the smell. Imagine now, how many particles of musk must radiate from such a body every second, in order to render the scent perceptible in all directions, and you will be aston- ished at their number and minuteness. oil of lavender. In like manner a single drop of oil of lavender evapo- rated in a spoon over a spirit-lamp, fills a large room with its fragrance for a length of time. § 6.—Elasticity is the name given to that property of bodies, by virtue of which they resume of themselves their figure and dimensions when these have been changed or altered by any extraneous cause. Different bodies possess this property in very different degrees, and retain it with very unequal tenacity. The measure of a body's elasticity, is the ratio obtained by dividing the capacity of resti- tution inherent in the body, by the capacity of the cause producing the change, both being supposed measurable. Thus, if R denote the capacity of restitution, F that of the extraneous cause, and e the elasticity, then will R e=F- When F and R are equal, the body is said to be perfectly elastic; when R is zero, the body is said to be non-elastic. These limits embrace all bodies in nature, there being none known to us which reach either extreme. The following are a few out of a large number of Examples of highly elastic bodies; viz., glass, tempered steel, ivory, elastic bodies. w^alebone) &c> Let an ivory ball fall on a marble slab smeared with Experiment with some coloring matter. The point struck by the ball ivory. shows a round speck which will have imprinted itself on the surface of the ivory without its spherical form being at all impaired Elasticity, its measure. INTRODUCTION. 19 Fluids under peculiar circumstances exhibit considera- ble elasticity; this is particularly the case with melted Elasticity of some .... melted metals. metals, more evidently sometimes than in their solid state. The following experiment illustrates this fact' with regard to antimony and bismuth. Place a bttle antimony and bismuth on a piece of Meited bismuth charcoal, so that the mass when melted shall be about an ^inD the size of a peppercorn; raise it by means of a blowpipe to a white heat, and then turn the ball on a sheet of paper so folded as to have a raised edge all round. As soon as the liquid metal falls, it divides itself into many minute globules, which hop about upon the paper and continue visible for some time, as they cool but slowly; the points at which they strike the paper, and their course upon it, will be marked by black dots and bnes. The recoil of cannon-balls is owing to the elasticity Recoil of of the iron and that of the bodies struck by them. FORCE. § 7.—Whatever tends to change the actual state of a body, in respect to rest or motion, is called a force. If a Forces. body, for instance, be at rest, tbe influence which changes or tends to change this state to that of motion is called force. Again, if a body be already in motion, any cause which urges it to move faster or slower, is called force. Of the actual nature of forces we are ignorant; we ignorant of their know of their existence only by the effects they produce, ™^;b^t^" and with these we become acquainted solely through the effects on bodies. medium of the senses. Hence, while their operations are going on, they appear to us always in connection with some body which, in some way or other, affects our senses. 20 NATURAL PHILOSOPHY. Universal forces, attractions, and repulsions. § 8.—We shall find, though not always upon super- ficial inspection, that the approaching and receding of bodies or of their component parts, when this takes place apparently of their own accord, are but the results pro- duced by the various forces that come under our notice. In other words, that the universally operating forces are those of attraction and of repulsion. Atomical action; attraction of gravitation. § 9.—Experience proves that these universal forces are at work in two essentially different modes. They are operating either in the interior of a body, amidst the elements which compose it, or they extend their influence through a wide range, and act upon bodies in the aggre- gate ; the former distinguished as Atomical and Molecular action, the latter as the Attraction of gravitation. Force of cohesion and of dissolution. § 10.—Molecular forces and the force of gravitation, often co-exist, and qualify each other's action, giving rise to those attractions and repulsions of bodies exhibited at their surfaces when brought into sensible contact. This resultant action is called the force of cohesion or of dissolution, according as it tends to unite different bodies, or the elements of the same body, more closely, or to separate them more widely. Inertia, Known by experience; passive in character. § 11.—Inertia is that principle by which a body resists all change of its condition, in respect to rest or motion. If a body be at rest, it will, in the act of yielding its con- dition of rest, while under the action of any force, oppose a resistance; so also, if a body be in motion, and be urged to move faster or slower, it will, during the act of changing, oppose an equal resistance for every equal amount of change. We derive our knowledge of this principle solely from experience; it is found to be com- mon to all bodies; it is in its nature conservative, though passive in character, being only exerted to preserve the rest or particular motion which a body has, by resisting INTRODUCTION. . 21 all variation in these particulars. Whenever any force acts upon a body, the inertia of the latter reacts, and this action and reaction are, as we shall see in the proper Action equal to place, equal and directly opposed to each other. reaction. §12.—Molecular action chiefly determines the forms Forms of bodies of bodies. All bodies are regarded as collections or detern"ned by o molecular action. aggregates of minute elements, called atoms, and are formed by the attractive and repulsive forces acting upon them at immeasurably small distances. Several hypotheses have been proposed to explain tbe constitution of constitution of a body, and the mode of its formation. Boscov'ich, The most remarkable of these was by Boscovich, about tbe middle of the last century. Its great fertility in the explanations it affords of the properties of what is called tangible matter, and its harmony with the laws of motion, entitle it to a much larger space than can be found for it in a work like this. Enough may be stated, however, to enable the attentive reader to seize its leading features, and to appreciate its competency to explain the phenomena of nature. 1. All matter consists of indivisible and inextended First postulate. atoms. 2. These atoms are endowed with attractive and repul- second postuiat sive forces, varying both in intensity and direction by a change of distance, so that at one distance two atoms attract each other, and at another distance they repel. 3. This law of variation is the same in all atoms. It Third postulate. is, therefore, mutual; for the distance of atom a from atom b, being the same with that of b from a, if a attract b, b must attract a with precisely the same force. 4. At all considerable or sensible distances, these mu- Fourth postulate. tual forces are attractive and sensibly proportional to tbe square of the distance inversely. It is the attraction called gravitation. 5. In the small and insensible distances in which sensi- ble contact is observed, and which do not exceed tbe Firth postulate. 22 . NATURAL PHILOSOPHY. 1000th or 1500th part of an inch, there are many alterna- tions of attraction and repulsion, according as the distance of the atoms is changed. Consequently, there are many situations within this narrow limit, in which two atoms neither attract nor repel. sixth postulate. 6. The force which is exerted between two atoms when their distance is diminished without end, and is just vanishing, is an insuperable repulsion, so that no force whatever can press two atoms into mathematical contact. Such, according to Boscovich, is the constitution of a material atom and the whole of its constitution, and the immediate efficient cause of all its properties. Molecule, Two or more atoms may be so situated, in respect particle, body. ^ p0g^on an(j Stance, as to constitute a molecule. Two or more molecules may constitute a particle. The par- ticles constitute a body. Add inertia. Now, if to these centres, or loci of the qualities of what is termed matter, we attribute the property called inertia, we have all the conditions requisite to explain, or arrange in the order of antecedent and consequent, the various operations of the physical world. Exponential Boscovich represents his law of atomical action by curve. what may be called an exponential curve. Let the dis Fig. 2. tance of two atoms be estimated on the line CA C, A being the situation of one of them while the other is placed anywhere on this line. When placed at i, for example, we may suppose that it is attracted by A, with INTRODUCTION. 23 a certain intensity. We can represent this intensity by the length of the line il, perpendicular to A C, and can express the direction of the force, namely, from i to A, Attractive because it is attractive, by placing il above the axis A C. ordinate3abONe- Should tbe atom be at m, and be repelled by A, we can express the intensity of repulsion by m n, and its direc- Repulsive ,.P , i/^n i • -ii ii • ordinates below. tion from m towards b by placing m n below tbe axis. This may be supposed for every point on tbe axis, and a curve drawn through the extremities of all the perpen- dicular ordinates. This will be the exponential curve or scale of force. As there are supposed a great many alternations of curve on opposite attractions and repulsions, the curve must consist of many m branches lying on opposite sides of the axis, and must therefore cross it at •€', D', C", D", &c, and at 67. All these are supposed to be contained within a very small fraction of an inch. Beyond this distance, which terminates at 67, the force Force of is always attractive, and is called the force of gravitation, &ravitatlon- the maximum intensity of which occurs at g, and is expressed by the length of the ordinate G'g. Further on, the ordinates are sensibly proportional to the square of their distances from A, inversely. The branch 67' 67" has the line A C, therefore, for its asymptote. Within tbe limit A C there is repulsion, which be- comes infinite, when the distance from A is zero; whence the branch C' Dn has the perpendicular axis, A y, for its asymptote. An atom being placed at 67, and then disturbed so as to move it in tbe direction towards A, will be repelled, the ordinate of the curve being below the axis; if dis- turbed so as to move it from A, it will be attracted, the corresponding ordinates being above the axis. The point position of 67 is therefore a position in which the atom is neither iIldifleren(* attracted nor repelled, and to which it will tend to return when slightly removed in either direction, and is called Limit of the limit of gravitation. gravitation. 21 NATURAL PHILOSOPHY. Limits of cohesion. If the atom be at C, or C", &c, and be moved ever so little towards A, it will be repelled, and when the disturb- ing cause is removed, will fly back; if moved from A, it Fig. 3. Permanent molecule. Positions of indifference. Limits of dissolution. Molecules of different orders; particles. will be attracted and return. Hence C', C", are positions similar to 67, and are called limits* of cohesion, C being termed the last limit of cohesion. An atom situated at any one of these points will, with that at A, constitute a permanent molecule of the simplest kind. On the contrary, if an atom be placed at D', or D", &c, and be then slightly disturbed in the direction either from or towards A, the action of the atom at A will cause it to recede still further from its first position, till it reaches a limit of cohesion. Tbe points D', D", &c, are also posi- tions of indifference, in which the atom will be neither attracted nor repelled by that at A, but they differ from 67, C, C", &c, in this, that an atom being ever so little removed from one of them has no disposition to return to it again; these points are called limits of dissolution. An atom situated in one of them cannot, therefore, con- stitute, with that at A, a permanent molecule, but tbe slightest disturbance will destroy it. It is easy to infer, from what has been said, bow three, four, &c, atoms may combine to form molecules of differ- ent orders of complexity, and how these again may be arranged so as by their action upon each other to form particles. Our limits will not permit us to dwell upon these points, but we cannot dismiss the subject without INTRODUCTION. 25 suggesting a consequence which the reader will find of interest when he comes to the subjects of light and heat. We allude to those characteristics of the sun by which he inference-light is the main source of these principles to the inhabitants of the earth. It results from the laws of gravitation, that every Attraction of atom in a spherical sobd body is attracted towards the sp enc centre by a force directly proportional to its distance from that point. The pressure towards the centre will, there- fore, increase as the magnitude of the sphere increases, and may ultimately become so great as to force tbe atoms near enough to each other to bring them within the last bmits of cohesion, in which case, the mass, composed of atoms thus urged into close proximity, becomes perfectly Production of elastic. The magnitude of this elastic mass will beelastlclty- greater in proportion as the whole sphere is greater. Every body falbng upon the sphere will, on reaching its position at the surface, send the motion with which it arrived towards the centre to agitate the atoms of the elastic mass. These being once disturbed will, under tbe Effect of a failing forces thus called into play, vibrate indefinitely about ° >- their positions of rest by virtue of their inertia. It is only necessary therefore to suppose, that the Nebular heavenly bodies bave been formed by the gravitation of >pot the particles of a vast nebula towards its centre, and to adopt the hypothesis which modern discoveries have revived and forced upon us, viz., that heat and light are Light and heat, ., . „ -i effects of motion. but the effects of vibratory motion, to account tor tne incandescent and self-luminous character of tbe sun. Tbe incandescence .,„., i .. n ,i • . i and luminosity of same principle furnishes an explanation of tne internal the sun beat of our earth which, together with all the heavenly bodies, would doubtless appear self-luminous were the acuteness of our sense of sight increased beyond its present limit in the same proportion that the sun exceeds the largest of these bodies. The sun far transcends all Those of the sun the other bodies of our system in regard to heat and bght, J[^^3t°f simply because of his vastly greater size. 26 NATURAL PHILOSOPHY. Effects of Veasure of cohesion. Threo states of aggregation. g 13.__The molecular forces are the effective causes which hold together the particles of bodies. Through molecular action. t]ienij the molecules approach to a certain distance where they gain a position of rest with respect to each other. The power with which the particles adhere in these relative positions, is called, as we have seen, cohesion. This force is measured by the resistance it offers to mechanical separation of the parts of bodies from each other. On the degree of this force, the three states or ag- gregate forms called solid, liquid, and gaseous depend. These different states of matter result from certain definite relations under which the molecular attraction and repul- sion establish their equilibrium; there are three cases, viz. two extremes and one mean. Tbe first extreme is that in which attraction predominates among the atoms; this pro- soiid, gas, liquid, duces the solid state. In the other repulsion prevails, and the gaseous form is the consequence. Tbe mean obtains when neither of these forces is in excess, and then matter presents itself under tbe liquid form. Let A represent tbe attraction and R tbe repulsion, then the three aggregate forms may be expressed by tbe following formulae: Forniulie. A > R solid, A < R gas, A = R liquid. These three forms or conditions of matter may, for tbe most part, be readily distinguished by certain external External peculiarities; there are, however, especially between sobds bTdielTrbjeato and li(luids, so many imperceptible degrees of approxima- change. tion, that it is sometimes difficult to decide where tbe one form ends and tbe other begins. It is further an ascer- tained fact that many bodies, (perhaps all,) as for instance water, are capable of assuming all three forms of aggrega- tion. INTRODUCTION. 27 Thus, supposing that the relative intensity of the change of molecular forces determines these three forms of matter, it molocuIar actlon ' in same body. follows from what has been said above, that this term may vary in the same body. Tbe peculiar properties belonging to each of these states will be explained when solid, bquid, and aeriform bodies come severally under our notice. § 14.—The molecular forces may so act upon the atoms Action or of dissimilar bodies as to cause a new combination or '""ecu ar between union of their atoms. This may also produce a separation dissimilar bodies. between tbe combined atoms or molecules in such manner as to entirely change the individual properties of the bodies. Such efforts of the molecular forces are called chemical action; and the disposition to exert these efforts, chemical action. on account of the peculiar state of aggregations of the ultimate atoms of different bodies, chemical affinity. chemical affinity. | 15.—Beyond tbe last limit of gravitation, atoms Attraction of bodies of se magnitude. attract each other: hence all the atoms of one body attract those of another, thus giving rise to attractions between bodies of sensible magnitudes through sensible distances, intensity of this The intensities of these attractions are directly proportional attiac to the number of attracting atoms, and inversely as the squares of their distances apart. The term universal gravitation is applied to this force universal when it is-intended to express the action of tbe heavenly sravitatl0U- bodies on each other; and that of terrestrial gravitation or Terrestrial simply gravity, where we wish to express the action of g the earth upon the bodies forming with itself one whole. The force is always of tbe same kind however, and varies in intensity only by reason of a difference in the number Effects of this of atoms and their distances. Its effect is always to gen- lorce• erate motion when the bodies are free to move. Gravity, then, is a property common to all terrestrial Gravity common bodies, since they constantly exhibit a tendency to ap- ^^^ II9 proach the earth and its centre. In consequence of this 28 NATURAL PHILOSOPHY. tendency, all bodies, unless supported, fall to the surface of the earth, and if prevented by any other bodies from doing so, they exert a pressure on these latter. This is one of the most important properties of terres- trial bodies, and the cause of many phenomena, of which a fuller explanation will be given presently. SECTION I, MECHANICS. § 16.—That branch of Natural Philosophy which treats of the action of forces on bodies, is called Mechanics. Me- chanics is usually divided into Statics, which treats of the mutual destruction of forces when applied to solid bodies; Hydrostatics, when applied to fluids; Dynamics, which treats of the motions of solid bodies; and Hydrodynamics which investigates the motions of fluids. Statics and Dynamics will be treated together, under the general head, Me- chanics of Solids, as will also Hydrostatics and Hydro- dynamics, under the head, Mechanics of Fluids. Mechanics, Statics, Hydrostatics, Dynamics, Hydrodynamics. Mechanics of solids, and of fluids. PART FIRST. MECHANICS OF SOLIDS. I. SPACE, TIME, MOTION, AND FORCE. § 17.—Space is indefinite extension, without limit, and space. contains all bodies. § 18.—Time is any bmited portion of duration. We Time; may conceive of a time which is longer or shorter than a given time. Time has, therefore, magnitude, as well as has magnitude. lines, areas, &c. To measure a given time, it is only necessary to obtain Time measured. equal times which succeed each other without intermission, to call one of these equal times unity, and to express, by a number, how often this unit is contained in the given time. When we give to this number the particular name of the unit, as hour, minute, second, &c, we have a com- units of time. plete expression for time. The fnstruments usually employed in measuring time Time are clocks, chronometers, and common watches, which are instruments' too well known to need a description in a work Hke this. The smallest division of time indicated by these time- pieces is the second, of which there are 60 in a minute, 30 NATURAL PHILOSOPHY. Performance of 3600 in an hour, and 86400 in a day; and chronometers, chronomeu.*. yfafa are nothing more than a species of watch, have been brought to such perfection as not to vary in their rate a half a second in 365 days, or 31536000 seconds. Thus the number of hours, minutes, or seconds, be- tween any two events or instants, may be estimated with as much precision and ease as the number of yards, feet, or inches between the extremities of any given dis- tance. Time represented Time may be rep- by lines. resented by lines, by laying off upon a Aa—±—>- 3 ± i—e—>-s=eff given right line A B, the equal distances from 0 to 1, 1 to 2, 2 to 3, &c, each one of these equal - distances representing tbe unit of time. Rest; absolute and relative. Example of relative rest. Motion, like rest, is relative. It is continuous. § 19.—A body is in a state of absolute rest when it continues in the same place or position in space. There is perhaps no body absolutely at rest; our earth being, without cessation, in motion about the sun, nothing con- nected with it can be at rest. In what follows, rest must, therefore, be considered but as a relative term. A body is said to be at rest, when it preserves the same position in respect to other bodies which we may regard as fixed. A body, for example, which continues in the same place in a boat, is said to be at rest in relation to the boat, although the boat itself may be in motion in relation to the banks of a river on whose surface it is floating. § 20.—A body is in motion when it occupies succes- sively different positions in space. Motion, like rest, is but relative. A body is in motion when it changes its place in reference to those which we may regard as fixed. Motion is essentially continuous ; that is, a body cannot pass from one position to another without passing through MECHANICS OF SOLIDS. 31 a series of intermediate positions; the motion of a point describes, therefore, a continuous line. When we speak of the path described by a body, Pathofabody. we are to understand that of a certain point connected with the body. Thus, the path of a ball, is that of its centre, &c. §21.—The motion of a body is curvilinear or recti- curvilinear and 7. -.. .n jii -t -i • rectilinear linear, according as the path described is a curve or motion- right line. When the motion is curvilinear, we may consider it as taking place upon a polygon, of which Direction of a the sides are very small and sensibly coincide with the ° y s mo ujl1, curve. The prolongation of any one of these sides will be a tangent to tbe curve, and will indicate the direction of the body's motion while upon this side. Conceive the time employed by a body to pass from uniform motion. one position to another, to be divided into a number of very small and equal parts. If the portions of the path successively described in these equal times be equal, the motion is said to be uniform. If otherwise, the mo- tion is said to be varied. It is accelerated when these varied motion; elementary paths are greater and greater; retarded, when accelerated and -i i n - retarded. less and less in the order of tune. §22/—Velocity is the rate of a body's motion. The velocity; rapidity or slowness of motion is indicated by tbe greater or less length of the path described by the body, during each of the small and equal portions of time into which the whole time is divided. This length is taken as the measure of tbe velocity when the small portion of time is its measure. made to denote the unit of time. Tbe velocity is constant in uniform motion : it is vari- constant and able in accelerated and retarded motion. variable. § 23.—In uniform motion, the small spaces described uniform motion. in equal consecutive portions of time being equal, it is obvious that the space described in any given time will 32 NATURAL PHILOSOPHY. Relation of space to the time. contain as many equal parts of space as there are equal parts of time. Consequently, in uniform motion, equal spaces will be described in equal times, whatever be the rate of motion, and the spaces will be proportional to the times employed in describing them. Denote hy S the length of space described during the time T; s the length of the space described in the small portion of time t, then, from what precedes, we have S : T :: s : t S T . . (1). a constant ratio. Velocity measured by the space described in any unit of time. Rule for finding velocity. § 24.—Since in uniform motion, the spaces are propor- tional to the times employed in describing them, the velocity may be measured by the space described in any time whatever, for example in a second, minute, an hour, &c. Thus we say the velocity is 2 feet a second, or 120 feet a minute, or 7200 feet an hour, or T2„ of a foot in j\ of a second, &c; all of which amounts to the same thing. since the ratio of the space to the time is not changed. When a body describes uniformly a certain space in a given number of units of time, as the second, for ex- ample, which is usually taken as the unit, the velocity is found by dividing the whole space by tbe whole time, for if we make t — one second in equation (1), s becomes the velocity, § 22, and denoting this by Fwe have V= - T (2). Example. Example: The space described in 1 minute and 5 seconds or 65s being 260 feet, the space described in 1*, or the velocity, is given thus : MECHANICS OF SOLIDS. 33 V- - ' T 260/ = 4/ Reciprocally, if the velocity be multipbed by the number of units of time, the space will result. § 25.—It frequently happens in practice that the ve- Periodical locity is not constant, although the spaces described at the mo lon" end of certain equal intervals are equal. Such for instance is the case in all periodical movements of which the dif- ferent changes are executed in the same interval of time, although the velocity is continually varying within this interval. The motion of a carriage and that of a pedes- instance— trian, are examples of this; the spaces certain intervals, are often the same, while the motion is sometimes accelerated and sometimes retarded. carriage and described m . .. pedestrian. §26.—Conceive a table consisting of two vertical Relation of space i • p i • i i j.7 1 and time, columns, m one of which are arranged the numbers ex- ' , ' o represented pressive of the intervals of time elapsed since any given geometrically. instant, and in the other, on the same horizontal fines, the numbers which designate the spaces described by any body in these intervals. Draw an indefinite right line in any kind of OB; assume any linear dimension, as an inch, to repre-motlon' sent the unit of time, and let tbe same length represent the unit of space; with a scale of equal parts, lay off a distance 0 tt representing Fig. 4. an interval of time given by tbe table; upon a per- pendicular to OB at the point t4, lay off a distance t4e4 representing the dis- tance passed over by the body in tbe time 0 lA. Do the same for the other times and corresponding spaces of the table, and we obtain the points ex, e2, e3, &c, 34 NATURAL PHILOSOPHY. In uniform motion. which, being united two and two by right fines, will give a polygon. This polygon will not differ sensibly from a curve when the intervals of time are small and differ very little from each other. 0% Ot2, 016, &c, are tbe abscisses, and 4*%, t2e2, ^e3, &c, the ordinates of this curve, of which the origin is 0. It is obvious that by means of the curve we may obtain, as by the table, the space de- scribed during any given interval; so that this curve gives the relation which connects the spaces with the times, whatever be tbe nature of the motion. In uniform motion the spaces increase in tbe direct ratio of the times, and the ordinates i\ e1, t2 <%, ^ e3, &c, are therefore proportional to the abscisses 0^, Ot2, Ot^, &c.; hence the curve becomes a right line. Let the axis OB, of times, be divided into any number of equal and very small parts; through the points of division draw the ordinates or spaces, and througb the extremities of the ordinates draw the fines e1 b2, e2 b3, e3 b4, &c, parallel to the axis of times, we shall thus form a series of small right-angled triangles Ot1e1, exb2e2, &c, similar to the triangle 0t4 e4, and because e3 b4 — ^ t4, we have Fig. 5. et /] es b <■ ' 1 6j / ei / A ti.e. 0t4 : : b4 e4 : % t4, whence OL h h Relation of to the times but b4 e4 is the space s, described in tbe small time tz t4 = t, and t4 e4 the space S described in the time 0t4 — T, and the above may be written MECHANICS OF SOLIDS. 35 S S_ T~ t' and making t = 1, s becomes the measure of the velocity V, and we have F=|, the same as before, equation (1). Or, 0t4 may be taken as the unit of time, in which case, t4e4 becomes tbe velocity V, and we have Velocity equal to the ratio of the space to the time. V= i. t Same for any space and time. Accelerated motion, represented geometrically. In varied motion, the spaces not being proportional to varied motion. the times, the fine 0 e^, el e2, e2 e3, &c, is not straight, and the small spaces e2 b2, e3 b3, &c, described in Fig. 6. the elementary times ti ^, ^ %, &c, are not equal. The velocity must, therefore, vary at every instant. For the case represented by the figure, the mo- tion is accelerated, because the spaces e2b2, e3b3, &c, described in tbe equal elementary times, continually increase. Now let it be supposed that at the point <% the motion ceases to be accelerated, and Motion ceases to that it becomes uniform with the velocity which the body bad at this instant. The law of the motion after- ward will be represented by the right fine -\ , i t ij-iy> • ,i -i, fixed or prevented by obstacles from moving, the result will be a compression and change of figure throughout the when none of the body. If, on the contrary, these extreme particles are particles are fixed. P .-, .-■•, ■> -, .. .•., , , -, free they will advance, and motion will be communicated by degrees to all tbe parts of the body. This internal motion, the result of a series of compressions, proves that < a certain time is necessary for a force to produce its entire a Definite velocity effect, and the absurdity of supposing that a finite velocity \ generated may ^>G generated instantaneously. Tbe same kind of instantaneously, action will take place when the force is employed to destroy the velocity which a body has already acquired; it will first destroy the velocity of the molecules at and nearest to the point of action, and then, by degrees, that of those which are more remote in the order of distance. \ LTcontrlrT1 ^ 34.—As the molecular springs cannot be compressed to action. without reacting in a contrary direction, and with the MECHANICS OF SOLIDS. 43 same effort, the agent which presses a body will experience an equal pressure. This is usually expressed by saying that reaction is equal and contrary to action. In pressing the finger against a-body, in pulling it with a thread, or pushing it with a bar, we are pressed, drawn, or pushed in a contrary direction, and with the same effort. Two Fig. u. Illustration. weighing springs attached to the extremities of a thread or bar, will indicate the same degree of tension, and in con- trary directions when made to act upon each other through the intervention of tbe thread or bar. § 35.—In every case, the action of a force is trans- Point of mitted through a body to the ultimate point of resistance, afan^nt*^611 by a series of equal and'contrary actions and reactions hue of direction. which destroy each other, and which the molecular springs of all bodies exert at every point of tbe right line, limited by -their boundaries, along which the force acts. It is in virtue of this property of bodies, that the action of a force may be supposed to be exerted at any point in its line of direction. 36.—Bodies being more or less extensible and com- nodies used to transmit the action of forces. pressible, a thread or bar, interposed between the power ln and resistance, will be stretched or compressed to a certain degree, depending upon the energy with which these forces act; but as long as the power and resistance remain the same, the thread or bar, having attained its new length, will cease to change. On this account, bodies, 41 NATURAL PHILOSOPHY. regarded as rigid which are usually employed to transmit the action of andi,,exteusible; forces from one point to another, may be regarded as perfectly inextensible or rigid, especially as such bodies are chosen and applied so as not to yield under this action. Inertia measured by means of forces; action of inertia on a thread; conduct of a spring when under the action of inertia; resistance to all changes of motion; Fig. 15. | 37.—^e have just seen that when a force acts upon a body to give it motion or to destroy that which it has, the body will react or oppose a resistance equal to the force. This' resistance measures the inertia of the matter of the body. It is obvious that for the same body, this resistance increases with the degree of velocity imparted or destroyed; we shall presently find that it is propor- tional to this velocity, and that it also increases in the direct ratio of the quantity of matter in the body. If a body, free to move, be drawn by a thread, the thread will stretch and even break if the action be too violent, and this will the more probably happen in propor- tion as the body is more massive. If a body be suspended by means of a vertical cord, and a weighing spring be interposed in the line of traction, the graduated scale of the spring will indicate the weight of the body when the latter is at rest; but if we sud- denly elevate the upper end of the thread, the spring will immediately bend more in consequence of the resistance opposed by the inertia of the body. The motion once acquired by tbe body and become uniform, tbe spring will resume and preserve tbe degree of flexure or tension which it had when the body was at rest. If, now, the body being in motion, tbe velocity of the upper end of the thread be diminished, the MECHANICS OF SOLIDS. 45 spring will unbend and the scale will indicate a pressure less than the weight of the body. The oscillations of the oscillations of a spring may therefore serve to measure tbe variations in ^"ateThe the motions of a bod}^, and the energy of its force of changes in i • i motion. inertia, which acts against or with a power exerted m the direction of the motion, according as the velocity is in- creased or diminished. § 38.—The effect of every force depends, 1st, upon its effect of a force; point of application; that is, the point to which it is *°p„*ion>line directly applied: 2d, upon tbe position of the line along of direction, and which it acts or the straight line which its point of appli- cation would describe if perfectly free: 3d, upon the direction in which it tends to solicit its point of application along this line, whether backward or forward : 4th, upon its absolute intensity, measurable in pounds or any other unit of weight. § 39.—Let A be the point of application of a force which acts upon the fine AB; from A, lay off upon Graphical representation the direction m of a force; which the force _,. _a Fig. 16. acts, a distance A P, containing as many linear units, say inches, as there are pounds in the intensity of the force; the force will be fully represented. Commonly the direction of the action is indicated by an arrow, and tbe intensity of tbe force by some letter as P, for tbe sake of brevity. Thus, we say a force P or AP, a, force Q or A Q, as we say a force of 5 pounds, a force of 8 pounds. In this way by length of line the investigations in mechanics are reduced to those of orbysymb( geometrical figures. 46 NATURAL PHILOSOPHY. Equilibrium of forces; statical and dynamical. lllustration- t vo men. No case of absolute rest. Earth's motion. Repose not necessary to equilibrium. § 40.—When the forces applied to any body balance, or mutually destroy each other, so as to leave the body in the same state as before their application, these forces are said to be in equiiibrio. The equilibrium may be statical or dynamical. In the first case, the forces finding the body at rest, will leave it so; in the second case, the forces being applied to tbe body in motion, will in no respect alter the motion. Two men pulling with equal strength at the op- posite ends of a cord, will be a case of statical equibbrium if tbe men be at rest, and a case of dynamical equilibrium if they be in motion. In reality there is no case of absolute statical equi- librium, since the earth's motion involves that of every body connected with it, in the same way that a boat moving over the surface of the water carries every thing on board along with it. The idea of repose is not neces- sary to that of an equilibrium, which only requires the mutual destruction of all the forces which act at the same instant upon a body. § 41.—When a body, subjected to tbe action of several extraneous forces, preserves its motion perfectly uniform, notwithstanding these forces, these latter will, from the definition above, be in equiiibrio. If the velocity however not in equiiibrio augment or diminish, the extraneous forces will not be in equiiibrio ; but if we take into account the force of inertia of the different particles of the body, and introduce among Effect of inertia the extraneous forces one equal to it and capable of pre- venting the modification of the motion, there will again be an equilibrium among all the extraneous forces. A horse which draws a carriage along a road, destroys at each instant all resistances which are opposed to his action ; if the motion is perfectly uniform, these resistances arise only from the ground, the different frictions, &c. If the velocity increases at each instant in consequence of an increased effort of the horse, the inertia of the carriage will come into action and add to the other resistances Forces in equiiibrio; when the motion changes on equilibrium of forces, Illustration- horse and carriage. MECHANICS OF SOLIDS. 47 above named, and the effort of the horse during this increase of velocity, will be in equiiibrio with all these forces; if, on the contrary, the velocity diminish, the inertia of the carriage, which tends to preserve its motion uniform, will add its action to that of the horse to over- come all the resistances, or to maintain the equilibrium. Thus inertia stands always ready to maintain an equi- inertia aiw.-.ys librium among forces of whatever nature; and hence the ready t0 establ,sh 0 ' an equilibrium distinction between the equilibrium of bodies and of among forces. forces. Forces are ever in equiiibrio, while bodies are not necessarily so. If, for example, a material point be acted upon by a force, it will move in tbe direction of this force, while the force itself is maintained in equiiibrio by the inertia developed during the yielding of the point. Action and reaction are equal and contrary. Fig. 17. § 42.—When an equilibrium exists among several Reaction equal forces, as 0, P, Q, &c, one of them, as 0, may be con- J^™7 to sidered as preventing the effect of all the others. If, then, we conceive a force R, equal and directly opposed to 0, at the same point of appli- cation C, this force will destroy of it- self the force 0, and will therefore pro- duce the same effect upon the body as the forces P, Q, &c, taken together. This force R is called the resultant of the forces, P, Q, &c, and these latter the components of the force R. Reciprocally, if to the resultant R of several forces P, Q, &c, an equal force 0, be immediately opposed, there will be an equilibrium between this force and the several Resultant of forces, components of a force. -t> NATURAL PHILOSOPHY. Resultant and forces P, Q, &c.: hence, the resultant is a single force components wiucn will produce the same effect as two or more forces ; the components are two or more forces which will produce the same effect as a single force. r.i-«-.iltant of several forces acting along the same line. § 43.—When several forces act along tbe same straight line and in tbe same direction, their joint effect will ob- viously be the same as that of a single force equal to their sum, which single force will be their resultant. If some of the forces act in one direction, and others in an opposite direction, the resultant will be a single force equal to the excess of the sum of those which act in one direction over the sum of those which act in the contrary direction; and it will act in the direction of those forces which give the greater sum, for when two unequal forces are directly opposed,.the smaller will destroy in the larger a portion equal to itself.- Three men pulling in the same direction a cord, with efforts 10, 17, and 25 pounds, and two others pulling in the opposite direction with efforts 12 and 19 pounds, the effect to move the cord will be tbe same as though it were solicited by a single force 52 — 31 = 21 pounds, acting in the direction of the first men. I Mechanical work of forces. Resistance overcome and reproduced. § 44.—'The most simple case of equilibrium, is that in which two equal and opposing forces destroy each other, and it is this to which the employment of force in the mechanic arts is always reduced. To work, is to destroy or overcome, in the service of the arts, resistances, such as the force of adhesion of the molecules of bodies, tbe strength of springs, the weight of bodies, their inertia, &c, &c. To polish a body by friction, to divide it into parts, to elevate weights, to draw a carriage along a road, to bend a spring, to throw stones, balls, &c, &c, is to work, to continually overcome resistances incessantly recurring. Mechanical work not only supposes a resistance over- come, but a resistance reproduced along the path described MECHANICS OF SOLIDS. 49 by the point at which tbe resistance is exerted, and in the direction of this path. To take away from a body a portion of its matter with a tool, for example, we must not only overcome the resistance opposed by the matter removed, but also cause the point of action of the tool to advance in the direction of the line along which the resist- ance incessantly recurs. The further the tool advances, the greater will be the length of the removed portion; on the other hand, the broader and tnicker this portion, the greater the resistance, and consequently, the greater the effort to overcome it. The work performed, therefore, work increases at each instant, increases with the intensity of the effort and W1. *?e °r the length of the path described by its point of application in described by the the direction of the effort. appUcaLn. § 45.—Let us suppose a constant resistance and, there- Measure of the fore, a constant effort which is equal and directly opposed work wbenthe ' J- j r± resistance is to it, that is, they are the same at each instant; it is constant. obvious, from what precedes, that the work produced will be proportioned to the length of the path described by the point of application of the effort—double, if the path is double, triple, if tbe path is triple, &c.; so that, if we take for unity the work which consists in overcoming a resistance over a length of 1 foot, the total work will be measured by the number of feet passed over. But if for another work, the constant resistance is double, triple, &c. of what it was in the first case, for an equal length of path, the work will be double, triple, &c. of what it was before. If, for example, the resistance were 1 pound in the first case, and 2, 3, 4, &c. pounds in the second, the work for each foot of path would be 2, 3, 4, &c. times that of 1 pound. In assuming, then, the work which consists in overcoming a resistance of 1 pound, through a distance of 1 foot, for the unit of work, we shall have for the measure of the work, of which the object is to overcome a constant resistance, the number of pounds which measures Rule. this resistance repeated as many times as there are feet in the 4 50 NATURAL PHILOSOPHY. path described by ilie point of application of the resist- ance. illustration. For example, suppose a motive force employed to draw a body on a horizontal plane; the work will be, to overcome the resistance of the constant friction exerted between the body and plane. Let this friction be 37.5 pounds, and the path described 64 feet, the total work will be 37.5 X 64 = 2400 pounds, or equal to 2400 pounds over 1 foot, or 1 pound over a distance of 2400 feet. In general, then, denoting by Q, the quantity of work performed; by P the constant resistance, or its equal, the effort necessary to overcome it; and by S, the' space de- scribed by the point of action, we shall have Equation of the Q = p gj.....(g\ quantity of work. To represent this geometri- Geometricai cally, assume any linear unit, representation of , i • i , ... Fig. 18. the quantity of as tne mcn> to represent 1 work. pound, and the same to repre- sent the unit of linear length; lay off from O on the indefi- nite right fine OB, the dis- tance 0 ej, equal to the length ' B of path described by the point of action, and at e^ the perpendicular exrx, containing as many inches as the constant effort contains pounds; then will the number of square inches in tlfe rectangle 0 ex rx r, express the quantity of work. work when the § 46.—If the resistance, or the equal effort which de- vTHabiTIS stroys it, instead of being the same at each instant, varies incessantly, as is most frequently the case, the quantity of work will not be given by the simple rule above; but, as the effort, however variable, may, during the descrip- MECHANICS OF SOLIDS. 51 tion of a very small portion of the path, be regarded as constant, the corresponding portion of work will still be Elementary measured by this constant effort into this small portion 'of' of the path. The total work, being composed of all its elements, will be measured by the sum of all these elementary products. Draw the curve r, rh r2, r3, &c, of which the abscisses Oe, 0e1, 0 e2, 0 e3, &c, shall represent the spaces described by the point of action of the resist- ance up to certain given successive in- stants of time, and of which the ordi- nates er, e1r1, e2r2, egr3, &c, shall rep- resent the corre- sponding resistan- ces. Let et\, e1e2, ~~°~l K k ^ \ j^-\—k~s e2e3, &c, be the equal and very small spaces described in successive portions of time. Tbe elementary portions of work during these intervals of time, having for their measures the products of the small spaces by the corresponding resistances, regarded as con- stant for each one, that is, by the products ee+Xer, ^ e2 X t\ rh e^e, X e2r2, these elementary portions of work are represented respec- tively by the elementary areas ers1e1, ^x^s^, e^s^, &c, and the total work will be represented by the sum of all these rectangles. But if we multiply suitably the points n S3 j n n s c ! Represented by geometry. 52 NATURAL PHILOSOPHY. Represented by an area. Rule for finding the area. of division e^ e2) <%, &c, by diminishing the distances ee1, e1e2, e2e3, &c, it is obvious that the sum of the rectangles will not sensibly differ from the area included by the curve rrx r2...r7, the whole path ee7 described by the point of action, and the two ordinates er and e1r1 drawn through its extremities. Hence we see, that when we know from experience, the law which connects the variable resistance with the length of path described by its point of action, to compute the amount of work performed, is but to construct by points, or otherwise, the curve of this law, and to calculate the area included by the curve, the total length of path described and tbe extreme ordinates. When the unit of length employed to construct the ordinates is the same as that by which the length of path is measured, it is plain that the unit of area will represent tbe work performed by a unit of effort, as a pound, through a unit of length, say a foot. To find this area, divide the path described into any even number of parts, and erect the ordinates at the points of division, and at the extremi- ties ; number the ordinates in the or- der of the natural numbers; add to- gether the extreme ordinates, increase this sum by four times that of the even ordinates and twice that of the uneven ordinates, and multiply by one third of the distance between any two consecutive ordinates. Demonstration: To compute the area comprised by a curve, any two of its ordinates and the axis of abscisses, by plane geometry, it is usual to divide it into elemen- tary areas, by drawing ordinates, as in the last figure, Fig. 20. MECHANICS OF SOLIDS. 53 the rule. Fig. 21. and to regard each of the elementary figures, «3 e2 r2 rh e2 e3 r3 r2) &cv as trapezoids ; and it is obvioUS that the Demonstration of error of this supposition will be less, in proportion as the number of trape- zoids between given limits is greater. Take the first two trapezoids of the preceding figure, and divide the dis- tance ex e3 into three equal parts, and at tbe points of division, erect the or- dinates mn, m1n1'J the area computed from the three trapezoids exmn rh m mx nx n, m^ e3 r3 nh will be more ac- curate than if computed from the two e1e2r2 rx, <% e3 r3 r2 The area by the three trapezoids is e, mcm. c„ e^r-i+mn , mn + m^n* , m-,m + eqrq exm X -LJ-^----+ mrn^-----—1—± + mxe3 1 n ' 3—3-. But by construction, e^m = mm^ = m1e3 = \exe3 = %exe2, and the above may be written, \exe2{e1rx + 2mn + 2^^ + e3r3), but in the trapezoid mm^n^n, 2mn + 2m^nx = 4<%r2, very nearly; whence tbe area becomes \exe2{exrx + 4e2r2 + e3r3\, the area of the next two trapezoids in order, of the pre- ceding figure, will be \exe2{e3r3 + 4e4r4 + e5r5); 54 NATURAL PHILOSOPHY. and similar expressions for each succeeding pair of trape- zoids. Taking the sum of these, and we have the whole area bounded by the curve, its extreme ordinates, and the axis of abscisses; or Algebraic expression of the Q = £ ^ e2 [ex rx + 4 e2 r2 + 2e3 r3 + 4 e4 r4 + 2 e5 r5 + 4 e6 r6 + e1 r7]; rule. whence the rule. § 47.—When the value of the mechanical work of §, variable resistance for any distance passed over by the Mean resistance; point of action, is found by the method just explained, if this value be divided by the distance, the quotient will equal to the entire be a mean resistance, or the constant effort which, exerted uiTenure^ath7 through tne entire path, will produce the same quantity of work; for we have seen that for a constant resistance, the quantity of work is measured by the product arising from multiplying this resistance into the path described by its point of action. Examples of § 48.—When a motive force is employed to bend a work?1"0 sPrm& i* wiH develop, at each instant, an effort which is greater in proportion as its point of action describes, that of a force in the direction of the effort, a greater'path; an effort bending a spring, ^^^ we kave geen may ^e measured for qq^ position of the spring or point of action. The curve which gives the law of these efforts may be constructed by the method just given, and the area determined by the rule in § 46 will give the total mechanical work performed by the force. We have already taken as an example the work pro- duced by a constant force in drawing a body over a horizon- tal plane, and above we have taken the work which arises from the action of a variable force in bending a spring; of the draft of a the reasoning appbed to these is applicable to all kinds onhl'effortof of work empioyed in the arts. Does a horse pull upon the shaft of a mortar mill; a man draw water from a well; a man MECHANICS OF SOLIDS. 55 an artificer saw, plane, file, polish; a turner fashion bis of the materials in the lathe; the quantity of work performed ^P^onsof is measured by the product of the effort, which is always obtained by the eqtal and contrary to the resistance opposed by the matter to the tool, into the path described by the point of action, if the resistance is constant, or by the sum of the partial products which measure the elementary por- tions of work, if the resistance is variable. § 49.—In seeking to appreciate different kinds of work, Distinction to be we must be careful not to confound that which is really ° expended by the motive force, with that which is actually effective in accomplishing an object. It is to this last that are to be applied the foregoing considerations and measure- ments. We shall presently examine the mode of action of motive forces, the circumstances which modify the result of this action, and the waste which may attend it. § 50.—To show the complication incident to certain complication kinds of mechanical work, take, for example, the work ^.teJnak|"d8 of of a filer: it is necessary 1st, to press upon tbe file to work. make it take hold; 2d, to support continually its weight; 3d, to push it along tbe surface of the body; 4th, to move it with a certain velocity back and forth, and therefore to overcome the inertia of the file as well as that of the matter removed. The quantity of work is the result of these different circumstances; but this complication may be made to disappear by separating from the result of the work, every thing not indispensable to it, in considering only what takes place where the metal is removed by the file: there, we only perceive a resistance which is op- posed to an equal and contrary effort in the direction of the path described by the points of action of the file,' and of which the quantity of work is measured in the The work reduced manner already described. The work of the operator bgf0™*a8ure may be reduced to this, by supposing the file placed upon a level surface, loaded with a given weight, and the 56 NATURAL PHILOSOPHY. What mws« oe understood by mechanical work of a force; operator or motive power only employed in drawing it uniformly in the direction of its length. § 51.—In general, then, we must henceforth understand by mechanical work, that which results from the simple action of a force upon a resistance which is immediately opposed to it, and which is continually destroyed in causing the point of action to describe a path on the fine of direction of this resistance. The force must be considered as a simple agent, producing an effort or pressure measurable in pounds, and acting in a single direction, as described in § 38; and we must be careful not to confound, as is frequently done, the terms work and force, with those by which we vaguely designate all the effects, more or less complicated, arising from the action of animate or inanimate agents upon resistances: thus we should not speak of the force of a horse, of a man, of a machine, without indicating the point of action of this force, its intensity, and its direction; we should not speak of the mechanical work of a force, without specifying the same things of the resistance which it overcomes at each instant, in each particular case of its application. § 52.—The most simple work, that which conveys at once an idea of its measure, is the elevation of a weight through a vertical height, if we omit the consideration invariable of inertia. The work in this- case obviously increases as standard by ■> . . , . . J which to estimatetne weight and vertical height increase, and is measured the quantity of by the product of the two, agreeably to what is said in § 45 and § 46; here the unit of work, is the unit of weight raised through a unit of height. n The utility of this measure is its great simplicity, and the ease it affords of estimating the pressure or effort in pounds, and the path described by the point of action in feet. We might, to be sure, take any other standard unit, as, for instance, the quantity of work necessary to grind 1, 2, or 3 pounds of corn, which is the old standard of work of the resistance. work; utility of this standard. MECHANICS OF SOLIDS. 57 millers and the proprietors of mills. But a given weight of corn will present different degrees of resistance, accord- standard of ing to its quality and the kind .of tool or machinemiUers; employed to grind it; so that not only is it impossible for people generally to understand what the millers mean by their standard, but for the millers to understand each other. It is hence indispensable to bave some objections to it. standard which does not admit of variation, and of being interpreted differently by different people; of such a nature is the standard which results from the considera- tion of the effort, and the path described by its point of action in the direction of the effort. It will remain to be found how many pounds of corn Means of this unit of'work is capable of grinding, how many square ^°^antng yards of boards it will saw, &c.: all this must come from standards. careful observation and experiment. It is, above all, essential that there shall be nothing arbitrary in the mode of estimating the quantity of mechanical work. § 53.—-Different authors have given different names to mechanical work, which should be carefully distin- guished from the object accompbshed, this latter being but its effect. Smeaton calls it mechanical power; Carnot, moment Different names of activity; Monge and Hachette, dynamic effect; Cou- glve"t0 , JJ' ' J JJ ' mechanical lomb, Navier, and others, quantity of action; and this work; last expression is now generally adopted. It will here- after be employed, and will always signify the quantity of work—mechanical work. Sometimes the mechanical work has been called quan- sometimes called tity of motion, and sometimes living force, both of which are ^"^y&°d livin but simple effects of mechanical work upon a body free force. to move. We shall explain, in the proper place, the mean- ing to be attached to these terms. All work is judged of by the quantity of each par- work judged of ticular species of result, or useful effect, which it produces; ^ *e useful but we have seen that this quantiy of result is propor- 58 NATURAL PHILOSOPHY. tional to the quantity of mechanical work necessary to its production, and hence mechanical work or quantity of action is what pays hi forces. to express the § 54.—When a motive force acts with a constant effort, continued wo in numbers; and its point of action moves uniformly during any con- siderable portion of time, it will be sufficient to express the work done in a unit of time, as a day, an hour, a minute, or second. This will avoid the use of multiplicity of figures in comparing the effects of different forces with each other, while it will enable us easily to obtain the value of the whole work, by simply multiplying the work work in unit in the unit of time, by the number of units of time during of time; note the ^ ^ ^ force ^ ^ op fe WQrkmg> The duration duration of the o effort. of the work must, therefore, be noted. Thus, we say the mechanical work of a particular horse is 120 pounds raised through a vertical height of 3 feet in one second, or 120 pounds raised through 180 feet in one minute, this work being continued during 8 entire hours each day. The path Ordinarily, we take for the length of path, that which is se^ndifuTuaii described in one second, this latter being taken as tbe unit taken; of time. But this distance, according to the definition of uniform motion, is the measure of the velocity of the point of action, which we have supposed constant; by this co- incidence, the mechanical work happens to be measured by the product of a constant effort into the velocity of its point of action: which has misled many persons in caus- ae consequences ing them, as we shall see further on, to confound the quantity of work or of action with the quantity of motion, although their measures are in fact very different. ah units In the same way that tbe unit of time is arbitrary, so arbitwy; also are tlie units of effort Qr weigilt an(i distance, and consequently the unit of work, which is always equal to unit of effort, one the unit of effort or weight, exerted through the unit of SmTne0' distance- We shall take for the unit of effort 1 pound, foot; and for the unit of distance 1 foot, so that the unit of mechanics of solids. 59 work will be, as before, the effort one pound exerted through unit of work, the a distance of one foot. pro uc ° ese' Suppose, for example, that the effort 75 pounds is exerted through the distance 4 feet, then will 4 X 75 = 300 units of work, of which each one is equivalent to an effort of one pound and has no exerted through a distance of one foot. This is ordinarily reference t0 tune* expressed thus, 300"». /■; and is read, 300 pounds raised through 1 foot. And this has no reference to the time in which the work is per- formed. § 55.—Mechanicians long felt the necessity of some Different units of well defined unit by which to express the work performed, ^°! v^°^ ' or capable of being performed, by a motive force, in a considered. given time, and several were proposed; but these ill according among themselves, there seemed as little likeli- hood of a general agreement in this respect as in regard to the unit of velocity, which depends upon the units assumed for time and space. After the introduction of the steam-engine, the horse- iiorse-power power was proposed, and is now generally adopted as the atl°Pted; measuring unit. By horse-power is meant, the quantity of work, measurable in pounds and feet, which a horse is capable of performing in a given time; but this would obviously be indefinite, since horses differ in strength and endurance, were it not that some fixed value has been agreed upon, according to the principle explained in § 51, as the standard of horse-power. This value is the mean of the results of a great many trials with different horses, and is set down at 550 pounds raised through a vertical 550 lbs. height of 1 foot in 1 second, or 33,000 pounds raised f™^.f00t ™ through 1 foot in 1 minute, or 1,980,000 pounds raised 60 natural philosophy. Example. through 1 foot in 1 hour; all of which amount to the same thing. When, then,, we are told that a machine or engine is of 30-horse power, or has a power equal to 30, for in- stance, we are to understand that it will do.work which is equivalent to raising 550 X 30 = 16,500 pounds through one foot in 1 second, or 33,000 X 30 = 990,000 pounds through one foot in 1 minute, &c. Error of considering the greatest effort alone; this effort may be replaced by a fixed obstacle; error of considering the path alone. § 56.—We can now appreciate the error we should commit, if, in estimating the productive power of a motive force or machine, we confine ourselves to the greatest absolute effort it is capable of exerting, without regard to the space described by its point of action; if, for ex- ample, in estimating the productive effort of a man, we only consider the greatest burden he is capable of sup- porting at rest under the action of its weight; or, of a horse, we consider alone the greatest effort, as indicated by a spring balance, he can exert while pulling against a fixed obstacle. We can conclude nothing from these in respect to the quantity of action; we must also have the path described in a continuous manner. Simply to support a weight or exert an effort, is not to work usefully; and this is rendered clear from the consideration that we may in all such cases replace the moter by an inert body, as a prop, a post, &c.; the action and reaction being equal and contrary, unaccompanied by any motion, there is no balance of work either in favor of the effort or resistance. It would be equally impossible to infer any work or quantity of action from the path described by the point of action, without taking into account the effort exerted at each instant. It is obvious that a man or horse, running at full speed, without exerting any effort except that which he is capable of impressing upon himself, is pro- ducing no useful effect; he overcomes no resistance ex- ternal to himself, which it can be an object to destroy. mechanics of solids. 61 In a word, the productive effect of every motive force is Productive measured, at each instant, by the product of the effort ml^ed Dy th0 into the path described in the direction of the effort; so product of the ■,.„.i i/v. it ,1 • p effort into the that, if either the effort or path be zero, the quantity of path> action will also be zero. § 57.—It must be remarked, however, that, since all Always some bodies are more or less extensible and compressible, a^'^™^. motive force cannot act against^ what are usually called fixed obstacles, without producing a certain quantity of action or mechanical work, such as we have defined it: for the point to which the force is applied will yield to a greater or less extent, and the body will be flattened or elongated; the molecular springs will oppose a resistance; there will be a small path described in the direction of the force. At first the efforts of the equal and contrary re- sistances are nothing; afterward they augment by degrees till the effort of the power attains its maximum, and the body its greatest change of shape; after this the action is reduced to maintaining the body or obstacle at its state of tension and repose, without producing .henceforth any mechanical action. Fig. 22. § 58.—Construct, in the manner before de- scribed, the curve Orx r2...r6, of which the abscisses Oex, exe2, &c, represent the spaces de- scribed by the point of action in each successive instant of time in the direction of the force, and the ordinates, the cor- responding pressures or resistances opposed by the body in a contrary direction. Tbe quantity of work destroyed while the point of action is describing any one of the small paths, as e2 e3, is the its value represented geometrically. 62 NATURAL PHILOSOPHY. area of the trapezoid e2e3rzr2, and the total quantity of action destroyed by the molecular action of the body during its entire change of figure, is the area comprised by the curve, its greatest ordinate es re, which denotes the maximum resistance, and the axis of abscisses. If, then, it should happen that the body or obstacle is either com- pressed or extended by any appreciably quantity as 0e6, which is the path described by the point of action, and the greatest resistance e6r6 should be considerable, this quantity of work must be taken into account in certain circumstances which will be explained. § 59.—But in general the bodies employed to receive This work may in and transmit the action of forces, are selected with special nTtcted'b6 reference to their capacity to resist all change of figure; so that when well chosen and judiciously disposed in com- binations, the work referred to in the preceding article, becomes so small a fraction of that developed by the force when it produces motion, or when the space described by the point of action is considerable in comparison with that which measures the linear change of figure, that it may, and indeed is in practice, neglected. It is under this point of view only that the work developed by a force, applied to a fixed obstacle can be said to be nothing. This work may also be neglected when the force which develops it, acts in a direction perpendicular to the path which the body is, by its connection with others, com- especiaiiy when pelled to describe. The force in this case will only mo"e aT comPress or stretch the body uselessly, without adding to right angles to or subtracting from the work in the direction of the motion. A man who pushes against the side of a carriage in a direction perpendicular to the path along which it is moving, neither aids nor hinders the horses: and although he actually develops a quantity of work by the compression of the carriage, it must be totally neglected in making an estimate of the useful effect. mechanics of solids. 63 § 60.—These considerations are important, as they Motive forces prove, in general, that forces may work without produ- ^^^ fu] cing any useful effect. If the different pieces, for example, effect; which compose a machine, and which serve to transmit motion and work, in acting upon each other, become com- pressed or stretched, it is obvious that, even though the point of action moves in the direction of the force, this latter must first expend a certain quantity of work in changing the figure of the pieces before the motion can become regular or uniform throughout. And it may happen that this first work of the power will be totally lost, if the pieces, on ceasing to be compressed or stretched, the pieces retain their altered shape: that is to say, if they be not ^ork^ot'be^n6 elastic, or, more generally, if the molecular springs do not perfectly elastic. contribute to augment tbe work when the effort of the force is relaxed, as they did to diminish it when the action began. § 61.—We also see that if the action of the force or loss greater in moter, or the resistance occasioned by the work, undergo p1'0?01'*10" as tne 1 J ' ° force changes. frequent alterations, in becoming sometimes feeble and sometimes stronger; in a word, if the pieces are often compressed and distended, the loss of work thence arising may bear a considerable ratio to the total work of the power, which could not take place if the action of the latter were constantly the same from the beginning to the end of the work. § 62. — The shock of bodies develops considerable stm greater in pressure, and produces sensible changes of figure; the quantity of action destroyed or generated will, therefore, always be appreciable. On this account it becomes in- dispensable, in the application of mechanics, to pay the strictest attention to the influence of concussions which may occur during the performance of mechanical work. the case of shocks. § 63.—And hence we perceive the advantage arising Advantages of 64 natural philosophy. stiff and elastic from the use of very stiff and Very elastic materials in the materials. construction of those pieces which are employed to receive and transmit the action of forces, and to regulate the mo- tions they produce. Elastic bodies restore,in expanding, the work absorbed in being compressed. Loss of work when the bodies are not perfectly elastic. Examples of elastic bodies; § 64.—To obtain a clear idea how the molecular springs of a body may develop or restore a certain quantity of mechanical work, we have but to consider what takes place at the instant when a body begins to resume, progressively, its primitive figure after it has been changed, and to recall what was said of the measure of the quantity of work of a force, employed to bend a spring, to compress or distend a body. Indeed, we have only to estimate, in pounds, the different pressures corresponding to each state of the body, from that of greatest compression or distention to that of restitution, or to some intermediate state which the body will retain of itself. If the body t resume, at last, precisely tbe form which it had before the change; if, also, the pressures which correspond to the same degree of tension—to the same shape and size of the body, are the same, if, in a word, the body be perfectly elastic, the quantity of work produced during the process of restitution against a resistance opposed to it, will be equal to that required to compress or distend it, since the curve, which gives the law of the pressures and spaces, will be the same in the two cases. If, on the contrary, the body be not perfectly elastic, it will not return to its former figure; the pressures will be less during tbe process of restitution, there will be a loss of space described by the point of action, and, con- sequently, less work performed than in tbe first change of figure, there will be a certain quantity of action lost. There are scarcely any perfectly elastic bodies except the gases and vapors, and these must be confined in a close vessel or reservoir and acted upon by a piston. Such contrivances, together with springs made of the most elastic solids, serve to store up mechanical work for mechanics of solids. 65 future use; forces are employed to compress or bend them, their use. in which state they are retained by mechanical contri- vances till the work thus expended is required for other purposes; the restraint is then removed and the work transferred to some other body, which, in its turn, com- municates it to something else, and so on to the ultimate object to be attained. The balistas, catapultas, and bows.Exampies- of the ancients, throwing arrows, stones, and other missiles balistas'binpute the circumstances ol this motion. describe in the same time, if it were to move uniformly with the velocity acquired during this time. Since the paths described during any two times, as 0^,015, are represented by the triangles 0 4 v3, 01-0 r6, respectively, and since these triangles are similar and their areas are to each other as the squares of their homologous sides, it also follows, 2d. In uniformly accelerated motion, the paths described at the end of any two times, are to each other as the squares of these tiiw-s. 3d. That these paths are to each other, as the squares of the velocities acquired at the end of the corresponding times. When in uniformly accelerated motion, the velocity tbv5, acquired at the end of a given time Otb, say one second, taken as the unit of time, is given, tbe law of the motion or the right line 0 v6, which represents it, is com- pletely determined, and we may compute the velocity and space which correspond to any other time. Denote by ex and vx, the length of path and velocity which correspond to the first secbnd, and by S and V, the path and velocity corresponding to any other time, as T; we have by the first law, P | 'ace in unit of time; relation of space, Luue, and velocity; ei = ^vx Xl! = \vx S=±VT • (4), • (5); and by the second law, ex : S : : Is x Is : T x T : : Is : T2- in any whence, S = exx T2 (6); MECHANICS OF SOLIDS. 71 and replacing ex by its value, Eq. (4), S= \vxT2. ..... (7). From the third law, ex, or irj : S :: v2 : V2; whence V2 = 2vxS.....(8). By the definition of uniformly varied motion, we have, Space in any time; Velocity due to any space. Vi whence V : V = vxT. T: (9). Fig. 24. Velocity due to any time. In what precedes, we have supposed the body to start The body has from rest, so that the right line, which gives the law of the olready an ' <_> / which corre- sponds to the time 0t3, is composed of two parts, viz. fc^'g, and t'zv3; tbe first is initial velocity. equarto the initial velocity Ov0, and the second to the 72 NATURAL PHILOSOPHY. Formulas to velocity which the body would acquire in the time v01'3, cuTuinltances of ecLuai to 0 ^, under the action of the constant force, had it the motion; moved from the point v0 with no initial velocity, as in the preceding case; for the line v0 vs gives, in reference to the line v0t'5, the law of acceleration. Knowing, then, the velocity which the force is capable of impressing upon the body in a unit of time when moved from a state of rest, it is easy to construct the line v0 v5, in relation to vQ t'5 or its parallel 015, and to deduce from it all the circumstances of the motion Let it be required, for example, to find the length of path described by the body in the time Ot4. This path will contain as many linear units as the trapezoidal area 014 v4 v0 contains superficial units. We perceive at once, that this length will be composed of two parts, viz.: that described uniformly in virtue of the initial velocity 0 v0, and represented by the rectangle Ot4f4v0, and that de- scribed in virtue of the constant force and represented by the triangle v6t'4v4. But, denoting by a the initial ve- locity, and by T the time, we have for the measure of the rectangle aT; and for the measure of the triangle, Eq. (7), \^T2; and if we denote by S the total length of path actually described by the body, we have S = aT + \vxT2 .... (10): and because the actual velocity at the end of any time, is the initial velocity increased by that due to the action of the constant force during this time, we have, Eq. (9), V = a + vxT.....(H). value of the space; value of the velocity. MECHANICS OF SOLIDS. Fig. 25. graphical representation. § 69.—If we now suppose the constant force, instead uniformly of increasing the initial velocity of the body, to diminish retar e mot'011; it, the motion becomes uniformly retarded, and the fine vQ v4 gives the law of the motion. By drawing vQ t'5 parallel to 0th, we see that the velocity v3t3, which corresponds to the time Otz, is nothing else than the initial velocity 0v0 dimin- ished by the velocity t'3 v3, which the body would acquire under the action of the con- stant force at the end of the time Ot^ had it moved from rest. The length of path de- scribed is now represented by the trapezoidal area Ot3v3v0; and is equal to that which would be uniformly described in the same time, with the initial velocity 0v0, diminished by that which would be described in the same time, if moved from rest under the action of the constant force, by a motion uniformly accelerated; that is to say, the length of path is represented by the rectangle 0 4 t'3 v0 diminished by the triangle v0 v31'3. The equations (10) and (11), which appertain to uni- Formulas to formly accelerated motion, become, therefore, applicable ^Turnstonc^sot- to uniformly retarded motion, by simply changing the this motion. sign of the velocity generated by the constant force, and that of the area of the triangle, which represents the path due to the action of this force; hence, S = aT - \vxT* (12), Value of space; V = a - vxT (13). of velocity. Let us suppose that, among other things, we desire the time required for the force to destroy all the initial NATURAL PHILOSOPHY. velocity; we have only to make V = 0, and equation (13) becomes a— vxT= 0, whence Time required to destroy all a rp __ __ fl4V body's velocity. Vx ' from which we conclude that the time required for a con- stant force to destroy all the velocity a body may have, is equal to the quotient arising from dividing the value of this velocity, by the velocity wnieh the force can generate in the body in one unit of time. To find the length of path described by the body during the extinction of its velocity, substitute the value of the time above found in equation (12), and we have The path a* described during ^ == ____ /1 q\ • the destruction of 2 Vx its velocity; that is to say, the space through which a body will move during the entire destruction of its velocity by the action of a constant force, is equal to the square of the velocity destroyed, divided by twice the velocity which this force can generate in the body during a unit of time. It is important to remark, that if the force continue to after the velocity act after having destroyed all the velocity, the body will is destroyed ttiG bodywiiireturn; return along the path already described, and pass in succession and in reverse order, as to time, through its previous positions, at each of which it will have the same velocity it had there before; for while the body is losing its velocity, it may be regarded as beginning its motion at any point of its path with its remaining velocity or that yet to be destroyed, which, in such case, is denoted by a, and when all its velocity is destroyed, it returns from a state of rest or begins to move backward with no initial velocity; so that equations (4) to (9) become applicable to MECHANICS OF SOLIDS. 75 this latter motion, while equations (14) and (15) are to the former. But from equation (8) we have V = V2v~S, and substituting for S its value given by equation (15) we get and have at its TT A [2VxaT x Previous y == \ I .------- == q . positions the & Vx same velocity a! before. that is to say, the velocity V, which the body has ac- quired in moving backward through a space S, is equal to the velocity a, with which it began to describe the same space in its forward motion. § 70.—One of the most important examples of uni- Motion of formly accelerated motion, is that presented by the verti- cal fall of heavy bodies; but, before discussing it, we will make known some of the circumstances which accompany and modify this motion at the surface of the earth. We bave already seen, § 32, that the force of gravity may be considered as constant within ordinary limits. But at the surface of our globe, all bodies are plunged into causes which the atmosphere, and this atmosphere is itself a material ™IqJJ * body, which, by its inertia and impenetrability, opposes with greater or less energy all kinds of motion of bodies; this opposition is named atmospheric resistance. Experi- ment shows us that this resistance increases as the velocity v__^... of the body and tbe extent of its surface increase; thus, in striking the air with a light flat board, the resistance which we experience is greater in proportion as the mo- influence of tion is more rapid, while it is scarcely sensible when the exlentofsurface; motion is very slow; and again, the resistance will be less if, instead of striking the air with the broad surface, we present to it the edge of the board. 76 NATURAL PHILOSOPHY. influence of air on the fall of bodies; bodies which weigh most and have least surface, fall most rapidly; Fig. 26. in vacuo all bodies fall equally fast; gravity acts on the interior and exterior particles of a body alike; distinction between the weight of a body and the force of gravity. It is plain, therefore, that the presence of the air must modify the laws of the vertical fall of bodies subjected to the action of their weight. In permitting bodies to fall through the air, from the same height, it is observed that those which weigh most under the same volume, or those which present the least surface in the direction of the motion, arrive soonest at the bottom; thus, a ball of lead will fall sooner than a ball of equal volume of common wood, and a ball of common wood sooner than one of cork, &c. But if made to fall in vacuo, or in a long hollow cylinder from which the air has been removed, experiment shows that all bodies fall equally fast, and therefore will reach the bottom at the same instant if they start together. This is called the guinea and feather experiment, from the fact that a guinea and feather will fall under the action of their respective weights in vacuo, with the same velocity and, therefore, will reach the bottom in the same time. From this it follows, that the force of gravity acts indis- criminately upon every particle of matter, and impresses upon each, at every instant, the same degree of velocity in vacuo, a fact which it is important to remember. We may easily assure ourselves that the force of gravity acts on the interior as well as on the exterior particles of all bodies, by observing that the same body weighs just as much by the weighing spring whether placed in the open air, or in a close chamber; which proves that the force of gravity acts through this chamber envelope without undergoing any change. The weight of a body, is the resultant of all the actions of the force of gravity upon its elementary particles; we must be careful, therefore, not to confound the weight with the force of gravity itself, which is, in fact, only the ele- mentary force impressed upon each particle. MECHANICS OF SOLIDS. 77 § 71.—Finally, it is important to remember that the denser bodies, such as gold, lead, iron, &c, are those Gold, lead. &c. t.i i i , i/i mi o -n fall most rapidly winch, under equal volumes, or equal surfaces, will fall intneair. most rapidly in the air, because the resistance of the latter is weaker when considered in reference to the weight; and this resistance may become relatively so small that we may neglect it, particularly when the fall of the body is not very rapid. Galileo, an Italian philosopher, was the first to investi- the motion of gate, experimentally, the laws which govern the motion of ^"^ly 'es bodies falling under the action of their own weight, in accelerated. vacuo; and he found the motion to be uniformly accelerated. The force of gravity is, therefore, within the limits of ex- periment, a constant accelerating force, acting with an equal intensity at each instant whatever be the velocity ac- quired. Atwood, an English philosopher, in resuming the experiments of Galileo, with greatly improved means, obtained the same results. Laws of the § 72.—Hence, when a body falls from rest through a motion of failing certain height, in vacuo, 1st. The velocities acquired are proportional to the first law; times elapsed since the beginning of the motion. 2d. The total spaces passed over, or the heights of the second law; fall, are proportional to the squares of the times elapsed. 3d. These heights are proportional to the squares of third law; the velocities acquired at the end of each. 4th. The velocity acquired at the end of the first unit fourth law. of time, is measured by double the height of fall passed over during this time. Although the force of gravity, may, without sensible Force of gravity error, be regarded as constant at the same locality, it yet ^y^g! varies, as we have seen, from place to place, in going southward or northward, and cannot, therefore, generate as much velocity in one latitude as another. From careful experiments, made with a pendulum at different places, it is found that the length of path described by a 78 NATURAL PHILOSOPHY. space a body will describe under its action in first second; velocity it can generate in one second; formulas which relate to the fall of bodies in vacuo; body in the first second of its fall from rest in vacuo, will be given by the following formula, viz: feet. ex = 16.0904 - 0.04105 cos. 2 + . . (16), in which ^ is the space, and 4* the latitude of the place. In works on mechanics, the velocity which the force of gravity can generate in a second of time at the surface of the earth, is usually denoted by g ; and as this velocity is equal to twice ex, Eq. (4), as given by the above equation, we have, feet. g = 32.1808 - 0.0821 cos. 2± . . (17); hence all the circumstances of the motion of falling bodies at any place, will be given by equations (4) to (15) after substituting therein g for vx. Let H represent the height, in feet, through which the body has fallen in a given time denoted by T, and V the velocity acquired at the bottom of this height; then, from equations (5), (7), (8), and (9), we have H = \VT.....(18), H = IgT*.....(19), V2 = 2 g H.....(20), V = 9 T......(21), in which, for all ordinary cases we may take g = 32.1808 feet . (22). application to examples; Suppose we are required to find the velocity acquired and the path described at the end of 7 seconds; from equation (21), we have V = 32.1808 X 7 = 225.2656 feet, MECHANICS OF SOLIDS. 79 from equation (19), H = ^-^ x (7)2 = 788.4296 feet; that is to say, at the end of 7 seconds, the body will have a velocity which would carry it over a distance of results; 225.2656 feet during the 8th second, were its velocity at the end of the seventh second to become constant, and the space described during the seven seconds of fall, will be 788.4296 feet. It must be remembered that, in the atmosphere, the influence of the body will not fall with the same velocity, on account of atmospnere' the resistance of this medium; but from what has already in the case of been remarked, this resistance will not have much in- metals'lf'the 7 surface of the fluence if the falling body be very dense, as iron, lead, body and height &c.; or if the surface of the body be small; or if the height of fall be not great, say sixty or seventy feet. We might, therefore, measure approximately, the height of application to towers, depth of wells, &c, &c, by noting the time, as JJtwentS' indicated by a watch beating tenths or fifths of seconds, depth of weiis. required by a body to fall through the height. If we have given the height through which a body has fallen, it is easy to find the velocity acquired; for from equation (20), we have V= VlfH. Suppose a body to fall through a height of 80 feet, then will V = V 2 X 32.1808 X 80 = 71.75 feet. This proposition is of frequent occurrence in practical mechanics. Velocity due to a The quantity V is called, the velocity due to a qiven Pve° bfgbt; a •/ i j o height due to a height H; and II, the height due to a given velocity V. given velocity. 80 NATURAL PHILOSOPHY. A body thrown vertically upward; § 73.—When a body, as the ball from a gun, for example, is thrown vertically upward, its weight acts at each instant with the same intensity to diminish by equal degrees its primitive velocity; the motion will be uni- formly retarded ; the velocity will be totally destroyed when the body attains a certain height, from which it. will descend, in taking successively the different degrees of velocity which it had at the same places in its ascent, all of which is obvious from what was said in § 69. Thus, at the distance of 1, 5, 7, &c. feet from the place of starting, the body will have exactly the same velocity in ascending and descending; it will only have the direction of its motion changed. When it returns to its point of depar- ture, its velocity will be the same as it was at starting. Denote by H, the greatest height the body will attain; and V, the primitive or initial velocity; then will, equa- tions (20) and (21), greatest height to which it will ascend; time required to reach its greatest height. H = -~ 2<7 T = -— (23), (24). Example; •That is to say, the greatest height to which a body will ascend, when thrown vertically upward, is equal to the square of its initial velocity, divided by twice the force of gravity; and the time of ascent will be equal to the initial velocity, divided by the force of gravity Let the body, for example, leave tbe earth with a velocity of 150 feet a second, then will n (150)2 T = 2 X 32,1808 150 32,1808 = 350.28 feet. = 4.658 seconds. MECHANICS OF SOLIDS. 81 This is on the supposition that the air opposes no resist- effect of ance. The body will not ascend so high in the air; and, atrao9Ph " o 7 J resist.nnr moreover, will fall with less velocity than in vacuo. eric resistance. § 74.—We may now appreciate the quantity of work Quantity of work or of action which the weight of a body will expend, in oftheweight> ./ ± 7 required to impressing upon itself a certain velocity, or in overcoming impress upon a its inertia. Denote by W, the weight of the body, express- ^/ci*/!ven ed in pounds, or, in other words, the absolute effort which gravity exerts upon the body, and which is equal and contrary to that necessary to support it in a given posi- tion; this will measure the constant effort exerted upon the body during its descent through the height II. The quantity of work consumed during this fall will, § 45, be quantity of work denoted by consumed during J its fall; W X H, and this quantity of work will have generated in the body the velocity V, computed by the equation V2 = 2gH; from which we have V2 H=irg> and multiplying both members by W, W WH =l-xF . . (25). 75.—Thus, the quantity of work developed by the work required to impress a velocity; weight of a body to impress a certain degree of velocity m upon itself, is equal to half the product obtained by multi- plying the square of this velocity, by the weight of the body, divided by the velocity g, which the force of gravity is capable of impressing upon all bodies during 6 NATURAL PHILOSOPHY. the first second of their fall. This product, W V2, living force; equal to double the quantity of action necessary to produce il; half the living force lost or gained, equal to the work that overcomes the inertia. is what mechanicians call the living force of the body whose weight is W. We see, therefore, that the quantity of action expended by the weight of a body, is half the living force impressed; or that the living force impressed, is double the quantity of action expended by the weight. It is to be remarked, that when a body is thrown ver- tically upward with a certain velocity, the quantity of action of the weight, which is always measured by the product of the weight into the height to which this body has risen, is employe'd, on the contrary, to destroy this velocity, so that in the two cases of ascent and descent, the half of the living force lost or gained, measures the quantity of action ,or of work necessary to overcome the inertia of the body, whether the object of this action be to impress upon the body a certain velocity, or to destroy that which it already has. This principle is, as we shall soon see, general, what-' ever be the motive force employed to communicate motion to a body, and whatever be the direction of the motion. But it is necessary first to remark upon certain terms employed in mechanics. meaning of living § 76.—As the expression of living force, employed to force; not a force, but the result of a force's action;. designate the product, 9 ' may lead to error, it is proper to remark here, that it must not be regarded as the name of any force, any more than the name given to the product W.II, or the quantity of action, designates a force; it is simply MECHANICS OF SOLIDS. the result of the activity of a motive force, expressible in pounds, which has been employed to overcome the inertia of a body, to impress upon it a certain motion—a certain velocity. Under this point of view, the living force is but a dynamic effect of a force, or rather double this effect, since a dynamic effect W — iV2 = 2 W. II. 9 A body in motion, or a certain dynamic effect, may a body in mo; • indeed become, in its turn, a source of work; as, for ma>, ' ' ' ' work; example, a body thrown vertically upward is elevated in virtue of its velocity to a certain height, as though it were taken there by the incessant action of an animated moter. But this is, in all respects, analogous to what takes place when a force has developed a certain quantity of work to bend or compress a spring; the inertia of the matter has been brought into play in the same manner that the molecular springs have in this latter case. This inertia, § 66, when it has been thus conquered, becomes capable of restoring the quantity of work expended upon but cannot he ^ m i i • ,• vi force anv mm-.; it, as well as a compressed spring; in a word, inertia, like tnanan"ele,a, a spring, serves to store up a quantity of action, to tody, or bent soring transform it into living force, so that living force is a true disposable quantity of action. The same may be said of a body elevated to a certain height; this body solicited by its weight is the source of a quantity of action, of which we may subsequently dispose to produce a certain amount of mechanical work. But as we cannot say that this body, elevated to a certain height, is a force, that a compressed spring is a force, neither can we say that a body in motion, or that W 2L y* 9 is a force. It is the same of men, animals in general, or animals of caloric, of water-courses, of wind, &c, &c.; these are but ^J'" agents of work, or moters—not simple forces. S4 NATURAL PHILOSOPHY. »V\ ICt of hanics as lied to the It is the object of mechanics, in its application to the arts of life, to study tbe different transformations or metamorphoses which the work of moters undergoes by means of machines and implements, to compare different quantities of work with each other, and to estimate their value in money, or in work of this or that kind. In short, when we speak of living force, communicated to, or acquired by a body, it is only necessary to remem- ber, that it relates to a real motion of the body, and is equal to the product of the square of its velocity into its weight, divided by the force of gravity. The mass of a bo'ty; force of gravity proportional to the velocity it ' may impress in on -econd. § 77.—Since the force of gravity acts indiscriminately upon all the particles of a body, and impresses upon them at each instant, the same degree of velocity at the same place, the weight of a body, which is the result of these partial actions, may give us an idea of the relative quantity of matter it contains, or of its mass, for it is plain that the mass must be proportional to the weight; often, indeed, the weight is taken for the mass. But as the intensity of the force of gravity varies from one locality to another, and as the quantity of matter in the same body or the mass remains absolutely the same, it is obvious that this latter would be but ill defined by its weight. Ex- perience shows that the velocity impressed by the force of gravity, in one second of time, is directly proportional to the intensity of this force, and that therefore the ratio W 9' must remain the same for all places, since the weight is also directly proportional to the force of gravity. Thus if W and W, be the weights of the same body at different places, and g and g' the intensities of the force of gravity at those places, respectively, then will W : W :: g : g>; MECHANICS OF SOLIDS. b'5 whence W = W_ 9 9' This invarible ratio —, is taken, in mechanics, as the 9 measure of the mass of a body. Ordinarily the mass is Measure of the represented by M, whence mass of a body M = W_ 9' or W = Mg.....(26), ~eof,h'' in which W expresses the effort or pressure exerted by the weight of the body, and g the velocity which this weight can impress upon the body in a second of time. § 78.—By substituting the value of the weight, as given by equation (26), in the expression for the fiving force, we find ~\y Living force in ---- V2 = MV2) terms of the ma-a 9 and velocity; that is to say, the living force of a body in motion, is equal to the product of its mass into the square of its velocity. Finally, mechanicians have agreed to call the product of the mass of a body, as above defined, into its velocity, or MV, the quantity of motion of the body ; and this it must be quantity of remarked is very different from the quantity of action or motIon; of work. To understand what is meant by this new expression, denote the quantity of motion by Q, then will Q = Ky = MV . . . (27); 86 NATURAL PHILOSOPHY. '.- meaning; or, which is the same thing, Q : W :: V : g. ce weight; But W, is the weight of the body, and g, the velocity which this weight can generate in this body, in one second >* a pressure, of time; hence Q must designate either a weight or an equivalent effort, which can generate in the body, the velocity V, in one second. We see also that the living force, MV2, or MVV = QV, ^ force is the product of this effort, by the velocity T^ or by j,titvof the path described uniformly by the body in a unit of •on into the time in virtue of its acquired velocity. These observations show the distinction between the quantity of motion of any body and its living force, and the identity between this latter and double the quantity of action. at) ''■• oftue § 79.—It is principally to abridge and simplify the d<-:-••urinations ,• i . n , , .. u*. and computations and reasonings, that the denominations mass quantity of and quantity of motion, are employed in mechanics; and '■■"'ion. , . .. they might easily be dispensed with. But as authors generally have used them, it becomes important to under- stand their precise significations. a force is § 80.—We have just seen that the force of gravity will ><"i>portional to • t -i -i • -, ti.e velocity it . imP*esri upon a body, during one second of time, velocities can generate in a which are constantly proportional to its intensity or to £:\*n time, only ,iii. ■ -i ,. i , -, vii..aronst3..t. the absolute weight of the body in each locality. But this property arises only from the fact, that the weight y .«the force remains constant during the fall, so that the total velocity \ ^'"niotai to at the end of tne fall> is proportional to the equal degrees r»,Mil degree of velocity impressed at each instant. When the motive '•parted at a force> instead of being constant, varies at each instant, it is rwninstant. obvious that its intensity can no longer be measured by MECHANICS OF SOLIDS. 87 the velocity which it impresses upon the same body during a unit of time, and that its measure must depend upon the small degree of velocity which it communicates at a given instant. By observing what takes place at the surface of tbe forces earth, and in our planetary system, it is found that the f^e smlu'de^eea motive forces or pressures are, in fact, proportional to the small of velocity they , /• 7 •, 7 • 7 ,7 • ,7 7 7- can impress in a degrees of velocity wnicn they impress upon the same body in small equal indefinitely small portions of time. This fact serves portion of time. as the basis of all dynamic investigations, and must be regarded as a general law of nature. § 81.—Accordingly, let F be the measure, in pounds, Measure of the of a force of pressure; let v be the small degree of ^ertiab^the010 velocity which it can impress upon a body at any velocity instant or epoch, during an indefinitely small interval g3dtime.m' of time, denoted by t; also, let W be the pressure exerted by the weight of a body at any given place, and v' the small degree of velocity which this weight can impress upon the body during the same short interval t. We shall have, from the principles already established, since F may be regarded as constant within the limited time t, whence F '. "W '. '. V '. V'; Consequences of this law; F = —r-v. V1 But from the first law of falling bodies v' : g : : t : l960-; whence V = gt; NATURAL PHILOSOPHY. measure for the intensity of any motive force; therefore *= JLx-f = Jf.-f ■ • W 9 t t inertia exerted, proportional to the product of mass into the velocity imparted; That is, the intensity of any motive force, is measured by the product of the mass into the velocity it can generate while acting with a constant intensity, divided by the duration of the action. Thus, when we know the small velocity v, im- pressed in the short interval of time t, by the force F, we may compute the value of this force, which is equal and contrary to the resistance opposed to motion by the inertia of the body. This resistance has been called by some the force of inertia, and by others dynamic force. The relation given by Eq. (28), shows us that the force of inertia, which is equal and contrary to F, is directly proportional to the mass, and to the velocity v which this mass receives during the elementary time t. Let F' be the measure of a second force, which acts upon the mass M, impressing upon it in the same time t, the small velocity v', then will v' F' = M'.-j, relation of any two motive forces. which, with Eq. (28), gives F : F' : : Mv : M' v'. That is to say, any two motive forces are to each other, as the quantities of motion they can impress in the same elementary portion of time. § 82.—From Eq. (28), we find Velocity impressed in any short time; F.t v " -If' MECHANICS OF SOLIDS. SO from which we perceive that the degree of velocity which proportional to a motive force impresses upon a body, during a shortthe intensilyof *■ *■ J ' ° the force divided elementary portion of time, increases with the intensity of by the mass. the force, and inversely as the mass, or weight. §83.—If now we suppose, at any instant, the force Measure of inertia suddenly to cease to vary, and to continue to act upon the an(!of the equal " *> i and contrary body with the intensity which it possessed at that instant, motive force; the velocity will increase or diminish, proportionally to the time, § 67, and the intensity of the force may be measured by the definite quantity of motion which it can impress upon the body during the first succeeding second. Designate by Vx the velocity generated in the body during the 'first second succeeding the instant in which the force becomes constant, then will Vi whence which, in Eq. (28), gives F = VXM.....(29); and, in general, the motive force, equal and contrary to the equal to the force of inertia, is measured, at each instant, by the quantity quantlty of J J ' > J 1 J motion the latter of motion it can impress during one second, if, instead of vary- can impress in a inq, it retain unaltered the intensity it had at that instant. J' J constant. When the mass becomes the unit of mass, Eq. (29) becomes F = Vx......(30); the force in this case is called the accelerating force, or, Accelerating more properly, the acceleration or retardation due to the force, force; 90 NATURAL PHILOSOPHY. measured by the and is always measured by the velocity it is capable of im- 0Clty , pressing on a unit of mass in a unit of time, acting with a impressed on a ■* J J J u unit of mass in constant intensity. And from Eq. (29), which gives, vx = M' is equal to the it appears that the acceleration or retardation due to the motive force _ . . , . , , divided by the force, is, in every case, nothing more than that portion mass- of the entire motive force which results from dividing the latter by the number of units in the mass acted on. Geometrical illustration; Fig. 27. § 84.—Trace, according to the method described for uniformly varied motion, § 68, the curve v0 vx v2 v3, &c, which represents the law of the times and veloci- ties ; let t3 v3 and t4 v4 represent the velocities which correspond to the end of the times Ot^ and Ot4, or at the beginning and end of the very small por- tion of time % t4 = t. Draw through v3 the line v3 b4, parallel to the axis OB of times, and produce it till v3m = l second; this line will meet the ordinate t4 v4, and b4 v4 will be the small portion of velocity = v, impressed by the force, during the small portion of time t. Now if, at the instant cor- responding to the end of the time 0 £,, the force become constant, it will subsequently impress upon the body equal MECHANICS OF SOLIDS. 91 increments of velocity during tbe equal intervals of time t, and the curve v3v4v5 will become the straight line v3n, tangent to the curve at the point v3. Drawing through m a line parallel to t4v4, the portion mn will represent tbe velocity Vx impressed in one second, and the two similar triangles, v3 b4 v4 and v3 m n, will give V, b, : bxv± : : v*m : m n; Jzu\ or whence Vu V the value of the Vi = ~~r ; velocity at any " instant; as before found. Thus, when we know the law which connects the velocity with the time, or the curve which represents this law, we may, at any instant, by drawing a tangent to the curve, determine the velocity Vx, and consequently found by the compute the value of the intensity of the force from the tansenthne; equation, —j .. „ It t_ the measure of I = M Vx = —— . Vx ; the motive force. or, which is the same thing, the value of the equal and contrary resistance, opposed by the inertia of the body, at each instant during the action of the force. § 85.—Eeciprocally, if we know the value of the in- tensity of the force F at each instant, we deduce from it vaiueofthe ,1 -i • i n accelerating force, the corresponding value of ^ equal to motive force divided by „ mass. F F> = it; 92 NATURAL PHILOSOPHY. inclination of or of the inclination of the tangent v3n, or that of the 'the element of the curve of velocities to the axis OB of times. curve. The tangent of this inclination is given by mn vzm = Vx and if the initial velocity 0 vQ be given, nothing is easier curve constructed ^han to construct the curve, of which the ordinates shall be by means of this tangent. the successive velocities acquired under the action of the force; since, by means of the inclinations of the tangents or elements of the curve corresponding to each absciss of time, those elements may be drawn one after the other, thus forming a polygon, which will differ less and less from the curve, in proportion as the number of values of the force between given limits is greater. work necessary § 86.—By the aid of what precedes, we may readily TeiTcu6-31181™11 comPute tae quantity of work which must be expended against a body, whose weight is W, by a force F, equal and contrary to the force of inertia, to impress upon it a certain velocity V, or, more generally, to augment or diminish its velocity by a given quan- tity. The quantity of work expended during any small interval of time t, has, for its measure, the product of the intensity of the force F, into the elementary portion of the path described by the body during this time. This small path is given by the area of the small rectangle v3 ^ t4 b4, whose base is the element % t4 = t, and whose altitude is £, v3 = V, § 67 and § 68; Fig. 28. MECHANICS OF SOLIDS. 93 that is to say, by the product Vt. quantity of work is F Vt, Hence the elementary for each instant of time, or for each small increment b4 v4 of velocity V. But from Eq. (28) we have replacing F by this value, in the preceding expression, we have, for the elementary quantity of work, MVv; and it is the sum of all these partial quantities of work which composes the total quantity of work; this sum may be found thus: From the point 0, as an origin, lay off the distances 0wx, u\w2, w2w3, &c, to represent the different incre- ments of velocity during the different successive elementary portions of time t, which have elapsed since the beginning of motion — increments that will not be equal in the case of a vari- able force; then will 0wx, 0w2, Oiv3, &c, represent the veloci- ties of the body at the corresponding instants: lay off these same lengths upon the ordinates wxvx, w^v^ wzvz, &c, so that we shall have elementary quantity of work; Fig. 29. geometrical method of finding the whole work. wxvx = 0wx, w2v2 = 0w2, wzvz = 0w3, &c.; 94 NATURAL PHILOSOPHY, The area of a triangle represents the sum of all the products V o. the series of points vx, v2, v3, &c, will lie on a right line, inclined to the axis 0 B, in an angle of 45°. Consider now the velocity vz w3 = V, Fig. 29. for instance, of which the in- crement w3 w4 or v3 b4 = v4 b4, is called v. The product Vv, will here be represent- ed by the small rectangle v3 w3 io4 b4, or by the trape- zoid v3 w3 w4 v4, to which it becomes sensibly equal when the increment of velocity or that of the time is very small. The sum sought, of all the partial products Vv, has for its measure the sum of all the corresponding elementary trapezoids, or the area comprised within the right fine 0 v-,, tbe axis 0 wn, and the ordinate wn t>7, which latter represent the velocity acquired from the beginning to the end of the time for which we wish to estimate the work done by the force. Work consumed when the body's motion is accelerated; § 87.—For example, if the body sets out from rest, and we desire to find the sum of the products of Vv, correspond- ing to tbe acquired velocity w4 v4 = V, this sum being rep- resented by the area of the triangle 0 w4 v4, we shall have \0w 4 X W4VA \{w4v4)2 = \V< hence the quantity of work corresponding to the velocity V, and consumed by the inertia of the body whose mass equal to half the is M, will be measured by \ MV'2, or by half the living iivmg oice force communicated from the beginning of the motion, § 76. communicated; o o ) o This principle obtains, therefore, for any kind of motion, or for a motive force different from the force of gravity. For another velocity, wn vn = V", the consumption of work will be in like manner measured by \ M V"2, and consequently for the interval between the positions in which the body had the velocities F'and V", the quantity MECHANICS OF SOLIDS. 95 of work consumed will be measured by the difference, or \MV"2 — ±MV\ corresponding to the trapezoid w4 w7 v7 v4. But M V'2 and work consumed i/F"2arethe living forces at the beginning and end of ^TZuL the interval of time during which we are considering the difference of work of the motive force; the expression above is, there- bl^nninglnd fore, one half the increment of living force, or half the living end- force communicated in this interval; so that the principle applies to any two instants of the body's motion, and thus the quantity of work expended has, in every case, for its measure, half of the living force communicated in the interval between these two instants. § 88.—Finally, it must be remarked, that the preceding supposes the velocity of the body to increase incessantly; if it were otherwise, the force would be opposed to the motion, and would be a retarding force. But the reasoning remaining the same, would be applicable to this case, and we should find that the quantity of work or action work developed developed by the resistance F, (equal and contrary to the J*™^™.1™ force of inertia now become a power,) during the time necessary, to reduce the velocity from V to V", would have for its measure, \{MV'2 — MV"\ or half the living force destroyed or lost. equal to half the Thus, the diminution of the living force of a body ?iffere"ce °f. ' o J living force at between any two given instants, supposes that a quantity the beginning and of work or of action equal to the half of this diminution, eE has been developed by the inertia of this body against obstacles or resistances, as its augmentation supposes, on the part of a power, a consumption of work equal to the half of this augmentation. Inertia serves t0 transform work §89.—We now clearly perceive how the inertia 0f in'0,living';orce' ° " x and living force a body, serves to transform work into living force, and into action; 96 NATURAL PHILOSOPHY. examples in the mechanic arts; example of the grist-mill; the air-gun; living force into work; or, to use the expressions em- ployed, § 76, on the occasion of the vertical motion of heavy bodies, we see that inertia will store up the work of moters by converting it into living force, and give this work out again when the living force comes to be destroyed against resistances. The mechanic arts offer a multitude of instances in which these successive transformations take place, in operating by means of machinery, implements, &c, &c. The water contained in the reservoirs of grist-mills, for example, represents a certain quantity of disposable action, or work, which is changed into living force when the sluice gates are opened; in its turn, this living force acquired by the water, in virtue of its weight and descent from the reservoir, is changed into a certain quantity of work; this is communicated to the wheels of the mill, and these latter transmit it to the millstones which pul- verize the corn. The air confined in the reservoir of an air-gun, represents the value of the mechanical work expended by a certain moter in compressing it; on open- ing the valve, the air acts upon the ball, impels it forward, and converts a certain quantity of work into living force. If this ball be thrown against a spring, or an elastic body, the latter will be compressed in opposing a greater or less resistance to the inertia of the former, and will finally have destroyed all its motion at the instant the quantity of work, Fig. 30. developed by the spring. the action of the spring, becomes equal ball against a to half ^ liying force of the ball; the spring being retained by any means in its compressed state, the living force will be stored up as a quantity of disposable work, so that when the restraint is removed from the MECHANICS OF SOLIDS. 97 spring, the ball will be thrown back with a velocity such, that the living force will be double the quantity of action or of work, restored by the spring in unbending or expanding. § 90.'—If, then, the spring be perfectly elastic, the Perfectly elastic velocity communicated to the ball, will be precisely equal th°e J^68,™.^ to that impressed upon it by the air-gun in a contrary lost during an direction. Thus, in the example before us, the quan- mpa tity of work has been alternately changed into living force, and living force into quantity of work, without any thing having been lost or gained. But if the spring be not perfectly elastic, a portion of the living force im- pressed upon the ball will be employed in destroying the molecular force of the spring, that is to say, in producing a permanent change in the arrangement of its particles. § 91.—Hence, in the collision of bodies, not perfectly elastic, there will always be a loss of quantity of work, and this, from what has already been said, must be equal to half the living force destroyed. Few, if any, solid Living force is /.ii. -i .i . • -j. always lost in the bodies are perfectly elastic, and as the vast majority are, comsionofbodiea to a considerable degree, deficient in this quality, the not perfectly quantity of work uselessly consumed by the molecular forces will, in general, bear an appreciable ratio to that developed by inertia during the compression; and it there- fore follows, that if this last force, or the velocity which occasions the collision, be considerable, there will take place, in a very short time, a great loss in the quantity of action; and this is why it is important, as before re- marked, to avoid all shocks in the motion of machinery. § 92.—We also see, from what precedes, that it is as The work impossible for the force of a spring to develop, in un- ^TetceTd bendino- a living force greater than that consumed in that consumed in °' n i o a or • creating the bending it, as for the force of gravity, § 65, to give moter> to a body while falling, a living force exceeding that destroyed in it, through the same height, while rising; indeed, the whole of the velocity will not, in general, 7 98 NATURAL PHILOSOPHY. / be restored, and as the corresponding living force lost in the shock, has really been employed to overcome a certain resistance, and therefore to produce a certain quantity of work, it is true, as before stated, that inertia does actually perform an amount of work equivalent to that which has been employed in putting it into action; only it happens, that, in certain cases, a portion of this work is diverted from the object we desire to accom- plish, and is not, on that account, regarded as forming a part of the useful effect, as was explained in § 50 with regard to tbe ordinary force of pressure. what takes place § 93.—We have shown, by examples, how the quantity m pei 10 ica ^ work or of action may be transformed alternately into motion; J J living force, and living force into quantity of action, by means of springs and machines which store up and give them out successively. These transformations take place, in general, whenever the motion of a body solicited by a motive force varies, by insensible degrees, so as sometimes to be accelerated and sometimes retarded. This occurs, for example, in the periodical motion spoken of in § 25, and, in general, in all cases of forward and backward movement, usually called alternating, and in which the velocity becomes nothing from time to time. The motion of the pendulum and that of the plumb-bob are evident when the examples of this last kind. W hen the velocity of a body velocity is . . . . , increased, inertia augments, it is a sign that some portion of the moter's opposes the work acts in the direction of the body's motion, and in- force; ....„, creases its living force by a quantity double this portion when the of work; the other portion being absorbed by resistances; velocity .-, , i i • diminishes, lh on tne contrary, the velocity of the body diminish, inertia aids the notwithstanding the power may be exerted in the direc- tion of the motion, a certain portion of the living force acquired will be expended against the resistances, and will augment the work of the moter by a quantity equal to half the living force thus expended, and so on, according to the number of alternations. MECHANICS OF SOLIDS. 99 § 94.—From which we see, that when the velocity or living force of a body oscillates between certain limits, it is a proof that inertia has alternately absorbed and given out portions of the moter's work. The work absorbed by inertia will be the same for all equal velocities, and for the interval between the instants of equal velocities there will be nothing lost or gained, and the power must be con- sidered as having been entirely employed to overcome resistances other than inertia. But, if in any interval of time, the velocity, after having undergone alternations, does not attain to what it was before, the half of the difference of the living forces which correspond to the beginning and end of this interval, measures the quantity of work which has really been consumed or given out by the inertia of the body. Consequently, if the body were to set out from rest, the quantity of work consumed by its inertia up to any instant, would be measured by half the living force possessed by the body at this instant; if the velocity had increased incessantly, the inertia of the body would have opposed the motive force without intermis- sion ; if the velocity bad, during any part of the time, diminished, the inertia would have aided the force. Within the intervals between instants of equal velocities, the moter is not employed to overcome inertia; work absorbed or given out by inertia, equal to half the living force acquired or destroyed. § 95.—All of which may be made manifest by means of the second figure employed in § 86, in observing that when the velocity of the body diminishes, after hav- ing augmented during a certain time, so will the abscisses and ordinates of the right line 0v7, which represent this velocity; the extreme ordinates wn v7, after receding from the point 0, while the velocity is increasing, will, on the contrary, approach this point while 100 NATURAL PHILOSOPHY. r;;rriage drawn by horses. the velocity is diminishing, to keep the triangular area 0 w*, v7, constantly proportional to the quantity of work absorbed by the inertia, or to its equal, one half the living xampieofa force. A carriage travelling at a variable rate, sometimes faster, sometimes slower, offers an example of this: at first, the horses exert a certain quantity of action to move the carriage with a trot; then, when the velocity is dimin- ished, by an increase of resistance, or by feebler action on the part of the horses, the inertia of the carriage develops against the resistances to its motion, a portion of the work it had at first absorbed, equal to half the diminution of its living force: and this alternation will continue till the carriage is brought to rest, at which instant, the work restored by the inertia will be exactly equal to the quan- tity of work consumed, so that nothing will be lost. In what is here said, it is understood, however, that no dimi- nution of velocity results from opposition or holding back of the horses, for in that case, the moter would be con- verted into resistance. The same § 96.—The same reflections are applicable to the reflections apply weight 0f a carriaore in ascending and descending a hill. to weight as well ° ° ° ° as to inertia. The quantity of work employed in overcoming the weight while ascending will be restored during the descent, pro- vided the latter be not so steep as to cause the horses to hold back, by which a quantity of work would be con- sumed uselessly. And this consideration shows us one of the many advantages which results from giving gentle slopes to roads. when a force is § 97.—When a moter is employed to raise a burden '^weight, inlrtuT through a vertical height, it takes the body from a state retain* nothing of of rest, and hence a quantity of work must be expended ' to overcome its inertia. Arrived at the desired height, the effort of the moter is relaxed to restore the body to a state of rest, and during this diminished action, a por- tion of the living force acquired is employed to destroy MECHANICS OF SOLIDS. 101 in part the effect of the body's weight, and the inertia will finally retain nothing of what it had absorbed. The same thing may be said of the operation of an the same is true artificer in filing, sawing, &c, since at the end of each of th" !nertia of, " ° ' an artificer s tool, oscillation of the tool, the velocity becomes nothing through insensible variations. This could not be the case if the motion were suddenly to change, or if concussions should take place between bodies not perfectly elastic; a portion of the living force would, in that case, be de- stroyed, or, which is the same thing, diverted from its intended purpose in producing a permanent change in the arrangement of the particles of the colliding bodies. § 98.—Finally, in order to give a fuller idea of tbe part Examples of the performed by inertia in the various operations of the part Performed L by inertia; mechanic arts, and to demonstrate how it may serve to explain an almost infinite variety of effects, we shall add a few special examples to those already mentioned. To take from a plane-stock its chisel, the carpenter the chisel of a strikes the plane a blow on the back; a velocity is thus plaue; suddenly impressed upon the stock which the chisel and its wedge only partake of in part, because of their inertia and imperfect connection with the body of the plane, and are, therefore, left behind. A bung is taken from a cask by striking, on either side the bung of a of it, the stave in which it is inserted; the resistance cask; which the inertia of the bung opposes to the sudden mo- tion communicated to the stave, causes the separation. We often see a handle adjusted to a tool, as an axe or handles of tools; hammer, by striking it on the end in the direction of its length; the inertia of the handle and that of the tool tend to resist the sudden motion impressed by the blow, but the former yielding more than the latter, by reason of the sbght connection, the handle becomes inserted. As an illustration of the agency of inertia, in trans- forming quantity of action into living force, take the com- the common mon sling, from which a stone may be thrown with much 8l,ng; 102 NATURAL PHILOSOPHY. motive force. greater velocity than from the naked hand. Here, living force is accumulated in tbe stone, by whirling it through . many accelerated turns about the hand before it is dis- the common charged. The common top turns and runs along the whirling top. groun(jj m yirtue of the living force acquired during an accelerated unwinding of the string from the coils of which it is thrown. § 99.—We would recommend to the reader, to con- sider attentively these examples, as well as all others of like nature which his observation and memory may inertia sometimes furnish. They will aid his conceptions of the manner in ;' pa9Sive which the inertia of bodies, like their weight and molecu- rosistance; ' <-J lar spring, sometimes acts as a mere passive resistance, sometimes a real and sometimes as a real motive force, according to the circumstances. It is, however, proper to remark, that the last example is mainly concerned with the inertia of a body having a motion of rotation, while, thus far, we have only spoken of the living force of a body possessing a motion of transla- tion, in which all the particles have the same velocity; but we shall soon see, that the principles which connect the bving force with the quantity of action, are universal and applicable to all kinds of motion. IV". OF FORCES, WHOSE DIRECTIONS MEET IN A POINT. rorces whose § 100.—Thus far we have only considered the effect of directions meet • i r t i n in ma point; a sinS*e force, directly opposed to an equal force, viz.: to molecular spring or elasticity, to weight, or to inertia. It often happens that several forces are applied to a body, in different directions, to overcome certain resistances MECHANICS OF SOLIDS. 103 through its intervention. When a body is thus subjected to the action of several forces, {powers, or resistances,) we forces in say these forces are in equiiibrio, when one of them ^hb™ destroys or prevents the effect which the others would prevents the produce, if the first did not exist. The body itself is in others" & equiiibrio, if the different forces applied to it, leave it at rest. This last kind of equilibrium can never be abso- no absolute lute, because all bodies connected with the earth partake JJJJ^fium °f of its continual motion through space, and there is, in fact, no rest for them. A body may, however, have relative rest, as when it retains the same place in reference to surrounding objects, such as mountains, houses, &c, which we are in the habit of regarding as fixed. Thus, the idea statical and of equilibrium is not alone related to rest, and by no dynamical x ' J equilibrium. means excludes motion. From this results the distinction of statical and dynamical equibbrium; the former relating to the repose of the body, and tbe latter to the mutual destruction of the forces which solicit it. Thus, a body may be in motion while the forces acting upon it are in equibbrio, or it may be at rest under the same circum- stances. § 101.—It has already been stated, § 43, that when Resultant of several forces act along the same right line and in the severalf01'ces; same direction, their effect will be equivalent to that of a single force equal to their sum, and which will therefore be their resultant. If these forces act in opposite direc- when acting tions, and along the same straight line, their resultant will alongthe same 70 a > jmg5 ,„ same or be equal to the excess of the sum of those which act in in different one direction, over the sum of those which act in the liectlous; opposite direction, and it will act in the direction of the greater of these sums. This is the case in which several forces are exerted in the direction of the same cord. The tension of the cord will be the same throughout, and it is not possible to draw its two ends with different efforts. The tension of a cord is the effort by which any two of tension of a cord; its consecutive portions are urged to separate from each other, 104 NATURAL PHILOSOPHY. the effect of unequal forces acting upon a cord. and this being the same throughout, the excess of the sum of the forces which act in one direction over that of those which act in the opposite direction, will be wholly employed in overcoming the cord's inertia and giving it motion. paths; relative or component paths; JP Fig. 82. S U1 ______ A. A"--- § 102.—When a body, or material point, moves from ' A to B, so as to describe the rectilineal path A B, each of the positions A and B may be projected upon the right lines 0 M and Parallelogram of 0 N, situated in the same plane with the line A B, by drawing paral- lels to these lines consider- ed as axes, the place A giving the two co-or- dinates A A' and A A", and the position B the two co-ordinates BB' and BB". The positions A' and A", on the axes, are simultaneous with the position A; and those of B, B", with the position B. The paths A' B' and A" B", on the directions 0 M and 0 N, are, therefore, described by the projections in the same time as the path A B by tbe moving point. The first are called component or relative paths in such and such directions. Prolong the co-ordinates of the points A and B, till the parallelogram AEBF is formed, and this principle will appear, viz.: the rectilineal path described by a point, may always be re- solved into two relative or component paths, in any two resolution of any directions, and these component paths will be the sides of a component parallelogram, constructed upon the path described by the paths; point as a diagonal, and parallel to the assumed directions. Eeciprocally, when we have the relative paths in any two directions, the true path, called the resultant, will be that MECHANICS OF SOLIDS. 105 diagonal of the parallelogram constructed upon the rela- composition of .. ,1 i-i ,i i j/i • • i. c j.- the relative paths. tive paths which passes through their point of meeting. §103.—It has been shown, that the velocity of a body ParaiieiogTam of in motion, is represented by the length of path described ve uniformly in any very small portion of time, assumed as the unit of time, and that it is only in the case of uniform motion, that the interval of time during which the velocity is estimated, may be taken as great as we please. The path A B, in the last figure, being described by the body in tbe same time that its relative paths true and relative A' B' and A" B" are described by its projections onve the directions 0 J/and ON, the first may be regarded as the point's true velocity, and the two last as its relative velocities. Hence the true velocity of a body, is the diagonal true velocity of a parallelogram constructed upon its two relative velocities, ^lve'°™ocitieg| estimated in any given directions whatever. and the reverse. § 104.—If the motion be curvilinear, the rectilineal di- Relative paths in agonal A B can no longer represent, in general, the path ^""tiont described. Nor, if tbe motion be varied, can its length measure the velocity, when the time of description is con- siderable. In such cases, conceive a given interval of time divided into a great Fig. 33. number of small and equal portions, and determine tbe relative paths described during each, by. the projec- tions of the moving point on tbe axes. Each pair of these relative paths will de- termine a parallelo- gram, of which the diagonal will be tbe corresponding -elementary path de- scribed by the point itself. Any one of these diagonals, T" ---------------------------j------------- -- , T \j/^a A' O i i I 1 1 A' T' / geometrical representation: 106 NATURAL PHILOSOPHY. construction of the direction of the body's motion. as A B, will sensibly coincide with an element of the curve, and its prolongation A T will be tangent to the curvilinear path. This tangent will determine the direc- tion of tbe body's motion at the instant, and may be drawn by laying off from the projections A' and A" of the body's place, the distances A' T' and A" T", equal respectively to double, triple, quadruple, or any number of times the body's relative velocities at the time, and drawing T T and T" T, respectively, parallel to the direc- tions 0 T" and 0 T. Roberval's method of constructing the tangent; results from the law which determines the path. Fig. 84. § 105.—When the law of a body's motion in two direc- tions is known, it is always possible by the preceding method to draw a tangent to the path described. Take, for example, the ellipse : this curve is generated by fixing at two points F and F', called the foci, the ends of a thread FA F', equal in length to a given line MM', called the trans- verse axis, and moving the point of a pencil A to all positions in which it will keep the thread stretched. Since, in the motion of the describing point, the sum of the lengths FA and A F' is always the same, the portion FA will increase just as much as the portion A F' will diminish, and therefore the point A tends to describe equal relative paths, or will have equal relative velocities, in the two directions A B and A F'. Hence, taking upon FA produced, and upon A F', the equal portions A B and A B', and completing the parallelogram A B CB', the diagonal A C, passing through the position of the point, will be a tangent line to the path described. This method, which, is due to Roberval, is very useful in MECHANICS OF SOLIDS. 107 all cases where we know the law by which the curve is described. Fig. 35. § 106.—We have seen that any single motion may be resolved into two others, and the reverse. This arises from the simple fact, that a body may, in reality, be ani- mated by two or more simultaneous velocities. To illus-illustration of the trate, let it be supposed that while a boat is crossing a ^J^™' river, a man walks from one side of the boat to the other, velocities; and that, starting from the point A, for exam- ple, he arrives at B at the moment the boat reaches a position such that the point A shall be at A', and the point B at B'. It is plain, that the man, though only conscious of hav- ing walked across the boat from A to B, will, in fact, have been carried from A to B' in reference to the surface of the river. He will have moved, at the same time, with the velocity which he impressed upon himself, and that impressed upon him by the boat. This being understood, it is easy to see that the result would be the same, if the boat example of four first move from A to A', and afterward the man walk s" across it from A' to B'; or if the boat were stationary, while the man is crossing it from A to B, and then were to move from B to B'. But this is not all; the earth turns about its axis, while the boat floats along the surface of the water, and the man walks across the deck of the boat; add now the motion of the earth about the sun through space, and we shall find the man ani- mated by four simultaneous velocities, of which it is easy to see that we shall find the resultant, in compounding, by the rule given in § 103, first, any two, then the resul- motions; 108 NATURAL PHILOSOPHY. resultant of several simultaneous velocities; rule; illustration. tant of these two with the third, and the resultant of the three with the fourth. In fact, when a body has several simultaneous motions, the effect is the same as if the body had re- ceived, one after the other, all the motions which it possesses at the same time. Hence, this rule, viz.: The resultant of several simul- taneous velocities is found by con- structing a polygon, of which the sides are equal and parallel to the component velocities, and by join- ing, with a right line, the point of departure with the extremity of the last side. This right line will represent the resultant required. Thus, let the point 0 have the simultaneous velocities OV, OV, 0 V", 0 V"; from the extremity V of 0 V, draw Vm parallel, and equal to 0 V; from m draw m m> parallel, and equal to 0 V"; from m' draw m' m" parallel, and equal to 0 V", and join 0 with m"; the line Om" will be the resultant velocity. independence of § 107.—The action of a force upon a body, whether at the action of simultaneous forces; rest or in motion, is always the same, and impresses upon it the same degree of velocity. Let a body fall, for exam- ple, under the action of its own weight, gravity will impress upon it the same velocity in a given portion of time, whether it set out from rest or is projected down- 'ward by the action of some other force. For Fig. 31. example, when a bomb- shell is thrown into the air, it describes a curve, under the joint action of the living force with which it leaves the mortar, and the incessant action of its MECHANICS OF SOLIDS. 109 weight, and its velocity at any instant is the resultant MR, of the velocity M Q, which it has received at the instant immediately preceding that we are considering, and the small velocity MP, which its weight can impress upon it during the very short interval of time between these two instants. Thus, when two forces are applied to two forces the same body, they impress upon it, at each instant, and ^uTneousiy, simultaneously, the same degree of velocity which each the same velocity would impress if acting alone. This degree of velocity, "pJJ^JJ we have said, § 81, is, from the general' law of nature, proportional to the intensities of the forces. § 108.—Accordingly, let a material point A be acted Parallelogram „ 1 ,„ n - ■ mi offorces; upon by the two forces P and Q, represented in intensity and direction by the fines A B and A C respectively. These forces will impress simultaneously, and in their respective directions, the same degrees of velocity A m and An, as though each acted separately. The resultant velocity will, § 107, be represented by the diagonal A r of the parallelo- gram A m r n. Conceive a force X, to act upon the point along this diagonal, but in the opposite direction, or from r to A, and with such intensity as to destroy this velocity; no motion can take place, so that the force X, destroying the effect of the forces P and Q, will maintain these forces in equiiibrio. Take, up- on tbe diagonal, the distance AD = X, and conceive it to represent a force that acts upon the point A, from A towards D; it will produce the same effect as the forces P and Q, and will, therefore, be their resultant. Now, the forces P and Q, and their Fig. 38. 110 NATURAL PHILOSOPHY. the resultant of any two oblique forces applied to a point; represented by the diagonal of a parallelogram; forces combined by the same rules as velocities; experimental illustration of the parallelogram of forces. resultant A D, equal in intensity to X, are proportional to the velocities Am, An, and A r, which they can simultaneously produce, and, therefore, A D must be the diagonal of the parallelogram constructed upon the fines A B and A C as sides. Whence results this important principle, known under the denomination of parallelogram of forces, viz.: the resultant of any two forces applied to the same point, is represented, in magnitude and direction, by the diagonal of a parallelogram, constructed upon the lines which represent,'in intensity and direction, the two forces. It must not be forgotten that a force is, in geometrical inves- tigations of mechanics, always represented by a portion of its line of direction, containing as many linear units as there are pounds in the intensity of the force. It is plain, therefore, that forces may be combined by the same rules as velocities; and this is confirmed by experiment. If, for example, we attach to a cord A C B, fixed at its two ends, a weight R = fifteen pounds, it is easy, by a balance- spring, to measure the efforts exerted in the directions C A and C B. Laying off up- on the vertical through C, and from the point C, a distance C D equal to 15 inches, and completing the parallelogram by draw- ing D a and D b paral- lel respectively to CB and C A, we shall find the number of inches in C a and Cb to be the same as the number of pounds indicated by the bal- ances A and B. Fig. 39. § 109.—By the same principle that two forces, applied MECHANICS OF SOLIDS. Ill to the same point, may, without change of effect, be re- Resolution of a placed by a single one, may a single force be replaced by J^jto1 two others, acting in given directions. Let a given force, applied to the point 0, be represented in direction and in- tensity by the line 0 r: its com- ponents, in any two assumed directions, as Oi and 0B, are thus found. Through the point r, the extremity of 0 r, draw r m and r n parallel, respec- tively, to 0 B and 0 A ; the portions 0 m and 0 n will repre- sent tbe components required. Make Om = P; On = Q; Or = R; the angle trigonometrical AOB=v=mB= 180° - rn 0. Then, in the tri- ^^ angle Orn, because Om = rn = P, we shall have two components; R2 = p* + Q> + 2PQ COS applied to a point, is equal to the algebraic sum of the work sum of the quantities of work of the two components. When the projection of a force falls on the same side of the point of application as the path described, and we give the corresponding elementary quantity of work the positive sign, then when it falls on the opposite side,, the work must have the negative sign. of it components. Motion about a fixed poiut. Fig. 46. § 113.—The small space A a, may be described in different ways. If we suppose, for example, that the point of application A is on an axle A 0, which turns horizontally about some point 0, taken arbitrarily in the plane of its motion, as in the case of a bark or mortar mill, the path A a becomes the small arc of a circle, which we may regard as a small right line perpendicular to A 0. From the point a, let fall the perpen- diculars ab, ad, and ac, upon the directions of the forces P, Q, and their resultant R; then will the elementary quan- tities of work due to these forces be respectively P x Ab, Q X Ad, and P x Ac; and from § 112. Px Ac = P X Ab ±Q x Ad. From the point 0, about which the motion takes place, let fall the perpendiculars Op, Oq, and Or, upon the directions of the forces P, Q, and P, respectively; the triangles A Op and A a b are similar, since each has a right angle, and the angle A Op, of the first, is equal MECHANICS OF SOLIDS. 117 to the angle a A b of the second, the sides A 0 and Op being, respectively, perpendicular to the sides A a and A b ; hence, Ab: Aa : : Op : A 0; whence, * i ^ Aa Ab = Opx -j0j and, in like manner, from the similar triangles Ada and 0 Aq, we have . Aa Ad = Oq . -j-q\ and from the similar triangles A ca and A Or, Aa Ac = Or X AO' these values, substituted in the above equation, give, after omitting the common factors, and making Or = r, Oq = q, and Op = p, Rr = Pxp± QXq. . . (35). The effective quantity of work which a force is capable Moment of a of performing, while its point of application is constrained 0i to describe an elementary path A a, about a fixed centre 0, is called the moment of the force; the fixed point 0 is called the centre of moments; and the perpendiculars the centre of p, q, and r, the lever arms of the forces P, Q, and R, "^m. respectively. The elementary quantities of work performed by tbe forces P, Q, and R, during the description of the path A a, are measured by the products Pp, Q q, and R r, A n multiplied each by the constant ratio -j-k', and if this lis tlu relative lilt i>ure of a Uln ineut; the moment of th< ivultaut of tWl forces. NATURAL PHILOSOPHY. constant ratio be omitted, these products may be taken as the relative measures of the elementary quantities of work. Hence, the relative measure of a moment, is the product of the intensity of the force into its lever arm; and from Eq. (35) we see that the moment of the resultant of two forces, applied to a point, is equal to the algebraic sum of the moments of the components. § 114.—In what precedes, the two forces, P and Q, have been supposed to be applied to the same point; if they wii. n the forces be applied to different points ui. notappie to q an^ d jj. js evident that we the -ame point; ' may suppose two rigid bars, Fig- 46- C A and B A, to be firmly attached to the body, and to coincide in direction with the given forces. These bars, if the forces act in the same plane, will meet at the point A, and the latter thus becom- ing invariably connected with • the body, may be taken as the common point of application, without changing the effect of the forces. The resultant A R will be obtained by means of the diagonal of the parallelogram A PR Q, and the point D, where it meets the surface, may be taken as its point of application. If, now, the body be con- strained to move around any point, as 0, the common point of application A, will describe the small arc of a circle, which may be regarded as a small right line, to tii.-moment of be projected on the directions of the forces, as in the last tii reMiltant is ntill equal to th( ai..-v,raie sum of case also, the moment of the resultant is equal to the th- moments of th«- components. article; and the same reasoning will show us, that in this ntill equal to the al^fViraic sum of th- moments of i algebraic sum of the moments of the components. § 115.—The relations which have just been established between the quantities of work, and between the mo- MECHANICS OF SOLIDS. 119 ments of forces and of their resultant, will always obtain These relations wherever the point 0 be taken, since its selection was * hTreveTthe entirely arbitrary; but these relations were obtained by centre of . i . , i . „ , . ,, moments be considering the motion of the point, common to tne taken. directions of the forces, this point being assumed as their common point of application. To show that they are equally true in regard to the motion of the true points of application B, C, and D, see the last figure, we have only to remark that the measure of the moment depends alone upon the intensity of the force, and the length of the perpendicular drawn from the centre of moments to its line of direction, and is wholly independent of the posi- tion of the point of application. The moment of the or wherever the force P, for example, will be the same whether it be sup- a^auon posed applied at A, or at the point B, where its direction meets the surface of the body. The theorem of moments will be true, therefore, when the forces P and Q are not applied to the same point. § 116.—If it be shown that the quantity of work of Extension of the a force is the same, whatever point be taken on its line1 eoremo^the ' r quantity of work; of direction as the point of application, it is obvious that the theorem of the quantity of work, estimated by the motion of the common point of union of two forces and their resultant, will be equally true of all cases in which the quantities of work of these forces are computed in reference to the motion of their respective points of application. Three cases may arise, according work estimated as the body has a motion of rotation, of translation, or y of both combined. First case. The body and the di- rection A P, of the force P, being supposed to have a motion of rota- tion about the point 0, any two points, as A and B of the Fig. 47. Aa BV application on line of direction; First—in motion of rotation; 120 NATURAL PHILOSOPHY. line A P, will de- scribe arcs which are proportional to their distance, 0 A and OB, from 0; and we shall have Fig. 47. .PS' Aa To Bb OB Second—in motion of translation; but the quantity of work of the force P, estimated by the motion of its point of application supposed at A, will have, § 113, for its measure, r> n Aa PX 0pX0~A] or estimated by the motion of its point of application, supposed at B, will be measured by PxOPx^. Hence, the quantities of work are equal, being measured by the product of the intensity P, the length of the per- pendicular Op, and the equal factors -77-r, and 77-5-. Second case. If the body only have a motion of translation, any two points of ap- plication, as A and B, will describe the equal and parallel paths A a and Bb, which will be projected upon the direction A P, in the equal paths A a' and Bb'; and the quantities of work in the two cases being P X Aa' and P X Bb', are equal to each other. Third case. Suppose the line of direction A P of the Fig. 48. *' •MECHANICS OF SOLIDS. 121 combined; jb b p force P, to take the position Ax Bx, in virtue of the com- Third-when the bined motion of rotation and translation, and the points ™ansiation°and of A and B to be transferred to the positions a and b. This rotation motion of the points A and B may be regarded as resolved into a motion of rotation around the point 0, the centre of a circle, tangent to the two positions Fig. 49. of the line of direc- tion, supposed in- definitely near each other, and of trans- lation along the sec- ond position of this line. By the first, the points A and B are carried in the arcs of circles to Ax and Bx, and by the second, from these latter positions to a and b, thus making A a and Bb the actual paths described. Projecting these latter paths on the primitive direction of the force by the perpendiculars a a' and b b', we shall have for the quan- tities of work, considered in reference to the motion of the points A and B, P x A a' and P X Bb', respectively. But by projecting the points Ax and Bx on the primitive direction, by the perpendiculars Ax Ax' and Bx Bx, we have Aa' = Ax'a' - AX'A, Bb' = Bx'b' - BX'B; multiplying each equation by P, PX Aa' = P X Ax'a' - P X Axf A, P X Bb' = P X B{V - PxB{B. Now P X A{ a', and P X Bx b', are the quantities of work, on the supposition of a simple motion of translation 122 NATURAL PHILOSOPHY. alone, in the direc- tion A B, and these have been shown3 in the second case, to be equal; whence, Ax'a' = Bx'b'. no matter where The products the points of application be P ^ A ' A taken on the lines of direction; i Fig. 49. PX B{B, measure the quantities of work due to the motion of A and B, on the supposition of a simple motion of rotation about 0, which have been shown to be equal, in the first case; whence, AX'A = B{B; and consequently, PX Aa' = P x Bb'. the work of the Thus, the relation given in § 112, between the quantity resultant, is equal 0£ work; 0f the resultant of two forces, and the total quan- to the algebraic ' ± sum of the tities of work of the components, subsists in all cases, work onhf whatever be the points of application, and whatever be components. the nature of the motion. § 117.—Eesuming Eq. (35), Rr = Pp ± Qq} when the in which r, p, and q, denote the lengths of the lever arm rrwhenifsune0' of the resultant R and of the two components P and Q, of direction we see that the moment R r, of the resultant, can only taedpo^r^* reduce to zero wnen the moments of the components P and Q are equal and have contrary signs. But the prod- MECHANICS OF SOLIDS. 123 uct R r, becomes nothing, either when R = 0, or r = 0. there win be an In the first case, the resultant is nothing, and there will eiullbuum- be an equilibrium independently of all other considera- tions. In the second case, the perpendicular r, which measures the distance of the line of direction of the resultant from the centre of moments, being nothing, indicates that the resultant passes through the fixed point. Again, the equality of the moments of the components, necessarily implies an equality in the quantity of work performed by each, and these quantities, having different signs, destroy each other; hence, there will be an equi- librium about a fixed point, when the resultant of the forces which act upon the body, passes through this fixed point. V. OF FORCES WHOSE DIRECTIONS ARE PARALLEL. 118.—It has been shown of two forces whose direc-Theorem of the tions intersect: 1st, that the line of direction of the resultant, will intersect those of the compo- nents in the same point; 2d, that the moment of the resul- tant is equal to the sum or dif- ference of the moments of the components, according as the components tend to turn the body upon which they act, in the same or in opposite directions about the centre of moments. Now, these properties, being entirely independent of the po- sition of the point of meeting 0, and of its distance from the body or centre of moments, will not cease to be true when quantity of work, and of the moments equally true, when the forces are parallel. 124 NATURAL PHILOSOPHY. Value of resultant when the components act in same direction; the point 0 is so far removed as to make the directions of the forces sensibly parallel: whence we must conclude, that the line of direction of the resultant of two parallel forces is in the plane of the forces, is parallel to the direc- tion of the forces, and that the moment of the resultant, taken in reference to any point in the plane of the forces, is equal to the sum or difference of the moments of the components, according as they tend to turn the system in the same or opposite directions about the centre of mo- ments. Fig. 51. Eesuming Eq. (31), and re- volving the directions of the forces P and Q about their points of application A and B till they become parallel, and the forces act in the same direc- tion, the angle

+ 2PQ = p+Q, Fig. 52. Again, revolving the directions as before, till they become par- allel and the forces act in op- posite directions, the angle

froma ° d L plane; Hence, the distance of the centre of gravity from any plane, is equal to the sum of the products obtained by multiplying the weight of each element of the body into its distance from this plane, divided by the whole weight of the body. Find the distance, given by this rule, from any three from three arbitrary planes, and the position of the centre of gravity a*sumed becomes known. This method, which becomes long and tedious in many instances, may be abridged according to circumstances, particularly when the object is to find the process may be centre of gravity of homogeneous bodies. A body is said *J^ °f to be homogeneous, when any two of its parts have the homogeneous . , , , -, bodies. same weight under equal volumes. 138 NATURAL PHILOSOPHY. Centre of gravity of regular and homogeneous bodies; of a bar; of a bar with equal spheres at the ends; centre of gravity of regular and homogeneous bodies, at the centre of figure; right prism; circle, &.c.; centre of gravity of a surface; of a line. Body symmetrical in reference to a plane; Fig. 62. 0 Q § 131.—Experience shows us that a bar A B, of wood, metal, or any other material, which is perfectly homo- geneous, will remain in equiiibrio in a horizontal position, if suspended by its middle point C; and hence the centre of gravity of this bar is situated at the middle of its length. The bar is also found to remain in equiiibrio when placed in a vertical posi- tion, if suspended by the central point of its end; and hence the centre of gravity is situated at the central point of its thick- ness. If the bar support at its ends equal spheres, it will still remain in equiiibrio when suspended by its middle point, if placed in a horizontal position. The centre of gravity of a sphere is at its centre of figure, for when suspended by any one of its points, the direction of the suspending thread always passes through that point. And it is a general principle, that the centres of gravity of all regular and homogeneous bodies are at their centres of figure. And, hence, a right prism or cylinder has its centre of gravity at the middle of its length, breadth, and thickness; a circle at its centre; and a right line at its middle point. By the centre of gravity of a surface, is understood that of a body of extreme thinness, such as paper, tin-foil, gold-leaf, &c.; and by the centre of gravity of a line, is meant that of a body whose breadth and thickness are very small as compared with its length. § 132.—A body is said to be symmetrical in reference MECHANICS OF SOLIDS. 139 Fig. 63. to a plane, when the latter cuts into two equal parts every perpendicular which is drawn to it, and which is termina- ted by the opposite extremes of the body. This plane is called the plane of symmetry. A body is symmetrical in reference to a fine, when it has two planes of symmetry passing through the line. This fine is called a line of symmetry. A surface is symmetrical in reference to a line, when the latter cuts into two equal parts, all the perpendiculars to it which are terminated on opposite sides by the con- tour of the surface. In all cases, the centre of gravity of homogeneous symmet- rical bodies, is situated in their planes, or lines of symmetry. Consider, for example, a curve having A B for its line of symmetry, and of which we have found the centres of gravity G and G, of the two halves A MB and A M B. These two halves being turned about the line of symmetry till one is ap- plied to the other, their centres of gravity will coincide; that is to say, the centres of gravity G and G, were, before the motion, situated upon a right line G G, perpendicular to the line A B. Hence, if the curves be supposed concentrated at their respective centres of gravity, G G becomes a right line, terminated by two material points whose common centre of gravity is at the middle point 0, on the line of symme- try. A similar reasoning may be applied to all bodies of symmetrical dimensions. The centre of gravity of a surface which has two axes of symmetry, is at the intersection of these axes. The transverse and conjugate axes of the ellipse, for ex- ample, being axes of symmetry, cut each other at the plane of symmetry; symmetrical in reference to a line; line of symmetry; surface symmetrical in reference to a line; centre of gravity in planes and lines of symmetry; illustration in case of a symmetrical curve; centre of gravity of a surface with two axes of symmetry; 140 NATURAL PHILOSOPHY. 0E±E0 axes of symmetiy case of the centre of gravity of the elliptical surface. For the same reason, the rectangle; centre of gravity of a rectangle is at the intersection of the right lines joining the middle points of its opposite sides. When a volume has a right line of symmetry, its centre of gravity is on this line. A volume with one right cylinder, with an elliptical base, has two planes of ^mmltry; symmetry, determined by the longer and shorter axes of the ellipse, its centre of grav- * ity is, therefore, on the line G G, joining the centres of gravity of the bases, and at its middle point 0. sphere many Other bodies are divided symmetrically, in an infinity of ways. Such, for example, is the sphere of which all the planes of symmetry pass through the centre of figure; it is for this reason that this point is also its centre of gravity. centre of gravity § 133.—If the regular homogeneous body contain homogeneous "within its boundary another homogeneous body of dif- bodies, one ferent density, the centre of gravity of the whole mass is within the other. r> -i i /* -,.. «.„ n. round, by first regarding it as of uniform density, and the same as that of the larger body ; the centre of gravity 0, obtained on this hypothesis, gives rise to a first approximation. We then conceive the weight w, of the Kg- 66. body supposed homogeneous, to be concentrated at the centre of grav- ity 0, and subtracting this weight w from the total weight W, we ob- tain a difference W— w, neglected in finding the point 0. Let 0' be the centre of gravity of the volume corresponding to this MECHANICS OF SOLIDS. 141 of a parallelogram; difference; join 0 with 0' by a right line, and divide this fine at the point K, so that w x OK = (W - w) KO'; the point K will be the common centre of gravity. § 134.—Whenever a body may be divided into parallel when the layers layers, and the centres of gravity of these are situated on ofabodynave " J their centres of a right line, the centre of gravity of the whole body is also gravity on a right upon this line. For compounding the weights of any two lme" of these layers, supposed concentrated at their respective centres of gravity, and the resultant of these with the weight of a third, &c, it is easy to see, from the principle of parallel'forces, that the point into which the whole weight must be concentrated will be on the line in question. §135.—If, for example, the parallelogram A B C D, centre of gravity supposed to possess a small thickness, be di- vided by planes par- Fig. 61. allel to CD, into an indefinite number of strata or layers, the centre of gravity of each one will be at its middle point, and therefore on the line FE, joining the middle points of the opposite sides CD and AB; the centre of gravity of the parallelogram will, § 134, also be on this line. In like manner, it may be shown to be on the line IN, joining the middle points of the opposite sides C B and D A ; it must, therefore, be at their intersection 0. A similar reasoning will show that the centre of gravity of a parallelopipedon and cube, will be at the common of a intersection of three right lines joining the centres of |^£f pedoa gravity of their opposite faces. 142 NATURAL PHILOSOPHY. | 136.—The triangle ABC, being divided into very thin layers, parallel to the side A C, it follows, from what has just been said, that centre of gravity the centre of gravity of ofatriangle; ^ ^^ ^ ^g. fore, of the whole tri- angle, will be situated upon the right linePP, drawn from the vertex B to the middle of the side A C For the same reason, the centre of gravity of the trian- gle will also be on the line A F, drawn from Fig. 68. the angle A to the middle of the opposite side CB; and hence it must be at the intersection 67. Join F D. Since the sides A C and B C, are divided proportionally at the points D and F, the line D F is parallel to A B; hence the triangles A G B and D G F are similar, and give the proportion AG GF AB FD; but, because the points F and D are at the middle of the lines B C and A C, it follows that F D is half of A B, and, therefore, from the above proportion, F G is half A G; or F G is one third of the whole line A F. Hence, where situated; the centre of gravity of a triangle, is on a line drawn from one of the angles to the middle point of the opposite side, and at a distance from this side equal to one third of the line. common centre This point is also the common centre of gravity of thre^equaibaiis. tnree equal balls, whose centres of gravity are situated at the angles of the triangle, for the centre of gravity of the balls A and C is at the middle point D, and this point being joined with B, the centre of gravity of the three balls will divide the line B D at the point G, so that B G shall be double G D. MECHANICS OF SOLIDS. 143 § 137.—To find the centre of gravity of any polygon, centre of gravity as A B CDEF, draw from any one of the angles, as^l, ^p01**™1- the diagonals A C, AD, A E, kc, and thus divide the polygon into triangles. Find the centres of gravity g, g', g", g'", kc of each of these triangles by the rule above; join the points g and g' by the right line gg', and de- note the areas of the triangles ABC and A CD by a and a', respectively; then will the centre of gravity of the area AB CD A, be found by the pro- portion a + a' : a : : g g' : g' G. In like manner, joining G and g" by a right line, and denoting the area of the triangle A D E by a", will the centre of gravity of the area AB CD E A be found from the proportion, a + a' + a" : a" :: G g" : 67 67'; and so on to the last triangle; the quantities g' G, 67 G>, kc, being the only unknown quantities become known from the proportions. § 138.—A series of planes parallel to the base DBC, of the triangular pyramid AB C D, will give rise to a series of strata or layers per- fectly similar to the base, and all their centres of gravity will be situated upon a right line joining the centre of gravity of the base and the vertex, because they are all similarly situated to the base. Fig. 70. A M\ A pyramid divided into 1 il \ layers parallel / |! \ to the base; 144 NATURAL PHILOSOPHY. its centre of gravity found; As either of the solid angles may be taken as a vertex and the opposite face as a base, and as the dividing planes may be passed parallel to each of the bases, it follows that the centre of gravity of the pyramid must be upon the four lines drawn from the solid angles to the centre of gravity of the opposite faces, and must, therefore, be at their common point of intersection. Let 67' and 67" be the cen- tres of gravity of the triangu- lar faces A B D and BCD; join these points with the opposite vertices by the right lines A 67" and C 67', their point of intersection 67, will be the centre of gravity of the pyramid. Join 67' and 67" ; then, because the lines A E and E C are divided proportionally at the points 67' and 67", the line 67' 67" is parallel to A C, the triangles 67 67' 67" and 67 A Care similar, and give the proportion, 67'67" 67 67" A C : .4 67; where situated. The common centre of gravity of four equal balls. but 67' 67" is one third of A C, and hence 67 67" is one third of A G, or one fourth of A 67". The centre of gravity of a triangular pyramid is, therefore, on a line join- ing one of the angles with the centre of gravity of the opposite face, and at a distance from this face, equal to one fourth of the line. The same result may be obtained for the common centre of gravity of four equal balls, whose centres of gravity are situated at the four vertices of the pyramid. § 139.—The foregoing reasoning is equally applicable to a pyramid, of which the base is any polygon. For the MECHANICS OF SOLIDS. 145 Fig. 11. centre of gravity is on a line drawn from the vertex Sto centre of gravity the centre of gravity of the base, because it contains the ofanypyramid; centres of gravity of all sections parallel to the base; and if we conceive the pyramid divided into triangular pyramids by planes through this line, and through the angles A, B, C, D, kc of the base, the centres of gravity of these elementary pyramids, and therefore of the whole pyramid, will be situated in a plane parallel to the base, and at one fourth the distance from the base to the vertex; it must, therefore, be at the intersection of this line and plane. Hence, to find the centre of gravity of any pyramid, join the vertex with the centre of gravity of the where situated. base, and lay off a distance from the base on this line equal to one fourth of its length. This rule is also applicable to a cone, which may be centre of gravity regarded as a pyramid of an indefinite number of sides. of a cone. § 140.—Since every polyhedron may be divided into of any triangular pyramids whose weights may be supposed to Polyhedron- act at their respective centres of gravity, and since, from the principles of parallel forces, the sum of the products which result from multiplying the weight of each partial pyramid into the distance of its centre of gravity from any plane, is equal to the product of the entire weight of the polyhedron into the distance of its centre of gravity from the same plane, the distance of the centre of gravity from three planes may be found, and thus its position determined. § 141.—When a body is terminated by curved surfaces, of a body of any by planes, or by curve lines, it may be divided into smallform; elementary parts, similar to the figures which have been already considered—as right lines, triangles, parallelo- 10 146 NATURAL PHILOSOPHY. the partial products found; grams, pyramids, parallelo- Fig. 12. pipedons, polyhedrons, &c.; the sum of the products which result from multi- plying the weight of each into the distance of its cen- tre of gravity from some as- sumed plane, or right line, must be found, and this sum divided by the entire the sum of these weight of the body ; the result will be the distance of the enure6 weight; centre 0I> gravity from the plane or line. Let it be required, for example, to determine the centre of gravity of any plane area Cab Fd c ; draw in its plane any right line A B, and divide the given area into a series of very thin layers, perpendicular to this right line. The layer acdb, may be regarded as a small rectangle, and, supposing its density uniform, its centre of gravity is at its middle point 0; denoting the density by D, and the force of gravity by g, one of the partial products will be _. ac + db . . _ ac+db ea -f ec illustration; D X g X---s---- X %0 = Dg X ---5--- X ----x--. The other partial products being found in the same way, and their sum divided by the product of Dg into the entire area CcdFbaC, determined by the method of § 46, will give the distance of the centre of gravity of this area from the line A B. Performing the same operation in reference to another line A E, the centre of gravity is completely determined, being the intersection of two right lines, parallel respectively to A B and A E, and distant from them, equal to the results obtained by the above process. when the force It is to be remarked, that when the force of gravity g consTanTand ^ constant, and the density D is uniform throughout the density uniform; body, these quantities strike out, and leave the distance MECHANICS OF SOLIDS. 147 Use of the centre of gravity in computing volumes and surfaces; of the centre of gravity from the line, or plane, equal to the partial the sum of the products arising from multiplying the ele- ^a of the mentary volumes into the distances of their respective volumes. centres from the line or plane, divided by the entire volume. § 142.—The consideration of the centre of gravity is very useful in computing certain volumes and surfaces, which are found with considerable difficulty by the ordi- nary process. The screw, the curbs of stair-ways, surfaces of revolution generated by the rotation of a plane curve CDE about an axis A B, Fig-73. situated in its plane, are ex- amples. Suppose, in the case of a volume, the generating area CD E to be divided into small rectangles, of which the sides are parallel and perpen- dicular to the axis A B. Each rectangle will generate around the axis an elementary ring, and the sum of all these rings will give the volume of the solid of revolution. Let r denote the distance of the centre of gravity of one of these small rectangles from the axis; we know that the volume of the ring, of which the profile is the rectangle, is measured by the product of the area a of the rectangle, multiplied by the mean circumference of the ring, 2 * r; for the annular base of such a ring being developed, will form a trapezoid, the half sum of whose parallel sides is equal to 2 * r, and hence we shall have for the value of the ring the expression 2 *r r a. The volumes generated by the other rectangles, whose areas are a', a", a'", kc, will be 2 * r' a,' 2* r" a", 2 * r'" a'", kc And de- noting by Fthe total volume generated, we shall have V = 2«{ar + a'r' + a" r" + a"'r'" + &c); 148 NATURAL PHILOSOPHY. relation of volume to generatrix and path of centre of gravity; Fig. IS. but the quantity within the brackets, is the sum of the products which result from multiplying the elementary volumes of the generating area CED, by the distances of their respective centres of gravity from the line AB, which we know to be equal to the product of the whole area CED, into the distance of its centre of gravity from the same axis. Denoting the area CED by A, the distance of its centre of gravity from A B by P, we, therefore, have V = 2«RA (38). If, instead of an area, we had considered a plane curve C E, the quantities a, a', a", kc, would represent the lengths of ele- mentary portions of this curve, A Fig. 14. would represent its entire length, P would be the distance of its centre of gravity G, from the line A B, and V would be the value of the surface generated by the entire curve about A B. Whence we derive this rule, viz.: The volume generated by the motion of any plane, or surface generated by the motion of any line, is equal to the generatrix, multiplied by the path described by its centre of gravity ; the direction of the motion being perpendicular to the generatrix. This rule supposes the body to possess a constant pro- file, of which the plane is perpendicular to the path of the centre of gravity. MECHANICS OF SOLIDS. 149 Example 1st. Let it be required to find the volume generated by the ro- tation of the right-angled triangle ABC, about the side A B. The cen- tre of gravity 67, being found by the rule already explained, draw 67 D perpendicular to A B. Then, in the triangles EGD and EBC, we have CB : GD :: CE : 67 P whence GD = ±CB; and 2 * 67D = i«CB, which is the length of the path described by the centre of gravity. The area of the triangle is \AB X CB; whence the volume V becomes V = \«CB2 X AB, which is the usual measure of the volume of a cone. Example 2d. Let it be required to find the surface generated by the rota- tion of the line CD, about A B. The centre of gravity of C D is at its middle point 67; and GD', CA, andDB being perpendicular to A B, we have GD' = l(AC + BD); and for the path described by 67, ei> a 3 : 1; Fig. 76. example—the surface of a conic frustum : 150 NATURAL PHILOSOPHY. and hence, V = 2«AC + 2-kBD 2 .CD; • exnmple—the volume of a stairway curb; excavation from ditches; rule holds for iuiy portion of an entire revolution. Fig. 11. which is the usual measure for the convex surface of a conic frustum. Example Sd. Let it be required to find the volume of the curb of a stairway, of a helical form. First, compute the area of a section abed, perpendicular to a mean helix gg, or that described by the centre of gravity; then multiply this section by the length of the mean helix. The excavation taken from a ditch, of which the profile is con- stant, may be estimated in the same way. In examples 1st and 2d, the centre of gravity is supposed to have described an entire circumfe- rence ; but had it moved through only an eighth, tenth, or any other fractional portion of a circumference, the volume generated would still, as in example 3d, have been given by the area of the generatrix into the extent of the path described. VII. MOTION OF TRANSLATION OF A BODY OR SYSTEM OF BODIES. Motion of translation; § 143.—A body, or system of bodies, is said to have a simple motion of translation, when all its elements describe, simultaneously, equal and parallel paths. MECHANICS OF SOLIDS. 151 Denote by v the velocity which any motive force com- municates to all parts of the system during any small interval of time t. The force of inertia f of an element whose weight is p, will be given by the equation p V the measure of 1' T; f — £L J — ^ * . j the inertia of an element; and the force of inertia /', of an element whose weight isp', by p' v 9 f = £• J ~ ~ t1 and so of all the others, provided the degree of velocity impressed upon all the elements is the same during the time t. Moreover, as each force of inertia is exerted in the direction of the path along which the elements respec- tively move, and aS these are supposed parallel, the forces of inertia are parallel, and give a resultantfcqual in inten- sity to their algebraic sum. Denoting the intensity of this resultant by F, we have v / v 4- v' 4- P" + P'" 4- kc\ F = f + f'+f"+kc = -(^P a ); and replacing the sum of the partial weights by the entire weight P, and — shall finally have p weight P, and---by the entire mass if of the system, we f=m.^.....(39). ::™ It remains to find the invariable point of application of F. This point is called the centre of inertia. The inten-centre of inertia; sities of the forces / /', /", &c, are proportional to the weights p, p', p", kc, to which they are respectively applied, and thus the point of application of F, will coin- cide with that of the resultant of the forces p, p', p", kc; 152 NATURAL PHILOSOPHY. measure of that is to say, with that of the entire weight P, which is inertia in words. t]ie centre of g^yj^y 0f tne System. Hence, the total force of inertia of a body, or system of bodies, having a simple motion of translation, is measured by the mass of the system, multiplied into the ratio which the small degree of velocity communicated bears to the time during which the velocity is impressed. And the total force of inertia has its point of appli- cation at the centre of gravity. This coincidence of the centre of inertia with the centre of gravity, results from the assumption that the force of gravity is the same in its action upon the different parts of the system. Had it been otherwise, that is to say, centre of gravity had the force of gravity varied in intensity from one ele- and of inertia . ment to another, the centre of inertia, being always at the centre of mass, would be different from the centre of gravity. The intensity of the force of gravity, being regarded as the same witm$ the limits of a body on the earth's surface, iTbodies on thTtne centre of inertia and of gravity may be regarded as earth. coinciding, and hence these terms will be used indis- criminately. The force of gravity being constant, the coincide; these centres Quantity of motion of a body; § 144.—Let V represent the velocity of a body having a motion of translation, supposed uniform at any instant; the quantity of motion of any one of its elements whose weight is p, is measured by JLy 9 ' and of an element whose weight is p', £-v, and so for the other elements; and as these motions are parallel, their sum will give the quantity of motion of the entire body. Designating this quantity by Q, we shall have MECHANICS OF SOLIDS. 153 Q = P+P'+y + kcv=iMV . (40). its measure. Thus the total quantity of motion, in any body having a motion of translation, is measured by the mass of the body into its velocity. § 145.—When a certain degree of velocity v, is im-Motion of a pressed upon all the elements of a body during a very dirlcti^o/the short interval of time t, we have seen that the total force motive force of inertia is given by, Eq. (39), fhe cltreT gravity; F = M x -f; Fig. 18. We have seen, also, that this force of inertia is exerted in the direction of the body's motion, and through the centre of gravity. If, therefore, we suppose that at the instant in which the body has acquired the velocity v, a force equal to F is applied in a direction contrary to the motion, and at the centre of gravity, it will destroy the motion. This being supposed, if we apply at the centre of gravity of the body, a motive force X, it will commu- nicate to it a simple motion of translation. For this force X will be equal and directly op- posed to the force of inertia F, which it develops. This latter force F will be resolved into as many partial forces of inertia /, /', /", &c, as there are elementary portions of the body, and the inten- sities of these partial forces will be proportional to the respective weights of these elements. Denoting the masses of the elements by m, m', m", kc, we shall have, m w /' = m ~M F m /" =m-F, &<> 154 NATURAL PHILOSOPHY. The degree of velocity which each of these forces im- presses upon the part on which it acts, will, § 82, be measured by f_± Lz± r± to. m m m or, replacing/ /', /", &c., by their values as given above, simply by the expression F. t M ' and as this measure is the same as that before deduced, Eq. (39), for the degree of velocity impressed on the centre win be of gravity by the force F, or its equal X, we see that, trans°ati1™P ° ^° imPress a simple motion of translation upon any body, it is necessary that the line of direction of the motive force, or the resultant of the motive forces, when there are several, must pass through the centre of gravity ; and, reciprocally, if the line of direction of the force, or that of the resultant, in the case of several forces, pass through the centre of gravity, the body will have a simple motion of translation. § 146.—If the force X, were applied along the right line A B, Fig. 79. Motion when the not passing through the centre force does not n •, /-» •, • , pass through the °f SraVU7 #» lt 1S eaSy t0 See centreof gravity; that the motion cannot be one of simple translation. For, if this latter motion obtained, the partial forces of inertia would have a resultant of which the line of direction would, from what we have seen, pass through the centre of gravity 67; and if this resultant were replaced by an equal force F, applied along the same line and directly opposed to the motion, the latter would be destroyed, and an equilibrium would result. But it is impossible that two forces X and F, applied to the ex- tremities of a physical line or bar A 67, can produce an MECHANICS OF SOLIDS. 155 Fig. 80. 12L position of the element having a motion of translation. equilibrium, unless they act in the direction of the bar. ww be that of TT i 11 . ., . • ps? £ l • i translation and Hence, when a body receives the action of a force, ot wfiicfi of rotation at the the direction does not pass through the centre of gravity, same time; its motion will not be that of simple translation, but will be compounded of a motion of translation and of rotation; that is to say, some one of its elements will move, for the instant, in a right line, while the others will rotate about it as a centre. To find this element C, conceive a plane to be drawn through it, parallel to the direction of its motion, and per- pendicular to the planes in which the other elements, for the instant, rotate, and let A B be its trace upon that one of these planes which contains the point C, and its rectilineal path. Let m^ be the projection of some one element m' upon this latter plane, and take C Cx to represent the velocity v of translation, and m2mz the velocity of rotation acquired by the element m', in the small time t. Make mx m2 equal and parallel to C Cx; then would m^ m2 represent the velocity acquired by m', had the body moved with a simple motion of translation; but by virtue of the motion of rotation, the actual velocity acquired by m', in the direction of C^'s motion, is m^ m^, diminished or increased by the projection of ra2 mz upon the line C Cx according to the direction of the rotation. Project the points m^, m2, and mz, upon A B, by the perpendiculars rn^k^, m^k^, mz kz; then will the actual velocity v', acquired by ml, he mxm2 — m^o, or cv *3*» v = v — m9o: but m2o = rag m2 x cos ra3m2o = mzm2 x cos Cxra2k2; 156 NATURAL PHILOSOPHY. denoting Crn^ = Cx m2 = Cxmz by r, mx\ = m2k2hy yn and the velocity of rotation acquired by a point at the unit's distance from C by Vx, then will and m2ra.3 = Vxr, cos Cxm2k2 = —, Relative velocity which substituted above, give of two elements of a solid body in motion; v' = v - yxy> (41). Moreover, ra3 o is the velocity of the ele- ment m' perpendicu- lar to the direction of 6"s motion; and calling this velocity v", and the distance Cxk2, xn we shall the same in a have Fig. 80. direction perpendicular to the former; v" = vxXj (42). Denoting, as before, the weight of the element m' by p', and its force of inertia in the direction CCx by/', we have // _. P' v' P' , jr s f -J-T = Jt^v~v^ and similar expressions for the inertia of the other ele- ments. Taking the sum of these, and representing the inertia of the entire mass by F, we have, from the princi- ple of parallel forces, F 1 /»' W' »"' N t \g a a / MECHANICS OF SOLIDS. 157 or, denoting the entire mass of the body by M, and the masses of the several elements by m', ml', m'", kc, this reduces to tt value for the force F = M-------- • (m'y. + m"' y„ + rn!" y... + kc). of inertia of a t t body; Now the first term of the second member, which alone involves the motion of the point C, is wholly independent of the figure of the body and of the distribution of its elements. It will, therefore, remain the same whatever changes take place in its figure and size, provided its quantity of matter remain the same. The place of C, as determined from any supposition consistent with this last condition, will, therefore, be its position generally. This being understood, conceive the whole body to contract gradually in all directions till it is concentrated in a single point; this point must, from necessity, be the centre of gravity which alone remains undisturbed during contraction, as it will during an expansion, being the centre of mass. The point C, and the centre of gravity, not being disturbed by this change of volume, must coincide, and hence must always remain one and the same point. . But when the plane in reference to which the products m' y„ m" yn, kc, are taken, passes through the centre of gravity, we have m'y, + m"y„ + ml"'yul 4- &c. = 0; and the above equation reduces to which is identical with Eq. (39). always equal to the mass into ratio of the increment of velocity to that of the time. 158 NATURAL PHILOSOPHY. The body will We conclude, therefore, 1st, that when a body is acted have a motion of 7 , ■,,»..„ translation; it uPon ty one or more Jorcesi ^s centre of gravity will move will also rotate as though the forces were applied directly to it, provided their about the centre ... . 7 7 , of gravity directions remain unchanged; 2d, that when the line of di- rection of the force, or that of the resultant of several forces, does not pass through the centre of gravity, the body will, in addition, rotate about this centre. The law which regulates the motion of the centre of gravity results from the above equation, for if X represent the resultant of all the forces, and F the total force of inertia, we have from the equality of action and reaction, X= F, which value of F, substituted above gives, after reduction, value of the V" / velocity of V = —^—.....(42)'J translation. ■"* in which v is the velocity impressed in the very short interval t, from which we may pass to the velocity ac- quired at the expiration of any time, and thence to the space described. § 147.—What has been before explained, applies also to the total living force possessed by a body having a sim- ple motion of translation. For v being the common velocity of any one element, — X v2, will be the living i) p' force of that whose weight is p; — x v2 the living force of that whose weight is p', kc; so that the sum of all these living forces, or the total living force, denoted by L, will be v2 x------——^-------'-', and representing the entire mass of the system by M, as before, Living force in a Bimple motion of T __ yir' * translation. If the body have a motion of rotation as well as of MECHANICS OF SOLIDS. 159 translation, then will the living force of ml, in the direc- if the body have also a moi rotation; tion of the motion of translation be, Eq. (41) m' v' = ra' (v — Vx y) = w' v2 — 2 v. Vx ra' yt + Vx m' y2; and in the direction perpendicular to the motion of trans- lation, Eq. (42), m'v"2 = m'Vxx2; and similar expressions for the elements whose masses are ra", m'", kc Taking the sum of these, denoting the living force, as before, by L, and reducing by the equa- tions ml yJ 4- ra" ytl + kc = 0, y» + x2 = r2, y2 + xu = rJ2, kc kc = kc, m' + ra" + ra'" + kc = M; we find L = Mv2 + Vx (m' r2 + ra" r2 + &c.); or, making ml r2 + ra" r2 4- &c. = 2 . rn r2, L = Mv2 4- Vx . 2 ra r2 . . . (43). § 148.—The considerations which have now been developed, show that in the motion of translation of a body or system of bodies, the computations may be great- ly simplified, since we are permitted to disregard the shape of bodies, to suppose them concentrated about the living force is equal to that due to translation, increased by that due to rotation. 160 NATURAL PHILOSOPHY. their centres of gravity, and to reason upon these points as upon the total masses. General theorem § 149.—We have seen that in all questions affecting -the quantity of ^ circumstances of simple motion of translation, we may wort of weights; x • j regard the mass as concentrated about its centre of grav- ity. But when the different parts of a body receive motions which differ from each other, this concentration is generally inadmissible, since the partial forces of inertia not being parallel, their resultant will no longer be equal to their sum. If, however, we desire, in any case of the coexistence of various motions, to estimate the work per- formed by the weights of the parts of a body, during a given time, the action exerted by these latter forces being parallel, and their resultant or the total weight always pass- ing through the centre of gravity, we may still reason upon the motion of this point as though the mass were concen- trated at it, and disregard the motion of rotation of the other parts of the body about it. In this case, the quantity of work expended in every instance, will be obtained Fig. si. by taking the product of ^^ equal to the the weight into the path weight, mto the described by the centre of projection of path , _ of the centre of gravity, estimated in a ver- gravity. ^caj direction. If, for exam- ( pie, the centre of gravity of any body, as a bomb-shell, pass from the position G to 67', describing the curve G G', we obtain the work done by the weight during the interval of time occupied in passing from one of these positions to the other, by mul- tiplying the weight of the shell into 67' P, the projection of the path G G' on the vertical through 67'. This theorem, in regard to the work performed by the weight, is by no means restricted to the motion of a single body, but extends to a collection of pieces, such as wheels, MECHANICS OF SOLIDS. 161 bars, levers, &c, connected with each other after the man- Applies to an ner of ordinary machinery. If the quantity of work ^.^ performed by each piece be computed, and the algebraic sum be taken, it will be found to be equal to the quantity of work performed by the weight of the whole system, acting at its centre of gravity, computed by the same rule. In general, let p, p', p", kc, be the weights of the several pieces connected together; h, h', h", kc, the verti- cal distances passed over by their respective centres of gravity, in passing from one position to another, by virtue of their connection; P, the sum of all the weights or the weight of the entire system; and z, the vertical space described by the common centre of gravity: then will Pz = ph+p'h' +p"h" + &c. (44), To demonstrate this, let ra, ra', ra", &c, be several bodies so connected as to be acted upon by each other's weights. Let P de- note the weight of the entire system ; p, p', p", kc, the weights of the sever- al bodies m, ml, ra", &c.; Z, the distance of the common cen- tre of gravity from a horizontal plane A B; and H, H', H", kc, the distances of the centres of gravity of the bodies m, ra', ra", &c, from the same plane.. Then, from the principle of the centre of gravity, will. PZ = pH + p'H' +p"H"+kc; and for a second position of the system, mathematical expression of the rule; demonstration of the rule; PZ, = pH, + p'H; + p"Ht" + &c; u 162 NATURAL PHILOSOPHY. conclusion and rule. and subtracting the first from the second, P{Z-Z)=p{II-H)+p'{H;-H')+p"{H;'-H")+kc And supposing the horizontal plane of reference to be below both positions of the entire system, Zt — Z is the vertical distance z, through which the common centre of gravity has ascended or descended, according as Z, is greater or less than Z; H-H, 11/ - IP, H," - II" kc, are the corresponding distances h, h', h", through which the centres of gravity of the bodies m, ra', m", &c, have ascended or descended. Moreover, the products P [Zl—Z), p {Ht — H), p' (H/ — H'), kc, are the quantities of work due to the entire weight and to the partial weights. Whence this rule, viz.: The total quantity of work due to the action of the entire weight of any system, is equal to the sum of the quantities of work of the iveights which ascend, dimin- ished by the sum of the quantities of work of the iveights which descend. VIII. EQUILIBRIUM OF A SYSTEM OF HEAVY BODIES. Equilibrium of § 150.—If the system of heavy bodies be so connected, heavy bodies; an(^ jn sncYx condition that the common centre of gravity continue on the same horizontal line, while the bodies are made to take different positions, then will Zt — Z = z = 0, and Eq. (44) becomes, ph+p'h' + p"h" 4- &c. = 0 . . (45); partial quantities hence, the partial quantities of work of the several bodies of work destroy destroy each other, and, therefore, there must be an equi- each other; J ' ' x bbrium in tbe system, and the least extraneous effort MECHANICS OF SOLIDS. 163 will impart motion. Such is the condition of equilib- rium of a system of bodies acted upon only by their own weights. This equilibrium presents itself under dif- ferent states according to the positions of the system. If the slightest the position be such that in a slight derangement the efrort sumcient t0 _ ° give motion. common centre of gravity descend, it will tend to descend more and more, and a certain quantity of work will be requisite to restore it to its primitive position. Such an equilibrium is said to be unstable, because the system unstable tends of itself, on slight derangement, to depart from it. eiuilibrium 5 On the contrary, if on slightly displacing the system, the common centre of gravity ascend, this displacement will require the expenditure of a certain quantity of work which the weight of the system tends to restore; the equilibrium is then said to be stable, because the system is stable urged by its own weight to return to its primitive state when abandoned or left to itself. Finally, if during a slight derangement, the centre of gravity neither ascend nor descend, the quantity of work expended by the sys- tem is always nothing, the system will have no tendency of itself to return to, or depart from its first position, and equilibrium of the equilibrium is said to be indifferent. equilibrium; indifference. Fijr. 83. § 151.—Take a rod sus- pended at one end so as to turn freely about a hori- zontal axis A, and support- ing at the other a body which is symmetrical in reference to a line drawn from the axis A to the com- mon centre of gravity 67. It is obvious that there will be an equilibrium when the rod is vertical. It is more- over stable; for in deflect- ing the system, the centre of gravity will ascend while Illustration of stable equilibrium ; 104 NATURAL PHILOSOPHY. illustration of unstable equilibrium; both kinds illustrated by means of the cone; Fig. 84. describing the arc 67 6?', about A as a centre, and a certain amount of work will be expended which the weight will restore as soon as the deflecting cause is re- moved. Indeed, the system will, when abandoned, per- form' a series of oscillations, whose amplitude about the vertical A 67, will diminish continually till it comes to rest. Now suppose the system inverted; if the "rod be per- fectly vertical, the line of direc- tion of the weight will pass through the point of support A, and there is no reason why the system should move one way rather than another. It will there- fore be in equiiibrio, but the equi- librium will be unstable; for, however slight the derangement, the centre of gravity 67 will de- scend along the circular path 67 67', described about A as a cen- tre, and a certain amount of work will be requisite to bring it back to its primitive position. When a cone ABC, resting upon its base BC, is inclined to the position A' B' C, its centre of gravity G will ascend and describe an arc 67 67', and if, in this inclined po- sition, it be abandoned by the disturbing force, it will return. When the cone is placed up- on its vertex, with its centre of gravity directly above that point, it will also be in equiiibrio as it was when resting on its base, but the slightest motion will cause the centre of gravity to descend. The first position is one of stable, the second of unstable equilibrium. An elliptical cylinder placed upon a horizontal plane Fig. 86. MECHANICS OF SOLIDS. l-"'0 Fig. 87. also by an ellipsoid of revolution; Fig. 89. equilibrium of indifference exemplified by the sphere; is in stable or unstable equilibrium, according as the smaller or longer axis of its elliptical base, is perpendicular to the plane. A spherical ball upon a horizontal plane, is an example of equilibrium of indifference. The centre of gravity re- maining at the same level however the ball may be displaced, provided it pre- serve its contact with the plane, the quan- tity of work necessary to displace it will always be inappreciable, and the ball will, in consequence, have no tendency either to recede from or return to its primitive position. A perfectly circular cylinder on a horizontal plane is an example of the same kind. Some varieties of draw-bridges are but collections of by some varieties pieces in a state of equilibrium of indifference. And to ofdraw-urid",JS: insure this state, it is only necessary that the common centre of gravity of the bridge and appendages, shall pre- serve the same level during the motion, in which case, the system will be in equiiibrio in all possible positions. Wagons and carriages should, in strictness, require no work to move them on a horizontal plane, except to over- come their inertia, and should, therefore, be so constructed as to preserve their centres of gravity always on the same level. H, during the motion of a wheel, it is seen sometimes to quicken and sometimes to slack- en its motion, it is because the centre of gravity 6? is out of the axis of motion A, and, there- fore, alternately rises and falls during the rotation. A wheel whose centre of gravity is out of the axis of motion, passes Fig. 90. effect of throwing the centre of gravity out of the axis of a wheel. 166 NATURAL PHILOSOPHY. The common balance; in the course of a single revolution through the conditions of stable and unstable equilibrium, the former occurring when the centre of gravity 67 crosses the vertical line B C, through the axis ^1, at the lowest point 0, and the latter when it crosses the same line at the highest point 0', of its path. The common balance consists of a horizontal arm J. B, mounted upon a knife-edge D, resting upon the sur- face of a circular opening made in the end of a vertical frame C, which is supported by a hook attached to a fixed point E. The ends of the balance carry basins of equal weights, one of which receives a sub- stance to be weighed, and the other the stand- ard weights previously determined. The bal- ance may be stable, unstable, or indifferent, the position of its according as it tends centre of gravity; tQ ^^ tQ & horizon. tal position when de- flected from it, to over- turn, or to retain any position in which it may be placed. Refer- ring the entire system Fig. 91. when stable; unstable; indifferent; to any horizontal plane A'B', and taking the sum of the products which result from multiplying the weight of each piece by the distance of its centre of gravity from this plane, and dividing this sum by the weight of tbe entire balance; the quotient will give the distance of the com- mon centre of gravity of the moveable part of the appa- ratus from the plane A' B'. If this distance be less than FD, the distance of the knife-edge above the plane of reference, the balance will be stable; if greater, the bal- ance will be unstable; and if equal to this distance, the MECHANICS OF SOLIDS. 167 balance will be indifferent. All of which supposes the centre of gravity common centre of gravity to fall somewhere on the vertical on the vertlcal ° J through the point line FD, passing through the knife-edge. of support. IX. EQUILIBRIUM OF SEVERAL FORCES, VIRTUAL VELOCITIES, AND MOTION OF A SOLID BODY. • § 152.—To find the conditions of equilibrium of several forces, P, Q, R, S, kc, applied to different points of a solid Equilibrium of body, take in the interior of the body three points a, b, c, ^^body. and regard these points as the vertices of an invariable triangle abc; resolve each force into three components whose directions shall pass through the given point of application and the vertices a, b, and c. In this way we shall be able to replace the given forces by three groups of components, the directions of each group having a com- mon point at a, b, §r c. Each of these groups, having a common point, may be Fig. 92. replaced by a single resultant, and thus, the equilibrium of the giv- en forces is reduced to that of three forces. Call the resultant of the group having the common point a, X; that of the group hav- ing the common point b, Y; and that of the group having the com- mon point c, Z. These three forces being in equiiibrio, three single the equilibrium will not be affected by supposing the three the given forces may be replaced by three groups of components; and these by 168 NATURAL PHILOSOPHY. an equilibrium requires these three to act in the same plane; the resultant zero; that is to say, the K the resultant be lines ab, be, and ca, to become fixed in succession. The line ab being fixed, the forces AT and Y, whose directions intersect it, will be destroyed by its resistance, and if the third force Z, does not act in the plane a be, it Fig. 92. will cause the system to turn about a b; the same may be shown of the forces X and Y. The forces, X, Y, Z, must, therefore, act in the same plane; and in order that they may be in equiiibrio, the re- sultant of either two of them must be equal and directly opposed to the third: resultant of the three must be zero zero, the quantity of work is zero. The quantity of work of X, Y, or Z, is equal to the algebraic sum of the quan- tities of work of the group of which it is the resultant, and thus the sum of the quantities of wo»k of X, Y, and Z, may be replaced by that of the quantities of work of the forces grouped about a, b, and c. But these last, taken three by three, give the quantities of work of the proposed forces P, Q, R, S, kc; so that the sum of the quantities of work of the forces X, Y, and Z, is the same as the algebraic sum of the quantities of work of the forces P, Q, the forces will be P, S, &c. Whence we conclude, that several forces, acting whence"0 uPon the different points of a free body, will be in equiiibrio, algebraic sum of when the algebraic sum of the quantities of work of the forces the quantities of . 7 work is zero. ™ equal to zero. Now suppose the body to be slightly deranged from its state of rest, and let A A' be A" the path described by the point of application A, of Fig. 93. A' MECHANICS OF SOLIDS. 169 the force P, in an indefinitely short time t. Draw A'n perpendicular to A P; An will be tbe space de- scribed by the point of application in the direction of the force, and the quantity of work performed by P during the derangement will be P X A n, or Pp, denoting A n by p. The path A A' is called the virtual velocity of the force P; virtual velocity, An=p, the projection of the virtual velocity; and the Pr°Jectionof ■" ' z J J i virtual velocity product Pp, the virtual moment of the force P. and virtual Denoting by q, r, s, kc, the projections of the virtual momen ; velocities of the forces Q, R, S, kc, the quantities of work, or the virtual moments of these forces, will be, respectively, Q q, R r, S s, kc; and if the system be in equiiibrio, we have, from the rule just demonstrated, Pp + Q q + Rr 4- Ss + kc. = 0 . . (46). This equation is but the mathematical expression of the principle of principle, known under the name—virtual velocities, which >utual ve,0Clties- consists in this, viz.: when several forces are in equiiibrio, the algebraic sum of their virtual moments is equal to zero. § 153.—Any mechanical device that receives the action of a force or power at one point, and transmits a machine; it to another, is called a machine. Conceive a machine, composed of wheels whose axes are sustained by supports, and which communicate motion to each other, either by teeth, chains, or straps, on their circumferen- ces. Suppose a force or power to be applied so as to turn the first wheel; this wheel will experience a resistance from the second; this resistance, in its turn, becomes, for the second composed of wheels; i;o NATURAL PHILOSOPHY. the process by which the action of a moter is transmitted; points of support replaced by active forces; Fiar. 94. wheel, a power which causes it to rotate also; the second will experience a resistance from the third wheel, which resistance becomes a power to give it mo- tion, and so on to the end. But each wheel experiences a reaction at the points of sup- port which keep it in position, and it is this reaction that becomes the means of transmit- ting the power to the following wheel; for if these points were unsupported, the wheels would cease to act upon each other and the power first applied could not be trans- mitted. Now, replace the supports, by the efforts of reaction which they exert: each piece or wheel will become a free body subjected to the action of the preceding piece, the resistance of the following, and the force of reaction by which we have replaced its point of support; and if the piece be in equiiibrio, the algebraic sum of the virtual moments of this action, resistance, and reaction, must be equal to zero. Represent the power applied to give motion to the first wheel A by Wh the resistance of the second wheel B by R2, and the reaction at the point of support of the first wheel, by Cx; the projection of the virtual velocity of Wx by ivx, that of P2 by r2, and that of Cx by cx; then will sum of the virtual moments for first piece; Wxwx + CxCx + R2r2 = 0; denoting the resistance of the third wheel D by P3, the reaction at the centre of the second wheel by C2, and the MECHANICS OF SOLIDS. 171 projections of their virtual velocities by r3 and c2, respec- tively, P2 r2 4" G2 C2 -\- P3 r3 = 0 ; - same for second; and thus we may continue throughout the entire com- bination till we finally arrive at the last wheel, to which is opposed, as a final resistance, the work to be done. De- noting this resistance by We, the resistance of the last wheel to the action of the preceding by Rn tbe reaction of the support of the last wheel by Ce ; anol the projections of the corresponding virtual velocities by ive, re, and ce, respectively, we shall finally have, Re 1"e 4" Ce Cc -{- We We = 0. also for the last; But from the nature of the connection, the points of sup- port must not move; their virtual velocities, and there- fore the projections, must be zero. Hence, Cl cx = 0, C2c2 = 0, . . . Ce ce — 0, and the preceding equations become Wxwx + R2r2 = 0, R2 r2 4- R3rz = 0, ......=0, Re re 4" W6 We = 0, Subtracting the second from the first, and adding the third, subtracting from this result the fourth and adding the fifth, and so on to the last, we finally obtain Wx Wx 4" We We = 0 . . . (48); relation of motive r>o force to the final which shows us that the quantity of work of the final resistance is equal to the quantity of work of the power, or that no work is lost. In other words, the quantity no work lost. of work of the forces which tend to turn the system in one direction is exactly equal to the quantity of work of those which tend to turn it in the opposite direction. virtual moments (47). of points of support, zero; 172 NATURAL PHILOSOPHY. Tool; work of power equal to that of tool; the work of friction and that employed to change figure; work estimated by a supposed displacement, to avoid inertia; the same in uniform motion; when the motion is variable, the work of inertia comes into the account. An examination of Eqs. (47), will show that the same remark is applicable to each piece of the combina- tion taken separately, and thus starting from the piece which first receives the action of the force, and pro- ceeding to that which does the work, and which, on this account, is called the tool, we see that the quantity of work of the power is equal to that of the tool. In a word, where forces work upon bodies through the medium of machinery, we must distinguish the powers from re- sistances, and we shall always find the work of the first to be equal to that of the second. If the bodies press against each other in a way to produce a change of figure and friction, new resistances arise which must be taken into account, and the work of these must be subtracted from that of the forces to obtain the work of the tool, and hence such resistances are, in general, a hinderance to the final work to be accomplished. If the equilibrium is to be maintained while the machine is at rest, then must the quantity of work be estimated by the aid of a supposed displacement, as in that case, the influence of inertia will be avoided. If the equilibrium is to exist during a uniform motion of the machine, the quantity of work must be computed from the actual motion of the points of application, for then the inertia will again be excluded. But if the equilibrium is to take place during an acceleration or retardation of the motion, the inertia of the pieces will no longer be zero, and must be compre- hended among the powers and resistances. The con- ditions of the motion must, however, always be the same; that is to say, the work of the powers must be equal to that of the resistances, augmented by the work of inertia when the motion is accelerated, and diminished by the- same work when the motion is retarded. 154.—Whenever the forces applied to a body accel- MECHANICS OF SOLIDS. 173 erate or retard its motion, the inertia of the body is developed; and by virtue of the principle that action is equal and contrary to reaction, this inertia must be in equiiibrio with the forces; that is, the quantity of work of inertia will be equal to the sum of the simultaneous quantities of work of the forces which urge the body in one direction, diminished by the quantity of work of those which urge it in the opposite direction. But we have seen, § 85, that when the body takes, at different instants of time, two velocities which differ from each other, the work of inertia is measured by half the differ- ence of the living forces possessed by the body at these instants, or by half the living force gained or lost in the interval, according as the motion has been accelerated or retarded. Hence, tlie total work of several forces acting upon a body, during any time, is always equal to half of the living force gained or lost by the body during the same lime. Suppose, for example, a projectile whose weight is P, to leave the point A with an initial velocity V. If its weight did not act, the body would pursue its primitive recti- lineal path A T. Relation of the work of inertia to the work of all other forces; the total work of several forces, equal to half the living force lost t or gained; Fig. 95. But by virtue of the weight, which would act alone in vacuo, the projectile is con- tinually deflected from this path, and will, in conse- quence, describe a curve line ABD; and we know, § 112, that when a body describes any curve under the action of its weight alone, the work is equal to the weight of the body into the difference of level of its two positions. Thus, in the case before us, while the projectile illustration—the case of a projectile; 174 NATURAL PHILOSOPHY. is passing from A to B, the work expended by its weight will be P X B C, or P X H, by making P C= II. Denoting by V, * two points; the velocity of the the living force at projectile at P, its living force at this point will be P -rrfi — x V, 9 and at A, it was P 9 V\ loss or gain of living force; and its loss of living force, in passing from A to B, P 9 (V2 - V'2), the half of which is the quantity of work of the extraneous forces, (in this case the body's weight,) in the same time, and hence equal to double the work of the force; i^_(r_ y^ = PHi or relation of velocity to difference of level; V2 - V'2 = 2gH (49). Thus the difference of the squares of the velocities in any two positions of the projectile, moving in vacuo, is equal to the difference of level of the two positions, multiplied by twice the force of gravity. When the projectile arrives at D, then will H = 0; and V2 - V'2 = 0; MECHANICS OF SOLIDS. 175 Fig. 96. that is to say, the velocity will be equal to what it velocity same on , a the same level; was at A. From Eq. (49), it is obvious that while the projectile velocity on is on the ascending branch of the curve, its velocity di- descending"1 minishes, and while on the descending branch its velocity, branch of curve; on the contrary, increases. The description of the trajectory or curve ABD in vacuo, is obtained by very simple considerations, founded upon the independence of the motions of the same body, the curve in and of the action of forces which solicit it in the directions ^a of these motions, (§ 105 and 108.) The body may be re- garded as animated by two motions, one horizontal in the direction .1 x, the other vertical in the direction A y. The initial velocities in the directions of these motions are the components of the initial velocity V, computed by the principle of the parallelogram of velocities. After the body leaves the point A, it will be Xl s subjected to the action of no motive force in the horizontal direction; the horizontal component of its velocity will be constant, and the spaces described in this direction in equal times will be equal. Denote the angle xA T by a; the space de- scribed in the horizontal direction A x by x, and the time required for its description by t, then will _ /kan PjT.ce described X = V COS a t.....(50). horizontally in the time t; But in the vertical direction, the weight will, during equal times, diminish the component of the initial ve- locity, in that direction, by equal degrees; the motion will be uniformly varied, and the spaces described in the direction of the vertical A y, in the time t, will be given by Eq. (12), after substituting Fsin a for a, t for T, and 176 NATURAL PHILOSOPHY. space in same y for T^. Hence, denoting the vertical space by y, we get time in vertical direction; y = F sin a if — % gt2 (51). of the projectile at any instant; the true position The true positions of the projectile, which are but points of the curve A B D, are given by the intersections of a vertical and horizontal line drawn at distances from A, equal to the spaces y = Ayh and x = Axx, simultaneously described in these two directions. To find these distances, it will be sufficient to substitute a given value of t, in equations (50) and (51). Eliminating t from these same equations, and re- ducing, we find 9 the equation of the curve described—a parabola; angle of. projection; y = tan a . x x2. . COS" a (52); which is an equation of a parabola. Hence, the curve described by a body when thrown in a direction oblique to the horizon, and acted upon alone by its own weight, is a parabola. The horizontal distance intercepted between the point of projection 'A, and the point D where the projectile attains the same level, is called the range. The angle x A T = a, is' called the angle of projection. To find the range, make y = 0, in Eq. (52), and find the corresponding value of x. Making y = 0, we have 0 = tan a x 9 2 V2 cos2 a ' whence x = 0, 2 V2 sin a . cos a 9 = AD = range; and representing by h, the height, due to the velocity V, we have V2 = 2gh.....(53); MECHANICS OF SOLIDS. 177 and denoting the range by R, and recollecting that 2 sin a . cos a = sin 2 a, we have, finally, P = 2 h . sin 2a. . . . (54). the value of the range; This value for the range will be a maximum when a = 45°, in other words, the great- est range corres- ponds to an angle of projection equal to 45°. Fig. 91. Since sin 2 a = sin 2 (90° - a), it follows that the same range may be attained by two angles DAT and D A T', which are complements of each other. If in Eq. (54), we make a = 45°, then will complementary angles give the same range; P = 2A, whence greatest range given by an angle of projection equal to 45°; h = \R; and this in Eq. (53)r will give V = Vrj (55). That is to say, if the range corresponding to an angle of 45° be measured, the initial velocity may be readily found, being equal to the square root of the product of this range into the force of gravity. Squaring the above equation, we obtain value of initial velocity in terms of greatest range; V2 = Rg; 178 NATURAL PHILOSOPHY. and denoting by W, the weight of the projectile, its living force on leaving the mouth of the piece from which it is thrown becomes W 2 IL . V2 = P W; 9 and the effective quantity of action impressed, denoted byC?, effective quantity of action impressed upon the projectile; eprouvette; examples of its use; these results but approximations in air; Q (56). It is from this relation that are obtained the results of the eprouvette, a small mortar constructed to test the relative strength of different samples of gunpowder. For this purpose, a heavy solid ball is projected from it under an angle of 45°, with small but equal charges of different kinds of powder, and the relative strength is inferred from the effective quantity of action impressed. For example, suppose equal charges of two different samples of powder, give P = 1050 feet, and P = 1086 feet; these values substituted successively in Eq. (56) give Q = W. 525 Q = IF. 543; so that, the weights of the projectiles being the same, the strengths of the two samples of powder will be to each other as 525 to 543. This supposes the motion to take place in vacuo. If the trajectory be described in the air, the resistance of this fluid will diminish the velocity of the projectile, the curve will cease to be a parabola, and the results above will be but approximations to the truth. But as the resistance to the motion of the same body in air varies as the square of the velocity, these approximations may be made as MECHANICS OF SOLIDS. close as we please by using small charges and very dense these projectiles. approximations made close by Taking the general case, without limitation as regards giving smaii the velocity of a body in air, the curve may still be ve described, provided we have a table giving, in pounds or general case in any other unit of weight, the resistances corresponding to w"^a°ta different velocities of different calibres. thrown into the Thus, knowing the initial velocity and its two com- air; ponents, find from this table, in pounds, the value of the initial resistance, and its horizontal and vertical com- ponents at the commencement of the motion. Of these components, one is the motive force in the horizontal, and the other, added to the weight of the projectile, the tables of motive force in the vertical direction. With these forces atmospheri0 ' resistance; supposed constant during a very short time, compute by the laws of uniformly varied motion, the loss of velocity in these two directions during this short interval; subtract from the primitive components of the initial velocity, the loss in their respective directions; the remainders will be new component velocities, of which, find the resultant, and take from the tables the corresponding resistance. This new resistance treated in the same manner as that successive steps due to the initial velocity, will give a third resistance, and by which t0 J ' ° ' obtain the place this a fourth, and so on indefinitely. We thus obtain a of the projectile series of components, forces acting for a short time with a*any iI18tant ai,d x ° the curve. constant intensity in the horizontal and vertical direc- tions ; with these compute, by the laws of uniformly varied motion, the corresponding spaces described in their respective directions by the projectile. The total spaces simultaneously described, obtained by adding together the spaces corresponding to the same number of con- secutive intervals from the beginning of the motion, will give the distances, A x1 and A yx, which determine the points of the curve. The actual space described by the trajectory will be the development of this curve. 180 NATURAL PHILOSOPHY. X. MOTION AND EQUILIBRIUM OF A BODY ABOUT AN AXIS. The work of § 155.—The principle demonstrated in § 113, of the forces which turn inn ,- in i ., i t , a body about a work of forces acting upon a body, may be extended to fixed axis any case whatever. Let us now apply it to that of a body which is free to turn about a fixed axis with which it is invariably connected. Conceive a force R, acting upon the point A of a body free to turn about a fixed axis LM; resolve this force into two others, the one Q, parallel to LM, the other P in a plane perpendicular to this line, and passing through the point of application A. Doing the same with regard to all the other forces acting upon the body, the system will be reduced to two groups of forces, of which one will be parallel to the axis, and the other in planes at right angles to it. The algebraic sum of tbe quantities of work of the components is equal to that of the resultants. But the work of the first group, is equal to the product of their resultant, multiplied by the path described by the body in the direction of this resultant, that is to say, in the direction of the axis; but as is reduced to that the body is invariably connected with the axis, it cannot colbp7nentsin move in tnat direction, and the path described by the planes point of application of the resultant of the parallel group tterPax^CUl ° is notnmg> and therefore the quantity of work is nothing. Thus, the total quantity of work of the given forces is MECHANICS OF SOLIDS. 181 reduced to that of their components, in planes perpen- dicular to the axis, and passing through the points of application. § 156.—The quantity of work of forces applied to a The work of the body which can only have a motion of rotation is always, eomP0"ent^ *^ J ' perpendicular to as we have just seen, reduced to that of their components the axis; in planes perpendicular to the axis, or, which is the same thing, to that of the projections of the forces on these planes. It remains to determine this work. Let P be one of these components, A its point of application upon the body, C the point of the axis in which it is cut by the perpendicu- lar plane containing the component P. Let fall upon PA, the per- pendicular CD, and recall what has been demonstrated in § 116, viz.: that the quantity of work of a force is always the same wherever its point of application be taken upon its line of direction. The quantity of work of P, estimated by the path described by the point D, is the same as that estimated by the path of A. But the point D describes, in the short interval of time t, an arc S, of which CD is the radius, and, hence, the quantity of work of P will be P. S. As all the points of the body are invariably connected with the axis and with each other, they will describe simultaneously equal angles, and consequently arcs pro- portional to their distances from the axis; hence if S^ denote the length of arc described at the unit's distance, and r the distance of the point D from the axis, then will S = rSx, and the quantity of work of P becomes PrS,, the quantity of work of a single component; 1^2 NATURAL PHILOSOPHY. Fig. 99. the same for other components; the effective work of all the components; conclusion; tigns of the moments. and for forces of which P', P", &c, are the projections, at distances from the axis equal to r', r", kc, respec- tively, we have the quantities of work measured by, P'r'S,, P"r" Sl, kc kc Knowing that the total quantity of effective work of the given forces, which we will denote by Q, is equal to the sum of the work of those which tend to turn the body in one direction, diminished by the sum of the work of those which tend to turn it in an opposite direction, we shall have Q = Sx{Pr + P'r' + P"r" 4- &c.) . . (57). But we recognize Pr, as the moment of the component P in reference to the axis, and the same of P' r', P" r", kc ; whence, the effective work of the component, and consequently of the force itself, is equal to the product arising from multiplying the arc described at the unit's distance from the axis, into the moment, in reference to the same line, of the projection of the force on the perpendicular plane; and Eq. (57) shows that the effective quantity of work of several forces, applied to turn a body about an axis, is equal to the arc described at the unit's distance multiplied by the algebraic sum of the moments of the projections of the forces on planes perpendicular to the axis. The sign of the moments of those forces which tend to turn the body in one direction, must be different from the sign of those which tend to turn it in an opposite direction; in other words, if the sign of the first be positive, that of the latter must be negative. § 15^-—If tne given forces be in equiiibrio about the axis, their total work will be zero, whether the body be MECHANICS OF SOLIDS. 183 at rest or in motion; a condition that can only be fulfilled by making, in Eq. (57), Pr 4- PV 4- P"r" 4- &c. = 0 . . (58); that is to say, several forces will be in equiiibrio about a Forces in fixed axis, when the algebraic sum of the moments, in reference t0 a reference to this axis, of the projections of the forces on per- fixed axis. pendicular planes, is zero. §158.—When forces are applied to a body to turn Extension of the it about an axis, the motion of its particles can only take f ™eCgPJa°m^n place in planes perpendicular to the axis ; if the forces be of rotation; not in equiiibrio, the motion will be either accelerated or retarded, and will give rise to forces of inertia which act in the direction of the motion, and of which the.quantity of work will be equal to that developed in the same time by the motive forces. When all the points of the body have simultaneously the same velocity, the total quantity of work of inertia is equal to the product arising from multiplying half the mass into tbe difference of the squares of the common velocity at the beginning and end of the interval for which the estimate is made. But when the different points have different velocities during the same time, which is always the case in a motion of rotation, it is necessary to estimate at the beginning and the end of the interval, the living force of each element of the body, to take the sum of those at the beginning, and the sum of those at the end; the difference of these sums will be the total increment or decrement of living force during the interval. The half of this living force being the work of inertia, and this latter being equal to that developed by the motive forces, or in rotation, the rather by their projections on planes perpendicular to the workofthe J i. o i. j. x perpendicular axis, it is easy to perceive that in the motion of rotation components, is of a body, the work of the perpendicular components of ^ *eent of the forces is half of the increment of the living force living force. of the body. The process of estimating the living force 184 NATURAL PHILOSOPHY. of a body having a motion of rotation will now be given. Estimate of the living force of a body turning about a fixed axis; Fig. 100. § 159.—Consider an element m of a body, situated at a distance r from an axis of rotation LM. Denote by Fthe velocity which it has at any in- stant, and by p its weight, m its P mass = £-. Then will its living 9 a force be —. V or m V2. 9 If S denote the space described by m during a very short interval of time t, and S± the space de- scribed in the same time by a point at the unit's distance from the axis, we shall have the angular velocity; S = r. Si, and dividing both members by % S t (59); but we have seen that, in any motion whatever, the velocity is equal to the space described, during a very short interval of time, divided by this interval, hence t = V 4. rl ) relation of angular to absolute velocity; in which Vx is the velocity of the point at the unit's dis- tance from the axis—in other words, the angular velocity; and Eq. (59) becomes V=r.Vx, MECHANICS OF SOLIDS. 185 and the living force of m becomes mr2 Vx. The simul- taneous living force of m', is m' r' Vx, and so on of others; and the total living force of the entire body, denoted by L, becomes „ 22 value of the L = Vx {mr2 4- mlV -f m" r" + &c.) . . (60). living force of a rotating body; In which it is to be remarked, that if the living force changes, the factor Vx will alone vary, while the factor 2 2 (mr2 + m!r' + m"r" 4- kc) will remain constant, and of course, appear in the estimate of the new living force. This quantity, which has been called the moment of inertia, let us designate by I, and we have I = mr2 4- m'r'2 4- m" r"2 + kc . . (60)' L = VXI......(60)"; whence we see, that the living force of a body which turns equal to the about an axis, is equal to the product of the square of its ^^L^iocit angular velocity, multiplied by its moment of inertia. into the moment Let us suppose that at the end of a certain interval, ofinertia; the .angular velocity becomes Vx, the living force L', will be L' = Vfl; and subtracting the preceding equation from this one, we get L' - L = I. {Vf - Vx) . . . (61), increment of living force during any for the increment of the living force during this interval, interval; which is double the quantity of work produced by the motive forces, or their perpendicular components, during the same interval. Denote by F, the resultant of these components, and by E, the path described by its point 186 NATURAL PHILOSOPHY. of application, estimated^ in its own direction during the interval in question; then will equal to twice the quantity of work of the motive forces in same time. I.(VX' - Vx) = 2F.E. (62). From this expression it is easy to deduce the nature of the quantity I. For if we suppose the change in the angular velocity to be unity, Vi Vx = 1, and /= 2F.E; Fig. 101. what is meant by whence we conclude, that the moment of inertia of any body, inertia™611 ° is twice the quantity of work exerted by its inertia, during a its measure? change in its angular velocity equal to unity. It is measured by the sum of the products which arise from multiplying each elementary mass into the square of its distance from the axis, Eq. (60)'. § 160.—By the aid of what has just been explained, we may find the intensity of a motive force which causes a body to rotate about an axis, when we know the angular velocity at any two given instants of time, and the path described by the point of application in the interval between them. And reciprocally, if the force and the path described by the point of application be given, we may deduce the angular accel- eration. Suppose a wheel, for ex- ample, mounted upon a horizontal arbor and turned around its axis by a weight P, suspended from a cord wound around the arbor; required Example illustrative of the preceding principle; MECHANICS OF SOLIDS. 187 the angular velocity Vx of the wheel when, moving from wheel turned by the action of a a state of repose, the weight shall have descended through a vertical height H. Let / denote the moment of inertia of the wheel, then will the living force acquired be / Vx, and we shall have, I. Vx2 = 2P.H; whence Vt = 2P.H I and consequently xr _ a /_ P • H value of the T ' angular velocity; § 161.—The fly-wheel is a large ring, usually of metal, application to the of which the circumference is thrown to a considerable fly"wheel' distance from the arbor upon which it is mounted by means of radial arms, and is used to collect the work of a moter when the effort of the latter is greater than that required to overcome a given resistance, to be given out again when the resistance becomes greater than the effort of the moter. It is a kind of store-house in which to husband work for future use. Conceive one or more forces to act upon such a wheel during an. interval separating two given instants at which the angular velocities are Vx and Vx. The increment of the living force of the fly-wheel will be equal to double the effective quantity of work of the moter, and we shall have, retaining the notation of § 159, I(VX'2 - Vx) = 2FE; increment of Or living force in any interval; y,2 y2 __ 2F.E '1 "l -jf J 188 NATURAL PHILOSOPHY. the value of the difference of the squares of the velocities; the motion may be made to approach uniformity; use of fly-wheel; exemplified in the common saw-mill. which gives the difference of the squares of the angular velocities. If the quantity of work developed by the moter remain the same during the interval, and, by changing the wheel, the moment of inertia increase, the fraction 2F.E I ' and consequently the difference of the angular velocities at the beginning and end of the interval, will be less. And, as the moment of inertia is in the direct ratio of the mass into the square of its distance from the axis, it is plain that it is always possible so to construct a wheel as to make its motion approximate to uniformity, even though the motive force be very great. While the motion is accelerated, it is obvious that the work of the moter will exceed that of the resistance; the fly-wheel will acquire an increase of living force which it will retain till, on the contrary, the motion is retarded, when it will be again given out in aid of the moter, which now becomes less than the resistance. There are certain machines whose tool cannot perform its work without the fly-wheel. This is strikingly ex- emplified in the instance of the common saw-mill, in which it is obvious that the work during the ascent and descent of the saw is very different; the work of the moter exceeds that of the tool or saw during one semi-oscillation, while the reverse takes place during the other; in the first case, the saw is merely elevated and the fly-wheel absorbs living force; in the second, this living force is given out to aid the moter in overcoming the resistance opposed to the saw, which, in its descent, sinks into the wood and is thus made to perform its work. § 162.—If the elementary mass m, receive in the short interval t, the velocity V, and we denote by / its inertia, we shall have, Eq. (28), MECHANIC'S OF SOLIDS. 189 1YI y Distance from / = —-—; the axis at which the resultant inertia of a and for any other masses m', m", kc, whose acquired rotating body is velocities in the same time are V, V", ml V J — 4 1 m" V" If, moreover, the masses m, m', m", kc, form parts of a body which has a motion of rotation, their velocities will be proportional to their respective distances from the axis. Denoting these distances by r, r', r", kc, and by Vx the small degree of velocity impressed upon the point at the unit's distance from the axis, we shall have V = rVx; V = r'Vx; V" = r" Vx; which in the above equations give, -r-r T7" T7~ value of the f — mr. — ' f = m'r'.—-; f" — m" r" -A ■ kc. partial forces of t t t inertia; But, if this increment of angular velocity Vx, has been impressed upon the body by a force F, whose direction is perpendicular to the axis, and applied at a distance from it equal to P, this force is the measure of the inertia of the body, and will be in equiiibrio with all the partial forces of inertia f /', /", &c. But these latter act in directions tangent to the circles described by the masses m, m', m", kc, about the axis, and hence, § 157, FR — (fr 4- f'r' + f'r" -f &c.) = 0; equilibrium of these with the motive force or w equal to their Ffi _ Yl-imr2 + m'/ 4- m"r"2 + kc.) = 0; v resultant; 190 NATURAL PHILOSOPHY. the moment of the inertia but the expression within the brackets is the moment of inertia I, and therefore F.R = -5../ . . . . (63); • actually exerted; t whence we see, that the moment of the inertia exerted by a body while receiving a motion of rotation about an axis, is equal to its moment of inertia in reference to the same axis, multiplied into the quotient arising from dividing the small degree of angular velocity communicated, by the element distinction of the time during which it is impressed. Notwithstand- audwtmtis mg tne close analogy which exists between the moment of usually called the the inertia of a body, and what has been called the moment moment of c • • l i n i -\ inertia; °f inertia, the j must not be confounded with each other. V The former is converted into the latter by making — equal to unity. From Eq. (63) we find F. R x t Vx =----j^- .... (64), value for the angular velocity; ' 1 I from which, having given the motive force that im- presses a motion of rotation upon a body about an axis perpendicular to its direction, we may find, at each instant of time, the angular velocity communicated, provided we can calculate the moment of inertia of the body in reference to the same axis. And from this, it is possible, how used. by means of a curve which has for its abscisses the series of times t, and for its ordinates the velocities Vx acquired, to determine all the circumstances of the motion of rota- tion. § 163.—The moment of inertia of a body with refer- Measuresofthe ence to any axis, we have seen, is measured by the sum of inertia"80 ^G products which arise from multiplying each elemen- tary mass into the square of its distance from the axis. MECHANICS OF SOLIDS. 191 Of all the moments of inertia of the same body, those are easiest obtained which refer to axes through the centre of gravity. It is, therefore, important to be able to find the moment of inertia with those in reference to axes through centre of gravity easiest obtained; Fig. 102. 2VT reference to any axis, by means of that taken with reference to a par- allel axis through the centre of gravity. Let 67 H be this latter axis, LM any parallel axis, m an ele- mentary mass of the body GKH, through which element conceive a plane to be passed per- pendicular to the axes, and cutting them at the points a and b. Join m with a and b, and let fall from m, the perpendicular me upon ab. Designate mb by r, ma by rt, ab by D, and a e by d ; we shall have r2 = r2 + D2 + 2Dd, and multiplying by the mass m, mr2 = mr2 4- mD2 + 2mDd; and for the masses m', m", ml", kc, that in reference to any axis, in terms of the moment in reference to a parallel axis through centre of gravity; m' r' = ml r/ + m' D2 + 2m' Dd', m"r"2 = m"r/f2 + ml'D2 4- 2m"Dd", kc, kc, kc D, which is the distance between the two axes, remaining obviously the same in all. 192 NATURAL PHILOSOPHY the sum of all the partial moments; Adding these equations together, and denoting the moment of inertia in reference to the axis 67 H by Ix, and that in reference to L M by /, we find 1= Ix + D2 (m 4- m' 4- m" + kc.) + 2D(md + m'd' + m"d" + kc); but ro+ro' + m" 4- &c. is the entire mass of the body, and ra d -f m'd' + m"d" + kc. is the sum of the prod- ucts which result from multiplying each mass into its distance from a plane through the centre of gravity, which sum is equal to zero. Hence, designating the mass by M, we have Fig. 102. resulting value; I = Ix 4- MD'< (65); conclusion. whence we conclude that, the moment of inertia of a body, taken with reference to any axis, is equal to the moment of inertia taken with reference to a parallel axis passing through the centre of gravity, increased by the product of the entire mass of the body into the square of the distance from the centre of gravity to the first axis. It follows from this theorem, that if the distances value, when the of the particles of the body from its centre of gravity be small in" comparison with the distance of this point from the axis of rotation, we may take, for the moment of inertia, simply the product of the mass into the square of the distance of the axis from the centre of gravity. Finally, if Eq. (65) be multiplied by the square of linear dimensions of the bodies are small, in comparison with their distances from the axis; » MECHANICS OF SOLIDS. 193 the angular velocity, Vx, with which the body turns about the axis LM, we shall have VX.I= VXIX + M.D2VX2 • • OT; vaiueofthe living force; but Vx I is the living force of the body; Vx Ix is the living force it would have, if it rotated about a parallel axis through the centre of gravity with the same angular velocity Vx; M.D2 Vx2 is the living force of the same body supposed concentrated at its centre of gravity. Whence, the living force of a body which rotates about any axis, is equal to the living force of the same body concentrated expressed in at its centre of gravity, augmented by that which it wouldwords; possess if it turned, with the same angular velocity, about a parallel axis through the same centre. Finally, when the body is so small that Ix Vx2 may be neglected in comparison with M. D2 Vx, we have simply Vx2I= M.D2VX . . . (66)'; value when the linear dimensions that is to say, the living force of the body is equal to the yerytmaii^r6 product of its mass into the square, D2 Vx2, of the velocity compared with n • n • *ts distance from of its centre of gravity. tne axi8. § 164.—Thus far the moment of inertia of a body has been expressed in terms of its elementary masses. If the Moment of inertia body be homogeneous and the specific gravity or weight of^heTnineaT1"8 of a unit of its volume be denoted by S, its elementary dimensions and volumes by a, a', a", kc, and masses by m,m',m", kc, ensiy' we shall have $ a . S a' .. 8 a" . m = ---; m = ---; ra = ---, &c.; 9 9 9 and these in the general expression I, of the moment of inertia, give 1= —(ar2 4- a'/ 4" a" r"2 4- &C.); its general value; 13 t 194 NATURAL PHILOSOPHY. that is to say, to find the moment of inertia of any homo- ruie; geneous body, find the value of a r2 + a' r'2 + a" r"2 + kc, and multiply it by the quotient arising from dividing the specific gravity, or weight of a unit of its volume, by the force of gravity. moment of inertia of a straight bar, in reference to a perpendicular axis through its middle; ^ 165.—1st. The moment of inertia of a straight bar whose length is a and cross section b, in reference to an axis passing through its mid- dle point A, and perpendicu- lar to its length, is given by Fig. 103. A $L. D that of a right cylinder, in reference to its own axis; Ix = b . (^ a3), — very nearly. 2d. The moment of in- ertia of a right cylinder having a circular base, with respect to an axis through its centre of gravity, and coinciding with its axis of figure, is given by the equation QZ Fig. 104. i) $ = " that of a right prism with trapezoidal base; b + b' I=ac(-2—)-\ ±11 24 + * b + W b + b °-) X Fig. 112. 9th. If the trapezoidal base of the above prism be replaced by a segment of a parabola, of which c is the length of the transverse axis, and b that of the chord per- pendicular to it, and which terminates the parabola, the moment of inertia, with reference to an axis parallel to the altitude and passing through the middle of b, is given by the same when the base becomes a segment of a parabola. /= \abc( 3 .5 . ft* + 16 c8' 70 ) x —• 9 §166.—We shall close this subject with an example Application to for the sake of illustration, and we shall first take that examPlea; of a trip-hammer, whose weight is P, mounted up- Fig j13 on a handle in the shape of a rectangular paral- lelopipedon which turns freely about an axis 0, at right angles to its length. Denote by P, the distance of the centre of gravity of the head B from the axis 0. that of a common trip-hammer; 198 NATURAL PHILOSOPHY. the moment of inertia of the head; If the linear dimensions of the head be small com- pared with this distance, its moment of inertia will not differ much from p — X P2, 9 Fig. 113. that of the handle with reference to its centre of gravity; and that of its handle is given in reference to an axis through its centre of gravity by the 7th case, or &abc(a2 + W.—; 9 and denoting by K, the distance of the centre of gravity of the handle from the axis, its moment of inertia, with reference to the axis 0, becomes, Eq. (65), or g lA g with reference to the axis; g \ ^ 12 /' since abc8 = P', the weight of the handle. The total moment of inertia is, therefore, given by the moment of the entire hammer; /= ^R2 + £L A 9 9 V , b2 4- a\ The process for finding the moment of inertia of the fly-wheel is much simplified by the fact that all its parts moment of inertia are nearly at the same distance from the axis. Thus, by of the fly-wheel; -,v D , J caning K tne mean radius of the wheel, we may take — P2 for mr2 + m' /+ m" r"*+ kc.; and hence, MECHANICS OF SOLIDS. 199 P lx = ---- R ; its value; 9 and denoting the angular velocity of the wheel by Vx, its living force will be, § 159, * y. D2 Y2 living force of A JX . V x • the flv.wneel. To find the angular velocity of the wheel, count the num- ber of its revolutions in a given time, multiply this number by 2 if, and divide the product by the number of seconds in the given time; the quotient will be the angular velocity angular velocity. Let Vx equal 9 feet; the weight P e°u°rimentall of the wheel 2000 pounds, and the mean radius P, 6 feet; omitting the fraction in the value of g, the expression for the moment of inertia becomes /l = ^x86 = 2250; and for the living force, example. 2250 X 81 = 182,250; the half of which, or 91,125 pounds, raised through one foot, is the quantity of work absorbed by the inertia of the wheel, to be given out when the moter ceases to act. § 167.—Besuming Eq. (60)', we may make o , , /2 . ,, ,,2 t nr v2 Centre and radius mr2 + m'r + m r + kc. = MK , „m .. ' of gyration; in which M is the entire mass of the body, and „ _ , . mr2 + ra'r'2 -f m" r" 4- &c. M But this is equivalent to concentrating the entire mass into a single point whose distance from the axis is K, without 200 NATURAL PHILOSOPHY. definition; changing the value of the moment of inertia. This point is called the centre of gyration, and the distance K, is called the radius of gyration. As the moment of inertia varies with the position of the axis, there will be an infinite number of centres and radii of gyration, or as many of each as there are possible positions for the axis. When the axis passes through the centre of gravity, they are principal centre called the principal centre and radius of gyration. and radius of -p. .• .-. ... ... gyration; Denoting the principal radius of gyration by K', we may write MK'2 for /„ in Eq. (65), and we have moment of inertia in terms of radius of gyration. / = MK'* 4- MD (66)". XL Central forces; a body made to revolve about a fixed point by means of a bar; CENTRAL FORCES. § 168.—Conceive a body, whose weight is P, attached to a fixed point 0 by a rigid bar A C, and suppose it to have any velocity whatever in the direction AT, perpen- dicular to the bar. If the body were free, it would, in virtue of its Fig. iu. inertia, move in the di- rection A T with a con- stant velocity. But not being free, the bar will c keep it at the same dis- € : tance from C and cause it to describe the circum- ference of a circle about this point as a centre. There are, then, during this con- strained motion of the body, two central efforts exerted in the direction of the bar, the one by the bar to draw the body from the tangential path A T, the other by the body MECHANICS OF SOLIDS. 201 to stretch the bar out to that path. These forces are equal and directly opposed, because action and reaction are always equal and contrary. The first, or that which tends to draw the body within the tangent, is called the centripetal force, and the second, or that which tends centripetal force; to stretch the bar, the centrifugal force. The centrifu- centrifugal force; gal force is, then, the resistance which the inertia of a body in centrifugal force motion opposes to whatever defects it from its rectilinear ans*s Cwm path. We will first suppose that the dimensions of the body are so small as compared with its distance from the fixed centre, that it may be regarded as a material point, animated with a velocity V. For the circle which it describes, we may substitute a regular polygon ABCDE, substitution of a of a great number of very small sides and having its po angles in the circumference. This being supposed, it is first to be shown that the material point will describe each of the sides of this polygon with the same velocity, or that there will be no loss of velocity in passing from one side to another. For this purpose, we remark, that if the body possess circle; Fig. 115. to prove there is no loss of velocity from the reaction of the curve; the velocity V at the moment of its arriving at P, the beginning of the side B C, it will be animated, while 202 NATURAL PHILOSOPHY. describing this side, with two simultaneous velocities. One of these is the primitive velocity V = B W, in the prolongation of AB; the other BU, in. the direc- tion BO, is a velocity due to the action of the centripetal the body has two force while the body is passing from the side A B to the velocities;. s^e B C of the polygon. But we have seen, § 106, that when a body receives two simultaneous velocities in differ- ent directions, its resultant velocity will be the same as if it Fig. 115. the resultant of which has the direction of the side next to be described; no loss of velocity; possessed them successively, and as though they were communicated one after the other in their respective direc- tions. Thus the resultant velocity BC, with which the side B C is described, coincides in direction with this side, otherwise the body would take some other path, which is contrary to the hypothesis. B U and WC are equal and parallel, from the parallelogram of velocities. The radius OB divides the angle ABC into the two equal angles ABO and OB C; the angle A B 0 is equal to the angle P WC, and the angle OB C is equal to the angle P C W; hence the angles P C W and B WC are equal, and the side B C is equal to the side B W; in other words, the resultant velocity P C, with which the side B C is described, is equal to the velocity P W which the body had at the .end of the side A B. Whence it MECHANICS OF SOLIDS. 203 results, that the velocity communicated to a material point is velocity not ,,..., . , o altered by the not altered in its circular motion; a result easy to ioresee, deflecting cause. since the centripetal force, acting in a direction perpen- dicular to the direction of the motion, cannot work effi- ciently; it can neither accelerate nor retard the motion, and, therefore, can neither increase nor diminish the living force of the material point. Now observe, that BU= W C, is the velocity genera- ted by the centripetal force in its own direction during the time the material point is passing to the side B C. Denote this time by t, and the centrifugal, which is equal though opposed to the centripetal force, by F, and the mass of the point by M, then will the value of F, be given, Eq. (39), by the equation F = M-----. Draw the radius CO; the triangles BO C and BW C to find value of are similar, because the angle OCB= OB C is equalcenl ug orce' to the angle B C W, and the angle OB C is equal to the angle B WC. Hence we have the proportion, BO : BC :: BW : WC; or, denoting the radius BO by P, and replacing B W by its equal V, R : BC : : V : WC; whence WC = BCx V R ! and this, in the value for F, gives 204 NATURAL PHILOSPHY. but B C, the element of the space, divided by t, the element of the time, is equal to the velocity V, whence the measure of the centrifugal force; F = M. V2 R (67). equal to the living force impressed, divided by radius of curvature; Such is the expression for the centrifugal force. The numerator is the living force of the body, and the de-. .nominator is the radius of the circular arc which the body is describing for the instant; whence we conclude, that the centrifugal force of a body of small dimensions, as compared with its distance from the centre about which it revolves, is equal to the living force impressed upon the body, divided by the radius of the circle described by its centre of gravity. Suppose, for example, that the weight of the body is 100 pounds, that its centre describes a circle whose radius is 3 feet, with a velocity of 12 feet. 100 32 M = ^; V = 12; V2 =■. 144; R = 3; illustration; expressed in terms of the angular velocity. Extension to a body of any dimensions; „ 100 X 144 * = 32 X 3 = 150 pounds; the body, therefore, tends to stretch the bar with an effort of 150 pounds. Denote by Vx the angular velocity; then will V2 = V2R\ and this, in Eq. (67), gives F= MV2R § 169.—Let us next take the case of a thin layer of matter DAB, rotating about an axis 0, perpendicular to its plane, with an angular velocity Vx. Taking any one of the elements of the layer whose mass is m, and de- (68). Fig. 116. P P MECHANICS OF SOLIDS. 205 noting its distance from the axis 0, by r, its living force will be mr2 Vx, and its centrifugal force, m r Vx -rr2 -------- = rar Ki ; r which will act in the direction Om oi the radius of the circle described by m about the centre 0. Through the point 0, draw in the plane of the layer, any two rectan- gular axes, as Ox and Oy. Resolve the centrifugal force into two components acting in the direction of these axes; these components and their resultant will be proportional to the sides and diagonal of the rectanglep Oqm, and we shall have, denoting Op by cc, Oq by y, the component parallel to the axis Ox by X, and that parallel to Oy by r; o r : x : : mr Vx : X, resolved into rectangular components; r : y : : rar Vx : Y; whence, X = mxVx Y = my Vx; and for any number of small masses ra', ra", &c, by using the same notation with accents, o X' = ml x' Vx , components parallel to the -<-ri, ,t i,tt2 axis i, for other X" = m"x'Vx, elements; centrifugal force of an element; &C. = &c.; 206 NATURAL PHILOSOPHY. Y' = ra' y' Vx, Y" = m"y"V2, kc = &c.; by this process all the centrifugal forces have been re- duced to two groups of forces acting upon the point 0, in the direction of the axes Ox and Oy, and from the principle of parallel forces, each group will have for its resultant, denoted by Xx and Yx respectively, Xx = Vx (m x + ra' x' + ra" x" + &c.), Yx = Vx (my + ra'y' + ra"y" + kc); Xx = Vx2M'x„ Yx = Vx2M'yr, in which M' denotes the entire mass of the layer, and x, and y, the co-ordinates 0 P and 0 Q of its centre of gravity 67. The resultant of the forces Xx and Yx is the entire centrifugal force of the layer; and this denoted by Fx, is, from the principle of the parallelogram of forces, Fx =\[vxiM'2y?+ VX centrifugal force at the equator; The angular velocity is equal to the absolute velocity, divided by the equatorial radius of the earth. The abso- lute velocity is equal to the circumference of the equator in feet, divided by the number of solar seconds in one siderial day: Diameter of earth in miles 7925..................Log. 3.8989993 * ...............3.1416................Log. 0.4971507 Feet in one mile..........5280.................Log. 3.7226340 Circumference of earth in feet....................Log. 8.1187840 Length of a sid. day in Sol: seconds, 86400X0.997269, Log. 4.9353259 Absolute velocity in feet........................Log. 3.1834581 computed; Radius of earth in feet..........................Log. 7.3206032 Angular velocity Vx............................Log. 5.8628549 .2 1.7257098 Square of angular velocity V .......=...........Log. Radius of earth in feet..........................Log. 7.3206032 / -------- Centrifugal force at equator. .0.1112.................. 9.0463130 Thus the value of the centrifugal force at the equator is us value; 0.1112 of one foot. By the aid of this value, it is very easy to find the angular velocity with which the earth should rotate, to to find angular make the centrifugal force of a body at the equator equal ^eslroSUfflCent to its weight; for by the present rate of motion we weights at the „ , equator; find 01112 = VX2R, 214 NATURAL PHILOSOPHY. and by the new rate of motion 321937 = VX'2R; f in which 32.1937 is the force of gravity at the equator. Dividing the second by the first, and we find 32.1937 I7/2 0ftQ -0lii2-= ^- = 289, nearly; whence result; Vx = 17Fi; that is to say, if the earth were to revolve seventeen times as fast as it does, bodies would possess no weight at the the weight of aii equator; and the weights of bodies at the various latitudes from the equator to the poles diminishing in the ratio of the squares of the cosines of latitude, the weights of all bodies, except at the poles, would be affected. bodies affected. Motion in a circular groove, when plane of groove is horizontal; Fig. 124. § 172.—If we now sup- pose the body, instead of being connected with the point C by means of a rigid bar, to move about the same point in a circular groove, the effects, as regards the centrifugal force, will ob- viously be the same, since the body will be constrained, by the resistance of the groove, to remain at the same distance from the centre. If the plane of the groove be horizontal, the pressure of the body against the side will be constant and equal to the centrifugal force, that is to say, to MECHANICS OF SOLIDS. 215 V2 But if the plane of the groove be vertical, the weight of the body will also exert its influence; for the weight being resolved into two components, one tangent and the other normal to the curve at the place of the body, the latter will sometimes act with, and sometimes in opposi- when vertical, tion to the centrifugal force, while the former will some-!he/fect of^e ° ' body's weight; times increase and sometimes diminish the velocity; so that the pressure becomes greater or less than the cen« trifugal force depending upon these two circumstances. Knowing one of the velocities which the body may have, from one velocity it is easy, by the principle of living forces, to find the toflndthedlhers; others. Take the body at its lowest point ra', and denote its velocity, supposed known, by V, and let it be re- quired to find its velocity at any other point ra, whose vertical height above ra' is H. Denote the velocity at this latter point by V, then will the loss of living force in passing from ml to ra be MV'2 - MV2; and this being equal to double the quantity of action of the weight denoted by W, in the same interval, which quantity of work is 2 WH, we have, M(V'2 - V2) = 2WH; W replacing M by its equal —, and reducing Vs - 7a = 2gH, ----5------------- value of velocity V — 2 g H. at any point; 216 NATURAL PHILOSOPHY. Denoting by H', the height due to the velocity V, we have V'2 = 2gH'; which in the above equation gives same in terms of difference of level of the points; V = V2g(H' - H). Fig. 124. velocity greatest at lowest point; least at the highest. Thus, the velocity of the body will be diminished by the action of its weight during its ascent, while, on the contrary, it will be in- creased during the descent, being always the same at points situated on the same horizontal line. The ve- locity will be greatest at the lowest and least at the high- est point. During the de- scent, the body will acquire living force by absorbing the gain and loss of work of its weight, which living force will again be living force. j , -, -. . ,, , u o1* "• "^ destroyed during the ascent because it is opposed to the weight. § 173.—When a body, in vir- tue of the motive forces which act upon it, describes a curve in centrifugal force space, the effect is the same as of a hndv whi^h -. though it passed over the arcs of the successive osculatory circles of which the curve is composed. If the positions of the centres C, C, C", kc, of these successive circular arcs be known, as well as their radii A C, A' C, A" C" kc, the curve will be given of a body which describes any curve; Fig. 125. \r" ^V T' i / ^ T / / 1 i>r i/° / // t-fir * Ad J>1 c MECHANICS OF SOLIDS. 217 by the series of arcs A A', A' A", A" A'", kc, de- scribed about these centres, and terminated by these radii. And it will be easy, from the consideration of the centrifu- gal and motive forces, to obtain for every point of the curve, the position of the centres and the magnitudes of the radii of the osculatory circles, and, consequently, to trace the path described by the body. Let P denote the resultant of the motive forces which to trace the curve act upon the body at any particular point as A; M tbe a° "centrifugal mass of the body ; V its velocity, of which the direction forces; is A T; and r the radius A C; then will the centrifugal force be measured by MV2 But the body, in describing the curve, does not abandon the small arc A A', and must therefore be retained on it by a force equal and directly opposed to the centrifugal force; in other words, the motive force AP, being re- solved into two components, one tangent and the other normal to the curve, this latter must be equal to the centrifugal force. Denote the normal component by p, then will whence MV2 m P = —T—.....(70) r = AZL.....(71). p value of the normal component of the motive force; radius of curvature : Such would be the radius of the initial arc A A', provided the velocity V were constant during its description. This condition cannot, however, be fulfilled, since the tan- gential component of the motive force will either increase or diminish the velocity. It will be sufficient to make V — n + n' 218 NATURAL PHILOSOPHY. Fig. 125. in which n and n' denote the velocities of the body at the terminal velocity beginning and ending of the arc. The former of these on the initial arc . i • t_ ■ . t_ • • to be found; mUSt be £1Ven> bemS the %Wlr tial velocity; the latter must be found, and for this pur- pose we remark, that as the are is described in a very short time, say the tenth of a second, the motive force, and therefore its tangential component, may be regarded as constant during this interval. Denoting the tangential component by q, and the time by t, we have, from the laws of uniformly varied motion, Eq. (11), its valr e; value of mean velocity; value of radius; n' = n -f and V = n -f n' which, in Eq. (71), gives M = »+ oV- • CT; 2M M(n + rwt) r = P (73). This distance being laid off from the point A, upon the perpendicular to the tangent A T, will give the centre C. The length of the arc, denoted by 5, is found from Eq. (10), or value of arc described; 2 M (74). The law of the motive force being known, the intensity of its action on the body at A' becomes known, and its MECHANICS OF SOLIDS. 219 component perpendicular to the tangent A'T', denoted by p', will give TLT ~\J value of normal p' r= --------, component of r motive force; or MV r = p' ' in which r' is the radius of the arc A' A", and V, the mean velocity with which it is described. Denoting the new tangential component by q', we find, in the same way as before, n" = n' + lft> n^±jnl__ _£_ V ~ 2 ~n +2Mh which in the equation above gives / 9' \2 M ( n' 4" ~ -.r- t ) radius of second \ 2 M / tenninal velocity on second arc; r = P and this being laid off, as before, upon the perpendicular to the tangent A' T', will give the centre C. The length of the arc A'A", denoted by s', will be found from r __ t j. i 9 f2 length of second s — n z "t" r\ ~\ir ' 2 M wc; Finding the value of the motive force at A", its nor- mal and tangential components p" and q", as well as the mean velocity V", we obtain the value of the radius C" A", and the position of the centre C"; the tangential the same process component and time will give us the length of the new forotherai'cs' 220 NATURAL PHILOSOPHY. application to the case of a bomb- shell thrown into the air; Fig. 126. osculatory arc, and thus the description of the curve may be continued to the end. To apply this general case to a particular example, take the instance of a bomb thrown into the air. The forces here are, that arising from the explosive action of the powder and which gives the initial velocity, the resistance of the air, and the weight of the bomb. Let A be the mouth of the piece, of which the axis coincides with the line A T. This line will be tangent to the path de- scribed by the bomb at the point A. Denote the weight of the bomb by W, the initial velocity by n, resistance of air; and the resistance of the air due to this velocity by / The value of / may be taken from a table giv- ing the resistances corre- sponding to different velocities and calibres. Through A draw A H parallel to the horizon, and denote the angle T A H by a; lay off upon the vertical through A, the distance A W to represent the weight of the bomb, and resolve this weight into two components: one, Ac — p, normal to the tangent A T; and the other, A ra = k, in the direction of this line. The angle WAc is equal to the angle TA H — a; and hence, _£-.-- fm- components of the weight of the bomb; p = WCOS a, k — W sin a; and since the resistance of the air is directly opposed to the motion, the force in the direction of the tangent, after the initial impulse, is retarding, and becomes MECHANICS OF SOLIDS. q = k + f = - (IFsin a 4-/); therefore . and , IF sin a 4- f n — n — -----^--— • t, M ' V = n - Wsin a + / 2M tangential component; terminal velocity; mean velocity; this value and that of p, in Eq. (71), give M (n — IT sin a + / ~2~M~ W COS a radius of initial arc; and writing, in Eq. (74), for q its value, we find s = nt x IF sin a 4- / 2 w~ length of initial arc: Through the point A, draw an indefinite perpen- dicular to tihe line A T, and lay off from A the dis tance A C, equal to r; with C as a centre, r as radius, describe the arc A A' equal to s. This will give the initial arc. The linear di- mension of an arc at the unit's dis- Fig. 127. construction; 222 NATURAL PHILOSOPHY. length of arc at unit's distance from the centre; tance from C, is and denoting the ratio of the circumference of the circle to its diameter by if, we have 2* — :: 360° its value in arc; in which z denotes the number of de- grees in this arc, or the value of the angle A CA'. But angle of the this angle is equal tangents at the tQ ^ made v ^ initial points of ^ two consecutive tangents A T and arca; • A'T at the extrem- ities of arc A A', and the angle which the tangent at the beginning of the second arc, A' A" makes with the ho- rizon, or the angle T A! H', will be angle of projection at the beginning of the second arc; Z = 360° X s 2ifr Fig. 128. s----------- a — z = a' Pursuing the same operation as before, we find p' = Wcos a', k' = IF sin a'; and taking from the tables the resistance /', corresponding MECHANICS OF SOLIDS. 223 to the new velocity n', we construct in the same way the second arc A' A", kc, kc. It is to be remarked, that as the angle denoted succes- sively by a, a', kc, diminishes in passing from arc to arc, it will presently become equal to zero, at the summit, and after- ward take the negative sign; in the first case, the tangential variation in the component of the weight of the bomb will be zero, its sign ^-ecuon- will then change, and instead of being a retarding, it will become an accelerating force. Hence, in this curve, three three parts of the portions are to be distinguished, viz.: the ascending branch, curve; the descending branch, and that immediately about the summit. The resistance of the atmosphere to the motion of bodies in it, is found to vary as the square of the velocity of the moving body, and some idea of the intensity of this resistance may be formed from the fact, that a twenty-four range in pound shot, projected under an angle of 45°, in vacuo, atmosPhere and with a velocity of 2000 feet a second, would have a range of 125000 feet, while the same ball, projected under the same circumstances in the atmosphere, would only attain to the range of 7300 feet; about one-seventeenth of the former. § 174.—The laws of the centrifugal force may be il- lustrated experimentally by means of the whirling-table. This consists of a frame- work upon which are mount- ed two vertical axes. Upon the top of each axis is fast- ened a circular block P, B, having a groove cut in the circumference for the recep- tion of an endless cord C, C, C, which also passes round a wheel IF. This wheel is Fig. 124. Whirling-table to illustrate centrifugal force; 224 NATURAL PHILOSOPHY. arrangement of the parts of the table; scale and moveable example first; Fig. 124. provided with a crank and handle H, for the purpose of communicating motion to the whole. The circular blocks are so made, that their circumferences, around which the cord passes, may be varied to change the velo- city of rotation. A piece of wood d d, is mounted upon each of the circular blocks, by means of screws, to support two polished horizontal metallic bars b, b, along which a small stage S may slide with as little friction as possible. This stage is connected with an- other S', which slides freely on a pair of vertical bars b', b', by means of a piece of flex- ible catgut passing over the pulleys p,p', in such manner as to lift the stage S' in a vertical, when motion is com- municated to S in a hori- zontal, direction. The stage S' is placed with its centre immediately over the axis of motion. On the piece d d is a grad- uated linear scale, having its zero in the axis, for the purpose of measuring the distance of the stage S from the centre of motion. A series of weights IF', IF', in the shape of small circular plates, complete this part of the apparatus. The weights, being perforated in the centre, are kept in place by a vertical pintle rising from the middle of each stage. Example 1st. Load one of the stages S, with the weight 5, and place it over the division 8 of the scale; load the other stage S with the weight 2, and place it over the di- vision 5; make the circumference of the first circular block double that of the second. The angular velocity of the first being VX, that of the second will be 2 Vx. When motion is communicated, the centrifugal forces will, Eq. (68), be, respectively, MECHANICS OF SOLIDS. 225 or 5x87/ and 2x5x4 VXl 40 Vx and 40 Vx; that is to say, the centrifugal forces will always be equal result; to each other. Hence, if the stages S' be loaded equally, they will be drawn up simultaneously. Example 2d. Eetaining the same ratio as before between example second; the angular velocities, viz., Vx and 2 Vx, load one of the stages S with weight 6, and place it over the division 8 of the scale; load the other stage S with weight 3, and place it over the division 7. When rotation takes place, the cen- trifugal forces will be, respectively, 6 X 8 Vx = 48 Vx, 3 x 7 x 4 Vx = 84 Vx, the ratio of which is 48 12 . 84 == 21 ' and hence, if the first stage S' be loaded with 12 weights, result. and the second with 21, they will rise together, and with a little care may be kept suspended by properly regulating the motion. If the particles of which a body is composed may move among each other, that is, if the body be soft, a change may be effected by the action of this force in its figure. Such a body of a spherical form, revolving about one when a rotating of its diameters, acquires a flattened shape in the direction a°dy0f8s8nerical of this diameter or axis, because the parts that fie in the fig^e*" acquires , «., ... n i • i ii a flattened shape; plane of the greatest circumference which can be drawn . perpendicular to the axis, that is, in the plane of the body's equator, have the greatest centrifugal force, while those 15 226 NATURAL PHILOSOPHY experimental illustration. Fig. 130. in the neighborhood of the poles have the least; the former will, therefore, recede from and the latter approach the centre. Hence the inference in regard to the causes of the flattened figure of the earth. Example 3d On the vertical axis ab, is an armillary sphere, composed of elastic wires, fitting round the axis by means of a ring, which holds them all together. By this contrivance it is possible for the elastic wires to assume an elliptical figure, having a shorter vertical diame- ter. Screw this apparatus into the middle of the circular block of the whirling table, and give to the whole a rotatory motion; the wires, instead of their original form represented by the dotted lines, will assume, in conse- quence of the centrifugal force, the figure shown in the dark lines. Fig. 131. § 175.—When a body principle of moves with uniform mo- the areas v ±- -1. i tion, it passes over equal spaces in equal times. Thus, suppose the body to start from A, and to move uniformly in the direction from A to B; the line A B being divided into equal spaces A ra', ra' ra", ra" ra'", &c, these spaces will be described a body in motion in equal times. If the several points of division be joined ofthe c!ntraTn witil an7 Point as 3 off tne line, a series of triangles A Cm', force; m'Om", ml'Cm'", kc, will be formed, all having a com- mon vertex and equal bases lying in the same straight line. The areas of these triangles will, therefore, be equal, and MECHANICS OF SOLIDS. 227 will have been described in equal times during the motion of the body by the line joining it with the point C. If when the body arrives at ra', it receive an impulse in the direction from ra' to C, which would cause it, if moved from rest, to describe the path ra' nt in the same time that it would have described ra' ra" ii unmolested, then will it describe, in the same time, the diagonal ra' mtl the forces first of the parallelogram constructed upon ra'' n, and ra' ra" imPul9ive; as sides. The line ra" mu being parallel to ra' G, the triangles Cm'm" and C m' mu will have the same base C ra', and equal altitudes; their areas will therefore be equal; hence the triangles C A ra' and C ra' mu will be equal. In like manner, if when the body arrives at mn, it receive another impulse directed towards C, which would cause it to describe munu, in the time it would have described mu 0 = ra' mu if undisturbed at mu, it will describe the diagonal rnn mlu of the parallelogram con- structed upon mu 0 and mu nu as sides; the triangle Cmu mul will be equal to the triangle C mu 0 = Cm' mu = CAm'. These equal triangles are described in equal intervals of time by the line joining the moving body with the centre C. If now the impulses towards C be applied at intervals of time indefinitely small, the force may be considered incessant, the sides of the polygon then incessant; A ra', ra' mn, mu mun kc, will become indefinitely small, and the polygon itself will not differ from a curve. The fine which joins the body and the centre C, is called the radius vector; and the incessant force acting in the direction radius vector; of this fine towards the centre, is called the centripetal force. Whence we conclude, that when any body having received areas described a motion, is acted upon by a centripetal force, of which the by rad,us vect01 , . x ° . . proportional to direction is oblique to that of the motion, its radius vector will the time of describe equal areas in equal times. description; And conversely, if the radius vector of a body moving in a curve, be found to describe equal areas in equal times about a fixed point, the body must be urged towards this fixed point by a centripetal force, for the equality of the triangles 228 NATURAL PHILOSOPHY. Fig. 181. the force must tend to the fixed point; conversely, the C m'm" and C' m' mu, areas being equal Q q and 0 in equal times, " " &c, depends upon the lines m" mlt, 0mlu, kc, being respectively paral- lel to ra' C, mn C, kc, drawn from the positions in which the body re- ceives the deflecting im- pulses to the centre C. Denote the area by A, and the time in which it is described by t; the ratio of A to t, must, from what has just been shown, be constant. Denote this constant by a, and we shall have ratio of areas to the times. or A — at (74)' and making t equal to unity, we find A = a; from which we conclude, that a denotes the area described in the unit of time. § 176.—Let a body de- scribe the curve A B under Measure of the the action of a centripetal centnpea orce, force directed to the cen- tre C; and suppose ra and ra' to be two positions of the body very near to each other. Draw the tangent ra Q to the curve at the place Fig. 132. MECHANICS OF SOLIDS. 229 ra, and draw ra' $ parallel to the radius vector Cm, and ml n parallel to the tangent. If the centripetal force had ceased to act at m, the body would have described ra Q in the time that it has actually described ra ra'. Again, if the components of body had been moved from rest at ra by the centripetal ^JJ^J™!1 force alone, it would have described the path mn = m'-Q, in the same time; the path ra n is, therefore, the path due to the action of the centripetal force. The places ra and ra' being very near each other, the centripetal force may be considered as constant during the passage of the body from the one to the other. Denote the velocity which the cen- tripetal force can generate in the body at ra, in a unit of time, by vn then, Eq. (7), will mn = \vtt2, whence 2 mn but, Eq. (74)', a and substituting this for t, we find 2«2 X ran vaiueofthe Vj — ---------r-g------. acceleration due ■"■ to the centripetal force; Multiplying both members by the mass of the moving body, denoted by M, we have ,, 2 Ma2 X mn Mv, = ----------. Draw from ra', the line ra' h perpendicular to Cm, then, because A is the area of the triangle Cm ra', will A — h Cm X ra' A. 230 NATURAL PHILOSOPHY. which in the above equation gives the intensity of the centripetal force; Mv, = 4Ma2 X - mn Cm x w! h (74)". versed sine; altitude of the sector; value of the intensity of the force in words. Fig. 132. The distance raw is called the versed sine of the arc rara', and ra' h the altitude of the sector; the first mem- ber, or Mvt, is the quantity of motion which the centrip- etal force can generate in a unit of time, and there- fore measures its intensity; whence we conclude that, the intensity of the centripetal force by which a body is made to describe a curve, is always equal to four times the mass of the body into the square of the area described by its radius vector in a unit of time, multiplied by the versed sine of the elementary arc and divided by the square of the radius vector into the square of the altitude of the sector. XII. Phenomena of the heavenly bodies; MOTIONS OF THE HEAVENLY BODIES. § 177.—The phenomena of the heavenly bodies may be divided into three classes: the first, comprehending the motion of revolution round the sun; the second, the mo- tion of rotation about their respective centres of inertia; and third, their figure and the oscillations of the fluids on their surfaces. It is only proposed to consider the force which produces the motion of revolution, and the orbits which the bodies would, if undisturbed, describe. Observation has established three laws respecting the MECHANICS OF SOLIDS. 231 motion of the planets, which, from their discoverer, are lawsofKepier: called Kepler's laws, viz.: 1st. The planets move in plane curves, and their radii 1st law; vectors describe round the centre of the sun, areas propor- tional to the times of their description. 2d. The orbits of the planets are ellipses with the centre of 2d law; the sun in one of the foci. 3d. The squares of the times of revolution of the different 3d law; planets are to one another as the cubes of their mean distances from the sun or semi-major axes of their orbits. These laws relate only to a motion of translation, and only relate to must, therefore, be limited to the motion of the centres of ^Xtum. gravity of the planets. § 178.—From the first of these laws, and the prin- ciple of areas proportional to the times, explained in § 175, it follows that, the centripetal force which keeps the planets in Consequences of their orbits is directed to the centre of the sun, and that this ' body is, therefore, the centre of the system. The consequence of the second law relates to the varia- tion which takes place in the intensity of the centripetal force arising from a change in the body's place, and may be determined thus. Let ra and ra', be two con- secutive places of the planet moving in an ellipse of which CA and CB are the semi- transverse and semi- conjugate axes, and having the sun, towards which the centripetal force is directed, in the focus S. Draw ra' n parallel to the tangent ra Q, and pro- duce it till it meets ra C, drawn to the centre of the ellipse, in the point v; let fall the perpendicular ra' h upon the radius vector Sm; join the body at ra with the other focus that of the second deduced; 232 NATURAL PHILOSOPHY. S'; draw S' N and CD parallel to the tangent ra Q, and produce ra C to the curve at 67. The tangent Q Q' construction of makes equal angles, QmS and Q' ra S', with the line drawn from the place ra to the foci, and because S' N is parallel to this tangent, the triangle the figure; mS'N is isosceles, making S'm — Nm; and because CD is parallel to S' N, and CS is equal to CS', the distance 772/ tSf "4" "T^i jlj NL is equal to P S; hence ra P =---------is equal to the semi-trans verse axis CA = A. Denote the semi- conjugate axis by P. In the similar triangles ra nv and ra P C, we have, ran mv : : ra P : ra C; value of the versed sine; whence, writing A for raP, we have A. mv mn = ra C Again, drawing ra P perpendicular to D C, we have, from the similar right-angled triangles m L F and ra' A w, —T72 ——2 ---=,2 ---?2 m h : m n :: mF : mL ; whence, writing A for ra L, we have value of the altitude o;' sector; —rr2 ra' n x mF rn h = - 1- MECHANICS OF SOLIDS. 233 and, dividing the last equation by this one, we have mn mv 1 ratio of the = A X ---77 X n----- o' versed sine to the ml h m v ml U X mF square of altitude of sector; The equation of the ellipse, referred to the conjugate x diameters Cm and CD, gives, because the points n and v will sensibly coincide for consecutive places of the body, —2 CD2 ra' n =---- X ra v x v G; Cm which, substituted for ra' n above, we find ran .„ Cm A3 x the same, in other m'h2 ~ CD2 x mF2 x v G ' tems; and, because the rectangle of the semi-axes is equivalent to the parallelogram constructed upon the semi-conjugate diameters CD and Cm, we have CD2 X mF2 = A2 X B2; moreover, the points ra and m' being contiguous, 67 v will not differ sensibly from 2 Cm. Making these substitutions, the above equation reduces to ran A ra h * n 4 Ma2 and, multiplying both members by 2, Cm . ,r 0 mn 2 Ma2 A 1 4 if a2 X ==----=5 = ------- X its final value; ra 'K X Cm B2 Cm 2" 234 NATURAL PHILOSOPHY. The first member we have seen, Eq. (74)", is the intensity of the centripetal force at ra. Calling this force F and writing r for the radius vector Cm, we finally have 2 Ma2 A 1 force; ^ ~~~ £2 X ^ value of the consequence of Every thing being constant in the second member but ' r, it follows that, the force which urges a planet towards the sun,- varies inversely as the square of the planet's distance from that body. The consequence of the third law is not less important, and may be evolved thus. Multiply both members of the last equation by V2 A2 B2, and we have to find the 1 consequence of F if2 A2 B2 = 2 if2 Mtt2 A3 X "j J the third law; ' divide both members of this equation by Fa2, and there will result «2A2B2 2if2M 1 a2 ~ - F X A X -,- Now, if A B is the area of the entire ellipse; a is the area described by its radius vector in a unit of time; hence periodic time; -----is the number of units of time in one entire revolu- a tion of the planet, called the periodic time. Denote this by T, and substitute it for-----, and we get the value of its m2 __ 2 if M g square; ~p "^2 ' -"- " In like manner for any other planet, whose mass is M', mean distance A', radius vector r', periodic time T, and centripetal force F', we have T,* 2if2M' 1 ,,3 1 ~ ~Fr~ ' P ' ; MECHANICS OF SOLIDS. 235 and dividing this equation by the one above T* M'F.r2 A^ T2 ~ MFr.r^ X ^l3' But, by the third law, T'% _ A'z T2 ~ Z3"' ratio of the squares of periodic times; whence or M'Fr2 MF' r'2 ~ *' Tlx r2 = F' ,2 M'xr' centripetal acceleration; F . Now -= is the velocity which the centripetal force can generate in one unit of time, or, which is the same thing, it is the measure of the acceleration due to the force F' which acts upon the planet M; so, likewise, -7^- is the acceleration due to the centripetal force which acts upon the planet M'; and resolving the above equation into the proportion F_ M M' r2 r, •A 1 consequence of the third law; we see that the forces which urge two different planets towards the sun, are to each other in the inverse ratio of the squares of the distances; so that the same law which regulates the intensity of the force in a single orbit, also extends to different planets revolving in different orbits. If r be made equal to r', then will the accelerations due to the centripetal force be equal; that is to say, if all the 236 NATURAL PHILOSOPHY. at same distance, the centripetal accelerations are equal; Newtonian hypothesis of universal gravitation; consequences of this hypothesis; the orbits might have been ellipses, parabolas, or hyperbolas. The angular velocity; planets were brought to the same distance from the sun, each unit of mass would be urged towards that body with the same intensity; and as the different planets might be inverted in respect to the order of their distances from the sun, without the relation of the periodic times as expressed by the third law being affected, it follows that the force which acts upon all the planets is absolutely the same in kind, and is only qualified, in intensity, by a change of distance. These considerations led Newton to adopt the celebrated hypothesis which laid the foundation of physical astronomy, viz.: that all bodies attract each other with an energy which is directly proportional to their masses and inversely proportional to the squares of their dis- tances from each other. Starting from this hypothesis, it is easy to solve by a process not suited to an elementary work like this, the converse problem of that which led to the consequence of the second law, and to show, that a heavenly body may describe any one of the conic sections having the sun at one of the foci, depending upon the relation which subsists between its velocity and the energy with which the body and the sun attract each other. The orbit will be a para- bola, an ellipse, or hyperbola, according as the square of the body's velocity is equal to, less, or greater than, twice the attractive force, multiplied by the distance from the sun. Fig. 134. § 179.—Let Cmm' be the sector described in the unit of time: take the dis- tance Cb equal to unity, and describe, with C as a centre and Cb as radius, the arc bd = sn which will measure the angular velo- city. With C as a centre, and Cm' = r as radius, describe the arc ra' h'; then will ra' h' — rsr MECHANICS OF SOLIDS. 237 Supposing the.unit of time small, in which case ra' will be very near to ra, m'h will be sensibly equal to m'h', ra C to w' C, and we have for the area of the sector Cm m', } Cm X m'h' = \i2st = a; whence s. = 2a 15-5 its value; Fig. 135. from which we find that, the angular velocity of a planet law of us about the sun, varies inversely as the square of its distance or vana lon' radius vector. Supposing the planet to describe the ellipse ABPD, having the sun at the focus S, the ex- tremities A and P of the transverse axis are call- ed, the former the Aphe- lion, and the latter the Perihelion. The angular velocity of the planet is the least at aphelion and greatest at perihelion. Again, denote the angle Cm Q by a, and suppose the velocity of the body on the small arc ra ra' uniform, which we may do without sensible error, the length of ra ra' will measure the ve- locity of the planet at ra, since it is described in a unit of time. Hence Fig. 136. aphelion; perihelion; angular velocity greatest at perihelion and least at aphelion; absolute velocity; mm ' sin a = ra' h = V. sin a; 238 NATURAL PHILOSOPHY. and the area of the triangle or sector Cmm' will be \ V. sin or x r; whence V. r sin a = a, or value of the absolute velocity; V = 2 a r . sin a Draw the tangent ra Q to F'g- 136- the curve at the point ra, and from C let fall the per- pendicular C Q, then in the right-angled triangle C Qm, will C Q = r . sin a = p, ' which substituted above gives the same in different terms; V = 2 a p its law of variation; greatest at perihelion and least at aphelion. that is to say, the velocity of a planet in its orbit, varies in- versely as the length of the perpendicular let fall from the centre of the sun upon the tangent drawn to the orbit at the body's place. From this it follows that the velocity of the planet will be greatest at perihebon and least at aphelion. § 180.—It will be found convenient when we come to discuss the nature of light, to know that when a body centripetal force describes an ellipse under the action of a force directed centTeVan ° towards the centre of that curve, the force will vary elliptical oi-bit; directly as the length of the radius vector, and that the periodic time will be the same for all ellipses, great and small. MECHANICS OF SOLIDS. 239 Fig. 137. Let the body, under the action of a force di- rected to the centre C, describe the ellipse of which CA and CB are the semi-axes, denoted respectively by A and B; and suppose ra and ra' to be two of its con- secutive places. Draw the tangent ra Q at the point ra, and parallel to this tangent draw the diameter D D', perpendicular to which, draw from ra the fine ra K. From ra' draw ra' n parallel to the tangent till it meets the radius vector Cm in n, and let fall upon the same radius vector the perpendicular ml h. The equation of the ellipse, referred to its conjugate diameters Cm and CD, gives to find the law; m n = whence CD' Cm X mn x n 67; mln X Cm' mn = = QJ)2 X n 67 ' value of the versed sine; Because ra' n and ra' h are respectively perpendicular to the lines ra K and ra C, the angles hm' n and Cm K are equal, and the angles at K and h being right angles, the triangles ra' nh and CmK are similar, and give the pro- portion ra n ra h Cm -T; whence , T721 m h = --1— J r^: m n X 'ni A Cm value of the square of sector's altitude; 240 NATURAL PHILOSOPHY. dividing the last equation by this one, we have 7T~4 ran 6 ra ratio of the v.'i-M'J sine to TY2 /i n2 •., t^2 ^,' , . , ra h CD X ra A X n 67 square of sector's altitude; But the rectangle of the semi-axes is equivalent to the parallelogram described upon the semi-conjugate diame- ters, hence CD2 X ~m~K2 = A2 x B2; moreover, n 67 is sensibly equal to 2 Cm; making these substitutions above, there will result same in different m U Cm terms; o := J ra7^2 2 A2 B2 multiplying both members by 4 Ma2, and dividing by Cm, we have, Eq. (74)", a ir 2 w mn n 2 Ma2 „ 4;Ma? X -=-2---==■ = F = ---- x Cm, Cm2 x nVh2 A2B2 in which M is the mass of the body. Finally, writing r for Cm, we find vaiueofthe et 2 Ma2 centripetal force; A2 D2 ^ ' ; the law of its that is to say, the centripetal force which will cause a body to describe an ellipse when directed to the centre of that curve, varies directly as the radius vector. to and the Multiply both members of the last equation by periodic time; +±2^ ^ ^ -^ F.if2A2B2 = 2if2Ma2 X r. Dividing both members of this equation by Fa2, and we have MECHANICS OF SOLIDS. 241 if2A2B2 . a M r = 2 if2 X r M if A It taking the square root, and recollecting that-----, is the a periodic time = T, we find T value of the periodic time; F The quotient -^ is the measure of the acceleration due to the centripetal force, which we have just found to vary directly as the radius vector. This makes the radical ex- pression constant; hence T must also be constant. Whence we conclude, generally, that when any number of bodies are solicited towards a fixed point by forces which vary directly as the distances of the bodies from that point, they conclusion. will describe ellipses, or circles, one of the varieties of the ellipse ; and that they will all perform their revolutions in the same time. XIII. THE PENDULUM. § 181.—A body M QN, suspended from a horizontal axis A, about which it may swing with freedom under the action of its own weight, is called, in general, a compound pendulum. When the body is reduced to a material heavy point, and the medium of con- nection with the axis is without weight, it is called a simple pendu- lum. 16 Compound pendulum; simple pendulum; 242 NATURAL PHILOSOPHY. has no real existence; effect of friction and resistance of air; figure of pendulum and mode of suspension; knife-edge and bob; The simple pendulum is but a mere conception, and yet the ex- pression for its length, which may easily be found in a manner soon to be explained, is of great prac- tical importance. When the pendulum is at rest, in such position that its centre of gravity 67 is below and on the ver- tical line passing through the axis A, it will be in a state of stable equi- librium, § 151; but as soon as it is deflected to one side, as indicated in the figure, and abandoned to itself, it will swing back and forth about the position of equilibrium, into which it will finally settle in consequence of the resistance of the air and friction on the axis. If these causes of resistance were removed, the pendulum would con- tinue its motion indefinitely; but this cannot be accomplished in practice, and hence such figure and mode of suspension are resorted to as to give these impediments the least possible influence. The pendulum is usually mount- ed upon a knife-edge A as an axis, resting upon a well-polished plate of metal, or other hard substance, P; and the figure of the pendulum is that of a flat bar C, supporting at its lower end a heavy lenticular- shaped mass D, called a bob. One entire swing of the pen- dulum, by which its centre of Fig. 139. g2T Fig. 140. Fig. 141. MECHANICS OF SOLIDS. 243 Fig. 142. to find time of a oscillation; notation; gravity is carried from the extreme limit 67 of its path, on one side of the vertical AL, to G" on the other, is oscillation; called an oscillation. To find the time of a single oscillation; call the weight of the entire pendulum, W; its mass, M; its angular velocity at any instant, Vx; its moment of inertia with reference to the axis of suspension, /; the dis- tance of its centre of gravity from the axis, D ; the vertical distance P 67', through which the centre of gravity must descend from its highest point 67 to arrive at any point The living force of the pendulum when the centre of gravity reaches the point 67' will, § 159, be I, Vx, living force; and the quantity of work of the weight will be z^G-" and hence Wy = Mgy, IVX2 = 2 Mgy. work of the weight; The point C on the line A 67 at the unit's distance from A, will, during the motion, describe an arc similar to 67 67', and the vertical distance 67, Pn denoted by yn . through which this point will fall while 67 is passing to G', will be given by y = Dyt; fall of the centre of gravity; 244 NATURAL PHILOSOPHY. square of the angular velocity; and this, in the above equation, gives I. Vx2 = 2MgDyt; whence TZ2 X.D . Vi2 = —r~ -^9Vr I Denoting by st the small distance described by the point C during the very short interval t, succeeding the instant at which the angular velocity is Vx, we shall have Vi = t' square of the time required to describe a very small arc; to find the arc described in the small time; which, in the preceding equation, gives s,2 M.D I %9vr, whence M.D 2gyf Taking A if equal to unity, let CB C" be the arc described in one oscillation by the point M, and MN the small arc s, described in the time t, immedi- ately succeeding the instant at which the angular velocity is Vx. Draw ME perpendi- cular to the vertical Fig. 143. MECHANICS OF SOLIDS. 245 one value for A B, and N Q perpendicular to ME: then, in the similar triangles A ME and MN Q, we have QN : EM :: MN : AM; and because A M is unity, and MN is sy, ' E M' the arc; But from the property of the circle EM =^ 2AB.EB - EB2 =^J 2EB - EB2, and if we take the arc CB 0" very small, the versed sine EB will be a very small fraction, and its second power may be neglected in comparison with the first. Whence EM = V 2 EB; which, in the value of s, above, gives s. = - «* another value for V 2EB7 .the arc; and this, in tht value for t2, gives I 1 Q N2 another value for t = W7D" 47' V BE' the square of the a -j ' time; Upon PP as a diameter, describe a semi-circumference DmnB, and through the points if and N, the extremities of the arc sn draw the horizontal lines Mm and Nn, cut- ting this semi-circumference in the points ra and n. Draw the radius Om, and the vertical nq. From the property of the circle we have ra~P2 = BE x ED = BE x PM = BE x yt; 240 NATURAL PHILOSOPHY. whence vertical distance from last point; BE = E* y< which, substituted for P E in the equation above, gives J value of the element of the time; QN2 M.D ±g mE2' and, taking the square root, I QN g .M.D ~ raP" The two triangles ra 0 E and ra q n are similar, and give qn = QN : raP :: nm : Om; whence QN mE nm Om' and this substituted above in-the value of t, gives 1 / I nm 1 ~ *V^7M7D X Om proportional to Such is the value of the time required to describe the arc oTthe circle elementary arc MN, which we see is proportional to whose diameter the arc mn, or to the projection of MN on the semi- arc of osculation; cfrcumference described upon PP as a diameter, every other quantity in the second member of the equation being MECHANICS OF SOLIDS. 247 constant; and hence, the time required to describe the whole arc CMB, which is obviously tbe sum of all the the time of elementary times of describing the elementary arcs MN, ^mi^c*iiation &c., must be equal to found; »v I x g .M.D Om' into the sum of all the projections of MN, kc, on the semi-circumference DmB; but this sum is the semi- circumference itself; and denoting the time from C to B, or that of a semi-oscillation, by ^ T, we have I DmB *T~ * V g.M.D X ~Um~' but DmB o^aw = if = 3.1416, Om the ratio of the circumference to the diameter; whence, rp _ I (n^\ time of a single X n M D ' ' ' ^ '* oscillation; From this formula we see that the duration is inde- pendent of the amplitude of the oscillation, when this amplitude is small; and a pendulum slightly deflected from its vertical position and abandoned to itself, will oscillate in equal times whatever be the magnitude of the arc, provided it be inconsiderable. Such oscillations are isochronal • i , i 7-7 7 oscillations: said to be Isochronal. If the number of oscillations performed in a given in- terval, say ten or twenty minutes, be counted, the duration of a single oscillation will be found by dividing the whole time of a single - , . . oscillation found interval by this number. from Thus, let d denote the time of observation, and N the observation; number of oscillations, then will 248 NATURAL PHILOSOPHY. the same for a second place; N v < I g .M.D' and if the same pendulum be made to oscillate at some other location during the same interval 6, the force of gravity being different, the number N' of oscillations will be different; but we shall have, as before, g' being the new force of gravity, w = * \v I N' ~ " V g>. M. D' Squaring and dividing the first by the second, we find 1T_ N2 (76); forces of gravity that is to say, the intensities of the force of gravity, at mTmbeTTr68 different places, are to each other as the squares of the osculations in number of oscillations performed in the same time, by the same time. n . __ . „ . same pendulum. Hence, if the intensity of gravity at one station be known, it will be easy to find it at others. Simple pendulum; § 182.—Eesuming the general value for I, Eq. (65), we have /=/, + D2M; which value of I, in Eq. (75), gives T = if \J Ix 4- D2M g .M.D (77). mass If, now, we suppose the entire mass of the pendulum to smXnpo.nt;ma ^e concentrated into a single point, and this point con- nected with the axis by a medium without weight, we have MECHANICS OF SOLIDS. 249 P = 2 ra T2 = 0 ; moment of inertia * in reference to the centre of since the centre of gravity must also go to that point, and gravity; r = r' — r" = kc. = 0; whence, writing I for the new value assumed by D, which now becomes the distance from the axis to the single heavy point, we have / J, lime of oscillation T = if V/ ---- .... (78) ; of the simple 9 pendulum; which is the expression for the time of oscillation of a simple pendulum of which I is the length. If the time of oscillation of the simple, be the same as that of the compound pendulum, we shall have, from Eqs. (75) and (78), * V a.M. T) = « VlT; or g .M.D _T_ _ lx 4- MP2 M.D ~ MD ' ' ^' in which case I is called the equivalent simple pendulum; equivalent simple that is to say, the length of a simple pendulum which will pendulum; oscillate in the same time as a compound pendulum whose moment of inertia in reference to the axis of suspension is I, whose mass is M, and of which the axis of suspension is at a distance from the centre of gravity equal to D. The point situated on a line drawn through the centre centre of of gravity of the pendulum, perpendicular to the axis of oscillatlon; suspension, and at a distance from that axis equal to I, is called the centre of oscillation; and is that point of which the circumstances of oscillation would in nowise be altered were the entire pendulum concentrated into it, or were it disconnected from the other points of the pendulous mass, its connection with the axis being retained. 250 NATURAL PHILOSOPHY. § 183.—A line drawn through the centre of oscillation, and Axes of parallel to the axis of suspension, is called the axis ofoscilla- suspension an fan. The axes of suspension and of oscillation are reciprocal. of oscillation are j ± j jt reciprocal; Let D' denote the distance of the axis of oscillation from the centre of gravity; then will I = D + D'. Invert the pendulum and make the axis of oscillation the axis of suspension, take I' for the new equivalent simple pendulum, then will 0 new equivalent T _i_ J^f T)' simple V = A—-—-—---; pendulum; -"* • ■*-' but we have, from the foregoing equation, D' = I - D; and this, in the preceding value for I', gives M. (I - D) ' Again, from Eq. (79), we have Ix I - D = MD ' substituting this in the above value for V, we finally get the simple ^ + Jf £2 pendulum the t — -----lrri---- = I, same; ■""■ ■*-' that is to say, when the axis of oscillation is taken as the MECHANICS OF SOLIDS. 251 axis of suspension, the old axis of suspension becomes the new axis of oscillation. In other words, these axes are conclusion; reciprocal. This furnishes an experimental method for finding the ISngth of any equivalent simple pendulum, equivalent which is the more valuable in view of the great difficulty found from of computing the moment of inertia of a compound pendu- experiment; lum by the ordinary calculus, owing to the peculiar forms of that instrument rendered necessary by the circumstan- ces under which it is employed. But before proceeding to the explanation of this method, it will be proper to premise, that the time of oscillation of a compound pendu- lum will be a minimum, when, in Eqs. (78) and (79), ^ 4- P2 Ix + D2M M ^ value of =: I equivalent simple JSl D D pendulum; is the least possible; or replacing -A by its value K', deduced from Eq. (66)' by making D = 0, the expression A"'2 4- P2 P ' must be the least possible. But it may easily be shown, either by trial, or by a simple process of the calculus, that this expression is a minimum when K' = D, and consequently 1=2 K' ; length of the shortest equivalent simple that is to say, the time of oscillation of a pendulum will pendulum; be the least possible when the axis of suspension passes through the principal centre of gyration, and the length 252 NATURAL PHILOSOPHY. usual form of the compound pendulum; Kg. H4. Ya device to change the position of the centre of gravity; position of centre of gravity; pendulum made to oscillate during same time; of the equivalent simple pendulum is twice the principal radius of gyration. Let A and A' be two acute parallel prismatic axes firmly con- nected with the pendulum, the acute edges being turned towards each other. The oscillation may be made to take place about either axis by simply inverting the pen- dulum. Also, let M be a sbding mass capable of being retained in any position by the clamp-screw H For any assumed position of M, let the principal radius of gyra- tion be 67 C; with 67 as a centre, 67 C as radius, describe the circum- ference CSS'. From what has been explained, the time of oscillation about either axis will be shortened as it approaches, and lengthened as it recedes from this circum- ference, being a minimum, or least possible, when on it. By moving the mass M, the centre of gravity, and there- fore the gyratory circle of which it is the centre, may be thrown towards either axis. The pendulum bob being made heavy, the centre of gravity may be brought so near one of the axes, say A', as to place the latter within the gyratory circumference, keeping the centre of this circum- ference between the axes, as indicated in the figure. In this position, it is obvious that any motion in the mass M would at the same time either shorten or lengthen the duration of the oscillation about both axes, but unequally, in consequence of their unequal distances from the gyra- tory circumference. The pendulum thus arranged, is made to vibrate about each axis in succession during equal intervals, say an hour or a day, and the number of oscillations carefully noted; if these numbers be the same, the distance between the axes is the length I of the equivalent simple pendulum; MECHANICS OF SOLIDS. 253 if not, then the weight M must be moved towards that axis whose number is the least, and the trial repeated, till the numbers are made equal. The distance between the distance between axes^may be measured by a scale of equal parts. measured • From this value of I, we may easily find that of the simple second's pendulum; that is to say, the simple pendu- simple second's lum which will perform its vibration in one second. Let pen um' N be the number of vibrations performed in one hour by the compound pendulum whose equivalent simple pendulum is I; the number performed in the same time by the second's pendulum, whose length we will denote by V, is of course 3600, being the number of seconds in 1 hour, and hence, from Eq. (78), and because the force of gravity at the same station is constant, we find, after squaring and dividing the second equation by the first, , I N2 V = (36007.....(80)- "SIensth; Such is, in outline, the beautiful process by which Kater determined the length of the simple second's pendulum at the Tower of London to be 39.13908 inches, or 3.26159 value at London; feet. • As the force of gravity at the same place is not sup- posed to change its intensity, this length of the simple second's pendulum must remain for ever invariable; and, basis of the on this account, the English have adopted it as the basis of^^ghTald of their system of weights and measures. For this purpose, measures; it was simply necessary to say that the 3-.2. If we make 4< = 40° 42' 40", the latitude of the City- Hall of New York, we shall find /'• in- length at City I = 3.25938 = 39.11256. mi of New York; The principles which have just been explained, enable us to find the moment of inertia of any body turning about a fixed axis, with great accuracy, no matter what its moment of inertia figure, density, or the distribution of its matter. If the found by means _ ^ of simple axis do not pass through its centre of gravity, the body pendulum; will, when deflected from its position of equilibrium, oscillate, and become, in fact, a compound pendulum; and denoting the length of its equivalent simple pendulum by I, we have, Eq. (79), M.D.I = I; 256 NATURAL PHILOSOPHY. or, since M = —, 9 its value; JUL 9 D.l = I (83); in which W denotes the weight of the body. simple second's Knowing the latitude of the place, the length V of the ftomMUudeT^ simPle second's pendulum is known from Eq. (82); and counting the number N of oscillations performed by the body in one hour, Eq. (80), gives the body's equivalent simple pendulum; distance from centre of gravity to axis found; value of the moment of inertia; I = I'. (3600)2 N2 Fig. 145. To find the value of D, which is the distance of the centre of gravity from the axis, attach a spring or other balance to any point of the body, say its lower end, and bring the centre of gravity to a horizontal plane through the axis, which position will be indicated by the maximum reading of the balance. Denoting by a the distance from the axis C to the point of support P, and by b the maximum in- dication of the balance, we have, from the principles of moments, ba = WD. The distance a may be measured by a scale of equal parts. Substituting the values of WD and I in the expression for the moment of inertia, Eq. (83), we get b.a.r.(S600y _ g.N2 (84). MECHANICS OF SOLIDS. 257 If the axis pass through the centre of gravity, as, for the moment of example, in the fly-wheel, take Eq. (79), ^Tl passes through Ix 4" MD2 _ the centre of gravity ; 1 = M D whence Ix = M.D.I -MD'< (85). Fig. 146. Mount the body upon a parallel axis A, not pass- ing through the centre of gravity, and cause it to vi- brate for an hour as before; from the number of these vibrations and the length of the simple second's pen- dulum, the value of I may be found as befBre; M is known, being the weight W divided by g) and P may be found by direct measure- ment, or by the aid of the spring balance, as already indicated; whence Ix becomes known. example of the fly-wheel; § 184.—When a body, B QN C receives a motion of rotation about an axis A, which is here supposed perpendicular to the plane of the paper, each elemen- tary mass ra, will develop a force of inertia whose di- rection is perpendicular to the shortest line connecting it with the axis, and whose intensity will be measured by Fig. 147. centre of percussion; Vx m . r . —, inertia exerted by an elementary mass during an elementary time; 258 NATURAL PHILOSOPHY. notation; co-ordinate planes; in which r is the distance FiS- 147- of ra from the axis, and Vx the elementary amount of angular velocity generated in the very small portion of time denoted by t. Through the axis A, draw two planes at right angles to each other, and let their traces on the paper be A X and A Y. Deno- ting the co-ordinates Ap and A q of ra, referred to these planes, by x and y, respectively, we shall have cos to A p a y cos m Aq = —. Eesolve the force of inertia, above given, into two compo- nents in the direction of these planes. The component parallel to the plane of which the trace is A y, will be component of the inertia parallel to the plane A y; that parallel to the plane A x; Vx x Vx mr ' — • — = to . x —, t r t and that parallel to the plane whose trace is A x, will be V\ y mr - — • —■ t r my • —; the same for other elementary masses; and for other elementary masses to', to", &c, of which the co-ordinates are x'y', x"y", kc, we shall have the com- ponents mix' . ^ to"V 5, &c. t* t m y m"y"^A, kc; MECHANICS OF SOLIDS. 259 the resultant of the components parallel to the plane A y, will be V V resultant of the (ra x + ra' cc' 4- ra" cc" 4- &c.) = — if a;,, components t N y * and of the components parallel to the plane A x, parallel to A y, -l(my + m'y' + m" y" + &c.) = —Myt; resultant of those Jj f pEU'ElllGl 10 .i X f in which M denotes the entire mass of the rotating body, and xJ ahd yt the co-ordinates of its centre of gravity. And the intensity of the general resultant will, from the parallelogram of forces, be '1 jr J---n ! if '1 -»r n resultant of the -- M V x2 -f V, = — M. D; 4- i > Ji . } whole; in which P represents the distance of the centre of gravity 67, of the whole mass, from the axis. The direction of us direction; this resultant will be perpendicular to A 67, drawn through the centre of gravity perpendicular to the axis, as will readily appear by reference to its components parallel to the planes A y and A x found above. The moment of this force, with reference to the axis, will therefore be its intensity multiplied into some dis- tance as A 0 = L, on this line, -— M. D . P. its moment; But, Eq. (63), the sum of the moments of all the forces of inertia actually exerted, in reference to the axis A, is equal to the product of the entire moment of inertia I, multiplied y by the ratio —, therefore V. V 11. M.D .L = 1.11. t t' 260 NATURAL PHILOSOPHY. or L = I M.D (86): point at which the resultant inertia of a rotating mass is exerted; shock experienced by the axis when the body is struck; Fig. 148. whence we conclude that, the point at which the resultant inertia of a rotating mass is exerted, is on a line drawn through its centre of gravity perpendicular to the axis, and at a distance from the axis equal to the moment of inertia divided by the product of the mass into the distance of the centre of gravity from the axis. This being understood, suppose a force F applied at the point C in a di- rection perpendicular to the line A 0, and immediately opposed to the direc- tion of the motion; this force would obviously tend to bend the line A 0, the point A being retained by the axis, and the point 0 being urged onward by the inertia concentrated at it. If the force be suddenly ap- plied, the axis must receive a shock, and to estimate its intensity S, de- note by X the distance A C; then, from the principles of parallel forces already explained, we have L- X F S; whence S= F—^= P(l l) ■ (87); or, substituting the value of P, Eq. (86). its intensity; s MD = F (l _ ~ • X). . . (88). If we suppose the body at rest, and desire to apply the MECHANICS OF SOLIDS. 261 force F so as to communicate no shock, we make S = 0, a condition that can only be satisfied by making whence 1---y- X X = 0; the blow applied bo as to communicate no shock to the axis; X = I MD = L = AO. distance from the axis at which it must be applied; defined; There being no shock to the axis, it can oppose no resist- ance to the motion of rotation, and hence we infer that this latter will be the same as though the body were per- fectly free. The point 0 is, on this account, called the centre of percussion, which may be defined, that point of centre of a body retained by a fixed axis, at which it may be struck in a Percussion direction perpendicular to the plane of the centre of gravity and axis without communicating any shock to the axis. ' The centre of per- cussion may be found Fig. i49_ experimentally thus:— lay the axis C upon a support A A, and per- mit the body to fall up- on a moveable edge P, resting on a horizontal plane; when this edge is placed in such position that the axis C will not move when the body falls upon it, the centre of percussion will be immediately above the point struck. Since the dis- tance of the centre of percussion from the axis is equal to 7 J/U' oentre of percussion found experimentally; 202 NATURAL PHILOSOPHY. to put a peudulum in motion, the force should be applied to centre of oscillation. it must be at the centre of oscillation. To move a pen- dulum without communicating action to its axis, the force must be applied at the centre of oscillation. The shock may be positive, nothing, or negative; centre of spontaneous rotation; Fig. 150. § 185.—Resuming Eq. (87), we see that the shock upon the axis A will be positive, that is to say, will act in the direction of the impressed force F, as long as X is less than L.• when Xis equal to L, there "will be no shock; when X is greater than L, there will again be a shock, but with a negative sign, which indicates that it will be exerted in a direc- tion opposite to that of the im- pressed force. Now these shocks in opposite directions, with a neutral point A, can only arise from an effort of the particles, which are situated on opposite sides of the axis, to move in contrary directions when the body is struck at the centre of oscillation; and as the effect upon the neutral point A is the same in this latter case, whether the body be retained by an axis or a force, it follows that every free body, when struck, in gen- eral, begins to move for the instant, but only an instant, about a single point. This point is called the centre of spontaneous rotation. If the blow be impressed at any point, as 0, the centre of spontaneous rotation will be upon the axis corresponding to the point 0 as a centre of oscil- lation, and hence its distance from the latter will be given by distance of centre of spontaneous rotation from axis; P = 7 MD (89); and since the centre of oscillation and axis of suspension are reciprocal, 7 will denote the moment of inertia taken MECHANICS OF SOLIDS. 263 with reference to an axis through the point 0, and D the relation of distance of the latter from the centre of gravity. spontaneous Referring to Eq. (88), if the axis be supposed to pass rotation to centre . -r^.-ii-i i^ J of oscillation; through the centre of gravity, P will be equal to zero, and S = F; that is to say, no matter where the force F be applied, its the entire shock entire effect will be communicated to the centre of gravity, ^3unicated to which is a confirmation of the result given in § 146. centre of gravity; If the line of direction of the force pass through the centre of gravity, P, in Eq. (89), will be zero, and the dis- if direction of tance of the centre of spontaneous rotation will be at a^XoS^nire infinite distance from the point of impact; in other words of gravity, the the body will not rotate, which is another result of § 146. ro°Jewl Fig. 151. § 186.—Let Q be a body suspended from an axis A perpendicular to the plane of the figure. This body be- ing at rest, suppose it to be struck at the point T by an- other body P, moving in the direction PP at right angles to the surface of contact, and in a plane perpendicu- lar to the axis A. Denote by ra and w the mass and weight of the impinging body, and by V its velocity before the impact. At the instant of meeting there will be developed a force of com- pression F, which will act equally upon each body along the line TL, but in oppo- site directions. The pressure upon both bodies, which is nothing when they begin to touch each other, will aug- Collision of a body having a motion of translation against another retained by a fixed axis; 264 NATURAL PHILOSOPHY. the action and ment by degrees as they ap- reaction variable; ^^ tQ ^ ^^ of great. est compression; so that P, although always represent- ing a number of pounds weight, is, nevertheless, not a fixed, but a variable quan- tity. We may disregard for a moment the body Q, and suppose the force F applied to the body P, considered as free; the force will deprive this body of a series of small degrees of velocity denoted by v, 6ach in the small time t, so that its measure at any instant will, Eq. (39), be given by Fig. 151. measure of the force of reaction; F = mv But the force F also acts upon the body Q, and turns it about the axis A, generating in it, during the same in- terval of time t, an angular velocity v,; and the forces of inertia thence arising, must be in equiiibrio with the force F; in other words, the sum of the moments of the first in reference to the axis A, must be equal to the product of the force F into the perpendicular A C, drawn from the axis to the line of direction PP. Hence, Eq. (63), moment of action equal to moment of reaction; F.AC = 7.^5 and substituting the value of F above, and dividing by A C, which we will represent by the single letter p, m . v Lv, p.t MECHANICS OF SOLIDS. 265 or, finally, O.m.V = 7. V,. result for a single -*■ ' instant of time; Denote by v', v", v'", kc, the small degrees of velocity lost by the body P, during the second, third, fourth, &c, intervals of time t, supposed to be always of the same length; and by v', v", v/", kc, the angular velocities ac- quired by the body Q during the same intervals; we shall have p . ra . v' = Ivl, x < ' the same for other instants of p . ra v" = Iv", time; &C. = &c.; by taking the sum of the whole, p(v+v' + v" + V" + kc.) m=I(v/ + v/ 4- v," 4- &c.); %£?ofthe and denoting by U the whole velocity lost by the body P, and by Vx the whole angular velocity gained by the body Q during the entire action, we shall have U = V 4" v' 4" v" 4" v'" 4" &C, velocity lost; Vx = v, 4- v/ 4- v," 4- v/" 4- &c.; whence, by substituting above, angular velocity gained; result for the p . m . TJ = IVX . . . . (90). entire duration of the impact; If the bodies be not elastic, it will only be necessary to consider the impact from the instant in which they first come in contact, to that in which the body P has lost its excess of velocity over that part of Q into which it be- comes imbedded; for, as soon as the body P has taken the 266 NATURAL PHILOSOPHY constitute a single one; if the bodies be angular velocity of the other about the axis, there will be not elastic, they ffort tQ in lost figure and the two bodies will will ultimately ° " , turn about A as though they constituted but a single one. But the angular velocity of Q about A being Vx, that of P will be p Vx, and we shall have U= V-pVx; substituting this value of U in Eq. (90), we find whence angular velocity generated by the impact. pm(V - pVx) = IVX; p ,m. V 1 ~~ ra. p2 4- 7 ■ (91); which gives the angular velocity of the body struck, after the impact, in terms of its moment of inertia, the mass and velocity of the impinging body, and the distance from the axis to the path described by its centre of gravity. Application to the balistic pendulum; its construction | § 187.—In artillery, the initial velocity of projectiles is ascertained by means of the balistic pendulum, which consists of a mass of matter sus- pended from a horizon- tal axis in the shape of a knife-edge, after the manner of the compound pendulum. The bob is either made of some unelastic substance, as wood, or of metal pro- vided with a large cavity Fig. 152. MECHANICS OF SOLIDS. 267 filled with some soft matter, as dirt, which receives the projectile and retains the shape impressed upon it by the blow. Denote by V and ra, the initial velocity and mass of the ball; Vx the angular velocity of the bafistic pendulum after notation; the blow, 7 and M its moment of inertia and mass. Also let r represent the distance of the centre of oscillation of the pendulum from the axis A. That no motion may be lost by the resistance of the axis arising from a shock, the the pendulum , ,, , -. • -i • , i t ,• n -i • • must be struck at ball must be received m the direction of a line passing centre of through this centre and perpendicular to the line A 0. oscillation; This condition being satisfied, we have p = r, and Eq. (91) becomes r ra V Vx — rar2 4- 7' from which we find T/. (rar2 4" I)VX ,Q9, value forth V = A_---------1—- . . . (92); velocity of ra r projectile; the velocity V becomes known, therefore, when Vx is known, since all the other quantities may be easily found by the methods already explained. To find Vx, denote by H the greatest height to which the centre of gravity of the pendulum is elevated by virtue of this angular velocity; then, since the moment of inertia of the ball is ra r2, we have, from the principle of the living force, whence (/ _J_ mr2\ yi = 2 (M + m) g II; equation of livin§ v ' force; (7 4- TOr2)!? = 2ff (M + m)g 268 NATURAL PHILOSOPHY. Denoting by T the time of a single oscillation of the pendulum after it receives the ball, we have, Eq. (75), time of a single oscillation of / 7 4" TO r2 balistic 1 = If V/ 777—j-----T~n----j pendulum; V (J/ + W) B • 9 D being the distance from the axis to the centre of grav- ity; whence, 7 + ra r2 DT2 (M 4- m)g if2 ' and this value, substituted in the equation of the living force, gives DT2 whence angular velocity 1-r * / 2 H of the pendulum; ' X ~" ~7p ' y j\ ; also moment of inertia of the whole; 7 + rar2 = (^+ rn)g.D.T\ if2 ' and because, Eq. (78), time of oscillation of the equivalent Bimple J- = * y ---j pendulum; " we find length of this T2 g pendulum; ' " ~ 2 Substituting these values of "Pi, 74- rar2 and r in Eq. (92), we find V= ~ VT7TD. 1 m MECHANICS OF SOLIDS. 269 or, replacing the masses by the weight divided by the force of gravity, V = L V2IPDx W + simpler value for velocity of projectile; Fig. 153. in which W and w denote the weights of the pendulum and ball respectively. Observe that H is the height to which the centre of gravity rises in describing the arc of a circle of which D is the radius. Let GG'K be half of the circumference of which this arc is a part, 67 and 67' the initial and terminal positions of the centre of gravity during the ascent; draw 67'P perpendicular to K67. Then, because AG = D, and GR = H, we have, from the proper- ty of the circle, to find the radical part of this value; P67' = VPr(2P - 70; and if the pendulum be made large, so that the arc 67 67' shall be very small, which is usually the case, H may be neglected in comparison with 2 P, and therefore P67' = V2H.D; value of radical part found; V2 HD is half the chord of the arc described by the centre of gravity in one entire oscillation. Denoting this chord by C, and substituting above, we have V j,-C W + w w velocity of projectile in terms of the chord of the arc of vibration; From this equation, we may find the initial velocity V; and for this purpose, it will only be necessary to have the 270 NATURAL PHILOSOPHY. duration of a single oscillation, and the amplitude of the arc described by the centre of gravity of the pendulum. The process for finding the time has been explained. To to And the arc of find the arc, it will be sufficient to attach to the lower ex- tremity of the pendulum a pointer, and to fix on a perma- nent stand below, a circular graduated groove, whose centre of curvature is at A; the groove being filled with some soft substance* as tallow, the pointer will mark on it the extent of the oscillation. Knowing thus the arc, denoted by 6, and the value of D, found as already described, § 184, we have R 67' = \ C = D . sin i 6; whence its value found; G = 2D .sin i 6 and finally final value of velocity. V = If D W + w sin£ (93). SIMPLE MACHINES. A machine defined. § 188.—A machine is any device by which the action of a force is received at one set of points and transmitted to another set, where it may either balance or overcome the action of one or more opposing forces and perform its effective work. The force impressed is usually called the power, and that overcome, the resistance. We proceed to discuss the simple machines, so named because some one or more of them enter as elements into the composition of all machinery. MECHANICS OF SOLIDS. 271 XIV. FUNICULAR MACHINE. § 189.—This consists of an assemblage of cords or bars; Funicular the former united by knots, and the latter by joints or machine; hinges. The cords are supposed, for simplification, per- fectly flexible, the bars perfectly rigid, and both inexten- sible, without weight, and devoid of inertia. The weight and inertia of the several parts of every machine, are usually small when compared with the intensity of the weight and power and resistance ; and when this is not the case, they inertia sma11 as 1 J compared with may be estimated and taken into the account by the the power and methods already explained. The hypothesis of inextensi-the re8istance '■> bility is also admissible, because when a cord or bar is ex- inextensibiiity tended or the latter compressed under the action of one or admis»s«>ie; of several forces, the maximum change of dimensions is soon attained, after which the figure remains unaltered during the subsequent action. Let the extremities of the straight cord AB be solicited by Fig- 154 several forces. Each \ / / force may be resolved * ^f...........................,b\-— 27a -°f * "^ into two components, A one in the direction of the cord, the other at right angles to it. Since the cord is perfectly flexible, if it be in equiiibrio, the perpendicular components at each conditions of end must destroy each other, otherwise they would pro- eiuilibrium» duce flexure. The components in the direction of the cord must reduce to two forces, which are equal in in- forces must act tensity and immediately opposed. They must also act to t0 8tretch the stretch the cord, for compression would only bend it, and 272 NATURAL PHILOSOPHY. in the case of a bar, the forces may also act to compress it; action of the molecular springs; the tension the same throughout, except when vertical; cords never equally strong throughout; in practice, cords and bars are weaker as they are longer. the action of one force could not be transmitted to the point of application of the other. If instead of a cord we suppose a bar, the conditions of equilibrium will be the same, only that the bar being inflexible, the forces in the direction of its length may act either to stretch or to compress it. By recalling what was said of the physical constitution of bodies, we may regard the molecular forces as so many springs which, as soon as an effort is made to disturb the particles from their posi- tions of rest, are extended or compressed everywhere equally by the equal and contrary forces which act at the ends of the cord or bar. Hence the tension, that is, the effort by which any two consecutive elements are urged to approach each other or to separate, in the direction of the cord or bar, must be equal throughout, and equal to one of the equal forces in question, except when the cord or bar is vertical; in which case, the tension at any point is increased by the weight of all the particles below it. When a cord or bar is subjected to a force of traction, it stretches, and may even break. If it be equally strong throughout, the rupture ought to take place simulta- neously at all its points, and yet this is never found to be the case in practice, and it is because bars and cords are not homogeneous, and break at the weakest point. When two pieces of cord of the same kind, are of the same length, no reason can be assigned why one should break rather than the other under the same resistance; but when of unequal length, the chance of rupture is greater for the longer; and this is the reason why cords and ropes, which to all external appearances are the same in kind, are generally found to be weaker as they are longer. § 190.—We have seen that when forces which act upon the extremities of a cord are in equiiibrio, the re- sultant of those acting at one end, must be equal and directly opposed to that of those acting at the other; and MECHANICS OF SOLIDS. 273 Fig. 155. A ,/,„;;///s.,,„ /j,,,,^ that their common line of direction must coincide with that of the cord. The work of these, resultants must The work of the be equal, and hence we conclude that the work of the f°T wh,ich,act x at the ends of a forces which act at one end of a cord is equal to the c1- And, although the to- effect of components perpendicular to tal quantity of work IS the main rope of transmitted to the main the bell; . rope, yet the disposi- tion of inclined cords effect of a hoop. is a source of real loss, which is the greater in proportion as the incli- nation is greater. It is for this reason that a rigid hoop ra n is so introduced as to sepa- rate the cords, and give the portions to which the efforts are immediately applied parallel directions. Equilibrium of several cords meeting in a point; equilibrium of a gliding knot; § 191.—When several forces act upon cords which meet in a point and are united by a knot, the tension of any one is equal to the resultant of the efforts exerted upon the others, and the equilibrium requires that this same tension shall be equal and directly opposed to the force which solicits the cord in question. Hence, when forces are applied to cords which meet in a knot, the condition of their equilibrium requires that the effort of any one shall be equal and directly opposed to the resultant of all the others. When a force P is applied to a point P, which may slide along a cord whose ends A and P are fixed, the equilibrium of the point D requires that the direction of the force P shall bisect the angle ADB formed by the portions of the cord separated by the bend at P; for MECHANICS OF SOLIDS. Fig. 158. direction of the force applied to the knot, must bisect the angle of the two parts of the cord; the force P must be equal and directly opposed to the resultant of the tensions on PA and DB; but the whole cord ADB be- ing continuous, these tensions must be equal, since the tension is the same throughout; if, therefore, the distance P C be laid off on P D produced, pro- portional to the inten- sity P, and from C, the lines Cm and Cn be drawn parallel to DB and DA respectively, the figure Cm D n will be a rhombus, because D ra and P n, which represent the tensions, must be equal. An example of this mode of action is furnished by the manner of suspending a common lantern P from a small pulley P, of which the groove receives the cord A D B, whose ends are fastened to hooks at A and B. The weight of the lan- tern will cause the pul- ley to move till the direction of the weight bisects the angle made by the branches of the cord; the pulley will then come to rest and remain in a state of stable equilib- the puiiey win rium. The equilibrium will be stable because, being a be in st:ibIe ° equilibrium when heavy system, the centre of gravity is the lowest possible; at the lowest and to show this, it will be sufficient to remark that the pomt; length of the entire cord being constant, the point D will, when in motion, describe an ellipse of which A and Pare the Fig. 159. example in the mode of suspending the common lantern; 276 NATURAL PHILOSOPHY. Fig. 159. position of the horizontal tangent; when the pulley is replaced by a knot; tension will not be the same throughout; Fig. 160. the conditions of the equilibrium will be the same as those of three oblique forces; foci, and as the direc- tion P C, of the weight of the lantern, bisects the angle ADB, it will be perpendicular to the tangent to the curve at D, which must therefore be horizontal, and no point of the curve can lie below it. If the pulley be re- moved and the lantern be attached by a knot arbitrarily to some point as P, the freedom of motion will be destroyed, the tension will no lon- ger be the same through- out, and the conditions of equilibrium will be those of forces applied to three cords meeting at a single point. Produce the vertical P D, and lay off DC to represent the weight of the lantern. Denote its weight by W; the tension on PA by a, and that on P B by b; the angle ADB by — 6, a will be greater than b; that is to say, the tension will be the greater upon that branch with branch most which the direction of the weight makes the least angle. J J J greatest tension; If the cord ADB be drawn into a straight horizontal line,

~D n J polygon and A £>, ±> 6, and tion of forces (jp^ will be in the plane of the Fig. 162. dii art- in same plane; VL M the polygon a collection of heavy bars; parallel forces Q and P. In the /'! \y same way it may be shown that the entire polygon and the forces applied to it are in the same plane. If the polygon be a collection of heavy bars, each side will be solicited by its own weight in addition to the weights applied to the angles. Denote by w the weight of the bar AB; this weight must pass through the centre of gravity of A B. Besolve it into two MECHANICS OF SOLIDS. 279 components acting at the extremities of the bar. If the bar have the same cross section throughout and be of homogeneous density, the components at A and P will be 2 u: In like manner, if w' be the weight of the side P C, weisht of sides the components at P and C will be \ ":\ and so on for the ^i™, other sides. Thus the angles P and C will be acted upon components; by the weights J (w + w') and \ (w' -\- w") respectively, that is, by the half sum of the weights of .the adjacent sides. The extreme ends will each be acted upon by half the weight of the adjacent side; and thus we have but to consider the polygon as without weight and solicited by forces applied to its angular points. Since all the weights P, Q, R, S, and the weights w, w', w", kc, are maintained in equiiibrio by the reaction N and N' of tbe fixed points, which are equal to the tensions of the sides A' A and DD' respectively, the resultant of these tensions must be equal resultant of and directly opposed to that of all the weights. If, there- g^™^nsi0ns' fore, the lines A A' and DD' he produced, their inter-opposed to th: section 0 will give one point through which the resultant ° eaghts! of the weights P, Q, R, S, and that of the polygon, will pass; and this resultant being vertical, if the distance OM be laid off, by any scale of equal parts, so as to contain as many linear units as there are pounds inP-f Q + R + S + w + w> + w"'? &c., and two lines MU and MVhe drawn value of extreme through J7 parallel respectively to A A' and DD', the dis- tances 0 V and 0 U will give, by the same scale, the tensions at A' and P', or the values of N and N'. If the polygon be only subjected to the action of its own weight, the line OM may be drawn vertically through its centre of gravity. §194.—It is often of great practical importance to Method of finding . . ,t • i r j? • i i the tensions of know the tensions on the sides of a funicular polygon the 8ide8. subjected to the action of weights, in order to proportion the dimensions of its several parts. Let A B CD E he a polygon in equiiibrio, under the action of the weights P, Q, R, S, T, including the 280 NATURAL PHILOSOPHY. Fix 163. single tension; weights of the sides, and the extreme forces N and N', of which the di- funicuiarpolygon rections are A A' and undeletion EE'i respectively. De- ofweights; note the tension of the side AB by ^, that of BChj t2, that of CDhy t3, kc Since the equi- librium subsists about each angle, as A for example, the force N which acts from A to A', is equal and directly opposed to the resultant of the two forces P and ^; and if A n he taken on the pro- longation of A' A to represent N, the parallelogram Apno, constructed on Aw as a diagonal, will give Ap for the determination of weight P, and p n for the value of the tension ^. This being understood, draw the horizontal line a' e, upon which lay off the distances a'a, ab, be, cd, de, proportional to the weights P, Q, R^ S, and T. From the point a' draw a'S perpendicular to A A', and proportional in length to the tension N, and join S with the sev- eral points a, b, c, d, and e; then will aS, bS, cS, d S, and eS, represent, respectively, the ten- sions tx, t2, ts, t4, and N'. For the two triangles Apn and a' Sa are sim- ilar, because a'S and a' a are respectively perpendicular to An and Ap; hence the angles Sa'a andp An are equal; moreover, the sides about these equal angles are propor- tional by construction and we, therefore, have Fig. 164. general construction for finding the tensions; ✓A K\ \ \ An = N : pn = t, : : a'S : Sa, MECHANICS OF SOLIDS. 281 and if a'S represent the tension N, Sa must represent the tension ^. For the same reason, a b being proportional to demonstration; Q, the third side bS, of the triangle aSb, will be propor- tional to t2, since the three forces £,, Q, and ,t2, are in equiiibrio about the point P. Finally, since a a' and a' S are perpendicular to the directions A p and A n of the forces P and N, a S will be perpendicular to the side A B of which it measures the tension ^. It will -be the same of BC and b S, and so on. Therefore, when a funicular lines which polygon is in equiiibrio under the action of weights, if a rePresent the . . tensions, series of distances be taken on a horizontal line propor- perpendicular to tional to these weights, the lines drawn through the points *he sides of the u ° r funicular of division perpendicular to the corresponding sides of polygon; the polygon will meet in a point, and the lengths of these perpendiculars, included between the common point of intersection and the horizontal line, will measure the tensions of the sides of the polygon. The point S is point of tensions. called the point of tensions. § 195.—The sides of the polygon may be very short and only subjected to the action of their own weight, The catenary; which would be the case with a heavy chain A CB sus- pended from its extremities. The polygon of equi- librium then be- comes a curve, called the catenary. This curve is em- ployed to give form to arches and domes. The use of the catenary for such purposes may be illustrated by conceiving a series of equal spherical balls held together by mutual attrac- 2c?2 NATURAL PHILOSOPHY. illustration by a string of balls; sides of the polygon, the chords of the balls; tions, but with perfect freedom to slide the one over the other. Such a collection of balls would resemble a string of beads, and if supported at the ends would, under the action of their own weight, as- sume the form of the catenary, or rather funicular polygon, of which the sides would be the chords of the spheres joining the points of contact. If the whole ar- rangement be re- versed, and the balls, instead of being suspended, be sup- ported upon the ends as fixed points, after the manner in- the stringofbaiis dicated in A' C B', the figure will remain unchanged and the balls will still be in equiiibrio; for, the action of the weights will be the same as before, and the reciprocal action of the balls upon each other will simply be changed from a force of extension to one of compression. If we now suppose the points of contact to be extended into tangent planes, and the spaces between filled up with solid matter, as wood, stone, or metal, we shall have a perfect sys- tem of voussoirs or arch-solids in equiiibrio under the action of their own weight, re- quiring no aid from friction or any other principle of sup- reversed ; points of contact extended to tangent planes; Fig. 166. arch-stones or vii tTjirs; f^4 MECHANICS OF SOLIDS. 283 port. The tangent planes or joints of the voussoirs will be position of the normal to the curve. The catenary is also employed injoinIs; suspension-bridges supported upon two or more parallel also used in chains stretched across a river. In the construction of »u?PCMion- bridges. such catenaries it is important to determine the tension at the ends, in order to secure an adequate resistance at those points. Fig. 161 § 196.—The catenary A CB, suspended from two General points A and P, is nothing more, as we have seen, than ^^°ttbe a heavy polygon in equiiibrio, and whose sides are indefi- nitely small; so that, if upon a horizontal line, a length A'B'he taken proportion- al to its weight, and this length be divided into a number of equal parts, there will exist a certain point S such, that all the right lines drawn from it to the points of division, will be perpendicular to the small successive sides or elements of the cate- nary, and that the lengths SA', SF', SC, kc, of these lines, are propor- tional to the tensions of the same elements. Of all the tensions, the least is given by the line S C, drawn perpendicular to the horizontal line A' P'. But the element of the cat- enary to which this tension corresponds being itself horizontal, it will occupy the lowest point of the curve. •A, construction to /?'f \ \\ find the tension /'/'/ j \ \ of the different VZ^l.S.'-..A^----^fli-^' points of the ,-•' /' / ! \ \ catenary; Jir'ff>""(k..........Q-r---p'.......ji! A least tension at lowest point; This length becoming greater 284 NATURAL PHILOSOPHY. tension greater as the element is at a greater distance from the lowest point; Fig. 167. elements of equal tensions form equal angles with the vertical through lowest point; catenary symmetrical in reference to this line; on same level the extreme tensions are equal; position of the point of tensions. and greater in proportion as the oblique lines SF', kc, recede from the perpendicular SC, the tensions of the elements of the catenary will increase in proportion as they are at a greater distance from the lowest point. Whence it follows, that the tension is the greatest possible at the extremi- ties A and P. Two equal tensions SF' and S Gr, appertain to two elements equally distant from the lowest point C: moreover, these elements form equal angles with the vertical L C passing through this point; hence, these ele- ments, M and N, are situ- ated on the same horizon- tal line MN, and the chord MN, as well as all similar chords, will be divided equally by this vertical line. The catenary is, therefore, a symmetrical curve in reference to a vertical line passing T through its lowest point. It follows, also, that when the extremities or attached points A and P are on the same horizontal line, the extreme tensions are equal, and that the point of meeting which determines the tensions is upon the perpendicular drawn through the middle of the horizontal line A'B', which is proportional to the weight of the catenary. A and D being, for exam- ple, the two points of suspension, and A' D' being the length proportional to the weight of the catenary A CD, SC, perpendicular to A'D' and passing through the V.:'dL$.'.___¥-—-/— >->.'?■ B'"j)'"Gv~ -y, "A' MECHANICS OF SOLIDS. 285 point S, will divide A'D' into two equal parts A' C and CD'. § 197.—Two catenaries, (last figure), A CB and acb similar are similar when the points of suspension A and P of the catenaries; one, and a and b of the other, are situated upon parallel right lines, and when their lengths A CB and acb are proportional to the distances A P and a b, between their points of suspension. If the equilibrium subsists in the catenary ACB, this equilibrium will not be disturbed if the length of its elements and its other dimensions be propor- tionally diminished indefinitely, § 192. Therefore, when equilibrium A CP is reduced to the size acb, the equilibrium will not indePendentof ' L size; only exist, but there will be no one of its parts which will not be parallel and proportional to the corresponding part of the original. But since the elements of the smaller catenary acb are parallel to those of the larger ACB, all the tensions of the former are comprised within the angle A' SB', which contains the different tensions of the latter. We have, then, but to find in this angle, the posi- tensions of one tion of a line a'V parallel to A' B', which represents the *;ateni"7 foimd r > rr from those of a weight of the smaller catenary, as A'B' represents the similar one; weight of the larger, and the slightest consideration will show that the two tensions Sf and SF' situated upon the same line converging to S will appertain to parallel ele- ments of the two curves. These are called homogeneous homogeneous tensions. But because A'B' and a'V are parallel, we tension8> have the proportion Sf : SF' :: a'V : A'B' whence we conclude that, in two similar catenaries, the ten- tensions of sions of elements similarly situated are to each other as the elements . J ° similarly situated weights of the catenaries. are as the weights of the entire 286 NATURAL P HIL 0 S O P H Y. Fig. 168. § 198.—Let A'B' To construct the be a horizontal line prO- catenary trom its ,. i , , i • n , weight, length, portional to the weight and the point of 0f the catenary, S the tensions; _ . „. point of tensions. Di- vide the line A' B', and the length of the cate- nary into the same and a great number of equal parts; those of the cate- nary may be regarded as its elements, and those of A[B' their corresponding weights. Draw the lines SA, SI', S2', SB' . . . SB'; these will be perpendicular to the different elements of the catenary. From any point A, on SA, draw A 1 perpendicular to SA and equal to an element of the catenary; from the point 1 draw 1-2 perpendicular to SV and equal to an element; again 2-3 perpendicular to S2', and equal to an element, and so on to the end. The polygon A-l-2-3 . . . B, will approximate to the re- quired catenary the nearer in proportion as the number of divisions is greater. to draw a tangent to any point of the catenary. The point of ten- sions S gives the means of drawing a tangent to the catenary at any point. Let E be the given point, and let A' e represent the weight of the portion A E of the catenary; through e and S draw the indefi- nite line e 67, and from E draw P67 perpen- dicular to e S, E G will be the tangent line. Fig. 169. MECHANICS OF SOLIDS. 287 Fig. 170. § 199.—The point of tensions in the cate- nary depends upon the intensity and directions of the extreme ten- sions. For A'B' being the horizontal line proportional to the weight of the entire catenary, if from the extremities A' and B' arcs be de- scribed with radii proportional to the extreme tensions, their intersection S will give the point of meeting. The process for find- ing the extreme, ten- Flg" 171, sions must of course depend upon the data given. Let us first sup- pose the catenary A CB to be given and traced out. It is evident from the conditions of equi- librium, that the verti- cal 0 L drawn through the intersection 0 of the extreme tangents A 0 and P 0, will pass through the centre of gravity of the catenary. If, therefore, a distance 0 G be taken on this line to represent the entire weight of the catenary, and the parallelogram OP' 67A' be constructed upon the tangents, the sides 0 A' and OB' will represent the tensions at A and B respectively. But if only the two Determination of the point of tensions; to find the extreme tensions from the curve traced; points A and P of suspension, the weight, and entire length of the catenary be given, the process for finding the extreme tensions is as follows, viz.: Take a small chain and sus- pend it against a ver- Fig. 17S to find the extreme tensions from the points of support, the weight, and length of the curve; 288 NATURAL PHILOSOPHY Fisr. r figure found by means of a small chain; the tensions found by construction: tical plane from two points a and b, situated upon a right line paral- lel to A P, and whose distance apart shall be to the distance from A to B, as the length of the smaller chain is to the length of the longer. The smaller chain being thus sus- pended, measure by means of a spring balance the tension exerted at the points a and b. The tensions on the points A and P produced by the larger chain, will be equal to the tensions at a and b, multiplied by the number of times which the weight of the larger chain contains that of the smaller. § 197. Instead of measuring with a spring balance the tensions at the ends of the catenary, we may proceed as follows: Draw through the low- est point of suspen- sion a, a horizontal line cutting the oppo- site branch of the small chain in the point d. Upon a horizontal line take the distance a' V to represent the weight of the entire chain, and lay off the distance a' d' proportional to the length acd. The por- tion a c d of the cat- enary would be in equiiibrio if the point d were fixed and the remainder db removed; the point of tensions for acd, and therefore for acb, will, from what has already been explained, be found MECHANICS OF SOLIDS. 289 somewhere on the perpendicular C K' drawn to the mid- dle of a' d'; assume it at 0, and by means of this point and the line a' V, construct a catenary after the manner construction of of § 198, and let a e he the resulting distance between its an aPProximate curve. points of support. Through 0 draw a perpendicular to C K', and lay off upon it from the point 0, the distance Og = ae — ab,to the right when ae is greater than ab, and to the left when the reverse is the case. Assume another point as 0' below 0, and do the same as before; we shall find a new point g', say to the left of C K'; repeat the process with points between 0 and 0' several times, and pass through the points g, g', g", kc, thus determined, a curve; its intersection S with C K' will be the true point of tensions. The distances Sa' and SV will repre- sent the extreme tensions. § 200.—We have seen that in the catenary the tensions at the different points are different, and that the small- est tension is at the lowest point. This is still true when the catenary becomes a vertical chain loaded with a weight. For the lowest link supports only the attached weight Q; the link C only supports the weight Q and link C, and so on to the topmost link, which supports all below it; so that if the chain were proportioned to the tension of its different parts, it would be made stronger above than below. Fig. 174. The smallest and greatest tension of a vertical chain. § 201.—The point S being the point of meeting of the tensions, and A' B' a hori- zontal line representing the weight of the catenary, we have seen that the tension at 19 Fig. 175. Direct measure of the tension on any point of the catenary; 290 NATURAL PHILOSOPHY. triangle; tension in horizontal direction; in vertical direction; Fig. 176. D is represented by the length D' S, and that at C, construction; the lowest point, by SC, perpendicular to A' B', the lengths A' D' and D' C representing, respectively, the weights of the portions AD and D C of the the tension at any Curve J that is to Say, hyTotheluseofa the tension &t **? right-angled point P, is represent- ed by the hypothe- nuse of a right-angled triangle, of which one side represents the ten- sion at the lowest point of the curve, and the other the weight of that portion of the cate- nary included between the lowest point and the point whose tension is to be found. Hence, the tension at any point of the curve, estimated in a horizontal direction, is constant and equal to the entire tension at the lowest point; and estimated in the vertical direction, is equal to the weight of that portion of the catenary included between this point and the lowest point. The horizontal tensions at A and P are therefore the same, although they may be situated on very different levels. H the catenary be suspended from the tops of piers, the vertical components will promote their stability tensions on piers. b^ pressmg tbem down, while the horizontal components will tend to overturn them. effect of these § 202.—It is comparatively easy to compute the ex- treme tensions of the catenary when the versed sine of its arc is small. Let A CB he a catenary, of which CD, the MECHANICS OF SOLIDS. 291 distance of the lowest point below the horizontal line BA, To find extreme is very small. The curve being in equiiibrio, the equi-tensionswhen ■*■ ' u the versed sine of the curve is small; librium of the part P C will not be disturbed by taking the point C as fixed, and regarding it and the point P as the points of suspension. But because of the smallness of P C, the curvature must be very small, and the centre of gravity of P C may, without sensible error, be regarded as at the middle point 67. The tangents CHand B 67', at the notation, points of suspension, will intersect at 67' on a vertical line drawn through the point 67. Denote by T, the tension at B; by T^ the tension at C; and by p, the weight of the portion P C. Because the three forces p, T, and T0, are in equiiibrio about the point 67', we have whence p : To :: BH : HG', p : T :: BH : BG', n HG' -P' T>rr j tension at lowest point; m BG' , . 1 = p .-------m tension at highest B H point; Observe that BH is the versed sine, which denote by /; and, because B 67 C may be regarded a right line, HG' is half the semi-space BD, which semi-space denote by I Then, since the triangle P 67'P" is right angled, 292 NATURAL PHILOSOPHY. Substituting these quantities in the above equations, we find horizontal tension or thrust; tension at highest point. Pn = pl_ 2/' T ^•v^T^V / 1 + i2 The first expresses the tension at the lowest point, which we have seen is equal to the horizontal thrust at the points of suspension. The second gives the entire tension at the same points, which must be known in order to adjust the dimensions of the chain. Application to suspension- bridge ; § 203.—To conclude the subject of the catenary, and show the application of the preceding principles, take the case of a bridge suspended from two parallel chains extended from one bank of a river to the other. To the different points, A, P, C, kc, of the catenaries, or rather to the angles of the funicular polygons thus formed, are attach- suspendingrods; ed vertical suspending rods, which are united at the bottom in pairs by transverse pieces called sleepers; these receive a set of longi- Fig. 178. 0 a joists; sections; tudinaiyoiste, which, in their turn, support the floor plank. The distances between the suspending pieces in longitudi- nal direction are supposed equal. These equal portions of the roadway included between two consecutive sleep- ers, are called sections. Each sleeper is loaded with half the section which precedes and half that which follows it; that is to say, with the weight of an entire section. This MECHANICS OF SOLIDS. 293 Fig. 179. weight is known, and determines the cross sec- tion of the suspending rods. The weight of the suspenders being small compared with that of the roadway, may be neglected, and thus the weight of the bridge will be equally distribu- ted. Draw a horizontal right line, and take uv proportional to the weight of the bridge; let S he a point such that Su shall be perpendicular to the side UA, and proportional to its tension. Take upon u v, the portions u a, a b, kc, proportional to the weights supported at the angles A, B, kc.; the converging lines aS, bS, kc, will be proportional to the tensions on the sides A P, &c, and the perpendicular Sd, to the tension on the horizontal side of the polygon. First, find the difference of level between any two consecutive angles, as A and P. Draw the horizontal line BA", and the two triangles A A"B and Sud, will be similar and give each pair of suspending rods supports the weight of one section; tensions on the sides of the funicular polygon; A A" A" B ud Sd; whence A A" = A" B Sd ud. difference of level between two consecutive angles; Because of the equality of distances between the sus- pending rods, A"B will be constant. Moreover, ud and Sd being proportional respectively to the weight of the portion A'D', and the tension t0 upon the horizontal side, 294 NATURAL PHILOSOPHY. if we denote by w the weight of a unit of length of the bridge, ratio of weight of half the bridge to the horizontal tension; ud ~Sd uA'D' tn which in the preceding gives uA"B.A'D' A A" = tn alue of the inference of level f two onsecutive .ingles; but wA"P is the weight of a section of the bridge. Denoting this by jp, we have A A" = 2- . A' D' tn and denoting the constant ratio of the weight p to the tension t0 at the lowest point by k, A A" = k.A'D'; from which we conclude, that the difference of level of two consecutive angles, is equal to the constant ratio k, multiplied by the horizontal distance of the higher of the two angles from the lowest angle of the funicular poly- gon. Denoting by I the constant length of a section, and beginning at the lowest angle K, the horizontal distances will be successively I, 2 I, 3 I . . . nl, for the 1st, 2d, 3d, . . . nth, angle to the right and left. Thus the difference of level between the lowest angle K and the next in order C, is k I; between C and P, 2 k I; between B and A, Ski, kc. The heights difference of level of the angles C, B, A, &C, of the angles of ■■ .-, -, , . , ^ .,, the polygon aD0Ve tne lo^est point K, Will above the lowest be respectively kI, kl-\-2kl, angle; kl + 2kl + Skl,kl + 2kl + 3 k I -f 4 k I, and, in general, Fig. 180. MECHANICS OF SOLIDS. 295 if there be n sections between the lowest angle and that un- der consideration, the height of the latter above the former will be given by the expression H(l + 2 + 3+4___+ n) = kl w+1 height of the nth angle above the lowest one; In this expression, if we make successively n = 1, n = 2, n = 3, n = 4, &c, we have kI, 3 kI, 6 kl, 10 kI, kc, for the heights of 1st, 2d, 3d, 4th, &c, angles above the horizontal side of the funicular polygon. The locus of all these angles is a parabola, for if y = KP = M U de- note the height of one of these angles above the lowest point K, n being tbe number of its place from the latter, we have Fig. 181. the locus of the angles is a parabola; K.J? y = kl. n n + 1 (96); and making nl = x = KM, , /» + l\ k . x or y = kx2 "27 4- equation of the locus of the angles; this is the equation of a parabola, of which the vertex is 2do NATURAL PHILOSOPHY. to the right of the point K, and at a distance from it equal to I x = — place of the vertex of the it is below the horizontal side by the distance locus curve; HK' = y= - —; y 8 ' a quantity so small that it may be neglected in practice. Moreover, from the property of the parabola, the squares of the ordi- nates are to each other to find the point as the abscisses; that is in which the , vertical through t0 Sa7> the vertex cuts the line of supports; IP" BQ' MA : NB; and from the similar triangles obtained by joining A and P, AP BQ' PC QO2; whence PC QO' MA : NB; or but PO2 x NB = QO x MA; PO = OK - KP = OK - MA, QO = QK - OK = NB - KO; MECHANICS OF SOLIDS. 297 which, substituted above, give (OK - MA)2 x NB = (NB - KO)2. MA; developing the squares and reducing, we get KO = MA x NB. Fig. 183. That is to say, the distance KO, at which the vertical line distance of this drawn through the vertex of the curve cuts the chord point above the ° vertex; joining any two of its points, is a mean proportional be- tween the heights of these points above the vertex. This property furnishes an easy method of finding the lowest point K on the level MN. For this purpose, join the points of suspension U and V, by the cord U V; draw the horizontal line UP through the lower point U, and produce it till it cuts the vertical VN in P'. Upon the distance P' V describe the semi- circle VTP', and from the point N draw the tangent NT; with N as a centre and NT as a radius, describe the arc TT' till it cuts VN in T', and through the point T' draw a hori- zontal line; this line will cut the cord U V in the point 0, through which draw a vertical line OP, and its intersection with the horizontal side will give the lowest point K. Taking this point as the extremity of the horizontal side, and laying off on the line MN the equal lengths of the and the abscisses sections; the points of division will correspond to the °ointesansu" 0 ^f.\ 1 \^ /JE- construction for finding the position of the lowest point; M jst vertical ordinates kl, 3 kl, 6 kl, n 4-1 kl. This last appertaining to the point U, whose height h is given, 298 NATURAL PHILOSOPHY. we have whence we have A ratio of the 2 h weightofa k = ( ,; .... (97); section to ,l VA ' -V 0 horizontal tension and hence the lengths of the several suspenders kl, BkI, from which the &c, are known. lengths of the We haye geen ^ suspenders are known; £ - *____Vt___ *r = k = t0 n (n 4- 1) I and therefore n (n 4- 1) . I. p /nox and horizontal tn =---------pr—f--------- . . . (98) J tension found; " "• the tension on the horizontal side is, therefore, also known. The tension on the side next in order to the horizontal side is tension on side vtn2 4" V2 next in order; that of the second in order that on the , 2 (t> ,2 Becond in order; v ^ ' \*P)) that of the third that on third; A2 4- (3^, and so on to that on the ?ith in order; Vt2 4- (npY, MECHANICS OF SOLIDS. 299 which is the tension on the nth side from the horizontal one. If the points U and V he on the same level, it is obvious that the curve or polygon becomes symmetrical in reference to the vertical 0 K, in which case it is only necessary to find the lengths of the suspenders for one data necessary to half the bridge. Having given the points of suspension, JJ^J,0*^ their horizontal distance apart, and the level of the lowest bridge. side of the funicular polygon, it is easy to determine the dimensions of every part of the bridge. XV. OF BODIES RESTING UPON EACH OTHER, AND UPON INCLINED PLANES. § 204.—When two bodies touch and compress each Action and , ■■ tl .. t.t i ■ -it • reaction of bodies other, there is immediately a depression or yielding in a foi.ced into direction perpendicular to the surfaces at the point of apparent contact; contact, which indicates that the reaction of the two bodies takes place in the same direction; that is to say, in the direction of the normal common to both surfaces. Let us suppose one of the two bodies as A to be solicited by forces of which the re- sultant shall coincide with this nor- F%- 184- mal, and that the other body A' is fixed; it is plain that the reaction of the latter body will destroy this resul- f A^\----^ action and tant, and that the body A will remain K_J ™f™<* ' J bodies; at rest. But the equilibrium will also subsist if the body A' be replaced by a force equal to the reaction which it exerts on the body A, while this latter body is perfectly free to move and acted upon by this new force in conjunction with the given forces. This property of all bodies, by which they resist the re- 300 NATURAL PHILOSOPHY. the reaction of two bodies extends to several. ciprocal action of each other in directions normal to both the principle of surfaces at the common point of contact, extends to the general case of a single body pressing upon two or more bodies at the same time. The reaction of these last are so many real forces which may be substituted for the resisting bodies at the several points of contact, and in virtue of this substitution, the conditions of equilibrium of the first body will be the same as though it were free to move in any direction whatever. Let us examine the circumstances of the simple case of a body resting upon .a plane, and having first but one point of contact, then two, three, &c. Fig. 185. illustration; § 205.—Let us consider a sphere subjected to the action of its own weight, and rest- the bodies having ing upon a level plane A B but one point of • ,t • i • , r- . . contact- w a smg^e P°mt of contact m. Since the reaction takes place in the direction of the perpendicular to the plane through the point of contact, and must be in equiiibrio with the weight W of the sphere, the centre of gravity 67 must be upon a vertical line, in order that the weight and reaction may destroy each other. In like manner, when a body rests upon any plane whatever, and is solicited by forces, no matter how directed, their resultant must be perpendicular to the plane, and pass through the point of contact; for if the resultant were oblique, it might be resolved into two components, one normal, and the other parallel to the plane ; the first would be destroyed by the reaction of the plane, while the latter would put the body in motion. In order, therefore, conditions which that a body, supported against a plane, and having a TresrSatJstY single Pomt of contact with it, shall be in equiiibrio, it is plane. necessary, 1st, that the resultant of the forces which act upon ii be perpendicular to the plane; and 2d, that this resultant pass through the point of contact. MECHANICS OF SOLIDS. 301 § 206.—But when the body has two points of contact, A Kg. 186- If the body have •T ' two points of and^P, with the plane, it is not ^------^ contact the , i , , -i -,, , o / >. resultant need necessary that the resultant of z'----^\ not pass through the forces shall pass through [S ____________^ either; either. It will be sufficient if it meet the line A P in any point between A and P, and be perpendicular to the plane. For the reaction of these points of support being both perpendicular to the plane, their resultant, which is parallel to tl^em, will also be perpendicular to it: this resultant and that of the forces acting upon the body must be in equiiibrio; they must, therefore, be equal and directly opposed; in other words, the resultant of the forces acting upon the body must admit of being resolved it must be normal , -, i t , i to the plane, and into two components, respectively equal and directly op- intersect the ljne posed to the resistances at the points of support. But J°iuius the points o of contact; these latter act in the same direction, so also must the for- mer, and hence their resultant will have its point of appli- cation between A and P; and this resultant being parallel to its components, will be perpendicular to the plane. If the body be laid on a horizontal plane, the equi- when the plane is librium will subsist whenever the vertical drawn through the centre of gravity intersects the line joining the points of support somewhere between them. g 207.—Now let us suppose three or more points of contact. The resistances of these points are perpendicular case of three or' to the plane, and cannot maintain the forces which act upon moreP°mts; the body in equiiibrio unless the resultant of the latter may be decomposed into components which are respectively equal and directly opposed to these resistances; this resultant must, therefore, be perpendicular to the plane, and as its components must act in the same direction, resultant sim its point of application will, from the principles of parallel ™™*;£d forces, be within the polygon formed, by joining the points polygon of of contact. If the line of direction of the resultant, pierce °° 302 NATURAL PHILOSOPHY. if the resultant pierce the plane without the polygon of contact, the body will overturn; the plane in a point m, ex- Fig- 187. terior to the polygon which connects the points of support, the body will tend to overturn around the edge a b of this polygon nearest to m; if the line of contact be a curve, the Fig. 188. body will overturn about the tangent nearest to m. The ef- fort by which the body will be urged to overturn is meas- ured by the intensity of the resultant of the forces, into effort by which the shortest distance from its L overturn.11^8 nne °^ direction to that about which the motion of rotation takes place. § 208.—The conditions of equilibrium of a heavy sphere, resting upon a horizontal plane, have already been considered. Let us ap- ply the same principles to other examples, and take first the case of a heavy body resting upon a table having but three feet. If the feet be upon a horizontal plane and when the feet are in the same right line, and the vertical line through the centre of gravity be not in the vertical plane passing wm overturn through this line, the table will overturn towards the side on which the centre of gravity is situated, and with an effort equal to the product of the weight into the distance A g of the projection of the Examples; table having but three feet: in same right line; unless the weight pass through this line; Fig. 190. MECHANICS OF SOLIDS. 303 Fig. 191 Fig. 192. centre of gravity from the line a a' of rotation. This product is called the moment of stability. If the distance Ag is zero, the weight will pass through the line of support, and there will be an equilibrium; but it will be unstable, since the centre of gravity will be at the highest point. If the three feet be not in the same right line, and the weight pass within the tri- angle formed by joining the feet, the table will be in equiiibrio. But if the line of direction of the weight pass without the triangle of the feet, the table will over- turn about the nearest edge ab. In the first case, the equilibrium is stable, because no derangement can take place about the line of either two of the feet without caus- ing the centre of gravity to ascend. And, generally, if the table have any number of feet, there will be stable equilibrium whenever the line of direction of the weight passes within the polygon formed by joining them. The effort with which the table or any other body will resist a cause which tends to upset it, is measured by the product of its weight into the shortest distance A g from the line of direction of the weight to the line ab about which the motion is to take place; moment of stability; Fig. 193. Fig. 194. if the feet be not in same right line; stable equilibrium; in case of any number of feet the resultant must pass within the polygon; effort by which a body resists a cause to overturn it; 304 NATURAL PHILOSOPHY. moment of stability of a heavy body; the same principles apply to solids resting on plane faces; Fig. 195. s>— Fig. 196. example of the cube and right prism; stability diminishes as the centre of gravity is higher; inclined prism; and this effort will be smaller in proportion as the dis- tance Ag is less. For this reason, the moment of stability of a heavy body is the smallest moment of its weight taken with reference to the different lines of its polygonal base. The conditions are the same if the body rest upon a plane face bounded by a polygon or curve. The equi- librium will exist when the line of direction of the weight passes within the base. Such, for example, is the case with the cube resting upon a level plane; also with a right prism, whatever its height, only that its stability diminishes as the height increases; for, in pro- portion as the centre of grav- ity 67 is more and more ele- vated, the angle GAB be- comes less and less, and the centre of gravity will not have to be raised so much above its position of rest when the body is overturned about the edge a a', as it would if the angle GAB were greater, or the centre of gravity lower. In proportion as the centre of gravity is placed higher and higher above the same base, the body will approach more and more to the condition of unstable equilibrium. An inclined prism will pre- Fi m serve its equilibrium as long as the direction of its weight falls within its base. The dif- ficulty of overturning it will be less in proportion as the / V c j /w \ /----a1......-y fA MECHANICS OF SOLIDS. 305 Fig. 198 A"* Fig. 199. distance A g becomes smaller. When g falls without the base, the prism will overturn of itself. The Tower of Pisa, though considerably inclined, preserves its equilibrium be- cause the line of direction of its weight passes within its base. A pile of dominos or bricks, in which each one pro- jects beyond that immediately i--------j below it, will preserve its I I equilibrium till the line of |----H direction of the weight of the entire pile falls without the domino or brick at the bottom, when it will overturn. We see, therefore, that the natural stability of bodies in- creases as their bases increase, and the heights of their centres of gravity decrease; and that it is the greatest possi- ble when the centre of gravity is at the centre of figure of the base. This is the reason why walls are usually made of elements like brick, cut-stone, &c, placed with their faces vertical, and laid upon large bases, called foundations. If the heavy bodies are solicited by other forces than their weights, the resultant of the whole, weight included, must act in the direction of a line passing within the base. The resultant of the extraneous forces may unite with the weight and increase the stability of the body. Thus an inclined prism, the direction Gg of whose weight falls without the base A P, would, if abandoned to itself, over- turn ; whereas, if it were act- ed upon by a force in the di- rection 67 E, of such intensity as to give, with the weight, a resultant which intersects will overturn when weight falls without the base; Tower of Pisa; inclined pile of brick; stability increases as the base increases and as the centre of gravity is lower; heavy bodies solicited by other forces than their weights; Fig. 200. these may act to increase the stability; 306 NATURAL PHILOSOPHY. equilibrium stable; the base at 0, it would be supported, and the equilib- rium would be stable. Be- ciprocally, the weight W of the prism is opposed to the force GE=F, when the latter acts to turn the solid about the edge A. The measure of this opposing effort is Fig. 200. moment of stability; W.Ag; foregoing in the construction of sustaining walls; principle of counterforts; moment of natural stability; Fig. 201. a and in this view, we see that the moment of the natural stability will increase as A g increases. In walls destined to support an embankment of earth or a head of water, in order to resist the thrust with greater illustration of the effect, the lower exterior edge A is thrown as far as con- venience will permit from the vertical line Gg of the weight. This is done either by an exterior slope P A, or by masses of masonry C, called counterforts, attached to the back of the wall. It will be sufficient, in general, for the stability of the wall, if the resultant of its weight W and the pressure against it, inter- sects the base A D. The moment of natural stability of such structures is always equal to the product of the weight into the distance A g ; and therefore the figure of the cross-section of the wall may be varied at pleasure without injury to the sta- ff d Fig. 202. MECHANICS OF SOLIDS. 307 Fig. 203. E__G- 1 W f n when the ground is compressible, the resultant should intersect middle of the base; Fig. 204. case of a loaded cart on a level; bility, provided this product remain the same. Hence the external slope may be suppressed, if the thickness of the external slope wall be so increased that its augmented weight shall com- an weig *' pensate for the* diminution in Ag. If the ground upon which the wall rests be compressible, it will not be sufficient that the resultant of the weight and pressure pass within the base; it must also pass through its centre of figure; otherwise there would be more pressure on one side of this point than on the other, and the wall would incline in that direc- tion. If the load of a two- wheel cart be such that the direction of its weight does not intersect the axle- tree, it will tend to overturn on the side of the weight, and will either exert a pres- sure upon the horse or an effort to lift him from the ground, according as the weight passes in front or in rear of the axle-tree. If the centre of gravity of the load be immediately above the axle-tree on a level road, then, when the cart is as- cending a slope, the weight will pass behind, and the ten- dency of the load will be to lift the horse; while, on the contrary, when the cart is Fig. 205. on an inclined road ascending the tendency is to lift the horse; Fig. 206. descending; 308 NATURAL PHILOSOPHY. the tendency is descending a slope, the tendency of the load will be to to press upon the tniw & pressure upon him. If the centre of gravity be • on the axle-tree, the horse will experience no effort of the kind referred to. of height to base; § 209.—Let A B represent Fig- 207- the section of an inclined inclined plane; plane in the direction of its greatest declivity. Although the plane be indefinitely pro- longed, it will be sufficiently defined by ratio defined by the relation of the base A C to the height CB, corresponding to a given length A P. Conceive a heavy body resting upon this plane, and of which 67 is the centre of gravity. The equilibrium of this body requires, 1st, that its weight shall intersect the plane within the polygon formed by joining the points of con- tact; 2d, that the weight shall be perpendicular to the plane. This last condition cannot be satisfied for any but a horizontal plane, since the weight is always vertical. If the weight be replaced by its two components, one perpen- dicular and the other parallel to the plane, the former will be destroyed by the resistance of the plane, while the latter will cause the body to move in the direction of its length BA. If the direction of the weight meet the plane within the polygon of contact, the parallel component will cause the body to slide, otherwise it will cause it to roll. This last will happen in the case of a spherical ball, since the weight will not meet the plane in the single point of con- tact m. Let a force P be applied in the direction 67 S, next figure, to prevent the body from moving down the plane. Since the body must be in equiiibrio under the action of its weight W and the force P, these must have a resultant, and this resultant must be perpendicular to the plane and intersect a body on an inclined plane; the body may slide or roll; conditions of equilibrium; MECHANICS OF SOLIDS. 309 Fist. 208. it within the polygon of contact, or in the case of the plane in which sphere, at the point m. The force P must, therefore, be 1^°^™** applied in a vertical plane which passes through the centre of gravity, and which is, at the same time, perpen- dicular to the inclined plane. Lay off on the vertical through the centre of gravity 67, the distance 67 67' to represent the weight W, through the same point draw 67 if perpendicular to the inclined plane, and through 67', the line 67' M parallel to the di- rection of the force P; from the point M draw MQ paral- lel to 67 67'; the distance 67 Q will represent the intensity of the force P, and 67 if that of the resultant, R, of W and P. From the principle of the parallelogram of forces, we have intensity of the W : R : P '.'. sin Q 67 M : sin G' 67 Q : sin 67'67M; force found but 67 67' and 67 M being respectively perpendicular to A C and A B, the angle A is equal to the angle 67' 67 M, and we have J} intensity of the force found by construction; analytically; sin G'GM= sin BAC = -r-f. BC AB and this substituted in the foregoing proportion gives, after reduction, W: R : P:: AB. sin Q G M : AB. sin 67' 67 Q : BC; from which we find • BC P = W A P . sin 0 67 if ' (99); value of the force; R = W sin 67' G Q sin QGM value of the (100). pressure against the plane; 310 NATURAL PHILOSOPHY. power applied parallel to the plane; If the power P be ap- plied parallel to the plane, the angle QGM = 90°; and fhe angle 67' 67 Q becomes the sup- plement of the angle A B C; whence we have value of force; value of the pressure against the plane; sin Q 67 M = sin 90° sin 67' 67 Q = sin A P C which, in the above equations, give P = W • ** r )V AB1 AC AB R = W A_C_ AB' weight, and resistance of plane; Fig. 210. That is to say, when the power is applied parallel to the relation of power, plane, 1st, the power will be to the weight as the height of the plane is to its length ; 2d, the resistance of the plane will be to the weight as the base of the plane is to its length. If the power be ap- plied parallel to the base of the plane, the angle QGM becomes equal to the angle ABC, because 67 $ and 67 M are respec- tively perpendicular to B C and AB; and the angle 67' 6? Q becomes 90°, whence power applied parallel to the base; relation of the angles; sin Q 67 M = sin A P C = -~, A P' sin .67' 67 Q = 1; MECHANICS OF SOLIDS. 311 which, in Eqs. (99) and (100), give BC P = W • R = W AC AB AC value of power; pressure on plane; That is to say, when the power is applied parallel to the base of the plane, 1st, the power will be to the weight as the height relation of power, of the plane is to its base ; 2d, the resistance of the plane will ^JtancTJ be to the weight as the length of the plane is to its base. In the application of the power parallel to the plane, the power will always be less than the weight. When applied parallel to the base, the power will be less than limits within the weight, while the inclination of the plane is less than ^teTe^iTn 45°. When the inclination is 45°, the power and weight weight. will be equal. When the inclination exceeds 45°, the power will be greater than the weight. Fig. 211. Motion of a heavy body on an inclined plane; § 210.—Let us now consider the motion of a heavy body on the in- clined plane. The body being acted upon by its weight 67 67' alone, this, may be resolved into two components, the one 67 if, perpendicular, the other 67 N, parallel to the plane. The first will be totally destroyed by the resistance of the plane, while the second will be effective in giving motion. Denote the weight of the bodv bv W, the height P C of the plane by h, and its to find the J J ' „ , ., 0 ., , • i component of length A P by I; then, from the similarity of the triangles tne weignt parallel to the plane; APCand 67 67'iV, will W GN h; 312 NATURAL PHILOSOPHY. whence its value; GN = j • W; and because the incli- nation of the plane is the same throughout, the ratio — will be constant, 6 from the top to the bot- tom ; whence we see that the motion is that the motion of the same body down the plane, is that arising from the ac- tion of a constant force. It will, therefore, be uni- Fig. 211. arising from the action of a constant force; it will be uniformly varied; formly varied, and the circumstances of motion will be given by the laws of constant forces. Substituting Mg for W, we have GN= jMg; and making M equal to unity, and denoting by g' the corresponding value of the component 67 N, we find component of the force of gravity in direction of the plane; 9 = Such is the intensity of the force of gravity in the direc- tion of the inclined plane. This may be varied at pleasure by changing the ratio y; in other words, by altering the inclination of the plane. Now, since the velocities im- pressed during the first unit of time on the same body, the motion may moved from rest, are proportional to the forces producing varying the them, the motion may be made as slow as we please by inclination of the fa plane; diminishing y. It was in this way that Galileo discovered MECHANICS OF SOLIDS. 313 the laws which regulate the fall of heavy bodies. These in this way being the same as for bodies moving on an inclined plane, diag(!0ev0ered the it was easy so to regulate the inclination of the plane as to laws of failing enable him to note and compare the spaces described, times elapsed, and velocities acquired, with each other. If the body be mounted upon wheels, Fig. 212. as in the case of the loaded cart referred to in § 209, it will be urged to roll along the inclined plane by an effort of which the measure is when the body is mounted on wheels it will roll; example of the loaded cart; W.D; in which W denotes the weight of the cart and its load, and P the perpendicular distance mb from the point of contact moment of the m, to the line of direction 67 b of the weight W. effort by which ' ° rotation is produced. XVI. FRICTION AND ADHESION § 211.—When two bodies are pressed together, expe- niction; rience shows that a certain effort is always required to cause one to roll or slide along the other. This arises manifested when almost entirely from the inequalities in the surfaces of two bodies are J -1 < pressed together contact interlocking with each other, thus rendering it and one is moved necessary, when motion takes place, either to break them overtneother; off, compress them, or force the bodies to separate far enough to allow them to pass each other. This cause of resistance to motion is called friction, of which we distin- 314 NATURAL PHILOSOPHY. sliding and rolling friction; the measure of its intensity. guish two kinds, according as it accompanies a sliding or rolling motion. The first is denominated sliding, and the second rolling friction. They are governed by the same laws; the former is much greater in amount than the latter under given circumstances, and being of more import- ance in machines, will principally occupy our attention. The intensity of friction, in any given case, is measured by the force exerted in the direction of the surface of contact, which will place the bodies in a condition to resist, during a change of state, in respect to motion or rest, only by their inertia. Fig. 213. intensity § 212.—The friction between two bodies may be meas- s^rinTbaiance; ured directly by means of the spring balance. For this purpose, let the surface CD of one of the bod- ies i7, be made perfectly level, so that the oth- er body M', when laid upon it, may press with its entire weight. To some point, as E, of the body M', attach a cord with a spring balance in the manner the indication of indicated in the figure, and apply to the latter a force F of such intensity as to produce in the body M' a uniform motion. The motion being uniform, the accelerating and retarding forces must be equal and contrary; that is to say, the friction must be equal and contrary to the force F, of which the intensity is indicated by the balance. The experiments on friction which seem most entitled to confidence, are those performed at Metz by M. Morin, the most valuable under the orders of the French government, in the years ZsToTSorin; 1831> 1832> and !833. They were made by the aid of a contrivance, first suggested by M. Poncellet, which is one of the most beautiful and valuable contributions that the balance, when the motion is uniform is the measure; MECHANICS OF SOLIDS. 315 theory has ever made to practical mechanics. Its details where these are given in a work by M. Morin, entitled uXouvelles™p^™™amay Experiences sur le Frottemeni." Paris, 1833. The following conclusions have been drawn from these experiments, viz.: The friction of two surfaces which have been for a conclusions from considerable time in contact and at rest, is not only differ- experiment8; ent in amount, but also in nature from the friction of sur- faces in continuous motion; especially in this, that the friction of quiescence is subjected to causes of variation and uncertainty from which the friction during motion is exempt. This variation does not appear to depend upon the extent of the surface of contact; for, with different pressures, the ratio of the friction to the pressure varied greatly, although the surfaces of contact were the same. The slightest jar or shock, producing the most imper- in machinery, ceptible movement of the surfaces of contact, causes the *^^i Jh,c friction of quiescence to pass to that which accompanies motion to be , . , it t_ ' • considered; motion. As every machine may be regarded as being subject to slight shocks, producing imperceptible motions in the surfaces of contact, the kind of friction to be em- ployed in all questions of equilibrium, as well as of motions of machines, should obviously be this last mentioned, or that which accompanies continuous motion. The laws of friction which accompanies continuous the laws of this motion are remarkably uniform and definite. These laws *n^.m"and g-j.0 . definite; 1st. Friction accompanying continuous motion of two arst law; surfaces, between which no unguent is interposed, bears a constant proportion to the force by which those surfaces are pressed together, whatever be the intensity of the force. 2d. Friction is wholly independent of the extent of the second law; surfaces in contact. 3d. Where unguents are interposed, a distinction is to be made between the case in which the surfaces are simply third law; unctuous and in intimate contact with eacli other, and that in which the surfaces are wholly separated from one another 316 NATURAL PHILOSOPHY. influence of inguents; an apparent exception to second law; three conditions of the surfaces in respect to friction; fourth law; remarkable instance of the uniformity of these laws; by an interposed stratum of the unguent. The friction in these two cases is not the same in amount under the same pressure, although the law of the independence of extent of surface obtains in each. When the pressure is in- creased sufficiently to press out the unguent so as to bring the unctuous surfaces in contact, the latter of these cases passes into the first; and this fact may give rise to an apparent exception to the law of the independence of the extent of surface, since a diminution of the surface of con- tact may so concentrate a given pressure as to remove the unguent from between the surfaces. The exception is however but apparent, and occurs at the passage from one of the cases above-named to the other. To this extent, the law of independence of the extent of surface is, there- fore, to be received with restriction. . There are then three conditions in respect to friction, under which the surfaces of bodies in contact may be considered to exist, viz.: 1st, that in which no unguent is present; 2d, that in which the surfaces are simply unctuous; 3d, that in which there is an interposed stratum of the unguent. Throughout each of these states the friction which accompanies motion is always proportional to the pressure, but for the same pressure in each, very different in amount. 4th. The friction, which accompanies motion, is always independent of the velocity with which the bodies move; and this, whether the surfaces be without unguents or lubricated witk water, oils, grease, glutinous liquids, syrups, pitch, &c, &c. The variety of the circumstances under which these laws obtain, and the accuracy with which the phenomena of motion accord with them, may be inferred from a single example taken from the first set of Morin's experiments upon the friction of surfaces of oak, whose fibres were parallel to the direction of the motion. The surfaces of contact were made to vary in extent from 1 to 84; the forces which pressed them together from 88 to 2205 MECHANICS OF SOLIDS. 317 pounds; and the velocities from the slowest perceptible motion to 9.8 feet a second, causing them to be at one time accelerated, at another, uniform, and at another, retarded; yet, throughout all this wide range of variation, in no result; instance did the ratio of the pressure to the friction differ from its mean value of 0.478 by more than ^ of this same fraction. Denote the constant ratio of the normal pressure P, to the entire friction F, by/; then will the first law of fric- first law tion be expressed by the following equation, expressed by an •*• ° o ± j equation; D — J (101); whence F = f.P. Fig. 214. This constant ratio / is called the coefficient of friction, coefficient of because, when multiplied by the total normal pressure, fnctlon! the product gives the entire friction. Assuming the first law of friction, the coefficient of friction may easily be obtained by means of the inclined plane. Let W denote the weight of any body placed upon the inclined plane AB. Resolve this weight 67 67' into two components, one 67 if perpendicular to the plane, and the other parallel to it. Because the angles 67' 67 M and BA C are equal, the first of these components will be its value found by means of the inclined plane; W. COS A, component of the weight perpendicular to the plane; 318 NATURAL PHILOSOPHY. that parallel to the plane; the friction on the plane; friction and parallel component equal; Fig. 214. and the second, IF. sin A, in which A denotes the angle BAG. The first of these com- ponents determines the total pressure upon the plane, and the friction due to this pressure, will be /. Wcos A. The second component urges the body to move down the plane. If the inclination of the plane be gradually increased till the body move with uniform velocity, the total friction and this component must be equal and opposed; hence By, G \ Tvy V 4. G' 0 f. W. cos A = W. sin A; whence value of the coefficient of friction; / = sin A cos A tan A. We, therefore, conclude, that the unit or coefficient of friction between any two surfaces, is equal to the tangent of the angle which one of the surfaces must make with the horizon in order that the other may slide over it with a uniform velocity, the body to which the moving surface belongs being acted upon by its own weight alone. This angle of ft-iction; angle is called the angle of friction or limiting angle of limiting angle of . , resistance; resistance. The values of the unit of friction and of the limiting angles for many of the various substances employed in the art of construction, are given in the following tables: MECHANICS OF SOLIDS. 319 TABLE I. Experiments on Friction, without Unguents. By M. Morin. The surfaces of friction were varied from .o3336 to 2.7987 square feet, the pressures from 88 lbs. to 22o5 lbs., and the velocities from a scarcely- perceptible motion to 9.84 feet per second. The surfaces of wood were planed, and those of metal filed and polished with the greatest care, and carefully wiped after every experiment. The presence of unguents was especially guarded against. SURFACES OP CONTACT. Friction of Motion.* Friction of QuiESCENCE.t 1.1 11 O O .5 O g J«3« ■S a 0 0 Oak upon oak, the direction of the ) fibres being parallel to the motion ) Oak upon oak, the directions of the" fibres of the moving surface being perpendicular to those of the quies- -cent surface and to the direction of the motion:):....... Oak upon oak, the fibres of both sur- ) faces being perpendicular to the >■ direction of the motion - - - -J Oak upon oak, the fibres of the") moving surface being perpendicular to the surface of contact, and those > of the surface at rest parallel to the direction of the motion - - - J Oak upon oak, the fibres of both sur- "1 faces being perpendicular to the I surface of contact, or the pieces [ end to end........ Elm upon oak, the direction of the ) fibres being parallel to the motion J Oak upon elm, ditto§..... Elm upon oak, the fibres of the mov-"] ing surface (the elm) being perpen-dicular to those of the quiescent > surface (the oak) and to the direc- j tion of the motion......J 0.478 0.324 o.336 0.192 o.432 0.246 o.45o 25° 33' 17 58 18 35 IO 52 23 22 i3 5o 24 16 0.625 o.54o 0.271 o.43 0.694 0.376 0.570 32° 1' 28 23 i5 10 23 17, 34 46 20 37 29 4i * The friction in this case varies but very slightly from the mean. t The friction in this case varies considerably from the mean. In all the experi- ments the surfaces had been 15 minutes in contact. t The dimensions of the surfaces of contact were in this experiment .947 square feet, and the results were nearly uniform. When the dimensions were diminished to .043, a tearing of the fibre became apparent in the case of motion, and there were symptoms of tbe combustion of the wood; from these circumstances there resulted an irregularity in the friction, indicative of excessive pressure. Cg It is worthy of remark that the friction of oak upon elm is but five-ninths of that of elm upon oak. 320 NATURAL PHILOSOPHY. TABLE I.—continued. SURFACES OF CONTACT. Ash upon oak, the fibres of both sur- faces being parallel to the direction of the motion....... Fir upon oak, the fibres of both sur-) faces being parallel to the direction >• of the motion.......) Beech upon oak, ditto..... Wild pear-tree upon oak, ditto - - Service-tree upon oak, ditto - - - Wrought iron upon oak, ditto* - - Ditto, the surfaces being greased and ) well wetted........j Wrought iron upon elm..... Wrought iron upon cast iron, the fibres of the iron being parallel to the motion........ Wrought iron upon wrought iron, the fibres of both surfaces being par- allel to the motion...... Cast iron upon oak, ditto - - - - Ditto, the surfaces being greased and wetted......... Cast iron upon elm...... Cast iron upon cast iron .... Ditto, water being interposed be- tween the surfaces -..... Cast iron upon brass...... Oak upon cast iron, the fibres of the wood being perpendicular to the direction of the motion - - - - Hornbeam upon cast iron—fibres par- allel to motion....... Wild pear-tree upon cast iron—fibres parallel to the motion - - - - Steel upon cast iron...... Steel upon brass ------- Yellow copper upon cast iron - - - Ditto oak - - - - Brass upon cast iron...... Brass upon wrought iron, the fibres of the iron being parallel to the mo- tion .......... Wrought iron upon brass - - - - Brass upon brass....... Friction of Motion. S1* O o 0.400 0.355 o.36o 0.370 0.400 0.619 o.256 0.252 0.194 o.i38 o.iq5 O. ID2 o.3i4 0.147 0.372 0.394 o.436 0.202 O.I52 o. 189 0.617 0.217 0.161 0.172 0.201 ,5cJ Friction of Quiescknce. a o ii 3 ( Greased, and 0.332 0.869 < saturated with the direction of the motion, ( water. and those Of the oak parallel to it........J NATURAL PHILOSOPHY, TABLE III.—continued. SURFACES OF CONTACT. Friction of Motion. Friction of Quiescence. UNGUENTS. 0) 9 "3 <~.2 o :-O til Coefficient of Friction. The same as above, moving ) upon cast iron.....) Ditto ditto - - - -Soft calcareous stone of Jau-~ mont upon the same, with a layer of mortar, of sand, and ► lime interposed, after from 10 to 15 minutes' contact J 0.194 o.i53 O.74 Tallow. Olive oiL conclusions in regard to olive oil and lard; tallow not so well suited to metal. A comparison of the results enumerated in the above table leads to the following remarkable conclusion, easily fixing itself in the memory, that xoith the unguents hogs' lard and olive oil interposed in a continuous stratum between them, surfaces of wood on metal, ivood on wood, metal on toood, and metal on metal, when in motion, have all of them very nearly the same coefficient of friction, the value of that coefficient being in all cases included between 0.07 and 0.08, and the limiting angle of resistance therefore between 4° and 4° 35'. For the unguent tallow the coefficient is the same as the above in every case, except in that of metals upon metals ; this unguent seems less suited to metallic surfaces than the others, and gives for the mean value of its coefficient 0.10, and for its limiting angle of resistance 5° 43'. Adhesion; § 213.—Besides friction, there is another cause of re- sistance to the motion of bodies when moving over one another. The same forces which hold the elements of bodies together, also tend to keep the bodies themselves together, when brought into sensible contact. The effort by which two bodies are thus united, is called the force of Adhesion. MECHANICS OF SOLIDS. 327 spring balance; Familiar illustrations of the existence of this force are illustrations of furnished by the pertinacity with which sealing-wax, wa- J^™™^° fers, ink, chalk, and black-lead cleave to paper, dust to articles of dress, paint to the surface of wood, whitewash to the walls of buildings, and the like. The intensity of this force, arising as it does from the Ms intensity affinity of the elements of matter for each other, must vary ^tPeennt *"f™the with the number of attracting elements, and therefore with surface of the extent of the surface of contact. This law is best verified, and the actual amount of ad- hesion between different substances determined, by means measured by the of a delicate spring-balance. For this purpose, the surfaces of solids are reduced to polished planes, and pressed together to exclude the air, and the efforts necessary to separate them noted by means of this instrument. The experiment being often repeated with the same substances, having different extent of surfaces in contact, it is found that the area of the surface di- vided by the effort necessary to produce the separation gives a constant ratio. Thus, let S denote the area of the surfaces of contact expressed in square feet, square inches, or any other superficial unit; A, the effort required to separate them, and a the constant ratio in question, then will Fig. 215. mode of operation ; A S = a, or, A = a . S The constant a is called the unit or coefficient of adhesion, coefficient of and obviously expresses the value of adhesion on each adhesion; unit of surface, for making S = 1, 328 NATURAL PHILOSOPHY. we have A = a, adhesion between Bolids and liquids; mode of ascertaining its amount in any case; precaution to be observed; Fig. 216. attraction of fluid elements for each other and for those of solids; diversity in the action of bodies in this respect; To find the adhesion between solids and liquids, sus- pend the solid from the balance, with its polished surface downward and in a horizontal posi- tion; note the weight of the solid^ then bring it in contact with the hor- izontal surface of the fluid and note the indication of the balance when the separation takes place, on draw- ing the balance up; the difference between this indication and that of the weight will give the adhesion; and this divided by the extent of sur- face, will give, as before, the coeffi- cient a. But in this experiment two opposite conditions must be carefully noted, else the cohesion of the ele- ments of the liquid for each other may be mistaken for the adhesion of the solid for the fluid. If the solid on being removed take with it a layer of the fluid; in other words, if the solid has been wet by the fluid, then the attraction of the elements of the solid for those of the liquid is stronger than that of the elements of the liquid for each other, and a will be the unit of adhesion of two surfaces of the fluid. If, on the contrary, the solid on leaving the fluid be perfectly dry, the elements of the fluid will attract each other more pow- erfully than they will those of the solid, and a will denote the unit of adhesion of the solid for the liquid. It is easy to multiply instances of this diversity in the action of solids and fluids upon each other. A drop of water or spirits of wine, placed upon a wooden table or piece of glass, loses its globular form and spreads itself MECHANICS OF SOLIDS. 329 over the surface of the solid; a drop of mercury will not do so. Immerse the finger in water, it becomes wet; in quicksilver, it remains dry. A tallow-candle or a feather illustration of this from any species of water-fowl remains dry though dipped lvei8iy' in water. Gold, silver, tin, lead, &c, become moist on being immersed in quicksilver, but iron and platinum do not. Quicksilver when poured into a gauze bag will not run through; water will: place the gauze containing the quicksilver in contact with water, and the metal will also flow through. Solids which become wet on being immersed in a fluid, effect of covering lose this property if covered with any matter not similarly ^^[^c • affected by that particular fluid. A drop of water placed upon a wooden table or piece of glass, smeared with oil or tallow, will not spread, but retain its globular shape and roll off, if the surface be sufficiently inclined. Pour water from a clean common glass tumbler nearly full, and it will run along the exterior surface; smear the rim with hogs' illustrated in the lard or tallow, and the fluid will flow clear of the tumbler. °wo (wat!', < lrom a tumbler ; The living force with which the elements of the water in contact with the glass tend to leave the tumbler by the pressure from behind, is, in a great measure, overcome by the attraction between the glass and water, and they are thus made to flow along the surface, while the viscosity of explanation; the water, or the attraction of the fluid particles for each other, drags the remote elements after them; and thus the water, under the combined action of its living force, adhesion for the glass and viscosity, becomes spread out into a sheet of which the plane is normal to the surface of the tumbler. When the tumbler is smeared with grease, the adhesion is so much reduced as to offer but feeble opposition to the living force with which the water reaches the edge of the "tumbler, it will, therefore, pass the edge after the manner of a projectile. Quicksilver poured out case of of a glass or wooden vessel will, in like manner, flow clear J^'1^ of the outer surface ; but the contrary will happen if a tin diirerent kinds of vessels; vessel be used. 330 NATURAL PHILOSOPHY. effect of interposing a fluid between surfaces in contact; it is difficult to find the adhesion between the rubbing surfaces of machinery; this adhesion may be disregarded; except in watches and the like. Friction on a plane; normal component of the weight; The adhesion of solids is apparently increased by intro- ducing a liquid between them. The fluid fills up the ex- isting inequalities of the surfaces, and thus, by increasing the number of points of contact, increases the adhesion by an amount equal either to that of the fluid particles for each other, or to that of the fluid for the solid for which it has the least affinity, depending upon whether the solids are wetted or not by the interposed fluid. This is strikingly exemplified by means of common window-glass, blocks of wood, metallic plates, and the like. It is difficult to ascertain the precise value of the force of adhesion between the rubbing surfaces of machinery, apart from that of friction. But this is attended with little practical inconvenience, as long as a machine is in motion. The experiments of which the results are given in the table of § 212, and which are applicable to machinery, were made under considerable pressures, such as those with which the parts of the larger machines are accustomed to move upon one another. Under such pressures, the adhe- sion of unguents to the surfaces of contact, and the oppo- sition to motion presented by their viscosity, are causes whose influence may be safely disregarded as compared with that of friction. In the cases of lighter machinery, however, such as watches, clocks, and the hke, these con- siderations rise into importance, and cannot be neglected. § 214.—Let any body M, rest with one of its faces in contact with the inclined plane A B. Denote its weight by W, and suppose it to be solicited by a force F in the direction 67 Q, making with the inclined plane the angle Q G q', which denote by . The total pressure upon the plane will be W. cos a — F. sin + Q" = W[ML-LB'-\ +fW[AM+ BB'], or quantity in the Q = W X BC+f.W.AC.. (107). ascentand descent; Now it is to be remarked, that every trace of the path actually described by the burden whose weight is W, has disappeared from this value for the quantity of work; this latter is, therefore, wholly independent of this path, and for the same burden, only depends upon the difference of level from A to B, and the horizontal distance A C between these points; so that, the work would be the same as quantity of work though the load had been transported from A to B along £* ™^"pBth one continuous plane. Nothing is said here of the resist- had been ance of the atmosphere, which, like the friction, would be 8' a cause of opposition to the motion. § 216.__We are now prepared to measure the tension Tension of cords; of a cord arising from the action of its own weight. For ooo NATURAL PHILOSOPHY. the tension of a its own weight; this purpose take the cord PA MF, resting upon any surface of cord arising from ^^ ± - ^ ^ , its own weitrht: *"«" O est point, and con- sider the part AF which tends by its weight to move in the direction from A to F. Omit the con- sideration of friction for the present, and the question will con- sist in this, viz.: to Fig. 221. weight of a given portion; this weight resolved into components; find a force which, acting in the direction of its length, will keep the cord in equiiibrio. This force must be equal and directly opposed to the tension on the part A F. Designate by W, the weight of a unit of length of the cord; then considering the element whose length is MN, its weight will be W.MN. Through the centre of gravity 0 of this element, draw the vertical 0 G to represent this weight, which resolve into two components 67 Q and Q 0, the one perpendicular and the other parallel to the cord. The first will be destroyed by the reaction of the surface; the second will act to move the cord in the direction of its length, and will determine its tension. Draw MN' perpendicular and' NN' parallel to the horizon; then will the triangles 67 Q 0 and MNN' he similar, both being right-angled triangles, and the angle Q G 0 of the one, equal to the angle MNN' of the other, because the side 67 Q is per- pendicular to MN, and 0 67 to NN'; hence the pro- portion, QO : 0 67 : : MN' : MN; MECHANICS OF SOLIDS. 339 whence n n - 0 G X MN' component of V U — ------If^V------' weight parallel ■***■ -*■* to the cord; Denote the tension by t, which will be equal to Q 0; 0 G represents the weight, equal to If x MN; and pro- jecting the points A, M, N, F, P, upon the vertical by the horizontal lines A a, Mm, Nn, Ff, and Pp, we have MN' equal to m n, and the last equation becomes, W X MN X mn ljr vaiueofthe t — -------MN-------= X mn' tension for a single element; The second member is the weight of a portion of the cord equal in length to the vertical projection mn of the element MN. Now the length A F is composed of a number of elements, each one of which produces, in hke manner, a tension equal to the weight of a portion of the cord of the same length as its vertical projection. The tension on each element is transmitted in the direction of the cord to the elements above. Hence, the entire tension at any point of the cord, is measured by the weight of a portion tension at any equal in length to the vertical projection of all the cord Pointismeasuie« , , . . by the weight of below it. Thus, if F he the end of the cord, the ten- the vertical sion at A will be measured by the weight of a portion of p™^11™0^11 ^ ° r the cord below it; the cord equal to afi provided no motion take place. In like manner, the tension at A, arising from the weight of A P, will be measured by the weight of a portion equal to ap, so that if the cord have no fixed point it will move in the direction of the lower end F, under the action of Fig- 222. a force equal to W(af — ap). Lf the ends of the cord be upon the W AV an endless cord is same level, or if the cord be endless, it m equilibria will be in equiiibrio. 340 NATURAL PHILOSOPHY. | 217.—We shall now take into consideration the fric- lvutiuu of a cord tion of a cord when shding around any body, say a fixed sliding around a t l • i 1_ • — nxed cylindrical cylindrical beam in beam: a horizontal position. Let the cord support at one end a weight W, and be subjected to the action of a force F applied at the other end. If the force communi- cate motion, it must not only raise the weight W, but must also overcome the friction between the cord and solid. If the surface were perfectly polished, the friction would be zero, and the force F would be equal to the weight W, in the case of an equilibrium. Divide the enveloping portion of the cord, a, ^, ^, ^, &c, into an indefinite number of very small and equal parts, and draw through the points of division, ^, t^ t^, &c, tangents to the cord; these tangents will intersect, two and two, at the points b, V, b", kc, and the extreme ones will coincide with the straight portions of the cord to which the force and weight are applied. The points of division being extremely close, the arcs will be sensibly confounded with their chords a tx, t^ t2, ^ ^, &c. The tension of the cord on the tangent ab, with which the cord sensibly coincides, is obviously equal to W, if we neglect the weight of the cord. Let ^ be the tension which acts at ^ on the second tangent b V; this tension must overcome the weight W and the friction on the arc ^ a, comprised between the points of contact. to and the Denote by p the pressure exerted by this element upon Sreiemtnt«.f the cylinder> and hJ f the coefficient of friction, then the cord; will coustniction of the figure and notation: MECHANICS OF SOLIDS. gl] k = W + fp. its value; To find the pressure, we will still disregard the weight of the cord, and remark that the two tangents a b and b ^ are equal. Moreover, if we construct the rhombus am^b, and consider a b as proportional to the weight W, this same side will represent the tension of the cord from a to tx. The diagonal b m, will be normal to the chord a ^ and therefore to the surface of the cylinder, and being the to end the normal resultant of the tensions at a and U will be the pressure pressm'e ai'isi!lg r. from the tension ; arising from the tension, and consequently equal to p. The triangles a 04 and mab are similar, because they are both isosceles, and the angle 0 of the one is equal to mab of the other; hence mb : a\ :: ab : 0 a; mb represents the pressure p; at^ may be taken equal to the arc of which it is chord, which denote by s; ab represents the weight W; and Oa is the radius of the cyhnder, which denote by R, and the proportion may be written p : s : : W : R; whence S . W_ value of this P T> J normal press 1 )■'; and this, substituted in the value of tx, gives value of the Denoting by t2, the tension along the third tangent ^^est b" t2, and at the third point of division 4, this tension must the resistance; 342 NATURAL PHILOSOPHY. overcome the tension ^ and friction pro- duced by the elemen- to and tension on tary arc t2 ty, equal in next element in - , -, , order; length to a ^, or S. In a word, ^ will be circumstanced in re- spect to ^ as ^ was in regard to W. Hence Fig. 223. ■is value ; -*( 1 + f_s R and if %, tA, t5, . . . tn he the tensions on the consecutive tangents, and at the points %, t^ t5, . . . tn in order around the beam, we shall have values for the successive tensions in order around the beam; *> = L = h(i+fi\ tn = tn—xil + JT J- value of the tension on the last element of contact; Multiplying these equations together and dividing out the common factor, we have tn = W (i+sr The tension tn, being the last in order, brings us to the straight portion of the cord to which F is apphed, and, therefore, tn must be equal to F; whence MECHANICS OF SOLIDS. 343 / f S \n relation between F = VV . ( 1 -f- -77 J . the power and the » £*> ' l*p.sistnn/»o ■ resistance; Developing this by the rules for the binomial theorem, we have P= Fri 4- «.^ 4- n(p__l) fj? tnis vahie L ^ R ^ 1.2 R2 developed; + n(n-l)(n-2)fl ^ 1.2.3 R* + ®c-j It must be remembered that 5 was taken indefinitely small, and therefore for any definite extent of contac between the cord and cylinder, n must be indefinitely great; hence the numbers 1, 2, 3, 4, &c, connected with n by the sign minus, may be neglected in comparison with n; this gives F = Wil + -Jl + ztP^ + ^f^ + &,]: -—« but n s is equal to the entire arc enveloped. Denote this by S, and the above becomes fS f2 S2 P S3 the quantity within the brackets is the development of the function eR> whence / s final relation F = yr X e^ . . . . (108), power and the resistance; in which e = 2.71825, the base of the Nap. system of logarithms. 344 NATURAL PHILOSOPHY. example to illustrate; Suppose the cord to be wound around the cylinder three times, and/= £, then will S = B*.2R = 6 X 3.1416 . R = 18.849 R, and F = W X e*xl8M9 = W X (2.71825)' 6.2832 . ) or F = IF. 535.3; importance of friction; its absolute necessity. that is to say, one man at the end W could resist the com- bined effort of 535 men, of the same strength as himself, to put the cord in motion when wound three times around the cylinder. This explains why it is that a single man, by a few turns of her hawser around a dock-post, is enabled to prevent the progress of a steamboat although her machinery may be in motion. Here friction comes in aid of the power, and there are numerous instances of this; indeed, without friction many of the most useful\ contrivances and constructions would be useless. It is j by the aid of friction that the capstan is enabled to do its / work; the friction between the rails of a railroad and the ^\ wheels of the locomotive enables the latter to put itself and JJ its train of cars in motion. But for the friction between^ the feet of draft animals and the ground, they could per- form no work; nor, indeed, could any animal walk or even stand with safety, if they were deprived of the aid of this principle. MECHANICS OF SOLIDS. 345 XVII. THE WEDGE. description and use: § 218.—Thus far we have only considered the cases of The wedge; a body pressing against a single surface. The same body may also act against two or more surfaces at the same time. Such, for example, is the case Fig- 224. with the Wedge, which consists of an acute right triangular prism ABC, usually employed in the operation of separating and splitting. The acute di- hedral angle A Cb, is called the edge; the opposite plane face Ab, the back; and the planes A c and Cb, which terminate in the edge, the faces. The more common application of the wedge consists in driving it, by a blow upon its back, into any substance which we wish to split or divide into parts, in such manner that after each common advance it shall be supported against the faces of the ^^!im opening till the work is accomplished. | 219.—The blow by which the wedge is driven for- ward will be supposed perpendicular to its back, for if it were oblique, it would only tend to impart a rotary motion, and give rise to complications which it would be unprofit- the wow upon able to consider. And to make the case conform still fur- beep™peg°d-cu7ar ther to practice, we will suppose the wedge to be isosceles, to the back; 346 NATURAL PHILOSOPHY. to find the resultant of the reactions on the faces; Fig. 225. The wedge ACB being inserted in the opening ahb, and in contact with its jaws at a and b, we know that the resistance of the latter will be perpendicular to the faces of the wedge. Through the points a and b, draw the lines aq and bp normal to the faces A C and B C; from their point of inter- section 0, lay off the distances Oq and Op equal, respectively, to the resistances at a and b. Denote the first by Q, and the second by P. Completing the construction and parallelogram Oqmp, Om will represent the resultant of the resistances Q and P. Denote this resultant by R', and the angle A CB, of the wedge, by 6, which, in the quadrilateral a Ob C, will be equal to the supplement of the angle a Ob = p Oq, the angle made by the directions of Q and P. From the parallelogram of forces we have, notation; R'* = P2+ Q2 + 2PQcospOq = P2+ Q*-2PQcos&; or value of the resultant; R' = VP2 + Q2 - 2PQcos&. The resistance Q will produce a friction on the face A C equal to fQ, and the resistance P will produce on the face to find resultant B C, the friction/P / these act in the directions of the faces of the wedge. Produce them till they meet in C, and lay off the distances Cq' and Cp' to represent their intensities, and complete the parallelogram Cq' O'p'; C 0' will repre- of frictions; MECHANICS OF SOLIDS. 347 sent the resultant of the frictions. Denote this by R", and we have, from the parallelogram of forces, R"a = fQ2 + fP2 + 2f2PQ cos 6; or R" = f V P2 + Q2 + 2 P Q cos 6. value of the resultant of the frictions; The wedge being isosceles, the resistances P and Q will be equal, their directions being normal to the faces will inter-. sect on the line CD, which bisects the angle C— $; and their resultant will coincide with this line. In like manner the wedge being the frictions will be equal, and their resultant will coincide 1S( with the same line. Making Q and P equal, we have, from the above equation, R' = P VT(1 - cos 6), R" = fP V 2 (1 + cos 6). these values result; But 1 — cos 6 = 2 sin2 £ 6, 1 + cos 6 = 2 cos2 £ 6; whence we obtain, by substituting and reducing, R' = 2 P. sin i 6, R" = 2f.P. cosjo-; and further, sin -S-d cos % & = AB AC CD AC' circular functions in terms of elements of the wedge; 348 NATURAL PHILOSOPHY. therefore, final value of these resultants; R' = P AB_ AC' ff n R'=*f-P-AO Denote by F the intensity of the blow on the back of the wedge. If this blow be just sufficient to produce an equi- librium bordering on motion forward, call it F'; the fric- tion will oppose it, and we must have, value of the blow when the wedge is on the eve of moving forward; F' = R' + R" = P ■ 45 + 2/. A. 6 CD A C (109). If, on the contrary, the blow be just sufficient to prevent the wedge from flying back, call it F"; the friction will aid it, and we must have, value, when on the eve of moving back; F" = P- AB A C -^•p--9rc ••■(llu> limits within which the blow may vary to produce no motion; The wedge will not move under the action of any force whose intensity is between F' and F". Any force less than F", will allow it to fly back; any force greater than F' will drive it forward. The range through which the force may vary without producing motion, is obviously, F> _ p" = ±fp. CD A 0 (in); which becomes greater and greater, in proportion as CD and A C become more nearly equal; that is to say, in pro- portion as the wedge becomes more and more acute. The ordinary mode of employing the wedge requires that it shall retain of itself whatever position it may be driven to. This makes it necessary that, Eq. (110), MECHANICS OF SOLIDS. 349 conditions that A Tt PT) A 7? CD p.^ = 2f.P-±-±L, 0I p±£<2f.p. • the wedge may AC J AC AC J AC retain the place to which it is driven; or, omitting the common factors and dividing both members of the equation and inequality by 2 CD, AB CD f, or \AB CD ' of power and resistance; If there were no friction, there would be no obstruction to the free transmission of the effect of the force F to the substance to be compressed. But, making/zero, we have, Eq. (112), A T> work of the Fx = P ' X ' —. n ', power without -A 6 friction; and, from the principle of virtual velocities, Fx — P • x • -j-*;- = Ss; A C 352 NATURAL PHILOSOPHY. whence Px = Ss x A C AB' relation of quantities of work, of power, resistance, and friction; Now substituting this value for P x, and the above value for x, in Eq. (112), and it reduces to F-s CD AB = Ss + 2f-S-s CD AB (113). The first term of the second member is, obviously, the effective quantity of work done, being the reaction of the yielding body multiplied into the distance through which this reaction has been exerted, or through which the body has been compressed. This, we see, is less than the quan- tity of work of the power F, by the quantity quantity of work absorbed and lost; 2f.S.s x CD, AB] which has been totally absorbed, and therefore lost, in consequence of the friction. This loss is often very great, and to illustrate, suppose the reaction S to be 1000 pounds, and that the back of the wedge A B is ^ of its length CD; then will illustration by an example; lbs. lbs. 20 . F. s = 1000 . s + 40 ./. s . 1000; and, taking/=j^, lbs. 20 .F.s = 1000. s + 4000 s. numerical loss; Assigning any particular value to s we please, it appears that the useful effect is only 1000 s, while the loss from friction is 4000 s, and that the work performed by the MECHANICS OF SOLIDS. 353 force F is 5000 s. Dividing the above equation by 20 s, we get W KC\S'\ OAa'" ocr^*' numerical value I = 50 + 200 = 250; of the power; which is much less than 1000, the value of the reaction S. Hence we see that the advantage of the wedge press con- defect of the sists in this, viz.: by its aid the work may be executed machme- with comparatively a feeble power. The machine is, how- ever, defective, on account of the large amount of work absorbed by its friction. § 221.—As before remarked, the wedge is driven Effect of the blow forward by a blow on its back. This mode of employing in the use of the n . r j o wedge; force is an additional source of loss of work. When a hammer strikes the wedge, two periods are to be dis- tinguished, viz.: the first corresponds to the duration of the shock, that is to say, from the instant the hammer touches the wedge to that in which the greatest com- pression of the wedge and hammer takes place; the sec- ond follows immediately and includes the interval during which the reaction of the body to be pressed gives rise to the resistance called S, and to the frictions due to the pressures P and Q. While the wedge is acquiring motion under the blow, during the first period, its inertia acts as a resistance; in the second period, the inertia becomes a power to overcome the resistance S. The blow develops explanation of at each instant, between the hammer and wedge, realtheeffect; pressures, which are measurable in pounds; and these pressures are greater, for the same effect, in proportion as the duration of the shock or blow is shorter. The wedge will, in the first period, have a motion from the action of these pressures in consequence of its lateral compression; the inertia due to this motion being opposed by the lateral parts of the machine will give rise to friction, which fric- tion, together with the inertia exerted by the wedge in 354 NATURAL PHILOSOPHY. loss of living force; its measure; loss of living force from permanent change of figure. acquiring an increased velocity under the continued action of the hammer, will be in equiiibrio with these pressures. The work of these frictions, during the first period, will be absorbed by the machine, and therefore lost to the inertia or living force of the wedge when this living force becomes, in the second period, a power to overcome the resistance S. The quantity of action, or half the living force, preserved by the wedge at the close of the first period, and with which it enters upon the work to be performed during the second, will be given by the rule furnished in Eq. (113), and from which it appears, that this quantity of action will be equal to the quantity of action of the hammer on the back of the wedge during the first period, diminished by that consumed by the fric- tion due to the wedge's inertia within the same period. But this is not all. A part of the work of the hammer is consumed by the permanent change of 'figure of the wedge arising from the violence of the action. Thus we see, that a considerable portion of the living force with which the hammer begins its work, is lost by change of figure, and by friction due to the sudden development of inertia; neither of which would take place under a force of gradual and ordinary pressure. The wedge in universal use; § 222.—Notwithstanding the disadvantages arising from the great and wasteful consumption of work which accompanies the employment of the wedge, this machine is in universal use. It has not, however, always the Fig. 228. its different MECHANICS OF SOLIDS. 355 prismatic figure. It sometimes has the form of a pyramid it may he a with three, four, or more edges; in which case, the rela- P>'ramid; tions between the power and resistance when in equiiibrio are altogether analogous to those of the prism; the power apphed to the back is equal and directly opposed to the resultant of the resistances and the frictions against the faces. The wedge may also have the form of a truncated a truncated pyramid or prism. Often it is nothing more than a cone pyramid'or even . . ° a cone; at the extremity of a cylinder. Examples in tools. Almost all the tools employed in examples in the arts have some relation to the wedge; such as the t00ls; different kinds of knives, axes, shears, scissors, files, chisels, saws, hoes, ploughs, &c, &c. All wedges, of what- rules m regard to ever kind and however employed, are destined to act bv the degree of x •/ ' j acuteness. their pointed ends, and the shape of this should be regula- ted with special reference to the object in view. If too acute, it will break off; if too obtuse, it will not penetrate; and the angle adopted is generally the result of a com- promise between these difficulties, determined by the nature of the material of which the wedge is made and that of the substance to be worked. If the substance to be worked be hard, Fie 229 as cold iron, copper, &c, the basil angle abc should be large; this angle in the chisel of a carpenter's plane, which is only intended for wood, is about 30°; it is made still more acute in knives employed to cut the softer substances, meat, bread, and the like. § 223.—All rotating pieces, such as wheels supported Friction of upon other pieces, give rise by their motion to friction. rotatin«pieces; This is an important element in all computations relating to the performance of machinery, and cannot safely •be neglected. It seems to be different according as the rotating pieces are kept in place by trunnions or by trunnions; 356 NATURAL PHILOSOPHY. pivots; definition and description of a trunnion; boxes; definition and description of pivots and their sockets; pivots. By trunnions are meant cylindrical projections a a from the ends of the arbor A B of a wheel; they are usually made as small in diameter as may be found consist- ent with the requisite strength, and are so placed that their axes coincide with that of the arbor which is perpendicular to the plane of the wheel. The trunnions rest on the concave surfaces of cylindrical boxes CD, with which they usually have a small surface of contact m, the linear elements of both being parallel. Pivots are shaped like the trunnions, but support the weight of the wheel and its ar- bor upon their circu- lar end, which rests Fia 230. 4 B .v.;- ,"x..x\^ against the bottom of cylindrical sockets, EFGHIK. If the forces which give motion to the wheel press its pivot against the cylindrical surface of the socket, the friction will partake of the nature of that due to the trunnion as well as the pivot; but this is usually / prevented by special arrangements in the mounting of the V wheel. Of the two frictions here referred to, one takes MECHANICS OF SOLIDS. 357 place between the end of the pivot and the circular bottom nature of the of the socket, and is in all respects similar to that of two friction on tIie surfaces sliding over each other. The friction due to the *" motion of the trunnion has been found by Coulomb to be much less than that of the pivot; and there is also less friction on adhesion on account of the smallness of the surface of tnnmious less , . a, ii n i ™ ;' ......- " ■•■..■■■ -..*•;■-■.*■..•.- than on pivot", contact. A table of the coefficient of frictions which ac- company the motion of trunnions will be given in its proper place. §224.—It is not sufficient in case of rotary motion, To find the value to know the ratio of the friction to the pressure; we must ofthefriction°f also know how the friction arising from the peculiar ar- the bouonfon* rangements of the rubbing parts as just indicated, acts with'socket; respect to the other forces. We shall first take the case of the pivot turning around its axis. Let N denote the force, in the direction of the axis, by which the pivot is pressed against the bottom of the socket. This force may be regarded as passing through the centre of the circular end of the pivot, and as the resultant of the partial pressures exerted up- on all the elementary sur- faces of which this circle is composed. Denote by A the area of the entire circle, then will the pressure sustained by each unit of surface be Fig. 238. N_ A pressure on unit of surface: and the pressure on any small portion of the surface 358 NATURAL PHILOSOPHY. denoted by a, will obviously be pressure on a single elementary surface; a.N A and the friction on the same will be the corresponding friction; f.a.N This friction may be regarded as applied to the centre of the elementary surface a; it is opposed to the motion, and the direction of its action is tangent to the circle described by the centre of the element. Denote the radius of this circle by r, then will the moment of the friction be moment of this friction; / a.N A to (bid the moment of the friction on a -iu-'l- sector; Fig. 233. If we now consider all the elementary surfaces within the sector ACB, of which the angle at C is very small, we may regard the frictions on these elements as parallel to each other, and perpendicular to the radius 0 C, which bisects the angle A CB; in virtue of their parallelism, their resultant will be equal to their sum; and, because of their equality on equal elementary surfaces, the line of direction of this resultant will pass through the centre of gravity of the sector A CB. But this sector being very acute, will not differ from an isosceles triangle, of which MECHANICS OF SOLIDS. ^59 the equal sides, and perpendicular drawn to the base from the vertex C, will be sensibly equal to each other and to R, the radius CA of the circle: whence the distance of the resultant friction on the sector from the centre C will be f R. Substituting the small sector for a, and § R for r, in the foregoing expression, and we have, for the moment of the friction on the sector, / • -4- • sect .ACB x IR; tbe value of this A A ' moment; and the same is true of any other sector. If the moments be taken for all the sectors which make up the circle, and these be added together, we shall have the moment of the N entire friction. The quantity / • -j- • \R, is constant, sumofaiithe •"• similar moments; and hence the sum of these moments will be N f - —j- • § R X (sum of the sectors A CB); but the sum of all the sectors is equal to the area of the circle, or A ; whence the moment of the friction on the entire base of the pivot is moment of the f.N.%R.....(114); friction on the entire end of the pivot; whence we conclude, that, in the friction of a pivot, we may regard tJie whole friction due to the pressure as acting in a single point, and at a distance from the centre of motion equal to two thirds of the radius of the base of the pivot. This mean Uvtr of distance is called the mean lever of friction. It may happen, that the extremity of the pivot, in- stead of rubbing upon an entire circle, is only in contact when the friction with a ring or surface comprised between two concentric gU1°f"c"nannular circles. This occurs when the arbor of a wheel is urged 360 NATURAL PHILOSOPHY. find friction against a ring; moment on a sector of the larger circle; in the direction of its length by a force N a- gainst a shoulder deb a. Denoting, as before, the area of the ring which sustains the pressure by A, the moment of the friction on the ele- mentary sector ABC is, as before found, N Fig. 234. f . JL . fiZ x sector A OB; J A in which R denotes the radius of the larger cfrcle. Again, the moment of the friction on the sector Al OB' is that on the smaller; N f . AL. .\R'x sector A' CB'; A and the difference, that on an element of the ring; |/ . J* IE X sector ACB- R' sector A'CB'], will be the moment of the friction on the surface, AlB'B A. Taking the moments for the remaining elementary surfaces which make up the ring, and adding them together, observing that the sums of the sectors make up the areas of the circles to which they respectively belong, we find, for the moment of the friction on the whole ring, moment for the entire ring; N *f.(Rx area of circle R— R' X area of circle R']. .a. But the area of the circle whose radius is R, is MECHANICS OF SOLIDS. gQl * 7?2. m -£lr j area of one circle; that of the circle whose radius is R', is D,2 * -it/ ; that of the other; and the area A of the ring, is * \R R' ). area of the ring; Substituting these values in the above expression, we find r>3 __ t>i* moment of \ f, N X ----------- friction on the R2 - R'2 ring; Finally, denote by I the breadth of the ring, that is, the distance A' A; by r, its mean radius or distance from Cto a point half way between Al and A, and we shall have R = r + \l, R' = r - %l; substituting these values above and reducing, we have I2 f'Nx [r + A -7] • • (H5); and making same in different form; I mean lever 127 T ' ' 1 n. ~ rn we obtain, for the moment of the friction on the entire ring, f-N.r......(115)'. The quantity r, is called the mean lever of friction for a ring. Since the whole friction fN, may be considered as applied at a point whose distance from the centre is f R, or 362 NATURAL PHILOSOPHY. I2 r — r j-----according as the friction is exerted over an ' 12 r entire circle or over a ring; and since' the path described by this point lies always in the direction in which the work consumed friction acts, the quantity of work consumed by it will be by friction; e(^al tQ t}ie pr0(iuct of its intensity fN into this path. Designating the length of the arc described at the unit's distance from C by sn the path in question will be either %Rsn or r,st; and the quantity of work either %R.s,.f.N for an entire circle, or f-N(r its value for an entire circle; its value for a ring; for a ring. Let Q denote the quantity of work consumed by friction in the unit of time, and n the number of revo- lutions performed by the pivot in the same time; then will s, = 2* x n; and we shall have work consumed byfrictionina Q = | * . R . f. N. U . '. . (116) unit of time; for the circle, and Q = 2«.f.N.(r + ^)-n. . (H7) for a ring; in which if = 3.1416. The coefficient of friction / when employed in either of the foregoing cases, must be taken from the tables in §212. for a ring; MECHANICS OF SOLIDS. 363 Fig. 235. From these expressions, it ap- pears that the quantity of work con- sumed by friction, in a given time, augments with the radius of the piv- ot, or mean radius of the ring ; and as this work is always opposed to the motion, there is an advantage in re- ducing these radii as much as possi- ble, consistently with the strength of the pivot. With this view, the pivots are sometimes made in the form of a truncated cone, and often with a convex ellipsoidal or spherical ter- mination, and the socket having a corresponding shape, it will only be necessary to consider the small cir- cle of contact which arises from the compression of the material. Referring to the expressions (114) and (115)', we see, that to obtain the moment of friction, in the case of the pivot, either for an entire circle or ring, we multiply the coefficient of friction, as given in the table of § 212, by the pressure, and this product by tJie mean lever. And referring to Eqs. (116) and (117) we find, that the quantity of work is obtained by multiplying the moment of friction into the path described by a point at the unit's distance from the centre of motion. Example. Required the moment of the friction on a pivot of cast iron, working in a socket of brass, and which supports a weight of 1784 pounds, the diameter of the circular end of the pivot being 6 inches. Here work consumed by friction proportional to mean lever; pivots should be as small as possible; the moment of the friction; quantity of work of the friction; example of the circular base; #=| = 3= 0.25, lbs. N = 1784, numerical value of elements; / = 0.147; 364 NATURAL PHILOSOPHY. which, substituted in expression (114), gives vaiueofthe q-^ x 1734#'X § X 0.25 = 43.708. moment; And to obtain the quantity of work in one unit of time, say a minute, there being 20 revolutions in this unit, we make n = 20, and it = 3.1416 in Eq. (116), and find, work consumed = 5492.80; Q = i X 3.1416 X 0.25 X 0.147 X 1784 x 20 in unit of time; " J that is to say, during each unit of time, there is a quantity of work lost which would be sufficient to raise a weight of 5492.80 pounds, through a vertical distance of one foot. Example. Required the moment of friction, when the example of the pivot supports a weight of 2046 pounds, and works upon annular base; a g^^er wnose exterior and interior diameters are respectively 6 and 4 inches; the pivot and socket being of cast iron, with water interposed. I =-------- = 1 inch, A r = 2 + 0.5 = 2.5 inches, (1\2 in. ft. numerical value r< = 2.5 + K ' - =2.5333 = 0.2111, of the elements; 12 X 2.0 N = 2046 pounds, / = 0.314; which, substituted in expression (115), gives for the mo- ment of friction, lbs. ft. vaiueofthe q gU x 204g x q^H = 135.62. moment; The quantity of work consumed in one minute, there MECHANICS OF SOLIDS. 3§5 being supposed 10 revolutions in that unit, will be found by making in Eq. (117), as before, * = 3.1416 and 71=10, Q = 2 X 3.1416 X 0.314 x 2046 x 0.211 x 10 = 8517.24; work consumed in unit of time ; that is to say, friction will, in one unit of time, consume a quantity of work which would raise 8517.24 pounds through a vertical distance of one foot. The quantity rule for finding of work consumed in any given time would result from ym*ork m any multiplying the work above found, by the time reduced to minutes. § 225.—The friction on trunnions and axles, which we Friction on i . • t • 'jiii trunnions and now proceed to consider, gives a considerably less co- gxleB. efficient than that which accompanies the kinds of motion referred to in the tables of § 212. This will appear from the following table, which is the result of careful experi- ment, viz.:— TABLE IV. Fhiction of Trunnions in theie Boxes. KINDS OF MATERIALS. STATE OF SURFACES. Ratio of friction to pressure when the unguent is renewed. By the 1 Or, con-ordinary tinuously. method. | Trunnions of cast iron and boxes of cast iron." Trunnions of cast iron and boxes of brass. Unguents of olive oil, hogs' lard, and tallow - - -The same unguents moist-ened with water - - -1 Unguent of asphaltum -Unctuous and moistened with water - - - -Unguents of olive oil, hogs' lard, and tallow - - -Unctuous and moistened I with water - - - -| Very slightly unctuous* - jo.07) (0.08 ) 0.08 o.o54 o.i4 o.i4 \°z\ (0.08) 0.16 0.16 0.19 o.o54 o.o54 o.o54 o.o54 • The surfaces began to move about. 366 NATURAL PHILOSOPHY. TABLE TV.—continued. KINDS OF MATERIALS. Trunnions of cast iron and boxes of lignum-- vitae. Trunnions of wrought | iron and boxes of cast ]• Trunnions of wrought iron and boxes of brass. Trunnions of wrought iron and boxes of Bg- num-vitse. Trunnions of brass and boxes of brass. Trunnions of brass and j boxes of cast iron. j Trunnions of lignum- vitse and boxes of cast iron. Trunnions of lignum- vitse and boxes of lignum-vitae. STATE OF SURFACES. Without unguents* - - Unguents of olive oil and ) hogs' lard.....J Unctuous with oil and hogs' lard - - - - - Unctuous with a mixture of hogs' lard and plum- bago ...... Unguents of olive oil, tal- low, and hogs' lard - - Unguents of olive oil, hogs' lard, and tallow - - - Old unguents hardened - Unctuous and moistened with water - - - - Very slightly unctuousf - Unguents of oil or hogs' lard....... Unctuous - - - - - Unguent of oil - - - - Unguent of hogs' lard- - Unguents of tallow or of) olive oil.....J Unguents of hogs' lard - Unctuous..... Unguent of hogs' lard Ratio of friction to pressure when the unguent is renewed, By the ordinary method. 0.18 o. 14 0.07 to 0.08 0.07 to 0.08 0.09 0.19 0.25 0.11 o. 19 o. 10 0.09 0.12 o.i5 Or, con- tinuously. .090 .o54 ,o54 .o45 to .052 0.07 Let us now examine the part performed by friction in to investigate the connection with the forces which give motion. We have trunnions- seen that the contact of the trunnion with its box is aljmr a linear element, common to the surfaces of both. A section perpendicular to its length would cut from the trunnion and its box, two circles tangent to each other internally. The trunnion being acted on only by * The wood being a little unctuous. t The surfaces began to move about. MECHANICS OF SOLIDS. 367 Fig. 236. its weight, would, when at rest, give this tangential point at o, the lowest point of the section poq of the box. If the trunnion be put in motion by the application of a force, it would turn around the point of contact and roll in- definitely along the surface of the box, if the latter were level; but this not being the case, it will ascend along the in- clined surface op to some point as m, where the inclina- tion of the tangent d umv is such, that the friction is just sufficient to prevent the trunnion from sliding. Here let the trunnion be in equiiibrio. But the equilibrium requires that the resultant of all the forces which act, friction in- cluded, shall pass through the point m and be normal to the surface of the trunnion at that point. The friction is ap- plied at the point m; hence the resultant N of all the other forces must pass through m in some direction as m d; the friction acts in the direction of the tangent; and hence, in order that the resultant of the friction and the force N shall be normal to the surface, the tangential component of the latter must, when the other component is normal, be equal and directly opposed to the friction. Take upon the direction of the force N, the distance md to represent its intensity, and form the rectangle adbm, of which the side mb shall coincide with the tangent, then, denoting the angle dma by , A , -1 , y-2 of friction; in which s, denotes the path described by a point at the unit's distance from the centre of the trunnion. Denoting, 370 NATURAL PHILOSOPHY.' as in the case of the pivot, the number of revolutions per- formed by the trunnion in a unit of time, say a minute, by n; the quantity of work performed by friction in this time by Q,; and making « = 3.1416, we have = 2* and quantity of work Q _ % v . R n-N------* . . (122). in a unit of time; When the trunnion remains fixed and does not form part of the rotating body, the latter will turn about the trun- nion, which then takes axie; the name axle, having the centre of motion at A, the Fig- 238- centre of the eye of the wheel; in this case, the lever of friction; lever of friction becomes the radius of the eye of the wheel. As the quan- tity of work consumed by friction is the greater, Eq. (122), in proportion as this radius is greater, and as the radius of the eye of the wheel trunnion better must be greater than that of the axle, the trunnion has the than the axie m 2Avantaae, in this respect, over the axle. regard to friction; /- ° ' ■■■ ' / The value of the quantity of work consumed by fric-1 friction same for /tion is wholly independent of the length of the trunnion or trimn^ns^d >■ ax^e> an(^ no advantage is therefore gained by making it axies. \ shorter or longer. § 226.—If we examine Eq. (122), we find that, all other things being equal, the value of the work consumed by friction will depend upon the radius R of the trunnion, and Trunnions should that as the latter diminishes, in the same proportion will possiMe aS *^ consumption diminish. The trunnion should, therefore, be made as small as possible, and of the hardest and MECHANICS OF SOLIDS. 371 strongest material, as steel. The consumption of work by consumption of friction may also be diminished by-lessening the force N; 1^^\^cUou but with these two exceptions there is no way of avoiding diminishing the the effects of friction for pleasure, given materials. When the trunnion is employed to Fig. 239. support a piece which sim- ply oscillates through an arc, as in the case of the pendulum and weighing- balance, the knife-edge may be used to great ad- advantages of the vantage, for, in that case, the radius R is reduced to the ^fe-edgeasan smallest conceivable length, and the work of friction to almost nothing. Fig. 240. Friction of rolling motion; miiiiiiini § 227.—There is ano- ther species of friction yet to be mentioned, viz: that which arises from the roll- ing of one body over another. As the surfaces of contact are in this case applied to each other, and separated in a direction perpendicular to that of the motion, there would, at first view, appear to be no friction, nor would there if the surfaces were perfect—that is to say, free from all irregularities. But there can, in practice, be no such surface; when bodies are brought in contact in the manner here referred to, the slight protuberances on the surface of one will enter into the corresponding cavi- but m actual _ practice there is ties on that of the other, after the manner o± so many friction. wedges, and cannot be again withdrawn without giving no friction in rolling motion if surfaces were perfect; 372 NATURAL PHILOSOPHY. this kind of friction, as well as adhesion, may be neglected; friction of the second kind; rise to an amount of friction due to their dimension, depth of insertion, and nature of material. Here adhesion assumes a value which is appreciable, as compared with this friction, but both together are found, in practice, to be exceedingly small, and generally, adhesion in rolling, as well as in shding friction, may, without much error, be neg- lected. In general this friction, called friction of the second kind, is less in proportion as the diameter of the rolling body is less, A wheel of two feet diameter, loaded with a weight of 100 pounds, and rolling over a piece of level and smooth ground, only gives rise to a friction of 0.03 of the pressure—that is to say, to only three pounds. The wheels of carriages meet often with considerable resist- ance when rolling over compressible or rough ground, but this is because the carriage must be raised over the in- clined planes formed in front by the sinking of the wheels, or over obstacles which project above the common surface. The little resistance to motion arising from friction of the second kind, is well illustrated by the comparative facility with which heavy blocks of stone are often transported upon rollers over considerable distances. A roadway is first usually made by placing straight pieces of timber along the ground to prevent the rollers from sinking into it; the stone illustration of its is then mounted upon the rollers, which are placed upon these pieces at right angles to their length, and drawn in the direction of the road by the applica- tion of any convenient power. As fast as a roller is de- tached from behind, it is brought forward and interposed, in time to prevent the stone from tipping forward in con- sequence of its centre of gravity getting in advance of the leading roller. The quantity of work necessary to con- Fisr. 211. practical substitution for sliding friction. MECHANICS OF SOLIDS. 373 vey a stone over any considerable distance, in this way, is incomparably less than if it were to rest with its face against the ground. § 228.—The different kinds of friction may be so com- bined as to diminish both its intensity and the quantify of Employment of rolling to diminish sliding friction; Fig. 242. its work. Thus, let a pair of wheels CD be mounted upon an axle, and suppose a force Ft applied to the latter, parallel to a level plane A B, to put it in motion. De- note the weight of the axle and its load by W, that of the wheels by w. _Su.ppos.ej_ for^moment, that the wheels are firmly connected with the axle and that they case of a common cannot rotate, but, when put in motion, must slide along ca AB; the force Ft requisite to impart motion and keep it uniform, will be given by the equation Fd = (W + w)f (123); in which / is the coefficient of sliding friction between the wheels and plane A B. Next, suppose the wheels capable of turning about the axle, and the force requisite to keep the motion uniform to be denoted by Fu. This force, acting to communicate motion, will give rise to friction between the circumference of the wheel and the plane A B, and also between the axle and the inner surface of the eye: the latter will yield first, and the whole will move forward, the wheels having a rotary as well as a progressive motion. The friction at the axle will, Eq. (119), be force necessary to keep it in uniform motion when sliding; N f, Vl +/2 friction on axle when rotating; 374 NATURAL PHILOSOPHY. in which / denotes the coefficient of friction at the axle. The weight W is thrown upon the axle and acts vertically; the force Fu applied also to 'the axle, acts horizontally, and hence normal pressure J\J — -y W^2 + F 2 J on axle; . and the friction at the axle becomes friction on the VW2 + F2 x axle; " y^T , ^2 Denote the radius of the wheel by R, and that of its eye by r, and the space described by a point at the unit's dis- « tance from the centre of motion by s,; then will the quan- tity of work of the friction be / quantity of work y/^W2~+~F 2 X ~ X T . S. of friction; Vl _|_ f* The path described by the point of application of the • power, and in the direction of the power, will be equal to • the development of the arc of the circumference of the wheel corresponding to the arc st, that is to say, to Rs„ and hence the quantity of work of the power will be work of the power; whence we have FnRSl = VW2 + F2 X r^~-j2 X r.st; from which we find power to keep the cart in _^ y _ uniform motion " u = W •J/ • -=j— • — _---- when the wheels -^ A //-1 1 f2\ (-| _ f? v 11 are turning; . V ^ ■>' ' \ 1 + f2 R- MECHANICS OF SOLIDS. 375 The value of/ being a small fraction, as will appear from Table IV., the fraction v^+^-rfj?-^) this factor equal to unity; will differ but slightly from unity, and hence may be replaced by unity, which will give F = W • / • — R (~\ OA\ practical relation ^ /'of power to load; Dividing this by Eq. (123), we find F u F, _ W.f, (W + w)f X r R (125). relation of the powers to produce sliding and rolling motion; Here W is less than W +w; / is, by the tables, less than / and r is usually very much less than R, so that the second member must be a small fraction, and Ft, conse- quently, much greater than Fn. This is the theory of theory of the carriage-wheels of every kind, of castors, rollers for smooth- carriage-wheei, ing ground, and the like. Example. Suppose a carriage with four wheels, whose joint weight is 50 pounds, to be loaded with 2040 pounds, the weight of the axle-trees and body being together equal to 320 pounds. Let the wheels be of cast iron, the axles of wrought iron, the radius of the eye half an inch, that of the wheel one foot and a half, and suppose an un- example— guent of tallow and the carriage placed upon a rail-track ™™^e°a of wrought iron. Here 376 NATURAL PHILOSOPHY. numerical value of the elements; lbs. W = 2040 + 320 = 2360, lbs. w =...... 50, lbs. W + w =......2410, / = Table IY. . . . 0.08, / = Table 1.....0.194, ft- r =.......0.042, nearly, ft- R =.......1.5; and Eq. (125), ratio of the forces; F F. 2360 0.08 X 2410 0.194 0.042 1.5 = 0.0113; value of one in terms of the other; conclusion; runners substituted for wheels on ice; equality of the work of friction and of power; or Ft = 88A9.F„; that is to say, the force requisite to put the carriage in mo- tion when its wheels are free to rotate, is only about one eighty-eighth part of that which would be necessary to drag it, were its wheels locked. If we examine Eq. (125), we shall find that by taking / equal to zero, the force Fu will be vastly greater than Fn and the wheels will not turn. Now, although this extreme case can never occur in practice, yet, when a car- riage is placed upon ice, we approximate to it; and this is why runners are usually substituted for wheels, under such circumstances. The same equation explains why it is that so much more advantage arises from large wheels than small ones. Multiplying both members of Eq. (124), by Rsn we find FllRsl = W.f.r.s,; MECHANICS OF SOLIDS. 377 the second member is obviously the work performed by friction, as the first is that performed by the power Fn. Denoting by n the number of revolutions performed by the wheels, we have st = 2* . n; which, in the above equation, gives path described by point at unit's distance from axis of axle; FuRSj = W.f x 2«r.n. Denote the distance travelled by d, then will and n = d 2* R' number of revolutions of the wheel; FtiRs, = W-fi.^L (1 9R\ work of power and of friction; If we make d equal to one mile = 5280 feet, and take the dimensions and other elements the same as in the last example, we shall find example; KB-.. = 2360x0.08x0.042x5280 = ^ ^. in words, the work expended in moving the carriage one mile, or the work consumed by its friction, is equivalent to that which would raise 27912 pounds through a vertical height of one foot. When a trunnion is destined to support considerable method by which weight, its dimensions must be proportionably large; but as friction may be the radius of the trunnion increases, the effect of friction will reduced; increase in the same ratio. To avoid the inconvenience 378 NATURAL PHILOSOPHY. device for diminishing friction; that would arise from this, when great freedom of motion is desirable, we may have recourse to the following device. Conceive the trunnion A to rest upon the circumference of two equal wheels, sup- ported upon smaller trun- nions C, C, whose distance apart is slightly greater than the radius of the wheels. Resolve, by the parallelo- gram of forces, the pressure upon the larger trunnion into two components, normal to the circumferences of the wheels; these will be transmitted to the smaller trunnions C and C, where they will be supported. Denote these components by iVand N'. If the wheels could not turn, the friction between their circumferences and the larger trunnion would be / Nand f'N ; and the quantity of work consumed by this friction would be work of friction when wheels cannot turn; (fN + f'N')RlSl; in which R, denotes the radius of the larger trunnion, and st the arc described by a point at the unit's distance from its axis. If, on the contrary, the wheels may turn, the frictions on the trunnions C and C will yield before that at the circumference of their wheels, and from what has just been shown, Eq. (124), the frictions there become friction on the smaller trunnions; Nf.^, and N'f.jpl in which r and r' denote the radii of the smaller trunnions, and R and R' the radii of their corresponding wheels; and thus the quantity of work of friction will become work of friction; (*7 — + N f R ^ J R I'I R.s. MECHANICS OF SOLIDS. 379 a quantity obviously much less than that obtained comparison of above. result85 If the wheels and their trunnions be of the same size, and the trunnions as well as their boxes be of the same material, the above expression becomes / (N + N') • -L R.s. work of friction, when wheels same size and of same material; the value of this expression may be made as small as we please, indeed inappreciable, in a practical point of view, by selecting surfaces and unguents for which / is the least possible, and making r very small. A beautiful applica- used m Atwood's tion of this principle is exhibited in Atwood's machine, machine- which will be referred to hereafter. XVIII. STIFFNESS OF CORDAGE. § 229.—Let us now con- sider a wheel turning freely about an axle or trunnion, and having in its circum- ference a groove to receive a cord or rope. A weight W, being suspended from one end of the rope while a force F is applied to the other extremity to draw it up, the latter will experience a resistance in consequence of the rigidity of the rope, Fig. 244. Resistance from stillness of cordage; 380 NATURAL PHILOSOPHY. measure of the rigidity of cordage; the value of this measure; which opposes every effort to bend it around the wheel. This resistance must, of necessity, consume a portion of the work of the force F. The measure of the resist- Fis- 244 ance due to the rigidity of cordage has been made the subject of experiment by Coulomb; and, according to him, it results that for the same cord and same wheel, this measure is composed of two parts, of which one re- mains constant, while the other varies with the weight W, and is directly propor- tional to it; so that, designa- ting the constant part by K, and the ratio of the variable part to the weight W by I, the measure will be given by the expression K + I. W; in which K represents the stiffness arising from the natural torsion or tension of the threads, and I the stiffness of the same cord due to a tension resulting from one unit of weight; for, making W = 1, the above becomes K + I. Coulomb also found that on changing the wheel, the stiff- ness varied in the inverse ratio of its diameter; so that if stiffness on a i*iu""v'™1 ? "^ «.**—« ~ wheel whose diameter is K + I. W unity; he the measure of the stiffness for diameter, then will a wheel of one foot stiffness on a wheel of any diameter; K + I. W 2R be the measure when the wheel has a diameter of 2 R. A MECHANICS OF SOLIDS. 381 table giving the values of ^and / for all ropes and cords data from which employed in practice, when wound around a wheel of one tofindthe foot diameter, and subjected to a tension arising from a measure of unit of weight, would, therefore, enable us to find the stiff-stiffness; ness answering to any other wheel and weight whatever. But as it would be impossible to anticipate all the dif- ferent sizes of ropes used under the various circumstances of practice, Coulomb also ascertained the law which connects these data the stiffness with the diameter of the cross-section of the abridsed; rope. To express this law in all cases, he found it neces- sary to distinguish 1st, new white rope, either dry or moist; 2d, white ropes partly worn, either dry or moist; 3d, tarred different ropes; 4th, packthread. The stiffness of the first class he kindsofroPe; found nearly proportional to the square of the diameter of the cross-section; that of the second, to the square root of the cube of this diameter, nearly; that of the third, to the number of yarns in the rope; and that of the fourth, to the laws which the diameter of the cross-section. So that, if S denote the govern them; resistance due to the stiffness of any given rope; d the ratio of its diameter to that of the table; and n the ratio of the number of yarns in any tarred rope to that of the table, we shall have for New white rope, dry or moist. K + I W S = d2 • Z ' . . . (127). ^t for new & Xb white rope; Half worn white rope, dry or moist 2R i K + I. W S = d2 • ----—g---- . . . (128). old white rope; Tarred rope. S = n • ----o~jzT--- • • • (129). tarred rope; Packthread. x + I - w S = d • Z p . . . (130). packthread; 382 NATURAL PHILOSOPHY. TABLES for new white ropes, dry; for new white ropes, moist; OF WEIGHTS NECESSARY TO BEND DIFFERENT ROPES AROUND A WHEEL ONE FOOT IN DIAMETER. No. 1. White Ropes—new and dry. Stiffness proportional to the square of the diameter. Diameter of rope in inches. Natural stiffness, or value of K. Stiffness for load of 1 lb., or value of /. 0.39 0.79 1.57 3.i5 lbs. 0.4024 I.6097 6.4389 25.7553 lbs. O.0079877 o.o3i95oi 0.1278019 0.5112019 No. 2. White Ropes—new and moistened with WATER. Stiffness proportional to square of diameter. Diameter of rope in inches. Natural stiffness, or value of K. Stiffness for load of lib.,or value of /. O.39 O.79 1.57 3.i5 lbs. o.8o48 3.2194 12.8772 5i .5ni lbs. 0.0079877 o.o3i95oi 0.1278019 0.5112019 Squares of the ratios of dianfeter, or val- ues of d2. Ratios d. Squares d2. 1.00 1.00 1.10 1.21 1.20 i.44 1.3o 1.69 1.4o 1.96 1.5o 2. 25 1.60 2.56 1.70 2.89 1.80 3.24 1.90 3.6i 2.00 4.00 old white ropes, dry; No. 3. White Ropes—half worn and dry. Stiffness proportional to the square root of the cube of the diameter. Diameter of rope Natural stiffness, Stiffness for load of in inches. or value of K. 1 lb., or value of /. lbs. lbs. O.39 0.40243 O.0079877 0.79 I.l38oi 0.0525889 » 1.57 3.21844 O.0638794 3.i5 9.ioi5o 0.l8o6573 No. 4. White Ropes—half w^orn and moistened with water. Stiffness proportional to the square root of the cube of the diameter. old white ropes, moistened; Diameter of rope Natural stiffness, Stiffness for load of in inches. or value of K. 1 lb., or value of /. lbs. lbs. O.39 0.8o48 0.0079877 O.79 2.2761 o.o525889 1.57 6.43a4 0.0638794 3.i5 18.2037 0.i8o6573 Square roots of the cubes of the ratios of diameters, or val- 3 ues of d 2. Ratios or Power -|, d. or C?2. 1.00 1.000 1.10 i.i54 1.20 i.3i5 1.3o 1.482 1.4o 1.657 1.5o 1.837 1.60 2.024 1.70 2.217 1.80 2.4i5 1.90 2.619 2.00 2.828 I MECHANICS OF SOLIDS. 383 No. 5. Tarred Ropes. Stiffness proportional to the number of yarns. [These ropes are usually made of three strands twisted around each other, each strand being composed of a certain number of yarns, also twisted about each other in the same manner.] No. of yarns. Weight of 1 foot in length of rope. Natural stiffness, or value of K. Stiffness for load of 1 lb., or value of /. 6 i5 3o lbs. 0.0211 O.0497 I.OI37 lbs. O.I534 O.7664 2.5297 lbs. 0.0085198 0.0198796 O.0411799 for tarred ropes; For packthread, it will always be sufficient to use the tabular values given above, corresponding to the least tabular diameters, and substitute them in Eq. (130). An example or two will be sufficient to illustrate the use of these tables. Example 1st. Eequired the resistance due to the stiff- examples to ness of a new dry white rope, whose diameter is 1.18 illustrate the use inches, when loaded with a weight of 882 pounds, and ° wound about a wheel 1.64 feet in diameter. Seek in Table No. 1 the diameter nearest that of the given rope ; it is 0.79 ; hence d = 079" = L5nearly; and from the table at the side, d2 = 2.25. From Table No. 1, opposite 0.79, we find K = 1.6097, / = 0.03195; which, together with the weight W = 882 lbs., and ft. 2 R = 1.64, substituted in Eq. (127), give elements obtained from the tables; • 384 NATURAL PHILOSOPHY. s = 2.25 • l-60i>T + 0-03195 X 882 = ^ which is the true resistance due to the stiffness of the rope in question. Example 2d. What is the resistance due to the stiffness of a white rope, half worn and moistened with water, example to having a diameter equal to 1.97 inches, wound about a illustrate Table wheel 0>82 of a foot in diameter, and loaded with a weight No. 4; of 2205 pounds ? The tabular diameter in Table No. 4, next below 1.97, is 1.57, and hence d = -^=- = 1.3 nearly; 1.57 the square root of the cube of which is, by the table at the side, data from the 7-| ___ -i 4.QO table; ' In Table No. 4 we find, opposite 1.57, K = 6.4324, / = 0.06387; ft. which values, together with W= 2205 lbs., and 2 R = 0.82, in Eq. (128), give lbs. lb. * iio,v 6-4324 + 0.06387 X 2205 0JisynQ result; S — 1.482 X ------------7755------------ = 266.109, which is the required resistance. Example Bd. What is the resistance due to the stiffness of a tarred rope of 22 yarns, when subjected to the action MECHANICS OF SOLIDS. 385 of a weight equal to 4212 pounds, and wound about a example to illustr: No. 5; wheel 1.3 feet diameter, the weight of one running foot of m data obtained from*the table; the rope being about 0.6 of a pound ? By referring to Table No 5, we find the tabular number of yarns next below 22 to be 15, and hence n — -rr- = 1.466 nearly. In the same table, opposite W, we find K = 0.7664, / = 0.019879; ft- which, together with W = 4212, and 2 R — 1.3, in Eq. (129), give a = 1.466 0-7664 + 0.019879 X 4212 = ^ ^ X.O Example 4fh. Eequired the resistance due to the stiff- ness of a new white packthread, whose diameter is 0.196 example to inches, when moistened or wet with water, wound about a ^ ™ e wheel 0.5 of a foot in diameter, and loaded with a weight packthread; of 275 pounds. The lowest tabular diameter is 0.39 of an inch, and hence , 0.196 nK , d = O390 = °'5 nearlj< In Table No. 2 we find, opposite 0.39, K = 0*8048, data from Table No. 2; / = 0.00798; 386 NATURAL PHILOSOPHY. which, with W = 275, and 2 R = 0.5, we find, after sub- stituting in Eq. (130), result. 0.8048 + 0.00798 X 275 __ /JJy9 Fig. 245. work due to § 230.—The resistance just found is expressed in stiffness of pounds, and is the amount of weight which would be cordage; r ' necessary to bend any given rope around a ver- tical wheel, so that the portion A E, between the first point of con- tact A, and the point E, where the rope is attach- ed to the weight, shall be perfectly straight. The entire process of bending takes place at the bending takes this first or tangential place at the first -^ j_ f tf motion point of contact; L he communicated to the wheel in the direction indicated by the arrow- head, the rope, supposed not to slide, will, at this point, take and retain the constant curvature of the wheel, till it passes from the latter on the side of the power F. When, therefore, by'the motion of the wheel, the point rn of the rope, now at the tangential point, passes to ml, the working point of the force /Swill have described in its own path described by direction the distance AD. Denoting the arc described the working i a point at the unit's distance from the centre of the point; J r wheel by s/} and the radius of the wheel by R, we shall have AD = Rs,; and representing the quantity of work of the force S, by MECHANICS OF SOLIDS. 387 L, we get L = S.Bs,; replacing Shy its value in Eqs. (127) to (130), K + I. W L = Rsldl 2R (131); work of the stiffness; in which dt represents the quantity d2, d2, n, or d, m Eqs. (127), (128), (129), or (130), according to the nature of the rope. Example. Taking the 2d example of § 229, and sup- examples; posing a portion of'the rope, equal to 20 feet in length, to have been brought in contact with the wheel, after the motion begins, we shall have L = 20 X 266,109 = 5322.18; result; that is, the quantity of work consumed by the resistance due to the stiffness of the rope, while the latter is moving m words. over a distance of 20 feet, would be sufficient to raise a weight of 5322.18 pounds through a vertical height of one foot. XIX. WHEEL AND PULLEY. § 231.—A plane wheel, free to turn about its trunnion or axle, support- ed in a fixed box, may be moved in either di- rection by two forces F and Q, which act in its Fig. 246. Wheel and pulley; 388 NATURAL PHILOSOPHY Fig. 246. a wheel; But plane, and tangent to its circumference at A and B. These forces, acting in the same plane, perpendicular equilibrium of to the axle, and tending two forces acting tQ tmn ^ whee} m Qpp0. upon the circumference of site directions, will be in equiiibrio when the ele- mentary quantity of work developed by each is the same with contrary signs. the points of application A and B, belonging to the same circumference, the paths which they simultaneously de- scribe will be equal; and since the product of these paths by the forces F and Q must be equal, it follows that when- ever the forces are in equiiibrio, they must also be equal. This supposes the wheel free to turn, without obstruc- tion of any kind. But if we consider the friction at the trunnion or axle, then, supposing the equilibrium still to exist, but the wheel on the eve of motion in the direction of the force I] the elementary quantity of work of the when friction is latter must be equal to that of the resistance Q, increased by that of the friction; in which case F and Q will not be equal; and denoting the radius of the wheel by R, that of its trunnion or eye by r, and the resultant of F and Q by N, we shall have tbe forces must he equal; taken into the account; relation of work of power, resistance, and friction; FRs, = QRst +fN.rst (132); in which /is the coefficient of friction at the axle or trun- nion, and s/ the arc described by a point at the unit's dis- tance from the axis during motion. Dividing by R s„ we find relation of these forces; F=G+fN'R (133). From which we might conclude the value of F, but that N is unknown, being the resultant of F and Q. MECHANICS OF SOLIDS. I> 389 Now, two cases may arise, viz.: either the value of r may be very small in comparison with R, or it may not. two cases In the first case, any error committed in the determination arise; of N would but slightly affect the value of F, since only T the small fractional portion -= of N is taken. We may, therefore, be content with an approximate value for N To obtain this, we first omit the consideration of friction, which will make /= 0, in the above equa- tion, which then reduces to F= Q. Denote by

!+(/^-)2cos0 ±/^-- V(X + cos Q, we may find the true tension corresponding to any erroneous tension, as i\, by the following proportion, viz.: to find the true from the approximate tension; example to iliustrate; Qx = Q Qxk' or, which is the same thing, multiply each of the tensions found by the constant ratio ~r, the product will be the true Six tensions, very nearly. The value of tA thus found, substi- tuted in Eq. (151), will give that of F. Example. Let the radii R^, R2, R3, and R4, he respectively 0.26, 0.39, 0.52, 0.65 feet; the radii rx = r2 = r3 = r4 of the eyes = 0.06 feet; the diameter of the rope, which is white and dry, 0.79 inches, of which the constant and coefficient of rigidity are, respectively, K = 1.6097 and / = 0.0319501; and suppose the pulley of brass, and its axle of wrought iron, of which the coefficient / = 0.09, and the resistance Q a weight of 2400 pounds. Without friction and stiffness of cordage, approximate value of first tension; h = 2J00 = 6Q0- Dividing Eq. (148) by i^, it becomes, since d, = 1, * = * + ^T7?— + £/(* + *)■ A ltx itj Substituting the value of i^, and the above value of ^, and regarding in the last term ^ as equal to ^, which we may do, because of the small coefficient -^- / we find MECHANICS OF SOLIDS. 405 + 600 1.6097 + 0.0319501 x 600 2 X (0.26) + ^- X 0.09 x (600 + 600) l_ tfOQ on aPProx™ate * — VAO.Oo. value of second tension; Again, dividing Eq. (149) by R2, and substituting this value of 4 and that of R2, we find lbs. ^ = 673.59. Dividing Eq. (150) by R3, and substituting this value of 4, as well as that of i^, there will result approximate value of third, tension; \ lbs. h = 709.82; whence Qx = h + t2 + % + t4 = - and Q 2400 600 -f 628.39 -f 673.59 + 709.82 . = 0.919; Qx 2611 which will give for the true values of approximate value of fourth tension; i —- 261180" resultant of these ' tensions; ratio of the approximate to the true resultant; ti = 0.919 X 600 t2 = 0.919 x 628.39 4 = 0.919 X 673.59 t4 = 0.919 x 709.82 551.400 577.490 619.029 652.324 2400.243 406 NATURAL PHILOSOPHY. The above value for tA = 652.324, in Eq. (151), will give, after dividing by R4 and substituting its numerical value, F = + 652.324 1.6097 + 0.03195 X 652.324 2 X 0.65 0.06 + tttS X °-09 x (652.324 + F); 0.55 and making in the last factor F — t4 = 652.324, we find numerical value of the power; F = 652.324 + 17.270 + 10.831 = 680.425. work absorbed by friction and stiffness of cordage. Thus, without friction or stiffness of cordage, the intensity of F would be 600 lbs.; with both of these causes of resistance, which cannot be avoided in practice, it becomes 680.425 lbs., making a difference of 80.425 lbs., or nearly one seventh; and as the quantity of work of the power is proportional to its intensity, we see that to overcome fric- tion and stiffness of rope, in the example before us, the moter must expend nearly a seventh more work than if these sources of resistance did not exist. Wheel and axle; description; and application of power and it'si.-iauoe; § 235.— Wheel and Axle is a name given to a machine, which consists of a wheel mounted upon an arbor, supported at either end by a trunnion resting in a box. The plane of the wheel is at right angles to the axis of the arbor; the power F is applied to a rope wound around the wheel; the resistance Q is applied to another rope, wound in the opposite direction about the arbor, and also acts in a plane perpendicular to the axis of motion. The power is generally applied in the plane of the wheel, other- wise, being oblique to the axis, it would be necessary to resolve it into two components, one perpendicular and the other parallel to that line; the latter compo- MECHANICS OF SOLIDS. 407 Fig. 255. process where power does not act in plane of wheel T when the power acts in the plane of the wheel; nent would press the shoulder of the arbor against the effect of an face of the box, and increase the effect of friction bv oblique tJ application increasing its '•'■lever arm." It may happen, however, that of the power; the particular object to be accomplished will sometimes make it inconvenient to satisfy this condition of keeping the action of the power in the plane of the wheel, in which event, it will be easy to find the pressure arising from the parallel com- ponent of the power or resistance, and to com- pute the friction by the rules already given. Supposing the power and resistance to act in planes at right angles to the axis, we remark, that the plane of the wheel in which the power acts, and the plane perpen- dicular to the axis, through the direction of the resistance, will cut from the arbor equal circles. Through the point E, at which the rope is tangent to the circle in the latter of these planes, and the axis, conceive a plane to be passed; it will cut the circle in the plane of the wheel on the opposite side of the arbor in E', and the line joining E and E' will intersect the axis in /, making EI = E' I. At the point E' apply two opposite forces Qx and Q2, construction. parallel and each equal to the resistance Q. These forces will produce no effect upon the system. The resultant of the two equal and parallel forces Q and Qx will be equal to their sum, will pass through I, will be resisted by the axis, and produce no work, except what may arise from the friction due to its action on the trunnion. The equi- 408 NATURAL PHILOSOPHY. forces which librium, if the machine be at rest, or its uniform motion, maintain the if , mugt thereforo be maintained by the power F, equilibrium or ' ' ' J *■ motion uniform; the force Q2, the friction, and the stiffness of cordage. To this end, the resultant of F, Q2, and stiffness of cordage must intersect the axis. At the point of intersection, conceive this resultant to be replaced by its primitive components, and there will then act upon the axis the fo^es F, Q2, Q + Qx, and the resistance due to stiffness of cordage. Each of these forces being resolved into two parallel components acting on the trunnions A and B, there will result two groups of forces, one apphed to each trunnion. Denote the resultant of the group acting on the trunnion A by M, that of the group acting on the trunnion B by M', then will the frictions be respectively fM and /' M'; and, employing the usual notation, the quantities of work will be fMr s, and /' M' r' s„ the radii of the trunnions, and their friction being unequal. The quantity of work of the power F, must be equal to that of the resistance Q2, augmented by the work of the stiffness of cordage and friction, and hence, denoting the radius of the wheel by R, and that of the arbor by R', pressure upon the trunnions; friction on the trunnions and its work; F- Rs, = Q2R'- 8t + d, ^±^R'st+fMrst+f'M' r s, but if the trunnions and boxes are supposed of the same size and material, quantity of work; FRs, = Q2R'Sj + d, ^+{^i?'s, + f(M+ M') r S,. friction of trunnions, same effect wherever applied; The quantity M + M', being the sum of the pressurea upon the trunnions, the last term shows that the friction is the same as though the resultant of all the forces were applied to a single trunnion in any arbitrary position, and, therefore, at the centre of the wheel. But this would re- duce all the forces to the same plane, in which case Q would take the place of Q2, and Qx and Q2 would disappear from MECHANICS OF SOLIDS. 409 the system. Hence, denoting the resultant of the entire system of forces by N, and writing Q for its equal Q2, the above equation becomes FRS, = QR'Sj + d, -y^T^ R'St+fN-T-S,.. (153); quantity of work; and, dividing by Rsn F=Q§ + d' ^YW^ f +/ff • i • ■ (154>™luaof""' Now, N being the resultant of all the forces of the system except friction, it is the resultant of F, Q, and d, tD,—', 2R or, since Qand d1 —oR~ act m tine same direction, it is the resultant of F and Q + d. tn, • To fin<* ^ we will pursue the method explained in § 231. Make power; 2R then will Q + d, ^o^ = Qx . . (155); find the resultant of all the forces but friction; F= a^+f-ir.^. . (156). If we neglect the consideration of friction for a moment, by and find the resultant Nx of F and Qx, or of approximation; Qx -g- and Qh we shall have, denoting the inclination of the power to the resistance by cp, Xi=\/Qii+Qi^+ZQi^™s

Qi therefore less than Fx ■= • and that if this latter be Jti combined with Fx, to obtain a second re- sultant N2, this last will be too large, and when substituted in Eq. (156), for N, will give a value F2 for ^ / ///\ )%£-, F, which will also be too large. The mean of the two values of Fx and F2 will be the practical value of F. The value of N2 is given by the equation, Fig. 256. /A second approximation for resultant; second approximation to value of the power; , n2 = fx v/i + ^(-§ + 2cos^)- • (159); and F.-Q, *.+/».$■. MECHANICS OF SOLIDS. 411 whence F_ F1+F2 _QR'fr Nx + A; mean of the (160). two approximations; To find the quantity of work, multiply both members by Rs, replace Qx by its value, and we have FRs=QR'si+d1^^R'St+fr.Si^+^ . . (161). quantityofwork; Fig. 257. example to illustrate; Example. Eequired the quantity of work necessary to raise two tons of coal from the bottom to the top of a pit which is 80 feet deep, by means of the wheel and axle. The diameter of the wheel is 4 feet; that of the axle, 1 foot; that of the trunnion, which is of wrought iron, working in cast- iron boxes and lubri- cated with hogs' lard, 1.5 inches; that of the rope, which is white, half-worn, and dry, 1.5 inches; and the power acts in a horizontal direction. Here R = 2 feet; R' = 0.5 feet; r 3 / 1 T \ 3 3 questic /= 0.07; d, = d* = {^Y§)2 = (1.9)2 = 2.619; tables; K = 1.13801; i" = 0.0525889; Q = 4000 lbs.; R's, = 80 feet; s, = ~ = 160 feet; and cp = 90°, O.o or cos

; subtracting Tx — H from both members of this equation, and we have TX+H-(TX-H) = (Tx - H) eR* - (Tx - H); the first member reduces to 2 H; that is to say, to the dif- ference of tensions on the two branches of the band, which must be equal to the effect of Q at the surface of the drum; whence difference of tensions; 2H= Q.^- .... (167), MECHANICS OF SOLIDS. 417 J?" fs r\ ^ t m tt\ / ~ET -,\ ^ same m terma of Q~R~ = ( X ~ ^H6 --1) ' (168)5 the friction, &c; tension at rest; from which two equations we may compute J7and Tx, and therefore, Eq. (166), ^and t; and, finally, the resultants N2 and Nx by the rules for the composition of forces. Example. Eequired the tension of a band necessary to produce friction enough to move a wheel, when subjected to a resistance of 1000 pounds, the radius of the wheel example; being 0.5 foot, and that of the drum 2 feet, and the arc of the drum enveloped by the band 180°. Let the band be of black leather, and the surface of the drum of oak. Here R2 = 2 feet; R" = 0.5 feet; Q = 1000 lbs.; ' / = 0.265, (see Table I, § 212;) s = * R2 = 3.1416 R2; Q R" 0 5 lbs-- " = 1000 X ~ = 250; z?" ibs. H=\Q -^ = 125; JX2 half difference of tensions; Q -C-R2 250 IL (2.7182818)°-265x3-1416-(eR* - 1) V J - 1' lesser tension; Tx - H = The first term of the denominator may be easily found by the aid of logarithms, as follows: Log [(2.7182818)0-83251] = Log 2.718281 X 0.83251 = 0.4342942 X 0.83251 TJT* = 0.361554 nearly; the natural number of which is 2.2991, whence T,-H=t= __!«L_ = J5L = 19&4. 2.2991 - 1 1.2991 logarithms; value; 418 NATURAL PHILOSOPHY Adding 2H= 250 lbs., we have greater tension; lbs. Tx + H = T = 442.44. to find the resultant; The arc of the drum enveloped by the band being 180°, the tensions T and t must be parallel, and their resultant T2 = :T+£= 634.88 lbs., which being combined with Q = 1000, according to the principles of the composition of forces, will give N2, and with F will give Nx, whence every thing required to determine the quantity of work in Eq. (165) is known. If Eq. (165) be divided by R'sn it becomes valueofpower; F = Q ■ ^ • $ + /• N2 • !JL - h + /A rA R velocity of the circumferences equal; but whence final value of power when the motion begins in its direction; tl2 $2 — -"l ^1 ! f2_ R\. and by substituting above, F= Q RT_ R' Rx R2 +/.JS;.^.|+/.J»i.ai which is the relation subsisting between F and Q, in caso of an equilibrium bordering on motion in the direction of F, or in the direction of uniform motion. If we disregard the friction, then will without friction; F= Q R . Rx R . Ra MECHANICS OF SOLIDS. 419 Fig. 259. When the motion from one wheel and axle is communi- combination of cated to a second machine of the same kind, by passing ^utoutfrtettaT the band about the axle of the wheel to which the power F is applied, and the wheel of that to whose axle Q is applied, then will R^ be the radius of the first axle, and R2 that of the sec- ond wheel, and the preceding equation gives us this rule, viz.: When the friction is so small that it maybe disregarded, relation of power the power F will be to the resistance Q, as the product of the radii of the axles to that of the radii of the wheels, in the case of an equilibrium or uniform motion. to the resistance. Fig. 260. § 238.—In the preceding discussion, no mention is made Rigidity of of the resistance arising from the stiffness of cordage. n|"L*™^ When the connection or gearing is made by bands, these are so thin as to possess considerable flexibility, and their opposition to bending may, in practice, be safely neglected. If the connection be made by an end- less rope, the op- position to motion takes place at the points where the rope bends in pass- ing on to the drums, and not at all at the points where it leaves the latter rigidity of ropes; i 420 NATURAL PHILOSOPHY. and becomes straight. Thus at the point a, the resist- ance is value of the resistance at one point; K + I. Q 2 R" ' at the point b it is at another; K' + It 2 R2 and at the point d it is at another; K' + IT 2RX ' and, finally, at the points / and e it is nothing. These resistances must be included among those to be overcome by the power F. rigidity of chains; If the connection be made by an endless chain, each link, as it turns in the next one in order, may be regarded Fisr. 261. ■Em .£' each link a as a trunnion revolving in its box; and each, as it comes box.101 to be applied to the drum, revolves about the next one through an angle E'HE, equal to DCD', the angle through which the drum revolves to produce the contact; and taking the sum of all these angles, it is obvious that, MECHANICS OF SOLIDS. 421 although each link revolves through a very small angle, yet this sum must be equal to the angle through which the drum has turned to produce it. Denoting by r the radius of the inner circular arc in notation, &c.; which the end of each link is shaped, s2 the arc described by the point at the distance of unity from the axis of the drum B, f the coefficient of friction, and T and t the ten- sions on the two branches of the chain, then will the work of friction among the links at the points b and/respec- tively, (figure before the last,) be work of friction fTrS2} and ftrs2; among the link: at one set of • i points; and denoting by sx the arc described by a point at the dis- tance of unity from the axis of the drum A, the work of friction at the points e and d will be, respectively, fTrSx, and ftrsx; the same for another set; and the whole amount of this kind of work will be fr (T + t) (S2 + SX). whole work of this friction; Eecollecting that the points on the surfaces of the drums must have the same velocity, viz.: that of the different links of the chain, we have ■q -j-. _ velocity of ±i2 S2 — It1Sx; circumference ol drums equal; in which R2 and Rx are respectively the radii of the drums B and A. From this relation we find R2 sx = -& S2 5 which, substituted above, gives frs2(T+ 0(l + §)• • • (169)- value of the worl of friction among the links; 422 NATURAL PHILOSOPHY. Example. Let T and t have the values of the last ex- ample, (that of the strap,) and suppose r = 0.03, the chain 'xampie; of wrought iron, for which we find in the table of § 225, (assuming that/ is the same for trunnions of wrought iron in boxes of the same material, as for trunnions of wrought iron and boxes of cast iron,) /= 0.07; also let the radius of the drum B be four times that of the drum A ; then will data; the expression (169) for a single revolution of the drum B, in which case s2= 2 X 3.1416 = 6.2832, become ofamcL°.fW°rk 0.07 X 0.03 X 6.2832 (442*44 + 192.44) (1 + 4) = 41.88; that is, the work lost in consequence of the friction among the links of the chain, during one revolution of the drum of the wheel to which the resistance is applied, is sufficient to raise a weight of nearly 42 pounds through one foot of vertical height. friction on the teeth of wheels; conditions of construction of the teeth; Fig. 262. § 239.—Let us now suppose the circumferences of the Resistance from wheels to be furnished with teeth, which interlock with each other, so that a force being im- pressed upon one wheel, it cannot move without com- municating motion to the other. The teeth are usually curved, and so shaped as to have a common normal Dx D2, at their point of con- tact m, where the action of one and the reaction of the other take place; and although the point of contact alters its position, as the wheels rotate, MECHANICS OF SOLIDS. 423 D. conditions of preserving a constant normal at the point of contact: c4-: ,#' --^c V />/ yet the place of Fig. 263. this normal does............-., not change, but re- /' V'\. • mains stationary, /' V\....-V^* and the point of i contact is always \ on it. We will not stop to explain the constructions by which this is ac- complished ; it will be sufficient for our present purpose to be assured of its practicability, and that we may pro- ceed on the supposition that it has been executed in the case under consideration. From the centres Cx and C2 of the wheels, let fall upon the normal Dx D2, the perpendiculars Cx Dx and C2 D2. The points Dx and D2 must, during the rotation of the relative velocities wheels, have the same,absolute velocitv, and therefore the ofthetwo J wheels; number of»revolutions of the wheel whose centre is Ch in a given time, must be to that of the wheel whose centre is C2, in the same time, inversely as the perpendiculars Cx Dx, and C2 D2; or, because of the similar triangles Cx B Dx and C2BD2, inversely as the distances CXB and C2B. The circles described about Cx and C2 as centres, with radii Cx B and C2 B, respectively, are called the primitive circles, primitive circles; These circles and their radii may be easily found from the consideration just named. It will be our object to find a force which, applied tangentially to 4hese circles at B, will produce the same effect as friction on the teeth. Denote by Q the resistance acting at the distance R from the axis of the wheel whose centre is C2. The effect of this # resistance acting at D2, in the direction of the normal Dx D2, will, from the principles of the wheel and axle, be Qx, given by the relation Qx= Q R CD, (169)'; eiTect of the resistance at the distance of common normal; 424 NATURAL PHILOSOPHY. value of the friction on the teeth; to obtain the and this & will be the pressure at the point ra. Its fric- tion will be /Qx, acting in the direction qx q2, tangent to both teeth at their point of contact. The elementary quantity of work of this friction will be equal to its intensi- ty, multiplied into the elementary dis- tance by which the rubbing points now at m, separate in the direction of this tangent; which dis- Fig. 263. C*f 8^ 1/ ;a quantity of work ^^ ^ ODVioUSly of this friction; the value of this work; equal to that by which the points qx and q2, the extremities of the perpendiculars let fall from Cx and C2 upon the common tangent, approach to or recede from each other. Denoting the elementary path described by a point at the unit's distance from Cx by sx, and that described at the same distance from C, by s2, the paths described by qx and q2 will be, respectively, Cx qx X sx, and C2q2X s2; and because the points qx and q2 must move in the same direction when the tangent q2 qx passes between the centres Cx and C2, the elementary path of friction will be equal to the difference of these paths, and its elemen- tary quantity of work will equal fQx [02q2 X s2 - Cxqx X sj. Designating the radii of the primitive circles whose cen- tres are Cx and C2 by Rx and R2, respectively, we have, because of the equal velocities of the circumferences of these circles, MECHANICS OF SOLIDS. 425 Rxsx = R2s2; relation of the whence Paths at unit's distance from the 25 two centres; S2 = -p SV JX2 Moreover, drawing through the point B the line z2zh parallel to the tangent q2 qx, and denoting the angle ra B Cx, which is the complement of the angle CxBzx, by cp, and the distance ra i? by h, we find C2q2 = R2 cos cp + h, lever arms of the friction; Cxqx = Rj^ cos cp — h ; these values of s2, C2 q2, and Cx qx, substituted in the ex- pression for the elementary work of friction, give f Ql h Sx (-g- + l). work of friction; Denote by w the intensity of a force which, applied tangentially to the primitive circles at B, will produce the same effect as the friction. Its elementary work will be w Ri sx, and hence or j-, -p tangential force u=fQ1hIA-±-p . . (170). atthe jfij Xl2 circumference of primitive circle; Eepresent the angle B Cxm by 6. In practice, the angle mBC^ does not differ much from 90°, and we may take h = Rx tan &; 426 NATURAL PHILOSOPHY. and because & is generally very small, the tangent may be replaced by the arc, and h = 1^6; another form for tangential force at primitive circumference; which, substituted above, gives Rx + R2 = fQx RxR-2 x Rx6 . . (171). The value of 6 varies from a maximum to zero on one side of the line of the centres Cx C2, and from zero to a second maximum on the opposite side of this line; the first maximum corresponds to that position of ra in which any two teeth come first in contact, and the second to that in which the contact ceases; the intermediate or zero value occurs when m is on the line of the centres. The quantity 6 being thus variable, it must be replaced by a constant, and this constant must be a mean of all the values between to find the mean the two maxima. Designating the first of these by 6X, and vaiueofthe ,-i -, ■, . i angular distance the SeCOnd hJ 6* % of point of off the distance A E contact; 0' .X E x erect at A the Kg. 264. perpendicular A G = Qx; draw CE: then will the ordinates of this line which are parallel to A C repre- sent the different val- ues of 6, and the area of the triangle EAG will be the sum of all the values of 6 between 6X and zero. Again, make EB = &2; erect at B the perpendicular BG'" = 62; draw G'"E: the area of the triangle EBG'" will be the sum of all values of 6 between zero and L. Make altitude of a mean $ 2 _i_ ^ 2 triangle; B O = *i + V MECHANICS OF SOLIDS. 427 complete the rectangle BO', and draw A 0; then will the construction; triangle ABO he equivalent to the sum of the triangles A EG and EBG'", and therefore equivalent to the sum of all values of 6 between dx and 82, the mean of which is obviously the middle ordinate. „„ id/)_V + ^ (0i + d2)2-2M2 h + 42 M2 Xy = g XS U = ^rj------ = v , -----—- = —__-----1 2 mean value of A (t>! + do) 2 (^ + 62) 2 6X+ 62' angular distance Neglecting the last term as insignificant, &x 4- 62 xy = -±——- y 2 Multiplying by Rx, we find that Rx (6X + 62) is the interval between the place of the first and last point of contact of the same pair of teeth, estimated on the circumference of the primitive circle; denoting this interval by a, and sub- stituting in Eq. (171), we find of point of contact; . ~ Rx + R2 a fnfa a \ tangential fo ~ J Vl ~R1~R2~~ * j ~ fvi [yrT + TK)' atPlimitive Denote the number of teeth on the wheel whose centre is Cx by wj, and the number on the wheel whose centre is C2 by n2; then, because the teeth and intervals between them must be the same on each circumference, in order to work freely, distance from the __ 2 if Rx __ 2 * R2 _ place of first to Y* ™ ' that of last point of contact; which, substituted above, gives \nx n2l J nx n2 + nx Eeplacmg Qx by its value given in Eq. (169)', and recollect- 428 NATURAL PHILOSOPHY. final value of tangential force which is equivalent to friction; its quantity of work; example; data; work; ing that, within the limits supposed, C2 D2 becomes 1^, we finally have R hi., + ??!N R2 \ nxn2 = f«-Q (172). To find the quantity of work, multiply both members of this equation by R2 s2, which will give _n*2 s2 = f«s2QR ■ \\\±Ah n2n^ (173). Example. Eequired the work consumed in each revo- lution by friction on the teeth of a wheel whose arbor is subjected to a resistance equivalent to 1000 pounds, the number of teeth on the wheel being 64, and that of the connecting wheel being 192; let the teeth be of cast iron, and suppose the radius of the arbor equal to 0.8 foot. Here, R = 0.8; Q = 1000 lbs.; s2 = 2 x 3.1416; * = 3.1416; /= 0.152; ^ = 64; ^ = 192 ; arid, therefore, lbs. 64 4- 19'> uRisi = 0.152 X 3.1416 X 6.2832 X 1000 X 0.8 „, . ^ ~ 50; that is to say, the quantity of work consumed in one revo- lution by friction on the teeth, in the case supposed, is suf- ficient to raise 50 pounds through a vertical distance of one foot. XX. THE SCREW. The Screw, regarded as a mechanical power, is a device by which the principles of the inclined plane are so applied as to produce considerable pressures with great steadiness and regularity of motion. MECHANICS OF SOLIDS. 429 Fig. 265. mode of generating; § 240.—To form a clear idea of the figure of the screw screw with and its mode of action, conceive a right cylinder a k, with Bquare flUet' circular base, and a rectangle abcm having one of its sides ab coincident with a surface element, while its plane passes ( through the axis of this cylinder. Next, suppose the plane of the rect- angle to rotate uniformly about the axis, and the rectangle itself to move also uniformly in the di- rection of that line; and let this twofold motion of rotation and of translation be so regulated, that in one entire revolution of the plane, the rectangle shall progress in the direction of the axis over a distance greater than the side ab, which is in the surface of the cylinder. The rectangle will thus generate a projecting and winding solid called a fillet, leaving between its turns a similarly the fiiiet, channel, shaped groove called the channel. Each point as m in the an eix; perimeter of the moving rectangle, will generate a curve called a helix, and it is obvious, from what has been said, that every helix will enjoy this property, viz.: any one of its points as ra, being taken as an origin of reference, as well for the curve itself as for its projection on a plane through this point and at right angles to the axis, the dis- tances d' ra', d" ra", &c, of the several points of the helix from this plane, are respectively proportional to the circu- properties of a lar arcs mdl, md", kc, into which the portions rara', rara", hellx; &c. of the helix, between the origin and these points, are projected. The solid cylinder about which the fillet is wound, is called the newel of the screw; the distance mm'", between newei; the consecutive turns of the same helix, estimated in the direction of the axis, is called the helical interval. The heiicai interval; surfaces of the fillet which are generated by the sides of the rectangle perpendicular to Xhe axis, are each made up 430 NATURAL PHILOSOPHY. relative position and inclinations of the different helices; Fig. 266. of a series of helices, all of which have the same interval, though the helices themselves are at different distances from the axis. The inclination of the different helices to the axis of the screw, increases, therefore, from the newel to the exterior surface of the fillet, the same helix preserving its inclination un- changed throughout. The screw is re- ceived into a hole in a solid piece B of metal or wood, called the nut; a nut or burr. The surface of the hole " through the nut is furnished with a wind- fiiietofthenut; ing fillet of the same shape and size as the channel of the screw, which it occupies; while the fillet of the latter fills up the channel of the nut, formed by the turns of its fillet, whose inner surface is thus brought in contact with the newel. From this arrangement it is obvious that when the nut relative motion of is stationary, and a rotary motion is communicated to the screw, the latter will move in the direction of its axis ; also when the screw is stationary and the nut is turned, the nut must move in the direction of the length of the screw. In the first case, one entire revolution of the screw will carry it longitudinally through a distance equal to the helical inter- val, and any fractional portion of an entire revolution will carry it through a proportional distance; the same of the nut, when the latter is moveable and the screw stationary. screw and nut; MECHANICS OF SOLIDS. 431 The resistance Q is applied either to the head of the screw, or to the nut, depending upon which is the moveable ele- ment ; in either case it acts in the direction D C of the axis. The power F is applied at the extremity of a bar application of the (reconnected with the screw or nut, and acts in a plane resistanceHnd • i i i power; at right angles to the axis of the screw. Denote the per- pendicular distance of the line of direction of F from the axis of the screw by R, and the helical interval by h; then will the quantity of work of the power F, in one revolu- tion, supposing it to retain the same distance from the axis, be F x 2«R; and the quantity of work of the resistance will be Q X h. work of the resistance; The power F and resistance Q, both act to press the fillet of the screw and that of the nut together, the first acting at right angles to, and the latter in the direction of the axis. To find the work of friction thence arising, it will be necessary to find a force Fx, parallel to F, whose effect at the fillet is the same as that of F, acting at the distance R from the axis, and to resolve both Fx and Q into two components, one normal and the other parallel to the common surface of the pressing fillets. But the surfaces being warped, the normals at their different points will be oblique to each other, and so inclined to the axis that the normal components of the resistance Q, near the newel, will be less than those towards the outer surface of the intermediate fillet, while the reverse will be the case with the power Fx. 1r1ix; The resolution must, therefore, be made with reference to a normal at a helix midway between the newel and outer sur- face. This helix, like all others, is situated upon the sur- face of a cylinder of which the axis coincides with that of the screw. Denote the radius of this cylinder C mlv by r work of the power in one revolution ; 432 NATURAL PHILOSOPHY construction; Conceive a tangent plane to this cylinder at any point, as raIV, and two cutting-planes normal to the axis, and at a projection of intermediate helix; Fig. 267. jxr distance from each other equal to a helical interval, and equally distant from mlv. If we now develop the portion of the cylindrical surface, included between the cutting- planes, on the tan- gent plane, the sur- face of the cylinder Kg. 268. will become a rectan- gle whose base A E development of is equal to 2*r, and the intermediate -, -.,., 1 717, • helix • whose altitude A B is equal to h; and the helix will become the diagonal A B. De- note the length of the resolution of the helix A B hj I. Then power and resistance into components; draw the normal mF L, and resolve Q and Fx as before stated. Since Q = miy K is perpendicular to A E, and L mF perpendicular to A B, the angles L mF K and EAB are equal; also, since Fx = Imlv is perpendicular to BE, the angles ImF L and ABE are equal, and the triangles ABE, ImlvO, and LmFK, being right angled, are similar, and give the proportions I 2«r Q Lmv I : h : : Fx : wIV 0; MECHANICS OF SOLIDS. 433 whence r ,„ 2 if r Q normal j. ynF = ______— 7 J component of resistance; h . F, normal ■,--- J component of mF 0 = power; and the total pressure, which is equal to the sum of mF 0 and mIV L, becomes 2_*rQ _^ hFx total normal I I pressure; and the friction J i and since in one revolution the path described by this friction is the diagonal A B = l, its quantity of work will be its quantity of /(2 If r Q + h Fx) ; work in one revolution; and because the work of the power F must equal' the work of the resistance Q, increased by that of the friction, we have work of power equal that of 2ifR.F= Qh + f(2«rQ + Fxh). resistance increased by work of friction; But the effect of F and Fx being the same, their quantities of work must be equal, and hence 2«RF = 2«rFx; whence F, = F±; 434 NATURAL PHILOSOPHY. which substituted in the preceding general equation, we get work of power; 2 if R F = Qh +f(2ifrQ + F^-h); and finding the value of F, value of the *■ - Q ,*■•?.2A, Si • • • <174> Power; ^ 2if Rr —fRh Multiplying both members by 2 if R; then adding and subtracting Qh, in the second member of this equation, we find workofpower; 2if R . F= Qh + / Q f + ^ ^f . . (175), Aif r — J Ii in which the work absorbed by friction is given by the last term ; that is to say, by work absorbed by fCl. friction; J* ' 2*r—fh ' If we neglect the consideration of friction, or make / = 0, we find, from Eq. (174), simply relation of power t and resistance F = Q X ______* without friction; 2 if R that is, the power is to the resistance as the helical interval is to the circumference described by the extremity of the perpendicular, drawn from the axis to the direction of the power. From which it is obvious that the power of the screw may be increased, either by diminishing the stated in words; distance between the thread or fillet, or by increasing the distance of the power from the axis. If we examine the expression MECHANICS OF SOLIDS. 435 f Q h2 + 4*y 2 if r zrfh\ intermediate helix should be small; Fig. 269. we shall find that the numerator of the fractional factor increases more rapidly than the denominator for any increment in the value of r, the radius of the mean helix. For this reason, r should be made as small as possible radius of consistently with sufficient strength. Let b 0 he the radius of the interior helix, or that of the newel, and a 0 that of the exterior helix; it is usual to make the projection a b, of the fillet, equal to the thickness ad, measured in the direction of the axis; and for facility of execution, the dimensions of the chan- nel are made equal to those of the fillet, that is to say, ab is made equal to ad; in which case, the helical in- terval a a' will be equal to 2 ad = 2 ab, when there is but a single fillet. Should there be two fillets, which are often employed to increase the helical interval without changing the size of the newel, and therefore of r, then rule; will the helical interval be 4 a b. Considerations affecting the union of sufficient strength with least friction, have suggested this general rule in regard to the projection of the fillet, viz.: make the projection a b equal to one third of the radius 0 b of the newel, or proportion of the different parts of the screw; ab = i Ob. This will give Ob = Bab; radius of the newel; 436 NATURAL PHILOSOPHY. and Ob + \ab = r - Bab + \ab = \ab; and because h — 2 ab, radius of intermediate r = -j fl ', helix; which substituted for r, in the expression for the friction, gives work of friction; J ' V ^ _ 7 / ' 22 and making * =-=- to which it is very nearly equal, the expression reduces to . n 122 its final value; / X Q n __ .. To apply this to a particular example, let the screw be example; made of wrought iron, and the nut of brass, and suppose an unguent of tallow, in which case /= 0.103, see Table III, § 212; hence the value of the friction becomes 1.152 X Qh; which, substituted in Eq. (175), gives result. 2ifR.F = Qh + 1.152 Qh = 2.152 Qh; whence we see, that friction occasions a loss of work greater than the whole work performed by the resistance. Endless screw; § 241.—The endless screw is employed to transmit a very slow motion, and, at the same time, to overcome con- siderable resistance. It is a short screw, with square fillet, MECHANICS OF SOLIDS. 437 Fig. 270. and so supported as to revolve freely about its axis, with no motion of translation. It is usually turned by means of a crank. The fillet pass- es between teeth on the circumference of a wheel of which the axis is per- pendicular to that of the screw. The resistance Q is apphed to the circum- ference of the arbor of the wheel. The rubbing faces of the teeth, instead of be- ing parallel to the axis of the wheel, are slightly in- clined to that line, so as to make them parallel to the surface of the fillet when the latter is brought in contact with the teeth. A rotary motion being communicated to the screw, its fillet presses against the teeth of the wheel; and as the screw can have no longitudinal motion, the wheel must turn about its axis. As the teeth are withdrawn towards one end of the screw, others are interposed towards the other end, and thus an endless motion may be kept up; hence the name of the machine. A plane through the axis of the screw and per- pendicular to that of the wheel, will cut from the rubbing surfaces of the fillet and teeth a profile; and if we confine our- selves to what takes place in this plane during the motion, we shall find that the circumstances will be use and description; surface of teeth inclined to axis of motion; operation and reason for the name; Fig. 271. section by a plane through the axis of the screw perpendicular to the axis of the wheel; 438 NATURAL PHILOSOPHY. Fig. 271. the same as those of two circumstances of wheels acting upon one action same as & through the inter- those of two ° wheels with vention of teeth; for, as the screw turns about its axis to bring different parts of the fillet in this cutting plane, the section a b will move in the direc- tion from A to B, driving the section be of the tooth before it. Let Qt he the force applied at b in the direction A B, which is tangent to the circumference whose centre is on the axis of the wheel, and whose radius is Ce = Rn and which will sustain the resistance Q in equiiibrio: then de- noting by N the resultant of Q, and Q, by r the radius of the arbor, and by r, that of the trunnion, will quantity of work; QiRjS, = Qrs4 -^f^riSi) in which st is the arc described at the unit's distance from the axis of the wheel. Dividing by R, s„ value of the power; Q, = Q^ + fN% • • • (176> friction; Find, by the process explained in §234, Eqs. (157) to (160), the value of Q, and N. The pressure upon the tooth at b will thus be known, being equal to Qr This pressure pro- duces a friction upon the teeth of which the value is wherein n denotes the number of teeth on the wheel whose centre is C, and n' the number on the other. But the cir- MECHANICS OF SOLIDS. 439 cumference of this latter wheel being a right line, is in- finite as well as the number of its teeth; hence _. reciprocal of the _ __ A . number of teeth fl on section of screw; and the foregoing becomes i value of the J n1 friction; which must be added to Q, to obtain the force necessary to turn the wheel and to obtain the total pressure on the fillet of the screw. This sum, which is 0,+/f Q.= Q,(i+f$, being substituted for Q in Eq. (175), will give ,.*,-ft(i+/i)»+/ft(i+/i)£t^, total pressure on the fillet; or URF = Q,{l +fl) [*+/.|±i*£] . . (m). *-»-«* of power; In the discussion of the screw, no reference has been made to the friction on the pivots and collars by which friction on pivots the screw is kept in position. It will always be easy to ^J^8 find this, in any particular case, by the rules for finding the friction upon pivots, sockets, and shoulders or rings, explained in § 223. 440 NATURAL PHILOSOPHY. XXI. THE LEVER. fulcrum; levers divided into different orders; first order; The lever; § 242.—The Lever is a solid bar A B, of any form, supported by a fixed point 0, about which it may freely turn, called the fulcrum. Sometimes it is supported upon trunnions, and fre- quently upon a knife-edge. Levers have been divided into three different classes, called orders. In levers of the first order, the power F and re- sistance Q are applied on opposite sides of the ful- crum 0; in levers of the second order, the resistance Q is applied to some point between the fulcrum 0 and point of application of the power F; and in the third order of levers, the power F is applied between the fulcrum 0 and point of application of the resist- ance Q. examples of The Common shears fur- different orders of • -. , . levers. msn an example of a pair of levers of the first order; the nut-crackers of the sec- Fig. 272. second; third; MECHANICS OF SOLIDS. 441 ond; and fire-tongs of the third. In all orders, the conditions of equilibrium are the same. Fig. 273. a % 243.—When the lever is supported upon a point, the Equilibrium of equilibrium requires that the resultant of the power and theleveria , r which the resistance shall pass through this point in order to be fulcrum is a destroyed by its reaction; to have a resultant, the power point; and resistance must lie in the same plane, and as the re- sultant will also be in this plane, the power, resistance, and ful- crum, must be in the same. If the result- ant pass through the fulcrum, its moment taken in reference thereto must be zero, which requires that the moment of the power shall be equal to that of the resist- ance. That is, when a lever A B is in equiiibrio and solicited by the power F moment of power and resistance Q, 0 being the fulcrum, if we draw fr0me resistance; a simple change in the point of support or fulcrum, which MECHANICS OF SOLIDS. 443 may be made at pleasure, gives the means of establishing any desired relation between the power and resistance. If, for example, the point of support 0 is placed so that the distance On is one thousandth part of Om, then will F = _ Q 1000 whence we see that with a very small power we may hold to effect a given in equiiibrio an enormous resistance; but as the quantity purpose'a . i- J diminution of of work of the resistance must equal that of the power, power increases the path described by the point of application of the latter itspath; must increase in the same proportion. To give an idea of the time necessary to raise a heavy burden through a moderate height with the lever, suppose the weight to be raised is 2000000 pounds, and that it is to be elevated five feet. The quantity of work will be 2000000 lbs. X 5 ft. = 10000000 lbs. Supposing a man to act by his weight =150 lbs. at the end of a lever, he would example to 10000000 illustatethi8; have to describe a path equal in length to —zr^---= 66Q66 feet, nearly. If in each second of time he move the point of the lever at which he applies his weight, through a distance of 0.2 ft., he will require —-^— = 333333 seconds nearly, = 92.6 hours nearly, = 9.26 days, suppo- sing the man to labor 10 hours a day: in fact a man left to his individual efforts would never accomplish such a task. This example shows us that the lever is only useful for practical use of momentary efforts, and when the burden, being considera-the lever" ble, is to be moved through a very small distance-. 444 NATURAL PHILOSOPHY. XXII. atwood's machine. Atwood's machine; objects of the machine; § 245.—We shall terminate this branch of our subject with a discussion of an instrument whose object is an ex- perimental verification of the laws of constant forces. This instrument is the invention of Atwood, an English philoso- pher, and bears his name. Before proceeding to describe it, let us first find the circumstances of motion under the general case of which the machine in question is but a particular instance. For this purpose, let A B and A D be two inclined planes having a common altitude AE; H and H', two wheels of different diame- the general case ters mounted Upon of which this .-, ^ the same arbor, to which they are firmly attached, and of which the axis is supported upon trunnions par- allel to the com- mon intersection of the two planes ; W and W two weights supported upon the inclined planes by means of cords c and c' wound, the first about the one body ascends wheel H and the second about the wheel H', the cords be- dicends3;0111 ing Parallel to tne inclined planes. Now if the weight W he made sufficiently heavy, it will overcome all opposition to motion and shde down the plane A B, while machine is a particular example; MECHANICS OF SOLIDS. 445 the weight W' must Kg. 276. from its connection move up the plane AD. It is required to find the circum- stances of motion. Denote the angle which the planes A B and A D make respectively with the verticals.i£, by

cos ,' + / W sin ,' = W (cos ,' + /sin <,') ■ which, substituted in the expression above, for the stiffness of the cord c', gives its stiffness; K + I. IF'(cosy' + f .sin cp') 2R' length of paths in direction of weights; At the instant motion be- gins, let the centres of grav- ity of W and W he at G' and Gn respectively, and in any subsequent instant at 67" and Gu; denote the dis- tance 67'67" by x, and 67,67,, by x', then will x and x' be the paths described by the centres of gravity parallel to the planes in the interval; and x cos -f W'x'Bmf') ,, ,JT4-/IT'(co3?'-f/8in^) ~r a'x o~p7 2R' + fNrS/ + V?I (178). The variables in this equation, for the same inclination of the planes, are V, V, Vx, x, x', and s,; but these, by the MECHANICS OF SOLIDS. 449 nature of the system, are connected by the following rela- tions, viz.: V : V :: R : R .-. V = x : x' :: R : R .\ X' = 1 : R VR s, : x s. = R ' . (179), xR' R " .'(180), X ~R~~ ' ■ (181), V . ri82V relation between the angular velocities and spaces; R : V : : 1 : Yx ,. Vx = -£- . . (182). These values of Vx, V, x', and st, being substituted in Eq. (178), will give R' Wx cos

+ f(Wx sin

R R' _/(IF. sin ? - IF • - • sin ) vaJueforthe square of the R' K+I.W'.jcost'+fsLnt') veIocity; 2i2' The coefficient of x containing no variable, we find that the space described by the body on the plane A B, 450 NATURAL PHILOSOPHY. varies as the square of its velocity. Llence the motion is, motion uniformly § 68, Eq. (8), uniformly varied; and the coefficient of x is twice the velocity which the force producing this motion is capable of generating in a unit of time. Making varied; value of the velocity generated in a unit of time; A = R'2 I w+w'w + *-r> R' W. cos f — W • — • cos

') R' jr+/.TP.(coaf+/Bin»') 'drR' 2R> ■'■"■T> square of the velocity; the foregoing equation may be written V2 = x. 2 A . (183). Since the motion is uniformly varied, if T denote the time of describing the space x, then will Eq. (7) become space described; x - \AT2 (184); writing A for Vx, and x for s: substituting this for x above, we find V2 = A2T2, or value of velocity; V = AT (185). Eqs. (183), (184), and (185), give the laws of uni- laws of uniformly formly varied motion, or, as it is usually expressed, the laws of constant forces. These laws are, 1st. The velocities are to each other as the times in which the force produces them; 2d. The spaces described, are to each other as the squares of the velocities acquired in describing them; or as the squares of the times in which they are described. MECHANICS OF SOLIDS. 451 Any device that will make the time in which the motion takes place com- paratively great, while the velocity acquired shall be small, will enable us to verify these laws from ob- servation. For this pur- pose, A must be small. By reference to Eq. (182), we find that A may be dimin- ished at pleasure by increas- ing the angle cp, or decreasing cp'; this will increase the effect of friction, which op- poses, while it will diminish the component of W, which aids the motion. Or A may be diminished, by diminish- ing the angles

J v removed; accelerated rather than retarded. To show that the motion will be uniform, repeat the experiment, placing the plane stage first at 1 foot 3 inches, then at 1 foot 9, then at 2 feet 3 inches, and so on, adding 6 inches each time, and it will be found that the basin will be arrested at the its proof; third, fourth, fifth, &c, beats of the clock after its motion begins; thus showing that the spaces described are pro- portional to the times, which is the characteristic of uni- form motion. Next adjust the ring so as to remove the bar when its basin reaches the 12 inch or 1 foot mark, and place the plane stage at the 2 feet mark, it will be repetition of the found that the bar will strike the ring at the second beat exPeriment; after its motion begins, and that the scale will be arrested at-the third beat. That the motion is uniform after the removal of the bar may be shown, as before, by repeating the experiment, and adding 12 inches each time to the space to be described after the bar is arrested. In the same way all the other results may be verified. If a bar-weight be placed upon the second ring, and the illustration of latter be so adjusted that the ascending basin shall take retarded motion; it up at the moment the bar on the descending basin is removed, the motion will become retarded, and we shall haYe the case of a body projected vertically upward from rest with a velocity equal to that of the basins. The plane stage being placed at a distance from the ring which takes up the descending bar, equal to that through which the latter has descended, it will be found that the aii the laws scale will iust reach this stage at the instant the motion ^h^h1I,e^late J o the fall of heavy is destroyed by the action of the ascending bar. All the bodies may be laws which regulate the fall of heavy bodies may be ^n-ne y thls verified by means of Atwood's instrument. 458 NATURAL PHILOSOPHY. XXIII. IMPACT OF BODIES. impact of bodies; § 246.—When a body in motion comes into collision with another, either at rest or in motion, an impact is said to arise. We have seen, § 204, that the action and reaction which take place between two bodies, when pressed to- gether, are exerted along the same right line, perpendicu- lar to the surfaces of both, at their common point of contact. When the motions of the centres of gravity of the two direct impact; bodies are parallel to this normal before collision, the im- pact is said to be direct. When this normal passes through the centres of grav- ity of two bodies which come into collision, and the mo- direct and central tions of these centres are Fig. 278. impact; along that line, the impact is said to be direct and central. When the motion of the centre of gravity of one of the bodies is along the com- mon normal, and the normal does not pass through the centre of gravity of the oth- du-ectand er, the impact is said to be eccentric impact; ^^ &nd ^^^ When the path described by the centre of gravity of one of the bodies, makes an oblique impact; angle with this normal, the impact is said to be oblique. Fig. 279. Fig. 280. m*—*■ MECHANICS OF SOLIDS. 459 Bodies resist, by their inertia, all effort to change circumstances of either the quantity or the direction of their motion, ^j*^!"1118 tbe When, therefore, two bodies come into collision, each will experience a pressure from the reaction of the other; and as all bodies are more or less compressible, this pressure will produce a change in the figure of both; the change of figure will increase till the instant the bodies cease to approach each other, when it will have attained its maximum, and the bodies will have the same velocity. The molecular spring of each will now act to restore the former figures, the bodies will repel each other, and finally separate. ■ In the impact of bodies, three periods must therefore three periods of be distinguished, viz.: 1st., that occupied by the process theimpact' of compression; 2d., that during which the greatest com- pression exis#, and in which it is obvious the bodies have the same velocity; 3d., that occupied by the process, as far as it extends, of restoring the figures. We are also carefully to distinguish the force of restitution from the force force of of distortion; the latter denoting the reciprocal action ex-^rut/n. erted between the bodies in the first, and the former that exerted in the third period. The greater or less capacity of the molecular springs of a body to restore to it the figure of which it has been deprived by the application of some extraneous force when the latter ceases to act, is called its elasticity. elasticity defined; The ratio of the force of distortion to the force of resti- tution, is the measure of a body's elasticity. This ratio is sometimes called the coefficient of elasticity. When these coefficient of two forces are equal, the ratio is unity, and the body is elastlclty '■> said to he perfectly elastic; when the ratio is zero, the body perfect elasticity; is said to be non-elastic. There are no bodies that satisfy non-eiastic. these extreme conditions, all being more or less elastic, but none perfectly so. | 247.—Suppose two bodies A and B to move in the same direction, the body A to overtake B, and the impact 460 NATURAL PHILOSOPHY. Direct impact of to be direct. The forces of distortion and of restitution, two bodies; arising as they do from the reciprocal action of the bodies upon each other, are real pressures, measurable in pounds, and are capable of generating in each body, in a given time, a certain quantity of motion. Denote the intensity of this force', at any instant of the impact, by F; the small velocity lost by the body A, in the short time during FiS- 281- which F may be re- notation; garded as constant, by v; and the small velocity gained by B, in the same time, by the action of the same force, by v'; also denote the mass of A by Jif, and that of B by M'; then will F, which may be called indifferently the action of one body or the reaction of the other, be measured by Mv, or M' v'; and, because of the equality of action and reaction, equality of action ir ~rr _. lhri tti and reaction; That is to say, the quantity of motion lost or gained by each of the bodies, during each indefinitely small portion of time, will be equal to each other ; and if we take the sum of all the quantities of motion lost or gained by each of the bodies, we shall have the whole quantity of motion gained or lost by the one, equal to that gained or lost by the other. Denoting the entire gain or loss of velocity of the body A by V„ that of the body B by Yn, we shall have gain and loss of ,, y yr, y motion equal; ""* ' i ""* ' u' But the force F acts in opposite directions upon the two bodies, and hence, if we give the positive sign to the velocity generated in one body, that of the other must be MECHANICS OF SOLIDS. 461 negative; that is, if Yt he counted positive, Yu must be negative, which will make forces producing _ _ __ , -t __ these act in ■"* ' i — ■"* ' //) opposite directions; or MY, 4- M' Y„ = 0. That is to say, the algebraic sum, or the whole quantity of quantity of motion lost and gained will be zero; and in every stage of the sy°tem constant impact the quantity of motion in the entire system will, there- fore, be the same as before the impact began. § 248.—At the instant the bodies have ceased to ap- to find the proach each other, they will have attained their greatest ^^^f^7 compression, and, considering their condition before the greatest ,.-,-, • n i ii compression; retrocession begins under the action of the molecular springs, it is obvious that they may be regarded as a single body, having a common velocity. Denote this velocity by U; also denote the velocity of the body A, before the impact, by V; that of the body B, before the impact, by V, the masses being M and M' as before. The whole quantity of motion before the impact will be MY 4- M'Y, and that at the instant of greatest compression will be (M 4- M') U. But these, by the last article, must be equal, or (M + M') U = MY 4- M' V; whence TT _ MY + M' V (190^ the value of this U------M + M' ' ' ' velocity; That is to say, when two bodies moving in the same 402 NATURAL PHILOSOPHY. expressed in words; value when bodies move in opposite directions. Velocity gained up to greatest compression; force of distortion; force of restitution; direction have a direct impact, the common velocity, at the instant of greatest compression, is equal to the sum of the quan- tities of motion before the impact, divided by the sum of the masses. If the bodies moved in opposite directions, either Y or V would be negative, say V', and MY — M' V jj = ^LAL---^L_L . , . (191). M + M' K J § 249.—The velocity lost by the body A, up to the instant of greatest compression, is obviously equal to Y- U, and that gained by the body B is equal to U- Y; the force of distortion will, therefore, be measured by M(Y - U), or by M'(U - Y'). Denote by Y, the velocity which A loses by the force of restitution; and by Y„, that which B gains by the action of the same force; the force of restitution will be measured by MY, or M' Yu; and if e denote the coefficient of elasticity, then, from the definition coefficient of elasticity; MY,___ M(Y - U) = e, MECHANICS OF SOLIDS. 463 whence wv„ M'(U - Y') ~ "' elasticity; Y, = e(Y - U) . . . (192), velocities losl Y„ = e(U- Y') . , , . (193). and gained; That is to say, the velocity which A loses by the force of restitution, is equal to the coefficient of elasticity, into the the same velocity which it lost by the force of distortion; and the velocity exPressed 'n gained by B by the same force, is equal to that which it gained ™ by the force of distortion, into the coefficient of elasticity. The total loss of velocity which A will experience by the impact will be jr _ 7T j_ 117" _ 77"\ *oss °^ velocity °f V U -f e { V ■ U); the impingmg body and the entire gain of B will be U - V' + e(U - V). gain of the other; Denote by v the velocity retained by A, and by v' that which B has after the impact; then, since the velocity retained by A, must be equal to that which it had before the impact, diminished by its loss, v = Y-Y + U- e(Y- U) = (1 + e) U-eY; and as B must, after the impact, have its primitive velocity increased by its gain, v' = Y'+ U -Y'+e(U-Y) = (1 4- e) U-eY'; and substituting for U its value in Eq. (190), we have n .MV+M'Y' Jr nQ.. velocity of the V = (1 4" e)---^----^,-----eV . . (194), impingmg body V J M + M' V 7' after the impact; 464 NATURAL PHILOSOPHY. velocity of the If Y -4- AF Y' other after the v' = (1 + e) ?L*+J?LJL -eV. . (195). impact; M -\- M Thus, the velocity of either body after impact, is equal to the coefficient of elasticity increased by unity, multiplied into the in words; common velocity at the instant of greatest compression, and this product diminished by the product of the coefficient of elasticity into the velocity of the body before impact. If the body B move to meet the body A, its velocity will be negative, and the above reduce to o=(i+e) Mli%v' -er ■ ■(i96)' when the bodies meet. MY — M' Y' v'=(l+e)-M L fr/— + eV . . (197). v ; M+ M' v ' § 250.—If the body B be at rest when the body A impinges against it, then will V be zero, and •= (1 + e) irrs> ~eV ■ ■ (198)- When one of the bodies is at rest; v' = v + e)irrM' ■ ■ ■ ■ <199)- From the last equation we find coefficient of _ v' (M + M') 1 forxCW elasticity; e ~~ ~MY--- * ' ' ' '' and when the masses of the bodies are equal, or if = M', its value when 2 v' the masses are e = -r==- — 1 .... (201)', equal; ' which suggests a very easy method of finding the co- efficient of elasticity of any solid body. For this purpose, MECHANICS OF SOLIDS. 465 Fig. 282. turn a pair of spherical balls of the same weight from the experimental body whose coefficient of elasticity is to be found; suspend determmation of J J > Xr the coefficient of them by silken strings, so that when the latter are vertical elasticity; the balls shall just touch each other, be upon the same level, and have their centres opposite the zeros of two cir- cular graduated arcs whose centres of curvature are at the points of suspension. The body A being drawn back to any given degree upon its scale and abandon- ed, will descend and impinge against the body B with a velocity due to a height equal to the versed sine of the arc which it describes before the impact; the body B will ascend on the oppo- site arc to a height due to the velocity with which it leaves A; this height will be the versed sine of the arc described by B before it begins to descend again. The arcs being known, their versed sines are easily computed from the properties of the circles. Denoting these versed sines by h and h', then will description of instrument, and mode of using it; Y = V2gh, v' = V2gh'; winch, substituted in the value of e, gives (202). velocity of impinging bod) and that of the body struck; coefficient of elasticity; Example. Two ivory balls of equal weights, and there- fore of equal masses, were made to collide in the manner 466 NATURAL PHILOSOPHY. above described. One descended through an arc of 20 example of two degrees, and the other ascended through an arc of 18 ivory balls; degrees ancl 30 minutes; required the value of e. By tables of natural sines and cosines, we find nat. cos 20° = 0.9396926; versed sin 20° = 1 - 0.9396926 = 0.0603074; and denoting the radius of the circular scale by R, we have ; SrSSSf * = 0-0603074 JJ. body; Again, nat. cos 18° 30' = 0.9483236; versed sin 18° 30' = 1 - 0.9483236 = 0.0516764; height due to the velocity of the ft — 0.0516764 R; body struck; and numerical value of the coefficient; due fi _ 2 . /0.0516764.Z? _-.„. /0.0516764 . lent; e ~ 2 V 0.0608074.1? * ~ 2 V O0603075 1-°-8oldy' whence we conclude that the coefficient of elasticity of the specimen of ivory employed, is about 0.85; that of glass will be found to be about 0.93, and that of steel about 0.56. Example. Two ivory balls, whose masses are repre- exampieofthe sented by 6 and 4, move in the same direction with ^11is8!onofivory velocities of 10 and 7 feet a second respectively. What is the velocity of each after impact? The conditions of the question require that the larger mass 6 shall over- take the smaller mass 4, because the former has the greater velocity. Hence MECHANICS OF SOLIDS. 467 M = 6; Y = 10; M'= 4; Y'= 7; e = 0.85. given data; These data, in Eqs. (194) and (195), give „ = 1.85 ^— - 0.85 X 10 = 7*78, V' = 1.85 ^~y0— - 0.85 X 7 = 1CK33. velocities after impact; Example. Let the same balls move in opposite direc- another example. tions so as to meet, each with the same velocity as before. The same data, substituted in Eqs. (196) and (197), give «A __ OO ft. v = 1.85 — ^^ - 0.85 x 10 = - 2.58, v'= 1.85 10 60 - 10 28 /'■ _^2 + 0.85 X 7 = 11.87. Fig. 283. description and notation; § 251.—Now suppose the bodies A and B to move, oblique impact; the first with a velocity Y in the direction from E towards F, and the second with a velocity Y' in the direction from C towards D; and let the collision take place at H. Through the point H, draw the common normal MN, and resolve each of the ve- locities Y and V into two components, one in the di- rection of the normal and the other in the direction of the tangent plane at H. For this purpose designate the an- gle F ON My cp, and DO,N 468 NATURAL PHILOSOPHY. normal velocities; tangential velocities; 22 by cp'; the components in the direction of the normal, will be Y cos ' _ v cos 6 = (1 -f e)-------m+W-----~~ 6 Vcoaf'- • ' (203)> ,, . . MVcoa

' „, , „ «' cos 6' = (1 4- e) -------^T -----r----e V cos 0'. . (204). tangential components of velocity after the impact; Moreover, because the effects of the impact arising from the components of the velocities in the direction of the normal will be wholly in that direction, the components of the velocities of each body before and after the impact at right angles to the normal, will be the same, and hence v sin 6 — Y sin cp . . . (205), v' sin 6' = Y' sin ,1 . .MY + M'Y' -, y = (1 + e)-^T-F'— ~eV' the same as Eqs. (194) and (195); and passing to the limits, non-elasticity on the one hand and perfect elasticity on the other, we have, in the first case, e = 0, and MY A M' Y' v = -rnr- ■ ■ ■ <218)' case of direct impact; MY 4- M' Y' "'-- u+h ■ ■ ■ <21i>; bodies non-elastic; and in the second, 6 = 1, consequently bodies perfectly elastic. 0 = 2 M+M'----F • • (216>' MY A M' Y' V' = 2 \,\ * - T . . (216). If 4- M' v y § 252.—The equations which have just been deduced, are sufficient to make known the circumstances of motion oblique and of the centres of gravity of the colliding bodies, for we «**»**> i™p««*5 have seen, § 146, that whenever a body is acted upon in a direction normal to its surface, its centre of gravity will move as though the force were applied directly to that point. But we have also seen, in the same article, 472 NATURAL PHILOSOPHY. that when the direction of in the eccentric the force does not pass impact the bodies trough tllG centre of grav- will rotate; a ° . ity, which is the case in the eccentric impact, the body will also have a ro- tary motion. Employing the same no- tation as before, and sub- tracting Eq. (203) from the identical equation, Y cos 9 = Fcos 9, we find loss of velocity of one body in direction of normal; „ ... lf'(Fcos9 - F'cos?). Fcos 9 — v cos a = (1 4- e)---*----;^—TF,------- r v ' M+ M the first member is the loss of velocity of the body A in the direction of the normal, during the impact; and mul- tiplying both members by the mass of A = M, we have, for the quantity of motion lost in the direction of the normal, motion lost in Ml YCOS 9 - V COS 6) = (1 + e) tnctt direction • MM'(Ycoscp - F'cos9') MA M' construction; If the force of which either member of this equation meas- ures the intensity, and of which the direction coincides with the normal, does not pass through the centre of grav- ity, it will give rise to rotary motion. From the centre of gravity G', of the body B, let fall the perpendicular G'C upon the normal, and denote its length by b; also denote the angular velocity of the body B by s„ and its moment of inertia with reference to an axis through the centre of gravity, and perpendicular to the plane of the normal and centre of gravity, by Ix; then, because the angular velocity MECHANICS OF SOLIDS. 473 L, equal to the moment of the impressed force divided by the moment of inertia, ,_, , . , MM' YCOS 9 — Y' COS Cp' fM n. angular velocity S, = (1 + e)6 ma. M> X --------T------- ' ' (217> ofoneofthe M + M A bodies; Also let fall from the centre of gravity G of the body A, the perpendicular G C upon the normal, and call its length a. Since the reaction of the body B, which is equal to the action of A, does not pass through the centre of gravity of the latter, it will communicate a rotary motion; and, denoting the angular velocity of A by s,,, we shall have, /I J. »\ r, MM' v V C0S ? ~ V' C0S ?' fOX R\. «>g«lar velocity S„ = (14- e)«^qrif X -------1,'------- ' ' ( 8)' of the other; in which Ix is the moment of inertia of the body A, in reference to an axis through its centre of gravity and perpendicular to the plane containing this point and the normal. In what precedes, no reference is made to friction, but thus far no it is obvious that this principle cannot be wholly dis- ^^"^ regarded; for the bodies acting upon each other in the direction of the normal with a pressure of which the measure is {1 + e)'WVM{Vcosc?~ F'C0S(f/)' this pressure will give rise to friction, whose intensity is measured by MM /T7-____ T7v „^„ „,/\. measure of the ir+w fa 4- e) • ——, (7cos 9 - V cos cp'); J \± Te; lf ^ ^, \ r ^^ r t / j friction; and this acting in the direction of the tangential com- ponents of the velocities will accelerate the one and retard • 474 NATURAL PHILOSOPHY. tangential force to overcome friction: limits within which friction may act lo produce rotation; Fig. 285. „'-•*# the other. Let U, denote the tangential velocity lost by the body A; then, the force exerted to overcome the friction will be measured by MU,. Now if the tangential velocities be equal, it is obvious that the bodies will move together in the direction of the tangent, M U, will be zero, the friction will not be called into action, and the bodies will not rotate from friction. If the tan- gential velocities differ by a quantity that will make M U, equal to the friction, then • will the whole* of the latter be exerted to produce ro- tation. If the tangential velocities be such as to give to MU, any value between these limits, a part only of friction will be exerted, and this part alone will determine the rotation. If the difference of the tan- gential velocities be such as to make M U, greater than the friction, the bodies will slide along each other and rotate at the same time; the latter motion being due to the entire friction, and the former to the excess of MU, over the value of this force. Denote by n, the ratio of the friction to M U„ then will quantity of tangential motion M U, = nf (1 -f c) • lost; MM' W+M ^ S 9 "" V cos 9 )' Let fall from the centres of gravity of the two bodies the perpendiculars G T and 67' T', upon the tangent TT'; 9 MECHANICS OF SOLIDS. 475 denote the length of the first by b, and that of the second by a,. Then will the angular velocity of the body B, produced by friction, be //1 \ 7, MM' Y COS Cp— Y' COS 9' angular velocity nJ V1 4- e)0, —-——- . -----------___--------; of one body due ' 1 to friction; and that of the body A, ,... MM' YCOS 9 — Y' COS 9' angular velocity nf(l 4" e) a, .---------_-------^; of the other, due iH -f- iki lx t0 friction; whence the whole angular velocities of the two bodies will become MM' Fcos 0 — V cos 0' ., , ., . s' = 0- + e) • ------2-=-------2- • (6 + nfbt), ■UJ- < jXL Ji whole angular velocity of the tr ^r, -r-r rr, i bodies; MM' Ucos0— V cos 0' . „ . •« = (!+«)• ^+^----T7— '(a + w/a/)- If the balls be spherical and homogeneous, the normal will always pass through the centre of gravity, b and a will reduce to zero, and the rotation will be due to friction alone. If the impact be direct, then 9 and 9' will be zero, particular cases there will be no tangential'components of the velocities ° gure' MU„ and consequently n will reduce to zero, and the rotation will be due to the eccentricity of the impact. PART SECOND. MECHANICS OF FLUIDS. I. INTRODUCTORY REMARKS. condition of aii § 253.—We have seen, § 12, that the physical condition u0ontheePend9 °^everJ body depends upon the relation subsisting among molecular forces; its molecular forces. When the attractions prevail greatly over the repulsions, the particles are held firmly together, a solid; and the body is called a solid. In .proportion as the differ- ence between these two sets of forces becomes less, the body is softer, and its figure yields more readily to external pressure. When these forces are equal, the particles will yield to the slightest force,. the body will, under the ac- tion of its own weight, and the resistance of the sides of a vessel into which it is placed, readily take the figure of the a liquid; latter, and is called a liquid. Finally, when the repulsive exceed the attractive forces, the elements of the body tend to separate from each other, and require either the applica- tion of some extraneous force or to be confined in a closed vessel to keep them together; the body is then called a a gas or vapor; gas or vapor, according to the greater or less pertinacity with which the repulsive retain their ascendency over the attractive forces. In the vast range of relation among the molecular forces, from that which distinguishes a solid to MECHANICS OF FLUIDS. 477 that which determines a gas or vapor, bodies are found in soiids, liquids, all possible conditions—solids run imperceptibly into ™toMother. liquids, and liquids into gases. Hence all classification of bodies founded on their physical properties alone, must, of necessity, be arbitrary. § 254.—Any body whose elementary particles admit of Definitions, &c; motion among each other, is called a fluid—such as water, a fluid; wine, mercury, the air, and, in general, liquids, gases, and vapors; all of which are distinguished from solids by the great mobility of their particles among themselves. This distinguishing property exists in different degrees in dif- ferent liquids—it is greatest in the ethers and alcohol; it is less in water and wine; it is still less in the oils, the sirups, greases, and melted metals, that flow with difficulty, and rope when poured into the air. Such fluids are said to be viscous, or to possess viscosity. Finally, a body may viscous fluids; approach so closely both a solid and liquid, as to make it difficult to assign it a place among either class of these bodies, as paste, putty, and the like. paste; Pu«y- § 255.—Fluids are divided in mechanics into two classification of classes, viz.: compressible and incompressible. The term in- compressible cannot, in strictness of propriety, be applied compressible and a ., _ incompressible; to any body in nature, all being more or less compressible; but the enormous power required to change, in any sensible degree, the volumes of liquids, seems to justify the term, when applied to them in a restricted sense. The gases and vapors are highly compressible. All liquids will, there- liquids * .111 1 incompressible; fore, be regarded as incompressible; the gases and vapors gasesandvaporg as COmpreSSible. compressible § 256.—There are many fluids that readily pass from the compressible to the incompressible class, when sub- jected to moderate increase of pressure, and reduction of temperature. These are called vapors, and are such as arise vapors; from the application of heat to liquids, particularly when 478 NATURAL PHILOSOPHY. vapors distinguished from mists and clouds; gases distinguished from vapors; atmosphere; its composition mechanical use of oxygen; proof of the existence of vapors and gases atmospheric resistance; used as a moter. confined in closed vessels, as in the instance of steam in boilers. Yapors are generally invisible, and must not be confounded with the mists and clouds which are often seen suspended above the surface of the earth, and which are nothing more than water, in the form of small vesicles filled with air, and supported by the buoyant action of the atmosphere. Others of the compressible fluids are more permanent, requiring very great pressure and reduction of temperature to bring them to a liquid form. All such fluids are called gases. The most familiar instance of this class of bodies is the atmosphere which surrounds us on every side and in which we live. It envelops the entire earth, reaches far beyond the tops of our highest mountains, and . pervades every depth from which it is not excluded by the presence of solids or liquids. It is even found in the pores of these bodies. It plays a most important part in all natural phenomena, and is ever at work to influence the motions and to modify the results of machinery. It is essentially composed of oxygen and nitrogen, in a state of mechanical mixture. The former is a supporter of com- bustion, and, with the various forms of carbon, is one of the principal agents employed in the development of me- chanical power. The existence of air, gases, and vapors, is proved by a . multitude of facts. Contained in a flexible and imperme- able envelope, they resist pressure hke solid bodies. The gas in an inverted glass vessel plunged into water, will not yield its place to the liquid, unless some avenue of escape is provided for it. Those winds, hurricanes, and tornadoes which uproot trees, overturn houses, and devastate entire districts, are but air in motion. Air opposes, by its inertia, the motion of solid bodies through it, and this opposition is called its resistance. Finally, we know that wind is •employed as a moter to turn windmills and to give motion to ships of the largest kind. 257.—Many bodies take, successively, the solid, liquid, MECHANICS OF FLUIDS. 479 or vaporous state, according to the heat to which they are subjected. Water, for instance, is solid in the state of ice change of state; and snow, liquid in its ordinary condition, and vapor when heated in a closed vessel. The process by which a body passes from a solid to a liquid state, is called liquefaction or liquefaction; fusion; from a liquid to a state of vapor, vaporization or vaporization; volatilization ; that by which a vapor returns to a liquid, condensation; condensation ; and a liquid to a solid, solidification or congela- solidification. tion. Some bodies appear to take but two of these states, while others constantly present themselves only under one, which is the case with the infusible solids and permanent gases, including among the latter, the atmospheric air ; but the number of these bodies is constantly diminishing in the progress of physical science. § 258.—The subject of the mechanics of fluids, is usual- Mode of ly divided, as before remarked, into hydrostatics and hydro- ™™e™.nz dynamics, the former treating of the equilibrium of fluids, hydrostatics; and the latter of their motions; and not unfrequently the y compressible fluids are discussed under a separate head called pneumatics. In the present instance, these divisions pneumatics; will not be adhered to, as it is believed the whole subject may be presented in a manner more connected and per- spicuous by disregarding them. And in the discussions which are to follow, the fluid will be considered as with- out viscosity; that is to say, the particles will be supposed to have the utmost freedom of motion among each other. Such a fluid is said to be perfect. The results deduced perfect fluid. upon the hypothesis of perfect fluidity will, of course, require modification when applied to fluids possessing sensible viscosity. The nature and extent of these modi- fications can be known only from experiments. 480 NATURAL PHILOSOPHY. II. MECHANICAL PRINCIPLES OF FLUIDS. Level surface; § 258.—From the nature of a fluid, it is obvious that when a force is applied to any one of its particles, the latter must move in the direction of the force, unless prevented by the reaction of the surrounding particles; but these being equally free, can only react to prevent motion, by being supported or acted upon by opposing forces. From this arises a general law, viz.: that when a fluid is in when at rest; equiiibrio, its free surface is always normal to the re- sultants of the forces which solicit each of its surface particles. For if the result- ant OF of the forces which act upon any one of these Kg- 286. particles 0 were oblique to f*~- —-*f normal to the, the Surface A B, this result- resultant of the ant • ht be resolyed intQ forces which act ° upon the surface two components, one OF' par ices, normal, and the other OF" tangent to the surface; the former would be destroyed by the reaction of the fluid mass supposed in equiiibrio, while the latter would move the particle along the surface, and with the greater facility in proportion as similar components tend to move the particles to which they are applied in the same direction. Hence the supposition of an oblique resultant is inconsist- ent with the equilibrium. This free surface which every level surface fluid in equiiibrio presents in a direction normal to the resultant of the forces which act upon each of its surface particles, is called a level surface. Hence every heavy MECHANICS OF FLUIDS. 481 fluid upon the earth's surface in a state of repose, presents its upper or free surface normal to the direction of the level surface of force of gravity. If the earth did not rotate about an heavy fluids; axis PP', thus giving rise to a centrifugal force, every such surface would be a portion of the surface of a sphere, having its centre at the centre of the earth; but its cen- trifugal force M C, combined with the weight M G of each element, giving rise to a re- Fig. 287. sultant MN slightly oblique _____P™ to the direction of the weight, /^ 7\T\ pP. / / i' \ figure of the level every free surface is m strict- / d'-'^r \ surface of heavy ness a portion of the surface V J fluids; of a spheroid of revolution, ^v,_______^s^ flattened at the poles and p' protuberant at the equator. The great size of the earth, and the limited field that may be brought under observation at the same instant, will scarcely permit us however to distinguish any visible visible portions portion of fluid surface from a plane. Instance, the 8ensibly plaue; ponds, lakes, ocean. The same is true of the atmosphere. This fluid being elastic, its elements tend to recede from each other and from the earth's surface; in proportion as it expands, the repulsive action becomes less; the weight of the elements tends to draw them towards the earth; at the upper surface of the atmosphere these opposing case of the forces, which act towards and from the centre of the earth, atmosPnere- become equal, and the further retrocession of the particles is impossible. The atmosphere would, under the opera- tion of these causes alone, come to a state of rest, and present an exterior boundary similar to that of the earth. § 260.—Let the vessel ABD C contain a heavy fluid, or a fluid acted on only by its own weight; the upper surface RS will, from what we have seen, be horizontal when at rest; and it is obvious that this position of the surface will not be disturbed, or in the least altered, if the 482 NATURAL PHILOSOPHY. A homogeneous heavy fluid in vessels communicating freely will stand in all at the same level; experimental illustration; Fig. 288. portion of the fluid indicated by the shaded parts of the second figure were to become solid, leaving the fluid portions E T, F H, H G, communicating freely with each oth- er ; that is to say, the surfaces at E, F; and G, of the communi- cating fluid would be upon the same level. Whence we conclude, that a heavy fluid, as water or mercury, poured into several ves- sels which communi- cate freely with each other, will, when in equiiibrio, have its upper surface in all the vessels on the same level. This important fact is easily illustrated by experiment. J. is a vessel at the bottom of which is a horizon- tal tube connecting freely with the vessels B and C, and having a stop-cock D inter- posed, so that the con- nection may be inter- rupted or established at pleasure. Fill A with water, the stop- cock being closed. When the water in A is at rest, open the cock D; the water will descend in A and ascend in B and C till it comes to the same level in all. If the vessel C be broken off at E, the water will over- Fig. 289. MECHANICS OF FLUIDS. 483 remote points; flow at this point till it sinks in the vessels A and B to the level of M To the operation of this principle we are indebted for this principle the transfer of water from remote locations to artificial determines the transfer of water reservoirs for the supply of cities and towns. Springs to artificial also owe their existence to it. The greater part of the 1-eservoirs from solid crust of the earth consists of various strata ranged one above another; many of these are of a loose and po- rous nature and are penetrated with clefts, whilst others are more dense and free from flaws. Through the former of these, rains and melted snows find their way to Fig. 290. the latter, where their fur- ther progress is for a time checked, till the water accumulates in sufficient quantity to force its way through the sides of hills and mountains, and often at points of considerable elevation. When the harder and impervious strata form the outer crust of mountain ranges, they often force the water to take an obhque underground course through porous strata, that extend to considerable it is also the cause of springs; depth and reach to re- mote districts. Here, if a channel be provi- ded for the water by boring through the hard crust which confines it, it will spout forth or over- flow, in its effort to gain the level of its source in the distant mountain. This constitutes an Arte- sian well, a name derived Fig. 291. and is the cause of the discharge from Artesian wells. 484 NATURAL PHILOSOPHY. Principle of equal pressures; a heavy fluid; several vessels communicating; several columns of unequal weights supporting each other; from the French province Artois, where, according to account, they were first employed. § 261.—From the principle of fluid level, it is easy to pass to that of equal pres- sure. Suppose a vessel, A BDCFE, in which the branches EF and BDC have a free communica- tion with the part AB; then if water, mercury, wine, or any other fluid, be poured in either at E, A, or C, and the whole be suffered to come to rest, the surface at IK of the fluid in the part A B, at L in the branch EF, and at M in the branch BD C, will be upon the same level. Through the point N, taken at pleasure below the sur- face of the fluid, conceive a horizontal plane to be passed. It is obvious that the weight of the fluid contained in the vessel below PNQ can contribute nothing to the support of the columns LP, 10, and MQ, since this weight acts downward; and the equilibrium would obtain if the fluid contained in the part of the vessel below PNQ were without weight. This fluid may therefore be regarded as solely a means of communication between the columns LP, 10, and MQ, in such manner that it will transmit the pressure resulting from the weight of the columns L P and M Q to support the weight of 10, and recipro- cally. If now, instead of the columns LP, 10, and MQ of the fluid, pistons were applied to the surfaces at P, N 0, and Q, and were separately urged downward by pressures respectively equal to the weights of these columns, the MECHANICS OF FLUIDS. 485 equilibrium would manifestly obtain in like manner. Or if a pressure equal to that arising from the column M Q weight of be applied to the surface Q, while the columns LP and col™offluid ■•• *• **' replaced by 10 remain, the equiiibrio will still subsist, and this, pressures upon whatever be the directions and sinuosities at D, F, kc. pistons' The weight Wof the column Q Mis measured hy b.h.d.g; in which b is the area of the base at Q, h the height Q M, d the density of the fluid, and g the force of gravity. The weight W of the column 10 is measured by b'. h.d.g, in which b' is the area of the base N 0, the other quantities being the same as before. Dividing the latter by the former, we find ratio of the W __ b .h.d.g _ b_ (219 V weights of W b . h . d . Q b columns of equal altitudes; hence, the weights are to each other as the bases b' and b. Now these weights act in the same direction, and are unequal; they cannot, therefore, maintain each other in equiiibrio, unless the pressure arising from the column 10 were transmitted by the fluid down the vessel NB, up the sinuous vessel BD Q to Q, and there diminished in the ratio of the base NO to that at Q. In like manner, the pressure from the column M Q must be transmitted by the fluid down the tube Q D H, up the vessel BN to the base N 0, and there increased in the proportion of the base at Q to that at N That is, the forces applied to two pistons in a vessel filled forces on two with fluid, will be in equiiibrio when their intensities are di- J^^^ rectly proportional to the areas of the pistons to which they are proportional to ,. „ , 7 -\ 11 e j_i • i. tne areas of the respectively applied. If the areas b and b of the pistons pistons. become equal, the forces will be equal, and this, whatever be the actual dimensions of the pistons. Whence we con- clude, that the force impressed upon a fluid, is transmitted by it equally in all directions; and that every surface exposed to pressure , . -. . 7 • .7 ,-7 transmitted the fluid will receive a pressure which is directly proportional equally __,_, to its extent. Moreover, this pressure will be perpendicular directions; to the surface, for if it were oblique, it might be replaced 486 NATURAL PHILOSOPHY. normal to the surface; pressure always by its two components, one normal, the other parallel to the surface; the former would be destroyed by the resist- ance of the surface, while the latter would give motion to the fluid, which is contrary to the supposition that the fluid is in equiiibrio. From Eq. (219) we find value of the pressure transmitted; 1st. rule; pressure transmitted when the pressure is applied to a unit of surface; W' = w V (220): whence we have this rule for finding the amount of pres- sure transmitted to any surface, viz.: Multiply the intensity of the pressing force into the ratio obtained by dividing the area to which the pressure is transmitted, by that to which the force is directly applied. Making b = l, W will be the pressure upon the unit of surface, and Eq. (220) becomes W' = W.V (221); 2d. rule; Fig. 293. whence we have this second rule for finding the pressure transmitted to any given surface, viz.: Multiply the intensity of the force applied to the unit of surface by the area of the surface to which the pressure is transmitted. The truth of these deductions is finely illustrated by the Anatomical Siphon. A short cylindrical vessel A, made of metal, and open at one end, is connected with an upright glass tube fh, say half an inch in diame- uiustration by the ter, open at the top. The vessel is filled with water, and closed by tying over it a bladder, on which a plate of wood or metal is laid to receive weights W'. Water is now poured down the glass tube fh; the water in A, with its superincumbent weights W', will be raised by the pressure anatomical siphon; MECHANICS OF FLUIDS. 487 arising from the weight of that portion of the fluid in the glass tube above the level of the bladder. Let this difference of level be 50 inches, then will the volume, in illustrated by a cubic feet, of the pressing water, be numerical example; rn. in. if R2 X 50 3.1416 X (0.25)2 x 50 <•/;• 1728 '" 1728 * Now one cubic foot of water weighs sixty-two and a half pounds, whence the weight of the pressing column or W becomes, in pounds, weight of the pressing column; area of a section of the tube; lbs. lb. W = 62.5 x 0.00568 = 0.355. The area of a section of the glass tube is in. b = ifR2 = 3.1416 X (0.25)2 = 0.196; or, in square feet, h = 9^. = 0.00136, nearly. 144 ' J Let the diameter of the vessel A he one foot then will i^lTeLl,° V = 3.1416 X (0.50)2 = 0.7854; and these values of W, b, and b', substituted in Eq. (220), give 0 7854 lh8, W' = 0 355—'----- = 204.8, nearly; weight sustained: * 0.00136 that is to say, the triflmg weight of three tenths of a pound sustains in equiiibrio a weight of more than two hundred hydrostatic and four pounds; a result usually denominated the hydro- pu" static paradox. 488 NATURAL PHILOSOPHY. if the bladder If the bladder were removed, and the vessel extended ZwateTwouid upward to the line ed, on a level with the fluid in the rise in the larger tube, the water would rise in it to that height, when it would come to rest. The volume of the added water, in cubic feet would be 50 12 X 0.7854 = 3.272; verification. and allowing 62J pounds to each cubic foot, the weight of distilled water at 60° Fah. gives 3.272 X 62i = 204.5, nearly, as before. III. WORK OF THE POWER AND OF THE RESISTANCE. Fig. 294. § 262.—It follows from Eq. (220), that a given power Multiplication of may be multiplied at pleasure by this principle of equal power by the transmission of pressure. It will be sufficient for this principle of equal L transmission of purpose, to provide a strong pressure; i p ,-i ,. p vessel for the reception of a fluid, and to connect with it a pair of pistons whose sur- faces bear to each other any desired ratio; the power F being applied to the smaller piston b will be transmitted to the larger b' and made to hold in equiiibrio or over- come almost any given re- sistance R applied to the latter. B ut we are not, there- MECHANICS OF FLUIDS. 489 fore, to infer that there is any gain in the quantity of work no work gained, performed, for if we multiply Eq. (220) by the distance howevei"> HI = s' through which the larger piston may have been moved by the pressure transmitted to it, we have, by writing R for W', and F for W, Rs' = F ■ s'b' (222). work of the resistance; The product s' V, being the area of the larger piston into the distance HI, is the measure of the volume of fluid which has passed into the chamber CE, by the action of the power F upon the smaller piston; and if we regard the water as incompressible, this must be equal to the volume of fluid which has been pressed out of the chamber A B. Supposing the smaller piston to have been depressed to /', and denoting the distance H' I' by s, this latter volume will be measured by s b, and, therefore, from what has just been remarked, whence s'b' = sb; s'b' volumes of the fluid equal; which, substituted above, gives Rs' = Fs work of power (223). equal to that of resistance; The first member of this equation is the work performed by the resistance, the second that performed by the power, whence we conclude, that in hydraulic machines depending conclusion; upon the transmission of pressure, as in other machines, the work of the power is equal to that of the resistance. If the friction of the pistons against the sides of their friction and respective chambers and the viscosity of the fluid be taken V1" into the account, the work of these must be added to the 490 NATURAL PHILOSOPHY. the hydraulic machine enables a feeble power to perform what it could not without it; general principle of all machines; this principle employed to prove that of equal pressure; term Rs', which would make the effective quantity of work, measured by Rs', actually less than the work of the power. What then is gained? The answer is the same as before, viz.: the machine gives to a feeble power the ability to perform, by a succession of efforts, an amount of work which it could not accomplish by a single one. It would be quite within the physical capabilities of an individual to raise to the summit of a wall a ton of bricks, by taking a few bricks at a time, wnereas an effort to ele- vate the whole at once by his unassisted strength would prove an utter failure. And this is true of all kinds of machinery; whenever a given amount of work is accom- plished by the application of a diminished power, the space through which the latter is exerted must be proportionally increased. Had this principle, together with the incompressibility of the fluid, been assumed at the outset, it would have been an easy matter to deduce Eq. (220), and therefore the principle of the equal transmission of pressure; for, the volume of the fluid remaining the same, we should have s'b' = sb, and the quantity of work of the power and resistance being equal, gives Rs' = Fs; dividing the first of these equations by the second, we find R b_ whence pressures are proportional to the surfaces. F R b' that is to say, the pressures are directly proportional to the areas of the pistons to which they are applied, when MECHANICS OF FLUIDS. 491 there is an equilibrium, or when the pistons have a uniform motion. Fig. 295. description, and mode of applying the power; § 263.—One of the most interesting and important applications of the principle of equal transmission of pressure is exhibited by the Hydraulic or Bramah's Press. Hydraulic pre The main features of this machine are the following: A large and small metallic cylinder A and a, are made to commu- nicate freely with each other by a duct-pipe r. Wa- ter stands in both of the cylinders, and each is provi- ded with a strong piston. The pis- ton S of the larger cylinder carries a strong head-plate P, which works in a frame, so as to move directly towards or from a plate R which is stationary. The substance to be pressed is placed between these two plates. The piston in the small tube a is worked by a lever c d, of the second order, having its fulcrum at c, the piston-rod being attached at b, while power is applied at d. The pressure exerted by the smaller piston on the water is transmitted by the latter to the piston S. Let the diameter of the cylinder a be half an inch, that its power of the larger 200 inches, then will initiated by an o ' example; bT=^= 160000; o (hf and suppose the distance c d to be equal to 50 inches, and 492 NATURAL PHILOSOPHY. c b to be one inch, and let a man throw his weight, say data; 150 pounds, on the point d; then from the property of the lever will the force F, applied to the smaller piston, be given by the proportion in. lbs. 50 : : 150 : F, whence value of resistance; lbs. power applied at F = 150 X 50 = 7500. smaller piston; 7/ Substituting these values for F and -r- in Eq. (222), and omitting the common factor s', we find lbs. lbs. R = 7500 X 160000 = 1200000000; thus an effort equal in intensity to a weight of one hundred and fifty pounds applied at d, is capable of holding in equiiibrio a power, or of maintaining in uniform motion a body subjected to a constant resistance, equal to one billion two hundred million pounds. Dividing both members of Eq. (223) by F, we find path of the power R . s' at smaller piston; S == j=j j substituting the above values for R and F, and suppose the piston-head to have been raised through the distance of one foot, we have its numerical value for one foot 1200000000 , nnnn£ of path of the S = ----——----= 160000; resistance; / OuU and because the power applied at d must pass over 50 times this distance, we find 160000 x 50 = 8000000*; MECHANICS OF FLUIDS. 493 or 8000000 5280 - = 1515, path of the power; for the distance described by the power to compress the resistance one foot, or to raise a weight equivalent to the resistance through that height. The hydraulic press is used in the arts to press paper, cloth, hay, to uproot trees, to test the strength of ropes, chains, building materials, „«» of the and guns; and two were recently employed with success hydraulic p1 to raise, through a vertical height of more than one hundred feet, the great iron viaduct-tube, weighing up- ward of eighteen hundred tons, over the Menai Straits. IV. PRESSURE OF HEAVY FLUIDS. Fig. 296. O' §264.—Let us now examine the pressure which a Pressure of heavy heavy fluid exerts on the base of a vessel in which it is flulds' contained. For this purpose, let ABD C he a vessel containing a heavy fluid, as wa- ter, in equiiibrio. The upper surface A B of the fluid will be hori- zontal. Conceive a horizontal plane G H to be passed, and sup- pose the fluid below this plane, or that contained in the fluid beiow the portion G CDH to be devoid of weight; then it is «*h°rizontal x ' ° ' stratum devoid obvious, from our previous principles, that the weight of of weight; any slender vertical column, as EI, will exert a pressure 494 NATURAL PHILOSOPHY. each elementary column sustaining all the others; Fig. 296. at /, which is distributed equally in all directions through the fluid GCDH, and that this pressure acts equally upward to oppose the descent of the other columns which stand vertically over the plane GH; the col- umn EI alone keeps, therefore, in equi- iibrio all the other columns of the mass A G H B ; conse- quently, the mass G CD H, being still supposed without weight, there will result no pressure upon the base CD, except that which arises from the weight of a single fila- ment EI, which being transmitted equally to all the points of the base CD, the pressure on the latter will be given by Eq. (220); that is, by pressure upon the base; W' = W • %-; b weight of the pressing filament; in which W is the weight of the column EI, b the area of its base, V the area of the base CD, and W the pressure which it sustains. Denoting the height of the column EI by h, its weight Wwill be given by W = h.b.D.g; in which D denotes the density of the fluid, and g the force of gravity. Substituting this above for W, we find pressure upon the base; W = h . b'. D . (224). If now the plane GHhe depressed so as to leave all the heavy fluid above it, this plane will coincide with MECHANICS OF FLUIDS. 495 the bottom, / will come to I', and h will become the ver- tical height EI' of the surface of the fluid above the base. But the product Vh is obviously the volume CO 0'D of the fluid contained in a right cylinder or prism having for its base, the base of the vessel; D.b'. h is the mass of this cylinder or prism, and D . b'. h . g is its weight. Whence we conclude, that the pressure exerted by a heavy fluid upon the horizontal base of a vessel containing it, is equal to the weight of a column of this fluid, whose base is the base of the vessel, and whose altitude is equal to the depth of this base below the surface of the fluid. In this measure for the pressure on the base of a vessel containing a heavy fluid, there is nothing at all relating to the figure or actual volume of the vessel, and we are, hence, to infer that this pressure is wholly independent of both, and will always be the same whenever the area of the base and altitude of the fluid are the same. The right cyl- inder, inverted and erect truncated cones, having equal inferior bases B, B, B, and the same altitude h, will, when filled, con- tain very different volumes of fluid, yet the bases will all experience the same amount of pressure from the weight of the fluid, if it be the same in kind, or of the same density. The experimental verification of this weight of the column which measiu-es this pressure; pressure independent of figure of vessel and quantity of the pressing fluid; Fig. 297. illustration; Fig. 298. right cylinder, truncated cone, both erect and inverted; 496 NATURAL PHILOSOPHY. experimental verification of this fact; apparent paradox is easy. AD CB is a glass tube, of which the ends are open and bent upward; the end B is furnished with a brass ferrule upon which a screw is cut for the reception of a mate-screw H around the bottom of the ves- sels F, F', and F", also open at both description of the ends. On the end A is a sliding ring of metal or wood. At E is a short wire that may be moved up and down, and is held in Fig. 298. apparatus for the purpose; details of the experiment; deductions; any desired position by friction. Pour mercury in either end of the bent tube till it rises to any desired level, say that of the dotted line;' next, screw either of the vessels, say F, on its place at B, and fill it with water. The water passing freely through to the surface of the mercury will press upon the latter by its weight and force it up the end A. When both fluids come to rest, move the ring on the end J. to a level with the mercury to mark its place, and press the wire E down to the surface of the water to determine its height. Now draw off the water by the stop-cock 67, remove the vessel F and replace it by F', and fill with water as before; when the level of the water reaches the end of the wire E, the mercury will be found to have reached the ring on the end A. The experiment being repeated with the slender vessel F", not even half as thick as the tube A D CB, the mer- cury will again be found at the ring. In all these ex- periments, the base pressed is the same, being a section of the bent tube at the level of the mercury; and the altitude is the same, being the difference of level of the mercury in the end B and lower extremity of the wire E, MECHANICS OF FLUIDS. 497 Fig. 299. A e 7i K Ik ---/n *\ ~7M _j when the mercury in the end A stands at the level of the ring. The quantities of water employed in the three cases conclusion. are very different, and yet the pressures exerted by their weights are the same. § 265.—The pressure of a heavy fluid upon a horizon- Pressure of a tal plane, enables us to pass to that on a plane inclined heavy fluid,. *■ r against inclined under any angle whatever to the horizon, and thence to surfaces; the pressure on a curved surface. Let ABD C he a vessel with plane or curved sides, and filled with a heavy fluid; suppose G H and 67' H' to be two horizontal planes in- definitely near each other. The layer of fluid between these planes may be consid- ered as without weight, and as transmitting the pressure of the superincumbent fluid to the surface of the vessel with which this layer is in contact; and the pressure upon this surface will be the same as though it. were in either of the two planes in question. Designating the ex- tent of this elementary surface by V, and the depth EI by h', the measure of this pressure will be pressure upon an D . g . 0 . fl ; elementary inclined surface; in which D and g denote respectively the density of the fluid and force of gravity. In like manner, the pressure upon any other elementary portions b", b'", b"", kc, of the surface at distances h", h'", and h"", kc, respectively, below the upper surface of the fluid, will be D . g . b" . h", D . g . V" . h"', kc.; similar pressures; and the pressure upon the entire surface will obviously be the sum of these; or, if the total pressure be denoted by 498 NATURAL PHILOSOPHY. total pressure upon the entire surface; P, then will P = Dg(b'h' 4- b"h" 4- b'"h'" A kc). But if we take the upper surface of the fluid as a plane of reference, and denote by b the entire area of which V, b", kc, are the elements, and of which the distance of the centre of gravity from this plane of reference is h, then, from the principle of the centre of gravity, will bh = b'h' 4- b"h" 4- b"'h'" + kc; which, substituted above, gives value of this pressure in weight; expressed in words; example first; P = D.g .b.h. (225); that is to say, the pressure exerted by a heavy fluid against the surface of any vessel in which it is contained, is measured by the weight of a column of the fluid having for its base the surface pressed, and for its altitude the depth of the centre of gravity of this surface below the upper level of the fluid. Example 1st. Eequired the pressure against the inner surface of a cubical vessel filled with water, one of its faces being horizontal. Call the edge of the cube a, the area of each face will be a2, the distance of the centre of gravity of each vertical face below the upper surface will be ^ a, and that of the lower face a ; whence, the principle of the centre of gravity gives, Fig. 300. \ |\ \ \ distance of centre of gravity below the surface; h = _ 4 a2 X \ a 4- a2 X a Again, o a* = fa surface pressed; b = 5 a2; MECHANICS OF FLUIDS. 499 and these, substituted in Eq. (225), give P = D.g.b.h = D.g .So?. Now Dgxl3 = Dg, is the weight of a cubic foot of water = 62.5 lbs. whence value of the pressure; P = 62.5 X 3 a3. Make a = 7 feet, then will in pounds; P = 62.5 X 3 X (7)3 = 27562.5. its numerical value The weight of the water in the vessel is 62.5 a3, yet the pressure is 62.5 x 3 a3, whence we see that the outward pressure to break the vessel, is three times the weight of conclusion; the fluid. Example 2d. Let the vessel be a sphere filled with mercury, and let its radius be R. Its centre of grav- ity is at the centre, and therefore below the upper surface at the dis- tance R. The surface of the sphere being equal to that of four of its great circles, we have whence and, Eq. (225), b = 4:ifR2; b.h = 4*ri235 Fig. 301. example second; surface pressed; volume whose weight is equal to the pressure; P - ±« .D.g.R*. whole pressure; The quantity Dg X V = Dg, is the weight of a cubic foot 500 NATURAL PHILOSOPHY. pressure in pounds; its numerical value; of mercury = 843.75 lbs., and therefore, substituting the value of if = 3.1416, lbs. P = 4 x 3.1416 X 843.75 . Rz. Now suppose the radius of the sphere to be two feet, then willf?3 =8, and lbs. lbs- P = 4 x 3.1416 X 843.75 X 8 = 84822.4. The volume of the sphere is %«RZ; and the weight of the contained mercury will therefore be £ if Rzg D = W. Di- viding the whole pressure by this, we find ratio of weight of pressing fluid to pressure; P - 3- W ~ d' whence the outward pressure is three times the weight of the fluid. example third; Example Sd. Let the vessel be a cylinder, of which the radius r of the base is 2, and altitude I, 6 feet. Then will b.h = ifrl(r 4- J) = 3.1416 X 2 X 6 X 8; value of pressure; weight of pressing fluid; ratio of weight to pressure. which, substituted in Eq. (225), P = 3*01.5936 X Dg, and W = 3.1416 X 22 X 6 X Dg = 75.398 X Dg; whence, P_ _ 301.5936 x Dg W == 75.3984. Dg == ' MECHANICS OF FLUIDS. 501 Fig. 302. that is, the pressure against the vessel is four times the weight of the fluid. § 266.—Although the pressure of a heavy fluid de- centre of pends upon the position of the centre of gravity of the pres surface pressed, yet the resultant of all the elementary pressures passes through a different point, the position of which for a plane surface may be thus found. Let EIF he any plane, and MN the intersection of this plane produced with the upper surface of the fluid which presses against it. Denote the area of any elementary portion n of the plane EIF by V; and let m be the pro- jection of its place upon the upper surface of the fluid; draw mM perpendicular to MN, and join n with M by the right line n M, the latter will also be perpendicular to MN, and the angle n Mm will measure the inclination of the plane EIF to the sur- face of the fluid. Denote this angle by 9, the distance m n by h', and Mn by r'; then will hi = r' sin 9. The pressure of the fluid upon the element n will, Eq. (225), be D .g .V .h' = Dgb'r'sincp; and its moment, in reference to the line MN as an axis,' geometrical representation and notation; distance of an elementary pressed surface below the fluid surface; pressure upon this element; Dgb'r' sin 9; its moment; and for any other elements of which b", V", kc, denote the 502 NATURAL PHILOSOPHY. areas, we have, in like manner, Dgb"r"2 sin y, moments of the elementary pressures: D g V" r'" sin 9, &C, &C. Denoting by h the depth of the centre of gravity of the area EIF below the surface of the fluid, and by r the dis- tance of that point from the line MN, we shall have depth of centre of gravity of the 7 whole area h = r Sm

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Pine, American - - -Deal, white - - - - 3 4-5ths ii 1.1 5.7 4 1 .i3 .o3 4 2.2 4i i3 9 8 7 5 5 4 5 4i 3i 6 6 6 Rennie Tredgold Guyton Barlow 3i i.4 1.6 2.4 2.7 2.8 3.7 4 4 5 .56 .8 i 1.4 .22 A'7 •57 .73 .86 Rennie • In the result just obtained for the value of t, no atten- tion has been paid to the pressure upon the ends of the boiler or pipe, but these are usually made thick enough to throw the chances of breaking altogether upon the cylin- drical portion of the surface. 510 NATURAL PHILOSOPHY. V. EQUILIBRIUM OF FLOATING BODIES. Equilibrium of floating bodies; a body wholly immersed in a fluid; first result; Fig. 305. § 268.—The rules for finding the pressure against the sides of vessels are equally applicable to the determination of the pressure on the surfaces of bodies, however sub- jected to the action of a homogeneous heavy fluid. But when it is the question to ascertain the circumstances that determine a heavy body to be in equiiibrio or in motion, when immersed in a heavy fluid, it is usual to employ the results deduced from the following considerations. Suppose a vessel A to contain any heavy fluid in a state of rest. All parts of the fluid being in equi- iibrio, it is obvious that this state will in no respect be altered by sup- posing any portion B to become solid without changing its density. This solid is entirely immersed in the fluid, with which it has the same density, and is in equiiibrio. Now this solid is urged downward by its weight, which passes through its centre of gravity. This weight can only be in equiiibrio with a single force when the latter is directed vertically upward through the centre of gravity of the body, which centre coincides with that of the fluid converted into a solid, or that of the displaced fluid. But the only forces that act upon the solid besides its weight, are the pressures of the surrounding fluid; whence we conclude that 1st. The pressures upon the surface of a body entirely im- mersed in a fluid, have a single resultant, and that this result- ant is directed vertically upward. MECHANICS OF FLUIDS. 511 2d. The resultant of all the pressures is equal, in intensity, second result; to the weight of the displaced fluid. 3d. The line of direction of the resultant, passes through third result; the centre of gravity of the displaced fluid. 4th. The horizontal pressures destroy each other. fourth result; Again, if without altering the volume of this solid, we give it an additional quantity of matter, it is obvious that the weight of this latter will cause it to descend, that is, sink to the bottom of the vessel. Or if, without altering its volume, we conceive a portion of matter taken from its interior, the equilibrium will again be destroyed, the weight of the solid will be diminished by that of the subducted matter, the resultant of the pressures will prevail, and the body will rise to the surface, through which it will con- tinue to ascend, till the weight of the fluid displaced by the part immersed, is equal to that of the entire body. In the first case, the density of the body will be in- creased, containing a greater quantity of matter under the an immersed same volume, and in the second the density will be dimin- J^J ^cording0' ished; and as the density of the original body was the as its density is same as that of the fluid, we see that when the density of than that of the an immersed body is greater than that of the fluid, it will sink fluid; to the bottom; when less, it will rise to the surface, and float. It follows, also, from what has been said above, that when a body is immersed in a fluid, it will lose a portion of its weight equal to that of the displaced fluid. This is beau- tifully illustrated by what is usually called the " cylin- der and bucket" experiment. Place a hollow cyhnder a, in one of the scales of a balance; suspend to this scale a second cylinder b, of solid metal, exactly fitting the former, and in the oppo- site scale put a weight c, that Fig. 306. the body will lose a portion of its weight equal to that of the displaced fluid; 512 NATURAL PHILOSOPHY. cylinder and bucket experiment; weight of the immersed solid transmitted to the vessel; experimental proof; this principle used to find weight of ships, &c.: shall restore the equilibrium Fig. 306. of the balance. Now im- merse the cylinder b in a vessel W of water, the scale of the weight c will de- scend; fill the cylinder a with water taken from the vessel W, the beam of the balance will return to its horizontal position. The weight lost by the solid is transmitted through the fluid to the vessel, in the same w#y that the weight of a person in bed is transmitted through the latter to the bed- stead, and thence to the floor. This is proved, experiment- ally, thus: Place a tumbler of water in one of the scales A of a balance, bring the beam to a horizontal position by means of the empty hollow cylinder a of the last experi- ment and a weight c; sus- pend the solid cylinder b by means of a thread from a detached ring R, and depress it till it is wholly immersed into the water of the tum- bler; the scale A will fall; fill the cylinder a with water of the same temperature and density as that in the tum- bler; the equilibrium will be restored. This important principle, which determines the circumstances under which a body will rest upon a fluid, is frequently employed to ascertain the weights of large floating masses, such as ships, boats, and the like, which are entirely beyond the capacity of our ordinary weighing machines. For this purpose the vol- ume, in cubic feet, of the immersed part is computed from Fig. 263. MECHANICS OF FLUIDS. 513 the known figure and dimensions of the body, and this is weight of a ship's multiplied by the known weight of a cubic foot of water, caTg0' which is 62.5 pounds avoirdupois; the product is the weight of the floating body, in pounds. By taking in this way the difference of weights of a ship, with and without her cargo, the weight of the latter may be ascertained. The upward action by which an immersed body appa- rently loses a portion of its weight, is called the buoyant buoyant effort of effort of the fluid; and as the line of direction of this effort a m ' passes through the centre of gravity of the displaced fluid, this point is called the centre of buoyancy. The vertical centre of line through the centre of buoyancy, is called the line of uoyancy' support. The weight of a body acting at its centre of grav- line of support; ity downward, and the buoyant effort at the centre of buoyancy upward, the body can only be in equiiibrio when the line joining these centres is vertical, for it is only then that the forces are directly opposed. When the line joining the centre of buoyancy and the centre of gravity of the floating body is vertical, it is called the line of rest, lineofrest; When the equilibrium exists, it may be stable, unstable, stable, unstable, tp i -i -i i i -ii i and indifferent or indifferent. If stable, the body will not overturn wnen equilibrium. careened; if unstable, it will; if indifferent, the body will retain any position in which it may be placed. Let MQN repre- sent a section of any body, as a boat at rest upon the water, of which the upper surface is AB, called the plane of floatation. When this plane is produced through the boat, it will divide her into two partial vol- umes, the lower of which being supposed for an instant to consist of water, would weigh as much as the entire boat and her load, and Fig. 308. plane of floatation; 514 NATURAL PHILOSOPHY. series of cutting planes; this whatever be her position, whether careened or erect. Whence it follows, that if a series of planes M' N, M" N", kc, he passed, making the volumes M'QN, M" QN", kc, respectively equal to MQN, these planes will, each in its turn, come to coincide with the plane of floatation, whenever the boat, in the process of careening, takes a suitable position. But these planes may be regarded as so many tangent planes to a curved surface a be, which may be conceived as invariably connected with the boat. Now the effect, as regards the careening motion, will be the same as though this surface were the boundary of a physical axis which is made boat; to roll back and forth osculations of the on the plane of float- ation, regarded as a physical surface, after the manner of the pendulum axis on its supporting plane, during an oscillation. When the boat has a position of equi- librium, the line of Fig. 308. position of the line of rest during the equilibrium; support and of rest coincide, and are normal to this surface at its lowest point c. As the boat careens, the line of support, being always vertical, will still be normal to this axis surface at its lowest point, being that in which it is tangent to the plane of floatation; hence each of these normal lines must in turn become a line of support. If two normals a 0 and a' 0, which lie in the same plane, be drawn at tangential points answering to two consecutive positions of the boat, these normals will intersect at some point 0, which point will, obviously, be the momentary centre of rotation, when the plane of floatation coincide.-; with M" N". When one of these normals coincides with the line of rest, the point 0 is called the metacentre, being the point of intersection of the line of rest, with an adjacent MECHANICS OF FLUIDS. 515 line of support. But we have seen that the equilibrium of defined; a heavy body which may turn about a fixed point, will be stable or unstable, according as the centre of gravity during a slight departure from a position of equilibrium is com- pelled by the connection to ascend or descend; and it is obvious that, in the present case, the centre of gravity will ascend or descend on making a slight derangement of the line joining the centres of buoyancy and of gravity from the line of rest, according as the centre of gravity is below or above the metacentre. Whence we see, that the equilib- rium will be stable when the centre of gravity is below the meta- the nature of the centre, unstable when the relative positions of these points are determined b reversed, and indifferent when these centres coincide, for then a the relative slight derangement will cause no motion in the centre of gravity. centre8. It is also obvious that the stability of the equilibrium will be the greater, in proportion as the centre of gravity of the floating body be at a greater distance below the centre of buoyancy. It is for this reason that ships sent to sea object of ship- without cargoes are provided with ballast of stone, sand, or other heavy mat- ter, to diminish the chances of upsetting. ___________^__'_ The buoyant effort of water is used to great advantage in raising heavy sunken masses. For this pur- pose it is usual to connect two or more boats A and B, by means of a substantial cross-beam; to fill them nearly full of water, that they may sink as low as possible, and while in this condition to attach the body to be raised to the cross-beam by means of a taught chain or a common mode rope, and then to pump the water from the boats; the ten- p^p"^1118 sion upon the chain will be equal to the weight of the water pumped from the boats. If it is the question to raise a sunken boat, one of the most effective means is to 516 NATURAL PHILOSOPHY. force empty and water-tight barrels between her deck and hull. Level strata in heterogeneous fluids; mixture of different fluids having no affinity for each other: will form level strata; the most dense lowest; the same results from the properties of the centre of gravity g 269.—We have just seen that when a body is im- mersed in a fluid, it loses a portion of its weight equal to that of the displaced fluid, and that it will sink or rise to the surface, depending upon its relative density. This is universally true whatever be the size and number of the bodies immersed. If, therefore, one fluid be poured into another for which it has no affinity, as oil into water, it will sink to the bottom or rise to the surface and float, according as its density is greater or less than that of the fluid into which it is poured. The elements of the lighter fluid will act as so many immersed bodies till they reach the surface of the heavier fluid, where, being freed from the buoyant action of the latter, they will arrange themselves, under the efforts of their own weight, into a stratum of which the upper surface will, like that of the fluid below it, be perpendicular to the direction of the force of gravity. What is here said of two, is equally applicable to three, four, or any number of fluids of different densities mixed together; whence we conclude, that such fluids will come to rest only after arranging themselves into LEVEL STRATA in the order of their densities ; the most dense being at the bottom and the least dense at the top. This is confirmed by daily obser- vation, and may be easily illustrated by pouring mercury, water, and oil, into a common tumbler. The mercury will come to rest at the bottom, the oil at the top, the upper surfaces of all being level. The same conclusion follows from the consideration, that these fluids when mixed constitute a heavy system, which, we have seen, can only come to a state of stable equilibrium when its centre of gravity is at the lowest point, a condition only fulfilled by the arrangement, in respect to density, just described. If the elements of one fluid have an affinity for those of another, this affinity will, when the fluids come into con- MECHANICS OF FLUIDS. 517 tact, counteract the buoyant action of the heavier fluid, they wm not and the lighter will be held in a state of mixture. In- °bt»in/hen the ° fluids have an stance wine and water, water and alcohol, brandy and aninity for each water, and the like. VI. SPECIFIC GRAVITY. § 270.—The specific gravity of a body, is the weight specific gravity of so much of the body, as would be contained under a e ne ' unit of volume. It is measured by the quotient arising from dividing the weight of the body by the weight of an equal volume of some other substance, assumed as a standard; for the its measure; ratio of the weights of equal volumes of two bodies being always the same, if the unit of volume of each be taken, and one of the bodies become the standard, its weight will become the unit of weight. The term density denotes the degree of proximity density; among the particles of a body. Thus, of two bodies, that will have the greater density which contains, under an equal volume, the greater number of particles. The force of gravity acts, within moderate hmits, equally upon all illustration; elements of matter. The weight of a substance is, there- fore, directly proportional to its density, and the ratio of the weights of equal volumes of two bodies is equal to the ratio of their densities. Denote the weight of the first by W, its density by D, its volume by Y, and the force of gravity by g, then will .p. T;r measure for the 11 = g . JJ . V ; weight of a body; and denoting the like elements of the other body by IF, 518 NATURAL PHILOSOPHY. weight of a second body -D„ and Y„ we have W, =g.D,.Y,. Dividing the first by the second, ratio of the W gDY DY weights; W, gD,Y, ' D,Y: and making the volumes equal, came when the volumes are equal; W D W, D,..... specific gravity; specific gravity and density expressed by same numbers for same standard. Choice of a standard; (227). Now suppose the body whose weight is W, to be assumed as the standard both for specific gravity and density^ then will D, he unity, and S = W_ W. = D . . (228); in which S denotes the specific gravity of the body whose density is D; and from which, we see, that when specific gravities and densities are referred to the same substance as a standard, the numbers which express the one will also express the other. § 271.—Bodies present themselves under every variety of condition—gaseous, liquid, and solid; and in every kind of shape and of all sizes. The determination of their specific gravity, in every instance, depends upon our ability to find the weight of an equal volume of the standard. When a solid is immersed in a fluid, it loses a portion of its weight equal to that of the displaced fluid. The volume of the body and that of the displaced fluid are equal. Hence the weight of the body in vacuo, divided by its loss of weight when immersed, will give the ratio of the weights of equal MECHANICS OF FLUIDS. 519 volumes of the body and fluid; and if the latter be taken as the standard, and the loss of weight occupies the de- nominator, this ratio becomes the measure of the specific gravity of the body immersed. For this reason, and in view of the consideration that it may be obtained pure at all times and places, water is assumed as the general stand- water assumed as -i o • r% •,• -in ... p hit the standard for ard oi specific gravities and densities for all bodies. g eciflc .avities Sometimes the gases and vapors are referred to atmo-and density; spheric air, but the specific gravity of the latter being known as referred to water, it is very easy, as we shall gases sometimes presently see, to pass from the numbers which relate to atmospheric air. one standard to those that refer to the other. § 272.—But water, hke all other substances, changes its varying density density with its temperature, and, in consequence, is not0 v>aei' an invariable standard. It is hence necessary either to employ it at a constant temperature, or to have the means of reducing the specific gravities, as determined by it at different temperatures, to what they would have been if taken at a fixed or standard temperature. The former is generally impracticable; the latter is easy. Let D denote the density of any solid, and S its specific reduction to a gravity, as determined at a standard temperature corre- J^]J^ sponding to which the density of the water is D,. Then, Eq. (227), ™ __ J-* specific gravity at P) ' one temperature; Again, if S' denote the specific gravity of the same body, as indicated by the water when at a temperature different from the standard, and corresponding to which it has a density D,„ then will D same at another d;; O — -^i—. temperature Dividing the first of these equations by the second, we 520 NATURAL PHILOSOPHY. have ratio of these specific gravities; S _ D» S' whence D.1 S = S' D, ■ (229); specific gravity reduced to a standard; expressed In words; density of water at different temperatures; and if the density D, be taken as unity, S = S' • D. (230). That is to say, the specific gravity of a body as determined at the standard temperature of the water, is equal to its specific gravity determined at any other temperature, multiplied by the density of the water corresponding to this temperature, the density at the standard temperature being regarded as unity. To make this rule practicable, it becomes necessary to find the relative densities of water at different temperatures. For this purpose, take any metal, say silver, that easily resists the chemical action of water, and whose rate of ex- pansion for each degree of Fahr. thermometer is accurately known from experiment; give it the form of a slender cylinder, that it may readily conform to the temperature of the water when immersed. Let the length of the cylin- der at the temperature of 32° Fah. be denoted by I, and the radius of its base by ml; its volume at this tem- perature will be, volume of a slender cylinder; itm2l2 X I = ifm2P. Let nl he the amount of expansion in length for each degree of the thermometer above 32°. Then, for a tem- perature denoted by t, will the whole expansion in length be its expansion; nl x (t - 32°), MECHANICS OF FLUIDS. 521 and the entire length of the cylinder will become I + n I (t - 32°) = I [1 + n (t - 32°)] ; which, substituted for I in the first expression, will give the volume for the temperature t equal to ifm2P [1 + n(t - 320)]3. its increased length; its increased volume; Fig. 310. experimental determination of the density of water at different temperatures; The cylinder is now weighed in vacuo and in the water, at dif- ferent temperatures, varying from 32° upward, through any desirable range, say to one hundred de- grees. The temper- ature at each pro- cess being substituted above, gives the vol- ume of the displaced fluid; the weight of the displaced fluid is known from the loss of weight of the cyl- inder. Dividing this weight by the vol- ume, gives the weight of the unit of volume of the water at the temperature t. It was found by Stampfer, that the weight of the unit of volume is greatest when the greatest density temperature is 38.75 Fahrenheit's scale. Taking the den-at sityx)f water at this temperature as unity, and dividing the weight of the unit of volume at each of the other tem- peratures by the weight of the unit of volume at this, o 38.75, the following table will result:— 522 NATURAL PHILOSOPHY. TABLE of the Densities and Volumes of Water at different Degrees of Heat, (according to Stampfer,) for every 2J Degrees of Fahrenheit's Scale. (Jahrbuch des Polytechnischcn Institutes in Wein, Bd. 16. S. 70.) t Di, Diff. V Diff. Temperature. Density. Volume. o 32. OO 0.999887 1.000113 34.25 0.99995o 63 1.oooo5o 63 36.5o 0.999988 38 1.000012 38 38.75 1.000000 12 1.000000 12 4i .oo 0.999988 12 1.000012 12 43.25 0.999952 35 1.000047 35 45.5o 0.999894 58 1.000106 59 47-75 0.999813 81 1.000187 81 5o.oo 0.999711 102 1.000289 102 52.25 0.999587 124 1.000413 124 54.5o 0.999442 i45 1.ooo558 i45 56.75 0.999278 164 1.000723 i65 , 59.00 0.999095 183 1.000906 183 61.25 0.998893 202 1.001108 202 63.5o 0.998673 220 1.001329 221 65.75 0.998435 238 1.001567 238 68.00 0.998180 255 1.001822 255 70.25 0.997909 271 1.002095 273 72.50 0.997622 287 1.002384 289 74-75 0.997320 302 1.002687 3o3 77.00 0.997003 3x7 1.oo3oo5 3i8 79.25 0.996673 33o 1.oo3338 333 81.5o 0.996329 344 i.oo3685 347 83.75 0.995971 358 1.oo4o45 36o 86.00 0.995601 370 1.004418 373 88.25 0.995219 382 1.004804 386 90.50 0.994825 394 I.005202 398 92.75 0.994420 4o5 1.oo56i2 410 95.00 0.994004 416 1.oo6o32 420 97.25 0.993579 425 1.006462 43o 99.50 0.993145 434 1.006902 44o With this table it is easy to find the specific gravity by means of water at any temperature. Suppose, for example, the specific gravity S' in Eq. (230), had been found at the temperature of 59°, then would D„ in that equation, be 0.999095, and the specific gravity of the body referred to water at its greatest density, would be given by S = S' X 0.999095. MECHANICS OF FLUIDS. 523 The column under the head Y, will enable us to determine relation of how much the volume of any mass of water, at a tempera- voll,tnesofthe J -t same amount of ture t, exceeds that of the same mass at its maximum den- fluid at different sity. For this purpose, we have but to multiply the volume temPeratures- at the maximum density by the tabular number corre- sponding to the given temperature. Fig. 311. § 273.—Before proceeding to the practical methods of instruments wed finding the specific gravity of bodies, and to the variations t0 flnd the specific gravity of in the processes rendered necessary by the peculiarities of a body; the different substances, it will be necessary to give some idea of the best instruments employed for this purpose. These are the Hydrostatic Balance and Nicholson's Hy- drometer. The first is simi- lar in principle and form to the common balance. It is provi- ded with numerous weights, extending through a wide range, from a small fraction of a grain to several ounces. Attached to the under surface of one of the basins is a small hook, from which may be sus- pended any body by means of a thin plat- inum wire, horse-hair, or any other delicate thread that mode of attaching will neither absorb nor yield to the chemical action of the thebody; fluid in which it may be desirable to immerse it. Nicholson's Hydrometer consists of a hollow metallic ball Nicholson's A, through the centre of which passes a metallic wire, bydp prolonged in both directions beyond the surface, and. sup- porting at either end a basin B and B'. The concavities hydrostatic balance; 524 NATURAL PHILOSOPHY. description, and conditions the instrument must satisfy. of these basins are turned in the Fig- 812- same direction, and the basin B' is made so heavy that when the in- strument is placed in water the stem CC shall be vertical, and a weight of 500 grains being placed in the basin B, the whole instru- ment will sink till the upper surface of distilled water, at the standard temperature, comes to a point C marked on the upper stem near its middle. This instrument is pro- vided with weights similar to those of the Hydrostatic Balance. Process for finding specific gravity of a solid heavier than waterby the balance; § 274.—(1). If the body be solid, insoluble in water, and will sink in that fluid, attach it, by means of a hair, to the hook of the basin of the hydrostatic balance; counterpoise it by placing weights in the opposite scale; now immerse the body in water, and restore the equilibrium by placing weights in the basin above the body,.and note the tem- perature of the water. Divide the weights in the basin to which the body is not attached by those in the basin to which it is, and multiply the quotient by the density cor- responding to the temperature of the water, as given by the table; the result will be the specific gravity. Thus denote the specific gravity by S, the density of the water by D,„ the weight in the first case by W, and that in the scale above the solid by w, then will specific gravity; w (2). If the body be insoluble, but will not sink in water, as when the body is would be the case with most varieties of wood, wax, and water'tht tbe like' attach to it some body, as a metal, whose weight in the air and loss of weight in the water are previously MECHANICS OF FLUIDS. 525 found. Then proceed exactly as in the case before, to find the weights which will counterpoise the compound in air process and restore the equilibrium of the balance when it is im- esC11 e ' mersed in the water. From the weight of the compound in air, subtract that of the heavier body in air; from the loss of weight of the compound in water, subtract that of the heavier body; divide the first difference by the second, and multiply by the density of the water answering to its temperature, and the result will be the specific gravity of the hghter body. Example. grs. A piece of wax and copper in air = 438 = W A W', example; Lost on immersion in water - - = 95.8 = w A w', tbe case of wax; Copper in air......= 388 = W, Loss of copper in water - - - = 44.2 = w'. Then W + W - W' = 438 - 388 = 50 = IT; w + w' - w' = 95.8 - 44.2 = 51.6 = w. o Temperature of water 43.25, D„ = 0.999952, S = D„ X -^ = 0.999952 X ^ = 0.968. ^^ W .. 01.o (3). If the body readily dissolve in water, as many of the salts, sugar, &c, find its apparent specific gravity in some liquid in which it is insoluble, and multiply this apparent specific gravity by the density or specific gravity when the body of the liquid referred to water at its maximum density as J^^J a standard; the product will be the true specific gravity, fluid; If it be inconvenient to provide a liquid in which the solid is insoluble, saturate the water with the substance 526 NATURAL PHILOSOPHY. and find the apparent specific gravity with the water thus saturate the saturated. Multiply this apparent specific gravity by the bodTanZ^ceed density of the saturated fluid, and the product will be the as before; specific gravity referred to the standard. This is a com- mon method of finding the specific gravity of gunpowder, the water being saturated with nitre. (4). If the body be a liquid, select some solid that when the body is will resist its chemical action, as a massive piece of glass a liquid; suspended from fine platinum wire; weigh it in air, then in water, and finally in the liquid; the differences between the first weight and each of the latter, will give the weights of equal volumes of water and the liquid. Divide the weight of the liquid by that of the water, and the quotient will be the specific gravity of the liquid, pro- ruie; vided the temperature of water be at the standard. If the water have not the standard temperature, multiply this apparent specific gravity by the tabular density of the water corresponding to the actual temperature. example; Example. i water at sulphuric acid, 277.5 = w, grs. Loss of glass in water at 41°, 150 = w' 977 ^ specific gravity g = f±^ X 0.999988 = 1.85. of sulphuric acid; 150 (5.) If the body be a gas or vapor, provide a large glass flask-shaped vessel, weigh it when filled with the gas; when the body is withdraw the gas, which may be done by means to be ex- a gas or vapo., p]_aj_ne(j presently, fill with water, and weigh again ; finally, withdraw the water and exclude the air, and weigh again. This last weight subtracted from the first will give the weight of the gas that filled the vessel, and subtracted from the second will give the weight of an equal volume process; of water; divide the weight of the gas by that of the water, and multiply by the tabular density of the water MECHANICS OF FLUIDS. 527 answering to the actual temperature of the latter; the result will be the specific gravity of the gas. The atmosphere in which all these operations must be influence of the performed, varies at different times, even during the same atmosPhere > day, in respect to temperature, the weight of its column which presses upon the earth, and the quantity of moisture or aqueous vapor it contains. That is to say, its density depends upon the state of the thermometer, barometer, and hygrometer. On all these accounts corrections must temperature; be made, before the specific gravity of atmospheric air, or pressure; that of any gas exposed to its pressure, can be accurately determined. The principles according to which these cor- rections are made, will be discussed when we come to treat moisture; of the properties of elastic fluids. To find the specific gravity of a solid by means of Nicholson's Hydrometer, place the instrument in water, mode of using and add weights to the upper basin till it sinks to the mark hydr°m°ter for on the upper stem; remove the weights and place the solid soiids; in the upper basin, and add weights till the'hydrometer sinks to the same point; the difference between the first weights and those added with the body, will give the weight of the latter in air. Take the body from the upper basin, leaving the weights behind, and place it in the lower basin; add weights to the upper basin till the instrument sinks to the same point as before, the last added weights will be the weight of the water displaced by the body; divide the weight in air by the weight of the displaced water, and multiply the quotient by the tabular density of the water answering to its actual temperature; the result will be the specific gravity of the solid. To find the specific gravity of a fluid by this instru- ment, immerse it in water as before, and by weights in the also for fluids; upper basin sink it to the mark on the upper stem; add the weights in the basija to the weight of the instrument, the sum will be the weight of the displaced water. Place the instrument in the fluid whose specific gravity is to be found, and add weights in the upper basin till it sinks to 528 NATURAL PHILOSOPHY. the mark as before; add these weights to the weight of the instrument, the sum will be the weight of an equal vol- ume of the fluid; divide this weight by the weight of the water, and multiply by the tabular density corresponding to the temperature of the water, the result will be the spe- cific gravity. The scale areometer; description; the principle of this instrument: Fig. 313. § 275.—Besides the hydrometer of Nicholson, which requires the use of weights, there is another form of this instrument which is employed solely in the determination of the specific gravities of liquids, and its indications are given by means of a scale of equal parts. It is called the Scale-Areometer. It consists, gen- erally, of a glass vial-shaped vessel A, terminating at one end in a long slender neck C, to receive the scale, and at the other in a small globe B, filled with some heavy substance, as lead or mercury, to keep it upright when immersed in a fluid. The appli- cation and use of the scale depend upon this, that a body floating on the surface of different liquids, will sink deeper and deeper, in proportion as the density of the fluid approaches that of the body; for when the body is at rest its weight and that of the displaced fluid must be equal. Deno- ting the volume of the instrument by Y, that of the displaced fluid by Y', the density of the instrument by D, and that of the fluid by D', we must always have conditions of equilibrium; gYD = gY'D'; in which g denotes the force of gravity, the first member the weight of the instrument, and the second that of the MECHANICS OF FLUIDS. 529 displaced fluid. Dividing both members by D' Y, and omitting the common factor g, we have D_ Y_ D' ~ V' In which, if the densities be equal, the volumes must be equal; if the density D' of the fluid be greater than D, or that of the solid, the volume Y of the solid must be greater than V', or that of the displaced fluid; and in proportion as D' increases in respect to D, will Y' diminish in lespect to Y, that is, the solid will rise higher and higher out of the fluid in proportion as the density of the latter is in- creased, and the reverse. The neck C of the vessel should be of the same diameter throughout. To establish the scale, the instrument is placed in distilled water at the standard temperature, and when at rest the place of the construction of surface of the water on the neck is marked and numbered escale; 1; the instrument is then placed in some heavy solution of salt, whose specific gravity is accurately known by means of the Hydrostatic Balance, and when at rest the place on the neck of the fluid surface is again marked and characterized by its appropriate number. The same pro- cess being repeated for rectified alcohol, will give another point towards the opposite extreme of the scale, which may be completed by graduation. To use this instrument, it will be sufficient to immerse it in a fluid and take the number on the scale which coin- cides with the surface. To bring into view the circumstances which determine use; the sensibility both of the Scale-Areometer and Nicholson's Hydrometer, let s denote the specific gravity of the fluid, sensibility of the c the volume of the vial, I the length of the immersed IU9trumen ' portion of the narrow neck, r its semi-diameter, and w the total weight of the instrument. Then will *r2, denote the area of a section of the neck, and ifr2l, the volume of fluid displaced by the immersed part of the neck. The weight, 34 ratio of densities equal to that of the volumes; 530 NATURAL PHILOSOPHY. weight of fluid displaced therefore, of the whole fluid displaced by the vial and neck will be sc A s if r2!; but this must be equal to the weight of the instrument, whence condition of the equilibrium; w = s (c A «f31), from which we deduce specific gravity ; w c A «r*V length of neck j w — sc immersed; ' 2 if r* s (231). Now, immersing the instrument in a second fluid whose specific gravity is s', the neck will sink through a distance I', and from the last equation we have length immersed for second fluid; V = w — s c difference of specific gravity; inference; sensibility of Nicholson's hydrometer; subtracting this equation from that above and reducing, we find I - V = w k r2 &)■ The difference I— V is the distance between two points on the scale which indicates the difference s' — s of specific gravities, and this we see becomes longer, and the instru- ment more sensible, therefore, in proportion as w is made greater and r less. Whence we conclude that the Are- ometer is the more valuable in proportion as the vial por- tion is made larger and the neck smaller. If the specific gravity of the -fluid remain the same, which is the case with Nicholson's Hydrometer, and it becomes a question to know the effect of a small weight MECHANICS OF FLUIDS. 531 added to the instrument, denote this weight by w', then will Eq. (231) become it w A w' — sc> if r2 s ' subtracting from this Eq. (231), we find From which we see that the narrower the upper stem of Nicholson's instrument, the greater its sensibility. TABLE of the Specific Gravities of some of the most important Bodies. [The density of distilled water is reckoned in this Table at its maximum 38J° F.=1.000.] Name of the Body. Specific Gravity. L SOLID BODIES. (1) Metals. Antimony (of the laboratory) - - -Bronze for cannon, according to Lieut. Matzka Copper, molten...... Ditto, hammered..... Ditto, hammered..... Ditto, white ------Ditto for cannon, a mean - - - -Platinum, native..... Ditto, molten...... Ditto, hammered and wire-drawn Quicksilver, at 32° Fahr. - - - -Silver, pure molten..... Ditto, much hardened - - - -Tin, chemically pure - - - - -Ditto, Bohemian and Saxon - - - 4.2 — 4.7 j 7.6 — 8.8 8.414 _ 8.974 8.758 7.788 — 8.726 8.878 — 8.9 8.78 19.238 — 19.253 19.361 — 19.6 7.207 — 7.788 7.25i 7.2 7-5 7.21 — 7.3o n.33o3 u.388 16.0 — 18.94 20.855 21 . 25 i3i568 — 13.598 io.474 10.5i — 10.622 7.919 7.840 7.818 7.833 7,291 / K 7.299 — 7-475 7.3i2 532 NATURAL PHILOSOPHY. TABLE—continued. Name of the Body. Tin, English - - - - Zinc, molten - - - - Ditto, rolled - - - - (2) Building Stones. Alabaster..... Basalt..... Dolerite ----- - Oneiss..... Granite..... Hornblende - Limestone, various kinds - Phonolite..... Porphyry ..... Quartz..... Sandstone, various kinds, a mean Stones for building Syenite..... Trachyte..... Brick..... (3) Woods. Alder..... Ash...... Aspen ..... Birch..... Box...... Elm...... Fir...... Hornbeam - Horse-chestnut - - - - Larch..... Lime - Maple..... Oak...... Ditto, another specimen Pine, Pinus Abies Picea Ditto, Pinus Sylvestris Poplar (Italian) Willow..... Ditto, white - - - - (4) Various Solid Bodies. Charcoal, of cork Ditto, soft wood . - . Ditto, oak .... Coal...... Coke..... Earth, common - - - - rough sand - - - - rough earth, with gravel moist sand - gravelly soil day..... clay or loam, with gravel Specific Gravity. 7.291 6.861 7.191 — 7.2i5 2.7 - - 3.o 2.8 - - 3.i 2.72 — - 2.93 2.5 - 2-9 2.5 - 2.66 2.9 - - 3.i 2.64 - - 2.72 2.5l — - 2.69 2.4 - 2.6 2.56 - 2.75 2.2 — - 2.5 1.66 - 2.62 2.5 - 3. 2.4 - 2-6 1.41 - 1.86 Fresh-felled. Dry. 0.8571 o.5ooi o.oo36 0.644o 0.7654 o.43o2 0.9012 0.6274 0.9822 0.5907 0.9476 o.5474 0.8941 o.555o 0.9452 0.7695 0.8614 0.5749 0.9206 o.4735 0.8170 0.4390 0.9036 0.6592 1.0494 0.6777 1.0754 0.7075 0.8699 0.4716 0.9121 o.55o2 0.7634 0.3931 0.7155 0.5289 0.9859 0.4873 O.I 0.28 - o.44 i.573 1.232 - - i.5io 1.865 1.48 1.92 2.02 2.o5 2.07 2.l5 2.48 MECHANICS OF FLUIDS. TABLE—continued. Name of the Body. Flint, dark - Ditto, white - Gunpowder, loosely filled in ,' coarse powder musket ditto Ditto, slightly shaken down musket-powder ... Ditto, solid .... Ice - -_ - Lime, unslacked ... Resin, common - - - - Rock-salt - - - - - Saltpetre, melted ... Ditto, crystallized Slate-pencil - Sulphur..... Tallow..... Turpentine - - - - Wax, white - - - - Ditto, yellow - - - . Ditto, shoemaker's ... II. LIQUIDS. Acid, acetic - Ditto, muriatic - - - - Ditto, nitric, concentrated - Ditto, sulphuric, English Ditto, concentrated (Nordh.) Alcohol, free from water - Ditto, common - - - - Ammoniac, liquid - - - Aquafortis, double - - - Ditto, single ... - Beer...... Ether, acetic - - - - Ditto, muriatic - - - - Ditto, nitric - Ditto, sulphuric Oil, linseed .... Ditto, olive - Ditto, turpentine - - - Ditto, whale - - - - Quicksilver - Water, distilled - - - - Ditto, rain - Ditto, sea - - - . - Wine..... III. GASES. Atmospheric air = YTo = Carbonic acid gas - - - Carbonic oxide gas Carbureted hydrogen, a maximum Ditto, from coals Specific Gravity. 2.542 2.741 0.886 0.992 1.069 2.248 — 0.916 — 1.842 1.089 2.257 2.745 1.900 1.8 — 1.92 — 0.942 0.991 0.969 0.965 0.897 1 .o63 1.211 I .521 1.845 1.860 0.792 0.824 o.875 1.3oo 1.200 1.023 0.866 0.845 • 0.886 0.715 0.928 0.915 0.792 0.923 13.568 - 1.000 1.ooi3 1.0265 - 0.992 Water =1. Temp. 385° F. o.ooi3o 0.00198 0.00126 0.00127 0.00039 0.00085 2.563 0.9268 2.24 1.99 -- 1.522 — O.79 — 1.o34 — 0.874 — 0.953 — 0.891 — 13.598 - 1.028 • i.o38 Barometer 30 In. Temp. = 34°, 1.0000 l.524o 0.9569 O.9784 o.3ooo 0.5596 534 NATURAL PHILOSOPHY. TABLE— Continued. Name of the Body. Specific Gravity. Barometer Water = 1. 30 In. Temp. 38J° F. Temp. = 34°. 0.00321 2.4700 0.00577 4.443o 0.0000895 0.0688 o.ooi55 1.1912 0.00162 1.2474 0.00127 0.9760 o.ooi43 1.1026 o.oou3 0.8700 0.00082 0.6235 0.00292 2.2470 Chlorine - Hydriodic gas - Hydrogen ... Hydrosulphuric acid gas - Muriatic acid gas Nitrogen - - - - Oxygen ... - Phosphureted hydrogen gas Steam at 212° Fahr. - Sulphurous acid gas - of a table of The knowledge of the specific gravities or densities of s' different substances is of great importance, not only for scientific purposes, but also for its application to many of the useful arts. This knowledge enables us to solve such problems as the following, viz.:— 1st. The weight of any substance may be calculated, if its volume and specific gravity be known. 2d. The volume of any body may be deduced from its specific gravity and weight. Thus we have always weight of any body ; weight of a cubic foot of distilled water at maximum density; W = gD Y; in which g is the force of gravity, D the density, Y the volume, and W the weight, of which the unit of measure is the /weight of a unit of volume -of water at its maxi- mum density. Making D and Y equal to unity, this equation becomes W, = g; but if the density be one, the substance must be water at 38.75 Fahr. The weight of a cubic foot of water at 60° is-62.5 lbs., and, therefore, at 3&\75, it is lbs. 62.5 0.99914 lbs. = 62.556 MECHANICS OF FLUIDS. 535 whence, if the volume be expressed in cubic feet, volume in cubic feet; lbs. weight of a body W = 62.556 x D Y. . . (232), m pounds, volume being in cubic feet; in which W is expressed in pounds; and if the unit of volume be a cubic inch, weight in 62 556 pounds, W = ~^- D Y = 0.036201 D Y, . . (233). vomme in cubic 11 Ao Also inches; VV,. re n a \ volume in cubic Y = "' . • • (234), lbs. 62.556 . D W. lbs. 0.036201 .D Y, = -rr-^----- . • • (235). feet; volume in cubic inches; Example 1st. Eequired the weight of a block of dry example first; fir, containing 50 cubic inches. The specific gravity or density of dry fir is 0.555, and Y = 50; substituting these values in Eq. (233), lbs. -weight of 50 W = 0.036201 X 0.555 X 50 = 1.00457. cubic inches of fir; Example 2d. How many cubic inches are there in a example second; 12-pound cannon-ball? Here W is 12 pounds, the mean specific gravity of cast iron is 7.251, which, in Eq. (235), give 1 c) in. volume of a 12- T7- ___________________________= 45.6. pound cannon- ' "" 0.036201 X 7.251 baa 536 NATURAL PHILOSOPHY. VII. COMPRESSIBLE FLUIDS. Peculiarities of gases and vapors contract and expand according to pressure■ conditions of rest; no marked variety of fluidity; usually transparent; small density; § 276.—The properties of liquids which have now been considered, are common to all fluids. But gases and vapors have, in addition, properties peculiar to themselves which we now proceed to consider. Gases and vapors differ mostly from liquids, in the readiness with which they yield a portion of their volume and contract into smaller spaces when subjected to an augmentation of external pressure, and diffuse themselves in all directions when this pressure is withdrawn. These distinguishing properties are due to the repulsive forces or molecular springs by which the particles are urged to separate from each other, and which make it impossible for compressible fluids, that are also highly elastic, ever to be at rest, unless these forces are opposed by the reaction of inclosing surfaces, as the sides of vessels, or the apph- cation of some other antagonistic forces acting inwardly, as in the case of the earth's attraction upon our atmosphere. Besides these essential peculiarities, there are other characteristics that distinguish compressible fluids, usually denominated aeriform bodies, from the other forms of aggregation. Between solids and liquids, a gradation is observable, and in the degree of fluidity of the latter, a strongly marked variety obtains—as in tar, oil, water, ether, and the like; but between compressible and incom- pressible fluids, no similar connecting links are found. Again, as a general rule, gases are highly transparent, for most part colorless, and therefore invisible, and are distinguished from all other bodies by their small degree of density and consequent low specific gravity. MECHANICS OF FLUIDS. 537 Fig. 314. The atmosphere, as being the most important of the aeriform bodies, may be taken as the representative of the whole class, as regards their mechanical properties. It is to this class of bodies, what water is to liquids. It exists all over the earth, and its ever-active agency in the pro- duction of phenomena, makes it not less interesting than important to determine the laws of its equilibrium and motion. (1) The compressibility and elasticity are easily shown by inclosing the air in a bag of some impervious substance, as india-rubber, and pressing it with the hand; the hand will experience a resistance, while the volume of the con- fined air will diminish: on removing the hand, the bag will be distended by the elasticity of the air, and restored to its for- mer dimensions. Air-pillows and cushions, in common use, are famil- iar illustrations. (2) A is a two-necked bottle con- taining some liquid, as water, B an inflated bladder, or india-rubber bag, attached by the neck to one of the mouths. A glass tube a b, open at both ends, is fitted air-tight to the other mouth, its lower end a reaching nearly to the bottom of the bottle. On compressing with the hand, the air in the bladder or bag, the liquid will be seen to mount up the tube. (3) Hero's Ball—A hollow globe a, from which the external air can be excluded by turning a cock b, contains a tube that reaches nearly to the bottom, and fits in the neck by a screw. Fill the vessel about half full of water, screw in the tube c d, atmosphere the type of the class; compressibility and elasticity shown; india-rubber bag; india-rubber bag connected with a two-necked bottle; Fig. 315. Hero's 'oall; 538 NATURAL PHILOSOPHY. principle of the fire-engine; Fig. 316. breathe through c, and close the Fig. 315. stop-cock b; the breath will ascend through the water, mingle with the air in the space a, take from it a portion of its volume and thus in- crease its elasticity, which, reacting upon the surface of the water, will force the latter up the tube cd on turning the cock b. On this princi- ple depend the operations of the air-chamber in fire- engines and similar machines. (4) Hero's Fountain.—In this apparatus, also, the com- pression of air and consequent increase of elasticity, are manifested in producing a water-jet. Two vessels a and g are united by a tube t, open at both ends, extend- ing from the upper surface of the lower vessel to near the top of the Hero's fountain; other. A pipe c d, provided with a stop-cock b, screws into the top of the vessel a, and extends nearly to its bottom, as in Hero's Ball. Upon the top of this vessel is a basin n 0, from the bottom of which a pipe efi open at both ends, passes clear through, nearly to the bottom of the vessel g. The tube cd, being unscrewed, is removed, and after pouring water into the vessel a till its surface comes nearly to the up- per end of the tube t, the pipe c d is replaced, and the stop-cock b closed. Water is now poured into the basin no; this will descend through the tube ef into the vessel g, and expel a portion of its air by forcing it up the tube t into the vessel a ; there, finding no means of escape, it will be compressed, and its increased elasticity made to act upon the water description; mode of action; MECHANICS OF FLUIDS. 539 precisely as in the case of the Ball. The water will con- conditions of tinue to descend through the tube ef from the basin, tillie the increasing elasticity of the air becomes equal to the pressure arising from a head of water equal to the differ- ence between the level of the water in the basin and that in the lower vessel, when the flow will cease, and every thing will come to rest. In this condition of things turn the cock b, and the water will spout through the tube c d. The fluid in the upper vessel being thus ejected, there will conditions to be room for more air; this will pass from the lower vesselca through the tube t, and the water will again descend from the basin to the vessel g. The water discharged by the jet falls into the basin n o, and is ready, in its turn, to pass down the tube ef. A constant flow is thus maintained as long as the fluid in the vessel a remains above the bottom of the tube c d. (5) The Cartesian Devil.—This is a well-known figure, constructed so as to float in a glass vessel of water, above cartesian devil; the surface of which a portion of air is confined in such manner, that if this air be com- pressed, the figure will descend, and rise again when the compression ceases. It is thus contrived: In the middle of the figure a is a small capillary tube b, through which so much water is admitted into the in- terior of the body as to make its mean density a little less than that of the water in which it is to float. Being thus adjusted, the figure is immersed in a wide-mouthed glass vessel, over which a piece of blad- der or sheet of india-rubber is then stretched to confine the air over the fluid. The finger being now pressed upon the bladder or india-rubber, Fig. 317. the air will be compressed, the increased elasticity thus pro- fi= motion of the fiurure; 540 NATURAL PHILOSOPHY. duced will be exerted upon the water, which will be forced by it through the tube b, the mean density of the figure will be increased, and it will sink to the bottom; on re- expianation of moving the finger, the air above the water as well as that in the figure, being relieved from the pressure, expands, the water is forced back through the tube b into the vessel again, and the figure will rise to the surface in consequence of diminished mean density. VIII. THE AIR-PUMP. Air-pump, or § 277.—Seeing that the air expands and tends to diffuse au-synnge; itself in all directions when the surrounding pressure is lessened, it may be rarefied and brought to almost any de- gree of tenuity. This is accomplished by an instrument called the Air-Pump or Exhausting Syringe, one of the most important pieces of apparatus used by the natural philosopher. It will be best understood by describing one of the simplest kind. It consists, essentially, of 1st. A Receiver R, or chamber from which the exterior air is excluded, that the air within may be rarefied. This the receiver; is commonly a bell-shaped glass vessel, with ground edge, over which a small quantity of grease is smeared, that no air may pass through any remaining inequalities on its surface, and a ground glass plate m n imbedded in a metallic table, on which it stands. 2d. A Barrel B, or chamber into which the air in the receiver is to expand itself. It is a hollow cylinder of metal or glass, connected with the receiver R by the com- the barrel and munication ofg. An air-tight piston P is made to move back and forth in the barrel by means of the handle a. MECHANICS OF FLUIDS. 541 Fig. 318. graphical representation; Fig. 319. 3d. A Stop-cock h, by means of which the communica- tion between the barrel and receiver is established or cut off at pleasure. This cock is a conical piece of metal fitting stop-cock, or air-tight into an aperture just at the lower end of the bar- valve' rel, and is pierced in two directions; one of the perfora- tions runs transversely through, as shown in the first figure, and when in this position the communication between the barrel and receiver is established; the second perforation passes in the direction of the axis from the smaller end, and as it approaches the first, in- clines sideways, and runs out at right angles to it, as indicated in the second figure. In this position of the cock, the communication between the receiver and barrel is cut off, whilst that with the external air is opened. Now, suppose the piston at the bottom of the barrel, and the communication between the barrel and the receiver mode of action; established; draw the piston back, the air in the receiver will rush out, in the direction indicated by the arrow-head, through the communication ofg, into the vacant space within the barrel. The air which now occupies both the description of stop-cock; 542 NATURAL PHILOSOPHY. mode of operation barrel and receiver is less dense than when it occupied the receiver alone. Turn the cock a quarter round, the com- munication between the receiver and barrel is cut off, and that between the latter and the open air is established; push the piston to the bottom of the barrel again, the air within the barrel will be delivered into the external air. Turn the cock a quarter back, the communication between the barrel and receiver is restored; and the same operation as before being repeated, a certain quantity of air will be transferred from the receiver to the exterior space at each double stroke of the piston. to find the degree To find the degree of exhaustion after any number of of exhaustion; fo^fe strokes of the piston, denote by D the density of the air in the receiver before the operation begins, being the same as that of the external air; by r the capacity of the receiver, by b that of the barrel, and by p that of the pipe. At the beginning of the operation, the piston is at the bottom of the barrel, and the internal air occupies the re- ceiver and pipe; when the piston is -withdrawn to the opposite end of the barrel, this same air expands and occupies the receiver, pipe, and barrel; and as the density of the same body is inversely proportional to the space it occupies, we shall have ratio of the densities; + p A b : r A p D in which x denotes the density of the air after the piston is drawn back the first time. From this proportion, we find first diminished density; x - D r A p A p A b The cock being turned a quarter round, the piston pushed back to the bottom of the barrel, and the cock again turned to open the communication with the receiver, the operation is repeated upon the air whose density is x, and MECHANICS OF FLUIDS. 543 we have r A p A b : r + p :: D ■--------£—- : x'; ratio of densities; r A p A b in which x' is the density after the second backward mo- tion of the piston, or after the second double stroke; and we find / j\ I r "T P \ . second \7" _l_ 7) -\- bl ' diminished; and if n denote the number of double strokes of the piston, and xn the corresponding density of the remaining air, then will the nth diminish - p A bl density; / f. _J_ rp \n == J) . I________" ) m diminished \r A V A bl ' From which it is obvious, that although the density of the air will become less and less at every double stroke, yet it can never be reduced to nothing, however great n may be; in other words, the air cannot be wholly removed from the receiver by the air-pump. The exhaustion will go on rapidly in proportion as the barrel is large as com- pared with the receiver and pipe, and after a few double the air can never strokes, the rarefaction will be sufficient for all practical exh™u°ted from purposes. Suppose, for example, the receiver to contain the receiver; 19 units of volume, the pipe 1, and the barrel 10; then will r A p _ 20 _ 2# r A p~A b == 30 ~ *' and suppose 4 double strokes of the piston; then will illustration; n = 4, and / r A V Y /,s4 16 AmT 1 density after 4th ---r^-4-T = (£) = ftT = 0'197> nearl7; doublestroke; \r A p A pi ol 544 NATURAL PHILOSOPHY. rarefaction by best pumps; gauies; objects, and construction; Bcale of the gauge, and position; first inventor; improvements; that is, after 4 double strokes, the density of the remaining air will be but about two tenths of the original density. With the best machines, the air may be rarefied from four to six hundred times. The degree of rarefaction is indicated in a very simple manner by what are called gauges. These not only indicate the condition of the air in the receiver, but also warn the operator of any leakage that may take place either at the edge of the receiver or in the joints of the instrument. The mode in which the gauge acts, will be better understood when we come to discuss the barometer; it will be suffi- cient here simply to indicate its con- struction. In its more perfect form, it consists of a glass tube, about 60 inches long, bent in the middle till the straight portions are parallel to each other ; one end is closed and the branch termina- ting in this end is filled with mercury. A scale of equal parts is placed between the branches, having its zero at a point midway from the top to the bottom, the numbers of the scale increasing in both directions. It is placed so that the branches of the tube shall be vertical, with its ends upward, and inclosed in an inverted glass vessel, which communicates with the receiver of the air- pump. Eepeated attempts have been made to bring the air- pump to still higher degrees of perfection since the time of Otto von Guericke, burgomaster of Magdeburg, who first invented this machine in 1560. Self-acting valves, opening and shutting by the elastic force of the air, have been used instead of cocks. Two barrels have been given to the air-pump instead of one, so that an uninterrupted and more rapid rarefaction of the air is brought about, the piston in one barrel being made to ascend as that of the other des<_ ds. The most serious defect in the air- MECHANICS OF FLUIDS. 545 pump was, that the atmospheric air could not be entirely the most serious defect of the older pumps; ejected from the barrel, but remained between the piston de and the bottom of the barrel. This intervening space is filled with air of the ordinary density at each descent of the piston ; when the cock is turned, and the communica- tion re-established with the receiver, this portion of air forces its way in and diminishes the degree of rarefaction Fig. 321. Bection of one of the most approved pumps; already attained. If the air in the receiver is so far rare- fied, that one stroke of the piston will raise only such a quantity as equals the air contained in this space, it is plain that no further exhaustion can be effected by continuing to pump This limit to rarefaction will be arrived at the limits to Xr l rarefaction due to sooner, in proportion as the space below tne piston is thedefectabove. larger; whence one chief point in the improvements has 546 NATURAL PHILOSOPHY. been to diminish this space as much as possible. A Bis a highly polished cylinder of glass, which serves as the bar- description of the rel of the pump; within it the piston works perfectly air- improved pump; tight The pigton consists of washers of leather soaked in oil, or of cork covered with a leather cap, and tied together about the lower end C of the piston rod by means of two parallel metal plates. The piston-rod Cb, which is Fig. 321. toothed, is elevated and depressed by means of a cog-wheel that is turned by the handle M. If a thin film of oil be poured on the upper surface of the piston the friction will be lessened, and the whole will be rendered more perfectly air-tight. To diminish to the utmost the space between the bottom of the barrel and the piston-rod, the form of a truncated cone is given to the latter, so that its extremity use of oil; shape of the lower end of piston-rod; MECHANICS OF FLUIDS. 547 may be brought as nearly as possible into absolute contact with the cock E; this space is therefore rendered indefi- nitely small, the oozing of the oil down the barrel con- tributing still further to lessen it. The exchange-cock E exchange-cock; has the double bore already described, and is turned by a short lever, to which motion is communicated by the rod cd. The communication Gil is carried to the two plates communication; i" and K, on one or both of which receivers may be placed; the two cocks iVand 0 below these plates, serve to cut off cut-off cock the rarefied air within the receivers when it is desired to leave them for any length of time. The cock 0 is also an exchange-cock, so as to admit the external air into the cock to readmit the air; receivers. Pumps thus constructed have advantages over such as advantages of work with valves, in that they last longer, exhaust better, " ^ and may be employed as condensers when suitable receivers are provided, by merely reversing the operations of the exchange valve during the motion of the piston. g 278.—The following are some of the most interesting- experiments performed with the aid of an air-pump, show- ing the expansive force of the atmosphere, and also the relations between air of ordinary density and that which is highly rarefied:— 1st. Under a receiver place a bladder tied tightly about Experiments with the neck and partly filled with air; exhaust the air in the air pump' receiver, and that confined within the bladder will gradu- first experiment; ally distend, proving experimentally the expansive force of atmospheric air. When the air is readmitted into the receiver, the bladder will resume its former dimensions. An analogous appearance will be exhibited if a jar, over which some india-rubber has been tied, be placed beneath second; a receiver, and the air be then exhausted. 2d. The expansive force of our atmosphere is further shown if a long-necked flask, or retort, be inverted so that its mouth shall be below the surface of some water con- tained in a vessel, and the whole be placed under the 548 NATURAL PHILOSOPHY. receiver of an air-pump; when the Fig. 322. air within the receiver is rarefied, showing also the that which was contained in the expansion of air; -,-,-, v ,-1 -i bulb, expanding, escapes through the water; and on readmitting the atmosphere the water will rise and occupy the space vacated by the air. 3d. The transfer of a fluid from one flask to another. Let there be a fluid in the flask A. The neck of this flask contains a glass tube fitted air-tight into it, and reach- third, illustrating ing almost to the bottom; the tube being bent twice ^ple; at riS]lt anSIeS' tlie 0ther eild passes freely through the neck of a second bottle B. Place this appa- ratus under the receiver of an air- pump, and exhaust; the fluid will mount up from the bottle A and pass through the tube over into the bottle B. Eeadmit the air, the fluid will pass back again. 4th. Place Hero's ball under the receiver when half fourth. filled with water, and exhaust; the expansion of the air within will send the water up through the tube in a jet. Fig. 323. n * tc~ Ii yl D h =idiLgF Atmospheric resistance illustrated; Fig. 324 § 279.—When a piece of metal and a feather are aban- doned to their own weight in the air, they fall with very different velocities. The cause is the great disparity in the extent of surfaces exposed to the resistance of the air as compared with the weights. Let a and b he two wheels re- sembling the arms of a windmill, with this difference only, that the vanes of b shall strike the air with their broad faces, whilst those of a shall cut it edgewise; each has a separate axis on which it revolves. By means of a mechanical contri- vance a rapid rotary motion is com- MECHANICS OF FLUIDS. 549 municated to them. In order that this may act under a receiver, a rod must be made to pass through an air- description and tight leather stuffing-box e; at the end of the rod is a "^J „t curved arm d, which drives the wheels. If the rotation take place in vacuo, the two wheels a and b will cease to revolve simultaneously ; whereas, if the motion take place in the ordinary atmosphere, the resistance of the latter will bring b to a stand long before a ceases to turn. § 280.—The atmosphere is the ordinary medium through Effects of which sound is transmitted to the ear. In proportion as ^*c; 101 the air becomes more rarefied, the transmission of sound through it becomes more feeble. Under a receiver furnished with a leather stuffing-box, place a bell whose clapper may be struck by a rod passing through the box, taking care to place the bell on some soft unelastic substance, to prevent its communicating sound to the plate of the pump and thus to the external air. The annexed figure represents such an apparatus, which may, however, be considerably varied: a is the bell, b the clapper attached by a spring to a thin plate of wood c, into which the support of the bell is screwed ; g is a leather drum stuffed with horse-hair, fitting into the upper wooden plate c, and into a lower plate d, by which the whole apparatus is fastened down to the plate of the pump; lastly, h is the lever by which the clapper is agi- tated. After about 10 strokes of the piston, the sound becomes sensibly more feeble, and if the exhaustion be continued long enough it will cease altogether. Air is necessary to respiration. Place a bird beneath air is necessary to the receiver of an air-pump; a few strokes of the piston willres,ma '0I cause it to make convulsive struggles, and death will soon Fig. 325. instrument by which this may- be illustrated; 550 NATURAL PHILOSOPHY. ensue unless air be admitted. Warm- Fig- 326. place a bud in blooded animals, as birds, die if the receiver of a p ,- i • -i -11 pump and rarefaction be carried to a small exhaust; degree; cold-blooded animals, on the contrary, endure a high degree of rarefaction. Many birds ascend to considerable heights in the atmo- sphere, and it may be hence inferred . that the density of the air at these altitudes is greater than that in the exhausted receiver of an air-pump. Air is necessary to combustion. Introduce a taper into a bell-shaped receiver full of atmospheric air, and observe air is necessary to the time it will continue to burn. Light the taper ao-ain, place it beneath the receiver and exhaust quickly, after it has been replenished with fresh air; the flame will expire much sooner than before. To the same cause it is owing that in vacuo no light is produced by striking a flint and steel together. IX. WEIGHT AND PRESSURE OF THE ATMOSPHERE. § 281.—From the resistance which the atmosphere opposes to the motion of bodies through it, we might infer that it has weight as well as inertia. That it has The atmosphere weight is obvious from the fact that the atmosphere hoe TiroinrVtt • , J- meases, as it were, the whole earth: if it were destitute of weight and subjected only to the repulsive action among its own particles, it would recede further and further and extend itself throughout space. But the existence of weight in the atmosphere may be shown experimentally, thus:— has weight; MECHANICS OF FLUIDS. 551 experiment to show this; Take a flask of some two or three inches in diameter, having an air-tight stop-cock. Suspend it from one end of the balance-beam and as- certain its weight when filled with air. Exhaust the air, *%. 321. by means of the air-pump, and the flask will be found = lighter than before; readmit \ the air, it will regain its for- j\ mer weight. .Force'into the flask an additional quantity of air, by means of the air- pump, used as a condenser, and the weight will be found to be increased. Since the atmosphere has weight, it must exert a pressure upon all bodies in it. To illustrate the truth the air exerts a of this, fill with mercury a glass tube, about 32 or 33 fT'6 u*on f ' J D ' bodies within it inches long, and closed at one end by an iron stop-cock. Close the open end by press- ing the finger against it, and invert the tube in a basin of mercury; remove the finger, the mercury will not escape, but remain apparently sus- pended nearly 30 inches above the level of the mer- cury in the basin. K we consider the cir- cumstances attending this experiment, it will be seen that the tube containing the mercury forms with the basin a system of communicating tubes, as in § 259. Now the atmosphere rests on the mercury in the basin, and is excluded by the glass from that in the tube, above which effect of -^__. , . i ■■ withdrawing the there is therefore a vacuum. Withdraw the atmosphere external from the surface of the mercury in the basin, and, by the atmosphere; Fig. 328. experiment to illustrate this; 552 NATURAL PHILOSOPHY. an instrument well suited to exhibit the facts of this experiment; description and use; inference from this experiment; law of equilibrium of fluids, the mercury will descend in the tube till it comes to a level with that without; restore the pressure of the atmo- sphere, and the mercury in the tube will again rise to its former height. This is well illustrated by the fol- lowing device. R is a re- ceiver closed air-tight at the top by means of a metallic plate; a is a tube filled with mercury after the manner just described, and termina- ting at the open end in an inverted vial-shaped vessel —this tube passes air-tight through the plate on the receiver ; b is a second tube bent in the manner indicated in the figure, and, like the tube a, it terminates at one end in a vial-shaped vessel, but is open at both ends; this tube communicates with the receiver by passing through the metallic plate at top, and thus a connection is established between the open air and the interior of the receiver. Mercury being poured into the vial of the tube b, it will rise to the same level on either side of the bend m, and the communication between the interior of the receiver and exterior air will be interrupted. The receiver being placed upon the plate of the air-pump and the air exhausted, the mercury will descend in the tube a, and ascend in the tube b to- wards the bend at the top; readmit the air into the receiver, the mercury will rise in the tube a and fall in the tube b. From this we see, that the atmospheric air presses on the mercury, and indeed upon the surfaces of all bodies exposed to it, with a force sufficient to maintain the quick- MECHANICS OF FLUIDS. 553 silver in the tube at a height of nearly 30 inches ; whence, the intensity of its pressure must be equal to t/ie weight of a column of mercury wlwse base is equal to that of the surface pressed and whose altitude is about 30 inches. The force thus atmospheric exerted is called the atmosplieric pressure. piessuie, The absolute amount of atmospheric pressure was first discovered by Torricelli, a pupil of Galileo; the tubes employed in the experiments are called, on this account, Torricellian tubes, and the vacant space above the mercury Ton-iceiiian in the tube is called, the Torricellian vacuum, to distinguish " es' . 7 ' o Torricellian it from that of a receiver, which is frequently called the vacuum; Guerickian vacuum, from Otto von Guericke, who first invented the air-pump. The pressure of the atmosphere at the level of the sea will support a column of mercury 30 inches high. Now, if we suppose the bore of the tube to have a cross-section of one square inch, the atmospheric pressure up the tube atmospheric will be exerted upon this extent of surface, and will sup- ^"onhesea- port 30 cubic inches of mercury. Each cubic inch of mercury weighs 0.49 of a pound—say half a pound—from which it is apparent that the surfaces of all bodies, at the level of the sea, are subjected to an atmospheric pressure of fifteen 'pounds to each square inch. The body of a man of ordinary stature has a surface of pressure upon _ , -ii-i . the surface of a about 2000 square inches ; whence, the whole pressure to nian; which he would be exposed, at the level of the sea, is 15 pounds X 2000 = 30000 pounds. The pressure of the atmosphere, resulting as it does from its weight, it is an easy matter to estimate the weight of the entire atmosphere of the earth. It will be sufficient weight of the to compute, from the known diameter of the earth, the !;"m'oe8pnej.e. extent of its surface in square inches, and to multiply this by fifteen ; the product will be the weight in pounds. When the height of the mercury in the Torricellian tube is 30 inches, the atmospheric pressure will support in column of water , . , .p -, p supported by the vacuo a column of water 34 feet, the specific gravity ot atmospheric mercury being 13.6 referred to water as a standard. This pressure; 554 NATURAL PHILOSOPHY. Magdeburg hemispheres; description and mode of using; has been verified by Hanson and* Sturm, who actually performed the experiment at Leipzig. The atmospheric pressure is exhibited in a most stri- king way by means of the Magdeburg hemispheres. These are two hollow hemispheres, of brass or copper, whose edges fit air-tight, each hemisphere being furnished with a strong ring or handle, one of them also having a tube with stop-cock. Place the two hemi- spheres together, connect them with the communication-pipe of the air- pump, exhaust the air, and turn the stop-cock, and disconnect from the pump. It will be found that great force will be necessary to pull the hemispheres asunder. If the diame- ter of the hemisplieres, as in the case of those employed by Guericke, in one of his experiments, were 2 feet, the number of square inches in a great circle would be Fig. 330 0 / examples of Guericke's hemispheres; f^J = 452.39, and the force, estimated in pounds to overcome the pres- sure, would be the forcing of fluid through pores of solids; 15 X 452.39 = 6785.85. In the experiment referred to above, there were successively from 14 to 30 horses harnessed to the hemispheres, without effecting the separation. The pressure of the atmosphere will force fluids through such solid bodies as are porous. Let R he a long receiver, provided with a tube Fi<*. 331. flUr £ it MECHANICS OF FLUIDS. 555 Fig. 332. atmospheric pressure is exerted in every direction; and stop-cock C at one end, for the purpose of connecting with the air-pump, and at the other a perforated metallic plate a a, into which fits, air-tight, a wooden cup b, whose instrument to pores are in the direction of the axis of the tube. This exhibitthis; cup being filled with mercury, and the air exhausted by the air-pump, the mercury will fall in a fine shower down the receiver. The tube below is made to enter the re- ceiver, and to curve over at the top to prevent the mer- cury from falling into the communication-pipe of the pump. The atmosphere presses not only downward, but upward, and later- ally in all directions. This is shown by the following experiment: The two hemispheres A and B, are con- nected by a tube in such manner that one of them may turn about a joint C, while the other is stationary. Place the hemisphere A upon the plate of the air-pump, and upon B lay a plane plate of glass or metal fitting it air-tight. Ex- haust the air, and the hemisphere B may be turned in any direction without its plate falling off. This equal pressure of the atmosphere in all directions, is of great practical utility, as we shall presently see when we come to speak of siphons and water-pumps. To this pressure it is owing that flies, and other insects, are enabled to support themselves upon smooth vertical walls, and in in- verted positions upon the ceilings of rooms, &c. The feet being flat and flexible, are brought close a- gainst the wall or ceiling so as to exclude the air, the centre of the foot is then drawn away, leaving the margin in contact; a partial vacuum is thus formed, and the external pressure of the 'air is sufficient to support the weight of the insect. Fig. 333. exemplification of this in the adhesion of insects to walls and ceilings. 556 NATURAL PHILOSOPHY. X. MARIOTTE'S LAW. connecting the pressure, densely, and elasticity ; the instrument for compressing the air; JL Mariotte-s law; § 282.—We have seen that the atmosphere readily contracts into a smaller volume when pressed exter- Fl£- 334 nally, that it as readily regains its former dimensions when the pres- sure is removed, and that it is, therefore, both compressible and clastic. Let us now consider the law which connects the pressure, density, and elasticity. For this pur- pose, procure a siphon-shaped tube A B D, open at A, the end of the longer branch, and hermetically sealed at the end D of the shorter branch. Place between the branch- es, and parallel to them, a scale of equal parts, say inches, having its zero on the line o o. Pour in, at the open end A, as much quicksilver as will fill the b horizontal part of the tube, and bring its upper surface to the zero line in both branches; a quantity of atmospheric air of mode of usiu-it; ordinary density will then be confined in the shorter branch. The expansive action of this air, resisting, as it does, the pressure of the external air, is measured by the weight of a column of mercury, whose base is a section of the tube and height 30 inches. Pour into the longer branch an additional quantity of mercury; it will rise in ^ • MECHANICS OF FLUIDS. OOi the shorter branch, and cause the air above it to be com- pressed into a smaller space, but. the heights at which it will stand in the two branches will be different. The difference between these two heights, added to 30 inches, details and will be the altitude of the column of mercury, whose experiment • * weight is just sufficient to resist the expansive action of the confined air. Now it is found by trial, that when the air in the shorter branch is compressed into half its primi- tive volume, the difference of level of the mercury in the two branches is just 30 inches, thus making the compress- ing force double what it was before; that when it is compressed into one third of its original volume, the dif- ference of level is 60 inches, thus trebling the pressure; when compressed into one fourth, the difference of level is 90 inches, thus quadrupling the pressure, and so on. Hence we see, that in compressing the same quantity of air into smaller spaces, the volumes occupied by it are in- volumes are versely proportional to the pressures. This law holds equally when the air, instead of being compressed, is per- mitted to expand. Let ab he a, glass tube, about 33 inches long, one end a, being fitted with an air-tight cock, and the entire length of the tube being graduated in inches. Open the cock a, immerse the tube with its open end downward into the vessel A, previously half filled with mercury, which will, of necessity, stand at an equal height within and without the tube. Now close the cock a, and so confine a portion of air at its ordinary density within the tube above the surface of the mercury. Elevate the tube any distance what- ever, taking care that its open end shall be below the surface; the air will ex- pand, and fill a larger portion of the inversely proportional to the pressures; Fig. 335. -U 6 instrument for expanding the air; description and mode of using; 558 NATURAL PHILOSOPHY. weight of the suspended column of mercury plus elastic force of confined aii', equal to atmospheric pressure; experiments made at Paris; tube, though a column of mercury will still stand at an ele- vation above the outer Jevel, so that the weight of this column, with the elastic force of the inclosed air, counter- balances the natural pressure of the atmosphere. The pres- sure therefore which the included air sustains, is equal to the weight of a column of mercury 30 inches high, minus that of the column supported in the tube. Let the space full of air above the mercury in the closed tube be 3 inches; lift up the tube so that this space shall be 6 inches, the mercury will be found to stand in the tube 15 inches above that in the outer vessel. Here the volume of the air is doubled, and the pressure upon it is 30 — 15 = 15 = one half of 30, what it was before. Again raise the tube till the volume of air be- comes 9 inches long, the mercury in the tube will be found to stand 20 inches higher than in the outer vessel; here the volume is three times its primitive volume, and its pressure 30 — 20 = 10 = one third of 30, its original pressure; whence the law is manifest. By experiments made at Paris, it has been found that this law obtains when air is condensed 27 times, and rare- fied 112 times. Other gases obey it also, till the pressure becomes a few atmospheres less than that at which they assume a liquid form. The density of the same quantity of matter is inversely proportional to the volume it occupies. If, therefore, P be the pressure upon a unit of surface necessary to pro- duce a density unity, p the pressure corresponding to a density D, then, according to this law, will, expression of Mariotte's law. p = PD (236). This law was investigated by Boyle and Mariotte, the former in 1660, and the latter in 1668, and is now known as Mariotte's law. MECHANICS OF FLUIDS. L>A XL LAW OF THE PRESSURE, DENSITY, AND TEM- PERATURE. § 283.—It is a universal law of nature that heat ex- Law connecting p? ads all bodies, and is ever active in producing changes ^^3' of density. We have now to consider the law of this temperature; change in air. It has been ascertained, experimentally, that air, sub- jected to any constant pressure, will expand 0.00208th of rate of the air's its volume at 32° Fahr., for each degree of the same scale expansi°' above this temperature; so that if Yx be the volume of the air at 32°, and Y its volume at any other temperature t, then will Y = Yx [1 + (t - 32°) 0.00208)]. . (237). If D, he the density at 32°, under a pressure p, and D that at the temperature t, under the same pressure, then, because the densities are inversely as the volumes, will value for any Vx : Vx [1 + (t - 32°) 0.00208] : : D : D,; Z^TZm pressure; whence T density at any 7) __ _____________■L'l________(238) temperature " ■ " 1 + ^ — 32°) . 0.00208 ' J' under a constant pressure; If p, denote the pressure necessary to restore this air to the density D„ we shall have from Mariotte's law 560 NATURAL PHILOSOPHY. whence pressure to produce at a given temperature a density at 32° under a given pressure; weight of a column of mercury; p, = p [1 + (t - 32°) 0.00208] . . (239). Again, let the pressure p he produced by the weight of a column of mercury, having a base unity, and an altitude h,,, taken at a given latitude, say that of 45°, in order that the force of gravity may be constant. Denoting the den- sity of the mercury by D,n its weight will be Du Ki g'; in which g' denotes the force of gravity at the latitude of 45°. Substituting this for p, in Eq. (236), we have Duh„g' = PD; whence pressure to produce a unit of density; P . DttKg'. D ' and substituting the value of D, given in Eq. (238), this becomes same in different form; P = iuK£ [i + (t - 32°) D 0.00208] . . (240). From Eq. (236), we have »-* and substituting the value for P above, we get density at 32° under a constant pressure; D = pDt Du K 9' [1 + (t - 32°) 0.00208]' MECHANICS OF FLUIDS. 561 Denote by h the height of the column of mercury at 32°, necessary to produce upon a unit of surface the pressure p, then will weight of a column of P = D,, h g' ; mercury equal to the constant pressure; which, substituted for p above, gives, after striking out the common factors, D, h density at 32u ~~ h„\lA(t- 32°) 0.00208]' 7^ZT^ Now, when h,, becomes 30 inches, then will D, take the value given in the table of § 275 opposite the name of the gas or vapor under consideration, and we have, for the practical apphcation of that table, density answering to a n — _/ y _________^ '_______ C240V • given 30 1 + (t - 32°) 0.00208 ' ' V ' ' temperature and barometric column; in which D, is the tabular specific gravity or density, h the height of the column of mercury expressed in inches, and D the density of the gas pressing upon the mercury. Example. What is the density of atmospheric air, when the barometer stands at 26 inches and thermometer at example to 42°? In this case, D, will be found in the table to be ^™^e 0.0013, whence D = *??!? X_________^-________= 0.0011. 30 1 + (42° - 32°) 0.00208 We are now prepared to understand how the values of D„ in the table just referred to, were obtained, and of which no explanation has, thus far, been made. It will be recollected that, when referred to the same to obtain the standard, the numbers which express the specific gravities *^'Jg^ of bodies also express their densities, and that the specific &<=.; 562 NATURAL PHILOSOPHY. specific gravity of any body; gravity of a body is the ratio obtained by dividing the weight of the body by that of an equal volume of the standard substance. The gases and vapors are incessantly changing their densities, on account of the varying pres- sures and temperatures to which they are subjected. Tabulated densities must, therefore, correspond to a standard of temperature and of pressure. Thirty-two temperature and d es Fahrenheit's scale is adopted for the former; and pressure for ° density of gases, the weight of a column of mercury, at the same tempera- ture, having an altitude equal to thirty inches, and resting upon a base whose area is a superficial unit, is taken for the latter. By a very simple transformation of Eq. (240)', we find standard &c tabular value for density; n 30 X [1 + (t - 32°) 0.00208] n D,--------------1 X JJ. Fig. 336. To make this formula applicable to any gas, it will only be necessary to observe h, by means of a barometer in the atmosphere; t, by a thermometer in contact with the gas; and to find D, corresponding to these quan- tities, by the following process: Provide a glass vessel A, whose mouth may be closed by a stop- cock B, air-tight, and of which the bottom terminates in a long vessel for finding narrow tube C, closed at the end. **ZT*^L-, Let tne capacity of this vessel be carefully ascertained by filling it with water, and pouring this water afterward into a graduated vessel; also let the tubular portion C be graduated and numbered by tenths, hundredths, &c., so that the num- bers shall increase towards the smaller end, and express that portion of the entire capacity MECHANICS OF FLUIDS. 563 of the vessel, regarded as unity, which is comprised be- tween its mouth B and these numbers. This being understood, denote the weight of this ves- sel by Wv; that of a volume of air, or of the gas under consideration, equal to the contents of the vessel, and notation; under the pressure h and temperature t, by Wa; the buoy- ant effort of the atmosphere, under the same pressure and temperature, by e; and the weight required to counter- poise the vessel filled with air by Wx, then will Wt = Wv A Wa --e. . . . (a). ™f\°J/es8el \ J filled with air; Connect with the air-pump, and exhaust as far as conve- nient ; close the stop-cock, disconnect and weigh again, and denote the weight necessary to counterpoise the vessel with its rarefied air by W% and we shall have weight of vessel W2 = Wv A Wa, — e; miedwith rarefied air; in which Wa, denotes the weight of the rarefied air re- maining in the vessel. Subtracting this from the equation above, we find Wx - W2 = Wa - Wa,; weight of the extracted air; which is obviously the weight of the extracted air. Now immerse the vessel in water, mouth downward, and open the stop-cock; the liquid will enter, and taking- care to keep its level on the inside and outside the same, the water will come to rest at or near some one of the graduated points on the tube. The air or gas within will then have the same elasticity as the external atmosphere, and the reading h of the barometer becomes applicable to the gas. This graduated point will make known the volume of the an -. t-7- p . , -i -i-i •, extracted under volume Y of air or gas extracted; and, knowing its the barometrio weight, that of a volume equal to the contents of the pressure; whole vessel, which we have denoted by Wa, may be 564 NATURAL PHILOSOPHY. found from the proportion V : W1 - W2 whence 1 .: Wa; weight of the vessel filled with air under the barometric pressure; Wl-_W_ v ~ a ' (b). weight of the vessel full of water; Next fill the vessel with water, and weigh again; denote the counterpoising weight by Wz, and the weight of the contained water by Ww, and we shall have W3 = Wv A Ww - e; and subtracting Eq. (a), we find Wz- Wl = Ww- Wa; adding Eq. (b), we find Wl + wL=_!5 = Wm. ratio of the weights of equal volumes of water and gas; and dividing Eq. (b) by this one> we get Wx - W2_______ (W3 - Wi) V A Wx - W2 Wa_ Ww Multiplying both members by the tabular density d of water corresponding to the temperature of that employed, and dividing both numerator and denominator of the first member by Wx — W2, we finally get density of the d______Wa W3 - Wy " Ww Wx - W2 ^ X d. But the second member is the specific gravity or density D of air or gas, under the pressure h and temperature t. MECHANICS OF FLyiDS. 565 Whence, to find the value of D, we have this rule, viz.: Weigh the vessel full of the gas under consideration; - exhaust, and weigh a second time; find, by admitting process for water, the volume of gas exhausted by the pump; fill ""nsuy Specific with water, and weigh a third time ; then divide the dif- gravity of a gas. ference between the last and first weights by the differ- ence between the first and second; multiply this quotient by the volume exhausted ; increase this product by unity, and divide the tabular density of water, corresponding to its observed temperature, by this sum. The value of D, thus found, and the observed values of h and t, being sub- stituted in the value for D„ this latter may be found and tabulated. XII. BAROMETER. § 284.—The atmosphere being a heavy and elastic fluid, The barometer; is compressed by its own weight. Its density cannot be the same throughout, but diminishes as we approach its density and upper limits where it is least, being greatest at the surface ^o^pLVaf6 of the earth. If a vessel filled with air be closed at the different places; base of a high mountain and afterward opened on its summit, the air will rush out; and the vessel being closed again on the summit and opened at the base of the moun- tain, the air will rush in. The evaporation which takes place from large bodies of water, the activity of vegetable and animal life, as well as vegetable decompositions, throw considerable quantities of aqueous vapor, carbonic acid, and other foreign ingre- foreign .-, ,i j. i.- c j.1 ingredients in the dients temporarily into the permanent portions of the ^ and its atmosphere. These, together with its ever-varying tern-change of perature, keep the density and elastic force of the air in a state of almost incessant change. These changes are indi- 566 NATURAL PHILOSOPHY. barometer; weather-glass; description of the barometer; column of mercury in equiiibrio with atmospheric pressure; common mountain barometer; De Luc's siphon barometer; Fig. 337. cated by the Barometer, an instrument employed to measure the intensity of atmospheric pressure, and frequently called a weather-glass, because of certain agreements found to ex- ist between its indications and the state of the weather. The barometer consists of a glass tube about thirty-four or thirty-five inches long, open at one end, partly filled with distilled mercury, and inverted in a small cistern also containing mercury. A scale of equal parts is cut upon a slip of metal, and placed against the tube to measure the height of the mercurial column, the zero being on a level with the surface of the mercury in the cistern. The elastic force of the air acting freely upon the mercury in the cis- tern, its pressure is transmitted to the interior of the tube, and sustains a column of mercury whose weight it is just sufficient to counterbalance. If the density and conse- quent elastic force of the air be increased, the column of mercury will rise till it attain a corresponding increase of weight; if, on the contrary, the density of the air diminish, the column will fall till its di- minished weight is sufficient to restore the equilibrium. In the Common Barometer, the tube and its cistern are partly inclosed in a metallic case, upon which the scale is cut, the cistern, in this case, having a flexible bottom of leather, against which a plate a at the end of a screw b is made to press, in order to elevate or depress the mercury in the cistern to the zero of the scale. De Luc's Siphon Barometer consists of a glass tube bent upward so as to form two unequal parallel legs: the longer is hermetically sealed, and constitutes the Torricellian tube; the shorter is open, and on the surface of the quicksilver & MECHANICS OF FLUIDS. 567 Fig. 338. n Sl_. 30 IrU 29 moveable or sliding scale; different devices for appreciating slight changes of barometric column; the pressure of the atmosphere is exert- ed. The difference between the levels in the longer and shorter legs is the barometric height. The most conve- nient and practicable way of measuring this difference, is to adjust a moveable scale between the two legs, so that its zero may be made to coincide with the level of the mercury in the shorter leg. Different contrivances have been adopted to render the minute variations in the atmospheric pressure, and conse-, quently in the height of the barometer, more readily perceptible by enlarging the divisions on the scale, all of which devices tend to hinder the exact meas- urement of the length of the column. Of these we may name Morland's Diagonal, and Hook's Wheel-Barometer, but especially Huygen's Double-Barom- eter. The essential properties of a good barometer are: width of tube; purity of the mercury; accurate graduation of the scale; and a good vernier. Heat affects the density of mercury as well as that of all other bodies. When its temperature is increased, it expands; when diminished it contracts. The same atmo- spheric pressure will sustain the same weight—in other words, the same quantity of mercury; but the same quan- tity of mercury will occupy different volumes, according effects of -i ,i i -n temperature; to its temperature, and the same atmospheric pressure will, hence, sustain a longer column when the temperature is high than it will when the temperature is low. The indi- cations of the barometer must, therefore, be reduced to what they would have been, if taken at a standard or fixed temperature, without which reduction they would be utterly worthless. From the experiments of Dulong and Petit, it is found 568 NATURAL PHILOSOPHY. expansion of mercury; barometric column reduced to standard temperature; attached thermometer; example for illustration; that mercury expands jn>Wtn Part of its volume for each degree of Fahrenheit's scale by which its temperature is increased, and that it contracts according to the same law as its temperature is diminished. If, therefore, T denote the standard temperature, and T the temperature of ob- servation ; b the altitude which the barometer would have at the standard temperature, and b' the observed altitude, then will, b = b' [l + ^-^] = V [1 + (T- T) 0.0001001] .. (241); when T' becomes T, V wall be equal to b. A thermometer is usually attached to the barometer tube for the purpose of observing the temperature of the mercury. Example. Observed the barometric column to stand at 29.81 inches, while its thermometer gave a temperature of 93°. What would have been the column under the same pressure, had the temperature of the mercury been 32° ? Here we have data; V = 29.81, T = 93°.00, T = 32°.00, and reduced column. O T - T' = - 61.00; = 29.81 [1 - 61 x 0.0001001] = 29.63. Barometer used § 285.—The barometer may be used not only to meas- e^ucTtrof1116 ure tbe Pressure of the external air, but also to determine confined gases, the density and elasticity of pent-up gases and vapors, and furnishes the most direct means of ascertaining MECHANICS OF FLUIDS. 569 Fig. 339. Fig. 340. as being admitted through d, presses on the surface of the mercury a and forces its use and application; the degree of rarefaction in the receiver of an air-pump. When thus employed, it is called the barometer-gauge, barometer gauge; In every case it will only be necessary to establish a free connection between the cistern of the barometer and the vessel containing the fluid whose elasticity is to be indicated; the height of the mercury in the tube, expressed in inches, reduced to a standard temperature, and multiplied by the known weight of a cubic inch of mercury at that temperature, will give the pressure in pounds on each square inch. In the case of the steam in the boiler of an engine, the upper end of the tube is sometimes left open. The cistern A is a steam-tight vessel, partly filled with mercury, a is a tube communicating with the boiler, and through which the steam flows and presses upon the mer- cury ; the barometer tube b c, open at top, reaches nearly to the bottom of the vessel A, having attached to it a scale whose zero coincides with the level of the quicksilver. On the right is marked a scale of inches, and on the left a scale of atmospheres. If a very high pressure were exerted, one of several atmospheres, for example, an apparatus thus constructed would re- quire a tube of great length, in which case Mariotte's manometer is considered preferable. The tube being filled with air and the upper end closed, the sur- face of the mercury in both branches will stand at the same level as long no steam is admitted. The steam scale of inches and another cf atmospheres; Mariotte's manometer; 570 NATURAL PHILOSOPHY. Fig. 340. its mode of action. Levelling by means of the barometer; it up the branch b c, and the scale from b to c marks the force of compression in atmospheres. The greater width of tube is given at a, in order that the level of the mercury at this point may not be materially affected by its ascent up the branch b c, the point a being the zero of the scale. § 286.—Another very important use of the barometer, is to find the differ- ence of level between two places on the earth's surface, as the foot and top of a hill or mountain. Since the altitude of the barometer depends on the pressure of the atmosphere, and as this force depends upon the height of the pressing column, a shorter column will exert a less pressure than a longer one. The quicksilver in the barometer falls when the instrument is carried from the foot to the top of a mountain, and rises again when restored to its first position : if taken down the shaft of a mine, the barometric column rises to a still greater height. effect of change At the foot of the mountain the whole column of the heighTonhe111 e atmosphere, from its utmost limits, presses with its entire barometer; weight on the mercury; at the top of the mountain this weight is diminished by that of the intervening stratum between the two stations, and a shorter column of mercury will be sustained by it. It is well known that the surface of the earth is not uniform, and does not, in consequence, sustain an equal atmospheric pressure at its different points; whence effects of the mean altitude of the barometric column will vary ^M^tmT at di£ferent places. This furnishes one of the best and most expeditious means of getting a profile of an ex- tended section of the earth's surface, and makes the barometer an instrument of great value in the hands of the traveller in search of geographical information. MECHANICS OF FLUIDS. 571 Fig. O 341. -«r-- • JL -<-- J3 -<--- - c -<-..... D .<--- X! To find the relation which subsists between the alti- tudes of two barometric columns, and the difference of level of the places where they exist, conceive the atmo- sphere to be divided into an indefinite number of elementary horizontal strata of equal thickness, and so thin that the relation between density from the top to the bottom of the same stratum cJUrona™nd'1C may without error be regarded as uniform, the density difference of level p J, of the places; varying from one stratum to another. Then, commencing at any elevated position 0, above the level of the sea, denote by p the pressure exerted upon the unit of surface by the wThole column of atmo- sphere above this point. The density of the stratum of air A, immediately below this point, will be due to this pressure; call this density D, then, from Mariotte's law, Eq. (236), will p = P D ; elastic pressure; in which P is the pressure necessary to produce, on a unit of surface, a unit of density. From this equation, we have T\ _ P_ density P corresponding; The weight of so much of this stratum as stands upon a unit of surface will be n Ji weight of a small g D h = p ' -yj ', column on unit of surface; in which h denotes the indefinitely small height common to all the strata, and g the force of gravity. The pressure upon the unit of surface of the second stratum B, will be p, transmitted through the first stratum, 572 NATURAL PHILOSOPHY. augmented by the weight of this stratum found above; and, denoting this pressure by p', we shall have pressure on unit n> _ „ + „ . lit = p (l + £?!.). of surface of p — p f p p L \ P ) second stratum ; Denoting by U the density of the second stratum B, we have again by Mariotte's law p' = PD', or D' = ?-• P' and for the weight of this stratum upon a unit of surface, weight of second g ll U = p ' —^zr-, stratum; ■* and substituting the value of p', found above, same under another form: The pressure upon the unit of surface of the third stratum C, will be the pressure p', transmitted through the second stratum, increased by the weight found above for this same stratum; hence, denoting this pressure by p", will pressure upon „h ghxeh / ghx , fAx / eh-J unit of surface of P" = P 0 + V) + ' 0 + V) V =P (l + V) 0 + ~p) = * i1 + ~P ) ' third stratum; and in the same way will the pressure p'", upon the fourth stratum, be given by the equation same for fourth tn „ M , <7 "• \ stratum; P = P \} + ^r) l and so on to the surface of the earth: and supposing n to denote the number of strata between these limits, then will MECHANICS OF FLUIDS. 573 / qh\n pressure upon In = P ( 1 + —p- J J unit of surface \ •*- ' of 7?.t.h st.i-nhim < in which ^?„ denotes the pressure at the lowest station. Developing the second member of this equation by the binomial formula, and dividing by p, we have P* _i . gh n(n-l) g2h2 n(n—l)(n-2) (fh? . ^tio of the upper __! + „_ + ___ ._ + ______._+&,, „^„„ The strata being indefinitely thin, the number in any definite altitude will be indefinitely great, and this being the case in the above series, it is obvious that the numbers 1, 2, 3, &c, connected with n by the minus sign, may be disregarded without sensibly impairing the result, which will give Pn_ _ -j , ngh , n2g2h2 nzg*hz p P 1.2P2 1.2.3P3 — i _l "»" j_ " y ,u j. ™ yAi____l km — X- "T —^7" T ^—^ r>0 ~r" ^---?;---n r,o "T <*C. same reduced; But the second member is equal to ngh e p ' in which e = 2.7182818, the base of the Naperian system of logarithms. Whence, pn __ p same under p different form; But n being the number of strata, and h the common height of each, nh will be equal to the difference of level between the first and last points. Calling this z, and taking the Naperian logarithm of both members, we find, after substituting z, , Pn 9 % Naperian P p ' logarithm; 574 NATURAL PHILOSOPHY. and passing to the common logarithms common if. Log ^ = 4£, logarithm; V P in which M denotes the reciprocal of the modulus of the common system; whence we have difference of level; MP T pn z = ---- • Log -*-—. g p Denote by bn the height of the barometric column at the lower station, where the pressure is pn, and by b that at the upper station where the pressure is p, then will ratio of pressures in terms of Pn On barometric ^ f, ' columns: and reducing the barometric column b to the temperature of V taken as the standard, we have, Eq. (241), same reduced to JO* a standard temperature T; p b\lA(T- T') 0.0001001]' in which T becomes the temperature of the mercury at the lower, and T' that at the upper station. Moreover, we have, Eq. (81)', value of force of /'• gravity; g ;= 32.1803 [1 - 0.002551 cos 2 +], or g = g' (1 - 0.002551 cos 2 >L); ft- m which g' = 32.1803, the force of gravity in the latitude of 45°. Substituting these values of £?, g, and the value of P given by Eq. (240), in the value for D above, and we find value for difference of _ MD„ h„ 1 + (t. - 32) 0.00208 - r A, 1 level; u, ' 1-0.002531^77" x Iog \-T x i + Kr- r1, o.oooiooJ' MECHANICS OF FLUIDS. 575 In this it will be remembered that t denotes the tem- perature of the air; but this may not be, indeed scarcely ever is, the same at both stations, and thence arises a dif- difficulty arising ficulty in applying the formula. But if we represent, for ^^n~ of a moment, the entire factor of the second member, into the two stations; which the factor involving t is multiplied, by X, then we may write difference of level Z = [1 A (t - 32°) 0.00208] X. for constant temperature; If the temperature of the lower station be denoted by t,, and this temperature be the same throughout to the upper station, then will temperature throughout the same as at lower station; %t = [1 + (*, - 32°) 0.00208] X. And if the actual temperature of the upper station he denoted by f, and this be supposed to extend to the lower station, then would temperature Z' = [1 + (f - 32°) 0.00208] . X. sameasupper station; Now if t, be greater than t', which is usually the case, then will the barometric column, or b, at the upper station be greater than would result from the temperature t', since the air being more expanded, a portion which is actually below would pass above the upper station and press upon the mercury in the cistern; and because b enters the de- mean value of nominator of the value X, z, would be too small. Again, *ff°™" *e by supposing the temperature the same as that at the one; upper station throughout, then would the air be more condensed at the lower station, a portion of the air would sink below the upper station that before was above it, and would cease to act upon the mercurial column b, which would, in consequence, become too small; and this would make z' too great. Taking a mean between z, and z' as the true value, we find 576 XATURAL PHILOSOPHY. true value for difference of level; value for difference of level; 2 = _____ = [1 + \(t, A t' - 64°) 0.00208] X. _ Replacing X by its value, M D„ A„ 1 + j («, + f - 64°) 0.00208 rft»__________1__________-, 1 = Di * —T^.oo^oTcTs -i * x g" La * l + ( t - 7") 0.0001001 J ■ The factor -—tt^—^, we have seen, is constant, and it ■^ i only remains to determine its value. For this purpose, measure with accuracy the difference of level between two stations, one at the base and the other on the summit of some lofty mountain, by means of a Theodolite, or levelling instrument—this will give the value of z ; observe the barometric column at both stations—this will give b and bn; take also the temperature of the mercury at the two stations—this will give T and T'; and by a detached end the value of thermometer in the shade, at both stations, find the values of t, and f. These, and the latitude of the place, being substituted in the formula, every thing is known except the coefficient in question, which may, therefore, be found by the solution of a simple equation. In this way, it is found that the coefficient; its value; MD" h" = 60345.51 English feet; final value for difference of level; which will finally give for z, ««_?« ! + *(', + «'- 64°) 0-°°208 r = 6034,-51-----_____ 0.00208 j-i^ _________Jl___ -. 24, X j°5- Ly X 1 + , T— T) 0X1001001 J* To find the difference of level between any two stations, the latitude of the locality must be known; it will then only be necessary to note the barometric data for its use; columns, the temperature of the mercury, and that of the air at the two stations, and to substitute these observed elements in this formula. MECHANICS OF FLUIDS. 577 Much labor is, however, saved by the use of a table labor saved by a for the computation of these results, and we now proceed table; to explain how it may be formed and used. Make 60345.51 [1 + (t, A t' - 64) 0.00104] = A, 1 1 - 0.002551 cos 2 + = B. mode of 1 „ computing one; 1 + (T - T') 0.0001 Then will = C. z = AB- Log.-^A, z = A B • [Log. C A Log. bn — Log. 6] ; abbreviated formula; and taking the logarithms of both members, Log. z = Log. A + Log. B A Log. [Log. C A Log. bn — Log. 6] . . (242). its logarithm; Making t, A f to vary from 40° to 162°, which will be variations of the sufficient for all practical purposes, the logarithms of the temPerature of corresponding values of A, are entered in a column, under the head A, opposite the values t, A t', as an argument. Causing the latitude -^ to vary from 0° to 90°, the variations in logarithms of the corresponding values of B are entered a'u e' ,in a column headed B, opposite the values of 4>. The value of T — T' being made, in like manner, to vary from — 30° to + 30°, the logarithms of the cor- responding values of C are entered under the head of C, and opposite the values of T — T'. In this way a table is easily constructed. That subjoined, was computed by variations m Samuel Howlet, Esq., from the formula of Mr. Francis *3£?" Baily, which is very nearly the same as that just described, there being but a trifling difference in the coefficients. 37 NATURAL PHILOSOPHY. Table for finding Altitudes Detached Thermometer. «,+ f A , t,+f A t,+ t' A t,+t' A 4o 4.7689067 75 4.7859208 no 4.8022936 i45 4.8180714 1 4i .7694021 76 .7863973 in .8027525 146 .8i85i4o 4- .7698971 i| 77 .7868733 112 .8032109 147 .8189559 43 .7703911 | 78 .7873487 u3 .8036687 148 .8193975 44 .7708851 : 79 .7878236 "4 .8o4l26l 149 .8198387 45 .7713785 1 80 .7882979 n5 .8o4583o i5o .8202794 46 .77l87II |l 8l .7887719 116 .8o5o395 i5i .8207196 47 .7723633 1 82 .7892451 117 .8o54953 l52 .8211594 48 .7728548 , 83 .7897180 118 .8059509 153 .8215988 49 .7733457 : 84 .7901903 119 .8o64o58 154 .8220377 5o .7738363 < 85 .7906621 120 .8068604 i55 .8224761 5i .7743261 86 .7911335 121 .8073144 156 .8229141 52 .7748x53 87 .7916042 122 .8077680 157 .8233517 53 .7753042 88 .7920745 123 .8082211 158 .8237888 54 .7757925 89 .7925441 124 .80867.37 159 .8242256 55 .7762802 90 .793oi35 125 .8091258 160 .8246618 56 .7767674 9X .7934822 126 .8095776 161 .8250976 57 .7772540 92 .7939504 127 .8100287 162 .825533i 58 •77774oo 93 .7944182 128 .8104795 163 8259680 59 .7782256 94 .7948854 129 .8109298 164 .8264024 6o .7787105 95 .7953521 i3o .8113796 165 .8268365 6! .7791949 96 .7958184 i3i .8118290 166 .8272701 62 • 7796788 97 .7962841 132 .8122778 167 .8277034 63 .7801622 98 . 7967493 i33 .8127263 168 .8281362 64 .7806450 99 .7972141 134 .8131742 169 .8285685 65 .7811272 100 .7976784 i35 .8136216 170 .8290005 66 .7816090 101 .7981421 136 .8140688 171 .8294319 67 .7820902 102 .7986054 137 .8i45i53 172 .8298629 68 .7825709 io3 .7990681 138 .8149614 i73 .8302937 69 •783o5ii 104 .7995303 i39 .8154070 174 .8307238 70 •78353o6 io5 .7999921 i4o .8i58523 i75 .83u536 71 .7840098 106 .8oo4533 i4i .8162970 176 .83i583o 72 •7844883 107 .8009142 142 .8167413 177 .8320119 73 .7849664 108 .8013744 i43 .8171852 178 .83244o4 74 4-7854438 1 109 1 4.8oi8343 144 : 4.8176285 179 4.8328686 ^ 00 ■-J OO co CO 03 cn cn cn cn cn cn cn cn cn ■b. 4-. 43* 4- Co CO Co •e NO o> CO o so 00 ^j OO cn 4-. CO IO M O no 00 cn IO no OO Co o o 4-. 00 cn S3 nO oo CO o o NO so o t-l NO NO $ NO NO NO NO NO NO NO NO nO so NO NO NO nO NO NO O o O o o o o o o o o o o o 0 0 NO NO NO >0 NO NO NO NO NO no NO Nf) NO NO NO NO NO NO 0 0 0 o 0 o 0 o o o 0 o o o n 0 CO 00 00 NO NO NO NO NO NO NO nO nO NO NO NO NO NO NO NO 0 0 0 o 0 0 o 0 o o bd 00 00 VO IO co 4~. ■lis ■U cn cn cn CO CO ^1 ^1 ^1 00 00 o 10 CO 45v oo ^.1 00 NO o o CO 00 00 S3 c_n 00 10 CO NO CO -~J cn ft CO -~.l o IO .Sn oo ^4 oo 00 no oo ■bs OO .b. rh oo O cn Cn NO OO .b. 0 oo LO NO 00 -~4 o> CO ^1 ^4 0 IO CO cn 4-. ^4 S3 cm cn S3 ^1 Co IO 00 o 4~. 10 CO cn 4-. to O) -J Co <-n to -~i OO --J 10 cn 0 CO Co cn 00 00 ■lis cn NO NO ■u NO ^J Co 4~. NO CO Co to S3 to IO S3 IO IO S3 to IO f o NO 00 v4 OO cn ■fcN CO 10 " 0 o 00 ^1 oo cn 4-. CO to " 0 SO 00 •-4 oo cn ■bs CO S3 o o 1 o 0 > o o O o o O 0 o n o o 0 o o o O O 0 o O o 0 o o o o o o o o o o o O a 0 0 0 o n 0 0 o o o o 0 o 0 n o o n o o o o o o 0 o o o o p o o o o o 0 0 0 O o 0 o o o o o o o o o o o o o + a & Co Co IO to o o NO NO NO OO 00 •vl ^1 oo OO oo cn cn 4~. ■b. 11) Co CO S3 S3 n o o 4_. 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NO co 00 S3 ■-4 OO 0 bd > o g w H -A___i z discharge-pipe; serted into the barrel; the piston is worked by means of a lever H, or other contrivance, attached to the piston-rod 67. The distance A A', between the highest and lowest points of the piston, is called the play. To ex- play; plain the action of this pump, let the piston be at its low- est point A, the valves E and F closed by their own weight, and the air within the pump of the same density or elastic force as that on the exterior. The water of the reservoir will stand at the same level LL both within and operation of the without the sucking-pipe. Now suppose the piston raised pump; 584 NATURAL PHILOSOPHY. to its highest point A', the air contained in the barrel and sucking-pipe will tend by its elastic force to occupy the action during the space which the piston leaves void, the valve E will, there- "teton-0'^ fore> ^e forced °Pen> and air wil1 Pass fi'om tlie P*Pe to tne barrel, its elasticity diminishing in proportion as it fills a larger space. It will, therefore, exert a less pressure on the water below it in the sucking-pipe than the exterior air does on that in the reservoir, and the excess of pressure on the part of the exterior air, will force the water up the pipe till the weight of the suspended column, increased by the elastic equilibrium; force of the internal air, becomes equal to the pressure of the exterior air. When this takes place, the valve E will close of its own weight; and if the piston be depressed, the air contained between it "and this valve, having its density augmented as the piston is lowered, will at length have its elasticity greater than that of. the exterior air; action during the this excess of elasticity will force open the valve F, and escentoithe a^r enou„}1 wj]i escape t0 reduce what is left to the same piston; ° a density as that of the exterior air. The valve F will then fall of its own weight; and if the piston be again elevated, the water will rise still higher, for the same reason as before. This operation of raising and depressing the piston being repeated a few times, the water will at length enter the barrel, through the valve F, and be delivered from the discharge-pipe P. The valves E and F closing after the water has passed them, the latter is prevented from the result of a returning, and a cylinder of water equal to that through plItonT eS° ^ wm?cn tae piston is raised, will, at each upward motion, be forced out, provided the discharge-pipe is large enough. As the ascent of the water to the piston is produced by the difference of pressure of the internal and external air, it is plain that the lowest point to which the piston may reach, should never have a greater altitude above the greatest altitude water in the reservoir than that of the column of this uLl0pi_y;,ia fluid wtlica tne atmospheric pressure may support, in vacuo, at the place. From a little reflection upon, what has been said of the MECHANICS OF FLUIDS. 585 Fig. 343. operations of this pump, it will appear that the rise of water, during each ascent of the piston after the first, depends upon the expulsion of air through the piston- valve during its previous descent. But air can only issue through this valve when the air below it has a greater density, and, therefore, greater elasticity, than the external air; and if the piston may not descend low enough, for want of sufficient play, to produce this degree of compres- sion, the water must cease to rise, and the working of the piston can have no other effect than alternately to com- press and dilate the same air between it and the surface of the water. To ascertain, therefore, the relation which the play of the piston should bear to the other dimensions, in order to make the pump effective, suppose the water to have reached a stationary level X, at some one ascent of the piston to its highest point A', and that, in its subsequent descent, the piston- valve will not open, but the air below it will be compressed only to the same density with the external air, when the piston reaches its lowest point A. The piston may be worked up and down indefinitely, within these limits for the play, with- out moving the water. Denote the play of the piston by a; the greatest height to which the piston may be raised above the level of the water in the reservoir, by b, which may also be regarded as the altitude of the discharge-pipe; the elevation of the point X, at which the water stops, above the water in the reservoir, by x ; the cross-section of the interior of the barrel by B. The volume of the air between the level X and A will be B X (b — x — a); fact upon which depends the rise of the water: to find the relation of the play to the other dimensions; hypothesis in respect to rise of water; notation; volume of the confined air, when the piston is at its lowest point; 5Stf NATURAL PHILOSOPHY the volume of this same air, when the piston is raised to A', provided the water does not move, will be volume of same air expanded B (b — X). when piston is at highest point; -|jepreseilt ty jz tae greatest height to which water may be supported in vacuo at that place. The weight of the column of water which the elastic force of the air, when occupying the space between the limits X and A, will support in a tube, with a bore equal to that of the barrel, weight of the is measured by column of water r> , j\ . which the first ' J ' > in which D is the density of the water, and g the force of gravity. The weight of the column which the elastic force of this same air will support, when expanded between the limits X and A', will be Bh'.g.D; in which h' denotes the height of this new column. But from Mariotte's law we have weight supported by the second; B(b-x -a) : B(b- -x) :: B whence ratio of the h' = h ■ b — x ■ — a heights; b - X Bh'gD : BhgD; But there is an equilibrium between the pressure of the external air and that of the rarefied air between the limits X and A', when the latter is increased by the weight of the column of water whose altitude is x. Whence, omit- ting the common factors, B, D, and g, , ,, , , b — x — a , x + h = x A h • —-------= h; equilibrium; q __ g. > or, clearing the fraction and solving the equation in refer- ence to x, we find altitude of point .---------------- of stopping; -=.&±JvtV — 4aA . . (243). condition of MECHANICS OF FLUIDS. 587 When x has a real value, the water will cease to rise, condition of but x will be real as long as b2 is greater than 4 a h. If, on st0Ppage; the contrary, 4 a h is greater than b2, the value of x will be imaginary, and the water cannot cease to rise, and the pump will always be effective when its dimensions satisfy this condition, viz.:— 4:ah > b2, , or -, \ v condition of ^ 4/l ' incessant flow; that is to say, the play of the piston must be greater than the rule for piay of square of the altitude of the upper limit of the play of the piston l e pi8ton' above the surface of the water in the reservoir, divided by four times the height to which the atmospheric pressure at the place, where the pump is used, will support water in vacuo. This last height is easily found by means of the barometer. We have but to notice the altitude of the barometer at the value of a the place, and multiply its column, reduced to feet, by !°und bfthe r ' Jr J ' i J barometer; 13-*, this being the specific gravity of mercury referred to water as a standard, and the product will give the value of h in feet. Example. Eequired the least play of the piston in a sucking-pump intended to raise water through a height of 13 feet, at a place where the barometer stands at 28 example; inches. Here b = 13, and b2 = 169. in. no Barometer, —r = 2.333 feet. data; h = 2.333 X 13.5 = 31.5 feet. —., . b loy __ i 'oh i . resulting limit for Play =a>TTl = rxTL5 - L341+ ' that is, the play of the piston must be greater than one and one third of a foot. 588 NATURAL PHILOSOPHY. The quantity of work performed by the moter during the delivery of water quantity of work through the discharge-pipe P, is easily of the moter in . . , the sucking- computed. Suppose the piston to have pump; any position, as M, and to be moving upward, the water being at the level L L in the reservoir, and at P in the pump. The pressure upon the up- per surface of the piston will be equal to the entire atmospheric pressure de- noted by A, increased by the weight of the column of water MP', whose height is c', and whose base is the area B of the piston; that is, the pressure upon the top of the piston will be Fig. 344. M pressure on top of piston; pressure on the under surface of piston; pressure to be overcome by the power; weight to be overcome; A A Be'gD, in which g and D are the force of gravity and density of the water, respectively. Again, the pressure upon the un- der surface of the piston is equal to the atmospheric pres- sure A, transmitted through the water in the reservoir and up the suspended column, diminished by the weight of the column of water NM below the piston, and whose base is B and altitude c; that is, the pressure from below will be A - BcgD, and the difference of these pressures will be A A Bc'gD -(A - BcgD) = BgD(cAc'); but, employing the notation of the sucking-pump just described, cAc' = b; whence the foregoing expression becomes Bb.g.D; MECHANICS OF FLUIDS. 589 which is obviously >the weight of a column of the fluid whose base is the area of the piston and altitude the height of the discharge-pipe above the level of the water in the reservoir. And adding to this the effort necessary to which friction to overcome the friction of the parts of the pump when in must be added' motion, denoted by _ 70 , vaiueofthe S=±iOgLJAcp. motive force; Denote the play of the piston by p, and the number of its double strokes, from the beginning of the flow through the discharge-pipe till any quantity Q is delivered, by n; the quantity of work will, by omitting the effort necessary to depress the piston, be Flip = Up [Bb.gD +, , valve E', opening, like the first, upward. Through the top of the air-vessel a dis- discharge-pipe; charge-pipe K passes, air-tight, nearly to the bottom. The water when forced into the air-vessel by the de- scent of the piston, rises above the lower end of this pipe, con- fines and compresses the air, and this, re- acting by its elastici- ty, forces the water up the pipe, while the valve E' is closed by its own weight and the pressure from above, as soon as the piston reaches the lower limit of its play. A few strokes of the piston will, in general, be sufficient to raise water in the pipe K to any desired height, the only limit being that determined by the power at command and the strength of the pump. During the ascent of the piston, the valve E' is closed and E is open; the pressure upon the upper surface of quantity of work the piston is that exerted by the entire atmosphere; the in forcing-pump; ,i i p , „ , pressure upon the lower surface is that of the entire atmosphere transmitted from the surface of the reservoir through the fluid up the pump, diminished by the weight of the column of water whose base is the area of the piston and altitude the height of the piston above the surface of the water in the reservoir; hence the resistance to be overcome by the power will be the difference of �9999999999� MECHANICS OF FLUIDS. 593 these pressures, which is obviously the weight of this resistance to be column of water. Denote the area of the piston bv B its overcome brthe i • ■, -i ' power; height above the water of the reservoir at one instant by y, and the weight of a unit of volume of the fluid by w, then will the resistance to be overcome at this point of the ascent be W . x> . y, its measure; and denoting the indefinitely small space described by the piston from this position by s, the elementary quantity of work will be wBy.S. elementary J quantity of work; In Hke manner, denoting by y', y", y'", &c, the different heights of the piston, and by s', s", s'", &c, the correspond- ing elementary spaces described by it, the elementary quantities of work of the power will be ■n t% —i same for different wBy s, ( wBy s , wBy'"s'", kc; positions of piston; and the whole quantity of work "during the entire ascent, will be _ [_»jr. + By' s' + By"," + By"','" + 4a] ; I2£__- but Bs is the volume of a horizontal stratum of the fluid in the barrel, and Bs x y is the product of this volume into the distance of its centre of gravity from the surface of the fluid in the reservoir; and the same of the others. Hence, if y, denote the height of the centre of gravity of the play p of the piston, in other words, of its middle point, then will •-,,,,, « equivalent Bpy, = Bys A By's A By s +&c.; expression for the same; and w . Bp . y, 594 NATURAL PHILOSOPHY. will measure the quantity of work of the moter during one ascent of the piston. During the descent of the piston, the valve E is closed and E' open, and as the columns of the fluid in the barrel and discharge-pipe, below the horizontal plane of the lower surface of the piston, will maintain each other in equiiibrio, the resistance to be work during one overcome by the power will, obviously, be the weight of a descent; colUmn of fluid whose base is the area of the piston, and altitude, the difference of level between the piston and point of delivery P; and denoting by z, the distance of the central point of the play below the point P, we shall find, by exactly the same process, its measure; W B p Zt for the quantity of work of the moter during the de- scent of the piston; and hence the quantity of work during an entire double stroke will be the sum of these, or wcrk -luring one w ]$p /„ _|_ - )# double stroke; But y, A z, is the height of the point of delivery P above the surface of the water in the reservoir, and denoting this, as before, by b, we have same; W Bp b / and calling the number of double strokes n, and the whole quantity of work Q, we finally have work for any number of double Q = UW.Bpb . . . . (245). strokes; If we make z, — y,, or b = 2 y„ which will give y, — —, _' motion made the quantity of work during the ascent will be equal to forcing-pump; ^at during tne descent, and thus, in the forcing-pump, the work may be equalized and the motion made in some MECHANICS OF FLUIDS. 595 degree regular. In the lifting and sucking pumps, the moter has, during the ascent of the piston, to overcome the weight of the entire column whose base is equal to the it is very area of the piston, and altitude the difference of level be-irregular in the x ' lifting and tween the water in the reservoir and point of delivery, and sucking-pumps. being wholly relieved from this load during the descent, when the load is thrown upon the sleeping-valve and its box, the work becomes exceedingly variable, and the motion irregular. XIY, THE SIPHON. § 291.—The siphon is a bent tube of unequal branches, siphc open at both ends, and is used to convey a liquid from a higher to a lower level, over an intermediate point higher than either; and although its discussion more naturally appertains to the motion of fluids, its analogy with the pumps, renders a descrip- tion of it here proper. The siphon having its parallel branches vertical and plun- ged into two liquids whose upper surfaces are at LM and L' M', the fluid will stand at the same level both within and without each branch of the tube when a vent or small opening is made at 0. If the air be withdrawn from the siphon through this vent, the water will rise in the branches by the atmo- spheric pressure without, and when the two columns description; mode of using; 596 NATURAL PHILOSOPHY. conditions of the unite and the vent is stopped, the liquid will flow from flow; the reservoir A to A', as long as the level L' M' is below L M, and the end of the shorter branch of the siphon is below the surface of the liquid in the reservoir A. The cause of this apparent paradox will be manifest from the following consideration, viz.: The atmospheric pressures upon the surfaces L M and L' M', tend to force the liquid up the two branches of the tube. When the explanation; siphon is filled with the liquid, each of these pressures is counteracted in part by the weight of the fluid column in the branch of the siphon that dips into the fluid upon which the pressure is exert- ed. The atmospheric pres- sures are very nearly the same for a difference of level of several feet, by reason of the slight density of air. The weights of the suspended columns of water will, for the same difference of level, differ considerably, in conse- quence of the greater density of the liquid. The atmo- spheric pressure opposed to the longer column will there- fore be more diminished than that opposed to the shorter, shorter branch; thus leaving an excess of pressure at the end of the shorter branch, which will produce the motion. Thus, denote by A the intensity of the atmospheric pressure upon a surface a equal to that of a cross-section of the bore of the siphon; by h the difference of level between the surface L M and the bend 0 of the siphon ; by h' the difference of level be- tween the same point 0 and the level L' M'; by D the density of the liquid; and by g the force of gravity: then will the pressure, which tends to force the fluid up the branch which dips below L M, be motion due to the excess of pressiu-e up the pressure up the Bhorter branch; A — ahD g. MECHANICS OF FLUIDS. 597 and that which tends to force the fluid up the branch immersed in the other reservoir, be A- ah'Dq; pressure up the v ' longer branch; and subtracting the second from the first, we find pressure which ttDg (h — h), determines the flow; for the actual intensity of the force which urges the fluid within the siphon, in a direction from the upper to the lower reservoir. Denote by I the entire length of the siphon. It is obvious that this will be the distance over which any one stratum will move, while subjected to the action of the above force, and that the quantity of action will be meas- ured Dy quantity of actior in passing a a D g {hi — h) I. siphon full from y the upper to lower reservoir; The mass moved will be all the fluid in the siphon which is measured by a ID; and if we denote the velocity by Y, we shall have, for the living force of the moving mass, a I U . V ; living force; and because the quantity of action is equal to half the living force, we find aDIY2 aDg(h' - h) I = --^—' whence -r- r-x---777-----7— velocity of the Y = V2g(h -h); flow. from which it appears, that the velocity with which the liquid will flow through the siphon, is equal to the square root of twice the force of gravity, into the difference of level of thefiuid 598 NATURAL PHILOSOPHY. flow will cease when the water in the reservoirs comes to same level; practical apphcation of the siphon; mode of using it for draining purposes; greatest elevation over which the water may be raised. in the two reservoirs. When the fluid in the reservoirs comes to the same level, the flow will cease, since, in that case, h' — h = 0. The siphon may be employed to great advantage to drain canals, ponds, marshes, and the like. For this pur- pose, it may be made flexible by constructing it of leather, well saturated with grease, like the common hose, and fur- nished with internal hoops to prevent its collapsing by the pressure of the external air. It is thrown into the water to be drained, and filled; when, the ends being plug- ged up, it is placed across the ridge or bank over which the water is to be conveyed; the plugs are then removed, the flow will take place, and thus the atmosphere will be made literally to press the water from one basin to another, over an intermediate ridge. It is obvious that the difference of level between the bottom of the basin to be drained and the highest point 0, over which the water is to be conveyed, should never exceed the height to which water may be supported in vacuo by the atmospheric pressure at the place. XV. MOTION OF FLUIDS. Motion of fluids; § 292.—The purpose now is to discuss the laws which govern the motion of fluids; and we shall begin with those that relate to liquids. Suppose A B D C to be any vessel containing a heavy fluid whose upper level is MECHANICS OF FLUIDS. 599 Fig. 849. flow of liquids from vessels, through apertures; AB. If a small opening a & be made in the vertical side of the vessel, the pressure from within will urge the fluid out, and this pres- sure being greater as we descend to a greater distance from the upper surface A B, the fluid will flow with a greater velocity and in greater quantity during a given time, in proportion as the opening is made nearer the bottom. The quantity of fluid discharged in a unit of time, as a second, is called the expense. The liquid on leaving the vessel forms a expense; continuous stream called the vein or jet, which takes the vein or jet; form of the curve described by a body thrown per- pendicularly from the side of the vessel with the velocity which the fluid has at its exit, and afterward acted upon by its own weight. This, we have seen, is a parabola. „ shape, a At every point of this parabola, the weight of the fluid paraboIa; tends to alter its velocity, but at the orifice, the ve- locity .is determined solely by what takes place within the vessel. If the orifice be in the horizontal bottom, as at a'b', the jet will be vertical, and the liquid will flow down- ward; if, as at d, the orifice be in a horizontal face pressed vertically upward, the jet will also be vertical, and the liquid will ascend on leav- ing the vessel. In general, when the sides of the vessel are thin, the direction of the vein will be per- pendicular to the surface through which the orifice is made. Fig. 850. direction of the vein determined by the face of the ft 293.—The interior surface of every vessel containing Motion through a heavy fluid is subjected, as we have seen, to a pressure therefrom, which depends upon the extent of surface and 600 NATURAL PHILOSOPHY. Fig. 351. to find the velocity of a fluid flowing freely through an orifice in a thin plate; A permanent flow; equal volumes flow through the different sections in same time; data; the distance of its centre of gravity below the upper level of the fluid. At the moment an orifice a & is made, the fluid at its mouth is urged by this pressure to leave the vessel, the neighboring particles crowd to- wards the opening, describing paths which converge towards and lead through it. This movement is soon propagated in some modified degree to all parts of the fluid, and speedily each point of space within the vessel becomes distinguished by the constant velocity which every particle of the fluid mass that passes through it will there possess. It is from this instant, when the motion of the fluid becomes permanent, that we are to consider the flow. If the fluid be incompressible, it is obvious that the same volume will flow through each horizontal section of the vessel above the orifice in the same time, and that this volume must be equal to that dis- charged through the orifice. De- note by A the area of the section NB of the interior of the vessel, at the upper surface of the fluid; by a the area of the orifice MO; by s the distance through which the up- per stratum NB descends in any indefinitely small portion of time; and by S the distance 0 0' through which the stratum at the mouth of orifice passes in the same time. The volume of the fluid which flows through the section NB in this time will be measured by As; and that through the orifice, by aS; and because these must be equal, we have equal volumes; As = a S; whence MECHANICS OF FLUIDS. 601 g r, ratio of spaces -~- -= —j-. and areas of ^ —L sections; But because the distances s and S are described in the' same time, they will be proportional respectively to the velocities of the strata which describe them; and denoting the velocity of the stratum at the upper surface by v, and that of the stratum at the orifice by V, we have S V ratio of spaces ~7 " ~Tr i and velocities; which, substituted above, gives V a ratio of velocities y _J_ * and areas; That is to say, the velocities of the strata are inversely proportional to the areas of the sections through which they flow, and from which we obtain Fa SOAG\ velocity through ' -j .... (246). anysection. Again, since the flow is permanent, it is obvious that the living force of the fluid mass N' B' MQ must always be the same. Denote this by L, and let w represent the weight of the fluid mass in NBB' N', equal to that in MM'O'O; then will the living force of the mass NBMQ be j , IV 2 living force of q ' the interior fluid; and that of the mass B'N' QM'O'Ohe . that of a portion LA — ~V j within and that g at the jet; and subtracting the first from the second, we find for the difference of living force of the same mass NBMQ, and 602 NATURAL PHILOSOPHY. difference of living force of the same mass; work of the weight of the entire fluid; work at the beginning of any short interval; that at the end; work during the interval; B' N' Q MM' 0' 0, moving with the velocities v and Y respectively, the expression -(Y2 - v2). g JW A- Fig. 352. B B' M M The quantity of work performed by the weight of this same mass in the interval between its oc- cupying the space NBMQ, and B'N'QM'O'O, is, as we have seen, equal to this weight multi- plied by the vertical. distance through which its centre of gravity may have descended in the interval. Let 67' be the centre of gravity of the whole mass when in the position NBMQ, and G" when it occupies the space B' N' Q M' 0' 0. Denote the vertical distance of 67' below the upper surface NB by h', that of 67" below the same surface by h", and the weight of the entire fluid by W, then will the quantity of work of this weight be W(h" -h')= Wh" - Wh'; and calling the distance of the centre of gravity of the mass MM'O' 0 below the upper surface, h'"; that of the centre of gravity of the mass N'B'MQ below the same surface, I; and the weight of each of these equal masses, W; we have, from the principles of the centre of gravity, Wh" = W'l Awh'", Wh' = W'lAw\s; in which \s denotes the distance of the centre of gravity of the mass NBB' N' below the surface NB; whence . Wh" - Wh' = w(h'" -l5); MECHANICS OF FLUIDS. 603 the same; but h'"—%s is the vertical distance between the centres of gravity of the masses NBB'N' and MM' 0' 0, and when these masses are considered as elementary, this distance becomes the depth of the centre of gravity of the orifice below the upper level of the fluid. Denote this distance by h, and the quantity of work of the weight of the fluid while the stratum NB is passing to N'B', and the stratum MO to M' 0', becomes w h. If the upper surface be subjected to any pressure, as that of a piston or the atmosphere, then will the quantity of work due to this pressure be elementary work P A S ; from external pressure above; in which p denotes the pressure exerted upon the unit of surface. If, moreover, the fluid at the orifice be also sub- jected to a like pressure inward, this pressure would be transmitted to the lower face of the stratum whose area is A, and its work would be measured by elementary work P A S ; from external pressure below; and taking the difference, we have, for the effective work of these pressures, effective work of (p — p) A S. external pressures; Now AsDg = w, from which a __ W . volume of the Do1 stratum; and, substituting this above, we have w (p' — p) As = (p' — p) yr— ; effective work; whence the whole quantity of work due to the weight of 604 NATURAL PHILOSOPHY. the fluid and the pressures at the upper surface and the orifice, becomes , total effective w h + ( , _ ) ™ work; vi Da and because the difference of the living force at the begin- ning and end of any interval, is equal to twice the quantity of action in this interval, we have quantity of work W 2 ( P' ~ P\ equal half gain ~ (Y —' V2) = 2 W ( h A ~f]---Jj of living force; g \ L) g I or, dividing out the common factor, multiplying by g, and substituting for v its value, given in Eq. (246), we have r-v'-i = ^h + pLir)' whence /«(qh I P> -p) velocity of egress / V D I through the Y = /-------------------- . . (247). orifice; y -| a A2 If p and p' denote the atmospheric pressures upon the unit of surface, they become equal when the altitude of the fluid above the orifice is not very great, in which case same when the ____________ pressures at top / 2 g h and orifice are Y — / --------— .... \&-±01 , the same; \l -j _ ^_ A' and if the area of the orifice be very small as compared with that of the upper surface of the fluid, the fraction a? —2 will be so small, that it may, without sensible error, be omitted; in which case, the fluid at the surface will be at MECHANICS OF FLUIDS. 605 comparative rest while it flows through the orifice, and ^_____ velocity of egress Y = V 2 g h ,' through a very small orifice; that is to say, when a liquid is flowing through a small orifice in the side or bottom of a large vessel, its velocity is equal to the square root of twice the force of gravity multiplied rule; by the depth of the centre of gravity of the orifice below the upper surface of the fluid. It is apparent from the form of the above expression, that this velocity is the same as that acquired by a heavy velocity same as body while falling, in vacuo, from a state of rest, through ^J^^^ the distance of the orifice below the fluid level. The failing through distance h is called, in the case of discharging fluids, the JJJJ^1 generating load. If a he equal to A, that is, if the bottom of the vessel be removed, then will, Eq. (246), v= Y The space described uniformly by the stratum of fluid at the orifice in a unit of time being Y, the expense, estimated in volume, will be expense in volume; aY; and in weight, AvDg. in weight; So that, if t denote the time of flow, expressed in seconds; Q the quantity in volume, and Q' the quantity in weight discharged, then will quantity in Q = a Yt.....(249), volume in a given time; quantity in Q' = aYDgt .... (250); weight in a given * time; in which D g is the weight of the unit of volume. 606 NATURAL PHILOSOPHY. example; Example. The upper surface of the water, which is 15 feet above the centre of gravity of the orifice, is pressed with an intensity equal to 20 pounds upon the square foot; the area of the orifice being 0.02 of a foot. What is the velocity of egress, and what the expense ? Here, the atmospheric pressure upon the piston and at the orifice being the same,. p' — p = 20 pounds, D= 1, data; h = 15, g = 32 nearly; a2 and neglecting the small fraction --,, we find, from Eq. __. (247), Y= V30 x 32 + 40 = 31.6 feet; and for one second, Q = 0.02 X 31.6 = 0.632 cubic feet, lbs. Q' = 62.5 X 0.632 = 39.5 pounds. XVI MOTION OF GASES AND VAPORS. § 294.—In the preceding case, we have supposed, 1st, that the volume of the fluid which escapes through Motion of gases the orifice, is equal to that which passes, during the and vapors; same time^ through an_ interjor horizontal section of the velocity of egress; quantity in volume in one second; quantity in pounds in one second. MECHANICS OF FLUIDS. 607 Fig. 353. vessel; 2d, that the density in all parts of the vessel remains the same: both of which suppositions are sensibly true for liquids, but are not so in the cases of gases and vapors. When fluids of this latter class are confined and sub- jected to any compressing action, as that of a piston, and are permitted to escape through an orifice at which the resistance of external pressure is too feeble to retain them, the density, tending as it always does to conform to Mari- otte's law, will be greater at the piston where the pressure is greatest, than at the place of egress where it is least. Again, the motion being permanent, the same amount, in weight, of gas will flow through any section A' B' of the vessel as through the orifice ab; but the den- sities at these places being different, the volumes of these equal weights will also be different. In these par- ticulars, the circumstances attending the motion of gases and vapors dif- fer from those of liquids. To find the velocity of egress at the orifice, we remark, that the fluid is subjected, as in the case of liquids, to the action, 1st, of its own weight; 2d, to that of the opposing pressures at the piston and orifice; and 3d, to the additional action arising from the repulsions of the particles for each other, this latter producing expansion whenever the pres- sure from without will permit it. The quantity of work upon the stratum issuing through the orifice, due to the weight of the fluid mass, is, as we have seen, measured by wh- in which w denotes the weight of the stratum, and h the height of the fluid above the orifice. To find the work due to the pressures, denote the pressure upon a unit of surface at the piston by p; that on the same both the volume through different sections and density vary; density greatest at piston and least at orifice; 1-3 3' L equal quantities, by weight, will flow through the different sections in the same time; the volume^ of these will be different; -*y the forces which act: weight, pressure from the piston, and molecular repulsion; work of the weight; to find the work due to the pressures; 608 NATURAL PHILOSOPHY. notation; the equal weights; extent of surface at the orifice by p'; the area of the piston by A; that of the orifice by a; the dis- tance between any two consecutive positions, as AB and A'B', of the piston by 5; the distance between the two corresponding positions ab and a' b' of the stratum at the orifice by S. Then, because the weights of the volumes ABB'A' and abb'a' of the fluid are equal, we have AsDg = aSD'g Fig. 353. -H^§____.B -----B Jy (251); in which D and D' denote the densities of the gas at the piston and orifice, respectively, and g the force of gravity. Whence volumes, inversely : densities; ___£. aiS D' But by Mariotte's law the densities are directly propor- tional to the pressures, hence densities directly as pressures; relation of volumes and pressures; loss or gain of work due to pressure; (252). J__. - ___ D p ' which substituted above, gives A s _ p' aS p ' Clearing the fraction and transposing, we find p . As — p'aS = 0. But pA is the pressure on the whole extent of the piston, and pAs is, therefore, the whole work of this pressure; also p a is the pressure on the surface of the stratum of MECHANICS OF FLUIDS. 609 fluid in the orifice, and p' a S is the quantity of work of this pressure; and as these quantities of work are pro- duced in the same time, we see that the loss or gain of this is zero; work, due to these pressures, is zero. The quantity of work due to the molecular actions, arises in consequence of the to find quantity expansion which takes place when the gas passes from the ^J"^^^. pressure p, within the vessel and near the piston, to the pressure p', at the mouth of the orifice. The amount of this work is directly proportional to the primitive volume expanded during the change of pressure; if the it is directly primitive volume to be expanded be doubled, tripled, j£° p°jm"tnive ° or quadrupled, kc, the quantity of the work will be volume to be doubled, tripled, quadrupled, &c. Hence, taking a cubic exp"n foot of the gas under the pressure p, and denoting the quantity of work due to the expansion, corresponding to a change from the pressure p to the pressure p', by E, then will the work due to the expansion of the volume ABB'A' to abb'a', he measured by work of the expansion from A . S . Mi. the volume at the piston to that at But since w denotes the weight of the gas in the volume ABB'A', we have a —, weight of the w = A sg D; stratum; whence and A W _LS == ——, its volume; A .s.E w* E < work (i„e t0 q J) ' expansion; whence the whole quantity of action or work due to the weight and expansion of the fluid will be work due to weight and Wh +W • —= = W [Jl A —n ). expansion of the g D \ g DI stratum; 610 NATURAL PHILOSOPHY. Denoting, as before, the velocity at the piston by v, and that at the orifice by Y, we have, from the principle of living forces, £ = »«(» + ,4); living force equal W to twice quantity — of action; 9 or Y2 - v2 = 2ghA 2E_ D (253). From Eq. (252) we have relation of elementary paths; s pa ~S == JA' and the spaces s and S, being described in the same time, they are to each other as the velocities v and Y, hence same as ratio of velocities; velocity at piston; v pa V ~ ~p~A' or v = V pa < p~A] which substituted in Eq. (253) for v, we find ^-£)-»'*■•■ Making 1 _ P a - jr2 p2A2 ' 2E D ' the above gives value for the velocity of F-l\/^7+-^ . . (264). MECHANICS OF FLUIDS. 611 It remains to find the value of E. For greater sim- to And the work plicity, let us take for the primitive volume of gas a unit due t0 the , . x ° expansion of a or cubic foot; and suppose this unit of volume to be con- unit of volume, tained in a tube, of which the area of the internal cross- pTsZto section is a unit of surface, or square foot, so that in its -«>_er; primitive condition, under the pressure p, the length of the tube it occupies will be the linear unit, one foot. When the pressure is reduced to p', the volume becomes dilated, and because the volume, and therefore the length, since the base is supposed constant, is inversely as the pressure, we have, calling the new length I, P : p' :: I : 1; whence 7 __ P new length of the rp1' volmneofgas; The path described by the moveable face of the cubic foot of the gas, during the expansion, will be I — \ = ___ — ]_ = ____£_ expansion during p' p' the change; Dividing this path into two equal parts, and adding one of them to unity, the original length, we have p __ pf p _|_ p' length when the 1 A —7i—;— == —^—;—> expansion is half 2p 2p for the length of the fluid when its expansion is half completed; and denoting the corresponding pressure by p„ we have, by Mariotte's law, completed; whence - p A p' __ *>PP corresponding ■** p _L p1' pressure; 612 NATURAL PHILOSOPHY. the three consecutive pressures; If we now observe that the consecutive pressures are and p'; 2pp' p' p~Tp'} spaces described by the pressure while its value is changing from the first to last; and that the constant space passed over, during the inter- val which separates the instants in which these pressures are exerted, is p -p\ 2p' the computation of the total work be- comes easy by the rule given in § 46. • For this purpose, take Fig. 354. AC=OB = ^f, and erect the perpendiculars AM = p, determination of the work; CM' = ^_4 P+P BM"= p'; join the points M, M', and M"; the area ABM"M will be the value of E: that is to say, the value of the quantity of work performed by the gas during its expansion. But this area is, by the rule just referred to, measured by its value; ± AC (AM A 4 CM' A BM"); and, substituting the values above, we have MECHANICS OF FLUIDS. 613 i.ifA-(p + i-^K+P') = z; same in other 2p' Kjr p A p' tertns; which, substituted for E, in Eq. (254), gives -• I ^ } o } value of ve! V=^J2ghA~^(pA^-^,Ap') . . (255). - terms of K V ^ SD p pAp measures: When the orifice is small, a*s compared with the area of the piston, the fraction >2 2 p or p2A2 may be neglected, and K will become equal to unity. Moreover, the term 2 g h, in the case of gases, is scarcely ever appreciable in practice; making these suppositions, Eq. (255) becomes = VrT)-^^^ + f+7 +y)•'(266)'" BD p' p A p of small orifices: determine the essures; The pressures p and p' are usually ascertained by means ' of gauges, or manometers, as they are sometimes called, and use of gauges to it will be convenient to express the velocity of egress in d^" terms of the indications of these instruments. For this purpose, denote by h the height of a column of mercury resting on a unit of surface, and whose weight is equal to p, and by h' the same for the pressure p'; then, denoting the density of the mercury by D,„ will p = ghD„, and p' = gh'D,,; which, substituted above, give velocity in terms of the indications F_ ./T^IT__EZ7(A + #^ Ah'). . (257); of*e~ V $ SD h h A h } manometer; NATURAL PHILOSOPHY. expense; quantity discharged in volume; density on leaving the orifice; quantity in weight discharged in unit of time; in which Y will be expressed in feet, g being equal to 32 feet very nearly, and Dt, equal to 13.5 nearly. The expense e, in volume, will be given by the equa- tion e = aY.....(258); and the quantity Q in volume, discharged in a given time t„ expressed in seconds, will be known from Q =' a Yt, (259); in which a must be expressed in square feet. The density D, it will be remembered, is that of the fluid in the vessel near the piston, where the pressure is p ; the density D', which the fluid assumes on leaving the orifice, is deter- mined by the pressure p', and is connected with D, ac- cording to Mariotte's law, by the relation v - nPl - h' D' = Z> -- = Dl±. p h Hence, the expense Q', in weight, will be given by Q' = D'gaY = DgaY^- . . (260); and the quantity Q" in weight, discharged in the time t„ quantity, in weight, in time example; Q" = DgaY~t, . (261); in which a must be express- ed in square feet, as above. The density D is com- puted by Eq. (240)'. Example. The open gauge, connected with a gasometer, containing heavy carbureted hydrogen, shows a difference of level in the mercury of 8 Fig. 355. MECHANICS OF FLUIDS. 615 whence inches; the barometer in the air stands at 28 inches; the thermometer of the gasometer, at 52°: required the conditions; velocity with which the gas will flow into the open air, and the volume and weight discharged through an ori- fice 0.02 of a square foot of area in 20 minutes = 1200 seconds. Here, h - h' = 8 inches = 0.666 feet, h' = 28 " = 2.333 " ; data; h = 36 " = 3.000 " D„= 13.5 g = S2 t = 52°; and from Eq. (240)', after substituting the value of h and t, above, and that of D, in the table, page 533, for heavy car- bureted hydrogen, we find _ 0.00127 36 nnm_ft*. D = ------- X----------------^-.,>-, = 0.001400 ; density; 30 1 + (52 f- 32) 0.00208 ' and these values, in Eq. (257), give y- x /32._-!-_-x--^ X (3 + 8*l*2Qf + S.888)_-=66&_ velocity; ' V ~ V 3 X 0.00146 * 2.333 A V ^ 3 + 2.333 T Substituting this and the numerical values of a and t, in Eq. (259), we find quantity in Q = 0.02 X 668.02 X 1200 = 16032.00 cubic feet. volume; The quantity Dg, in Eq. (261), is the weight of a cubic 616 NATURAL PHILOSOPHY. quantity in weight. Veinal contraction theoretical suppositions results of experience; foot of the gas, whose density in this case is 0.001465; and as a cubic foot of water weighs 62.5 pounds, the value of Dg becomes 62.5 X 0.001465 = 0.0916, nearly; whence lbs 98 s "*' Q" = 0.0916 X 0.02 X 668.02 X ^ X 1200 = 1142.4. § 295.—A stream flowing through an orifice is called a vein. In estimating the quantity of fluid discharged through an orifice, it is supposed, 1st, that the orifice is very small, as compared with a section of the vessel at the upper surface of the fluid; 2d, that there are neither within nor without the vessel any causes to obstruct the free and continuous flow; 3d, that the fluid has no vis- cosity, and does not adhere to the sides of the vessel and orifice; 4th, that the particles of the fluid reach the upper surface with a common velocity, and also leave the orifice with equal and parallel velocities. None of these conditions are fulfilled in practice, and the theoretical dis- charge must, therefore, differ from the actual. Experience teaches that the former always exceeds the latter. If we take water, for example, which is far the most important of the liquids in a practical point of view, we shall find it to a certain degree viscous, and always exhibiting a tendency to adhere to ununctuous surfaces with which it may be brought in contact. When water flows through an opening, the adhesion of its particles to the surface will check their motion, and the viscosity of the fluid will transmit this effect towards the interior of the vein; the velocity will, therefore, be greatest at the axis of the latter, and least on and near its surface; the inner particles causes which tend thus flowing away from those without, the vein will v0e.cn°n' increase in length and diminish in thickness, till, at a certain distance from the orifice, the velocity becomes the same throughout the same cross-section, which usually takes place at a short distance from the aperture. This effect will be increased by the crowding of the particles, arising from the convergence of the paths along which MECHANICS OF FLUIDS. 617 they approach the aperture, every particle, which enters near the edge, tending to pass obliquely across to the opposite side. This diminution of the fluid vein is called the veinal contraction. The quantity of fluid discharged veinai -it -1-1 -, • j contraction; must depend upon the degree of vernal contraction, and the velocity of the particles at the section of greatest diminution; and any cause that will diminish the viscosity and adhesion, and draw the particles in the direction of the axis of the vein as they enter the aperture, will increase the discharge. Experience shows that the greatest contraction takes place at a distance from the vessel varying from a half to place of greatest once the greatest dimension of the aperture, and that the amount of contraction depends somewhat upon the shape of the vessel about the orifice and the head of fluid. It is further found by experiment, that if a tube of the same its amount shape and size as the vein, from the side of the vessel to epen the place of greatest contraction, be inserted into the aperture, the actual discharge of fluid may be accurately computed by Eq. (261), provided the smaller base of the tube be substituted for the area of the aperture; and that, the actual generally, without the use of the tube, the actual may be ^-nXrom deduced from the theoretical discharge, as given by that the theoretical; equation, by simply multiplying the theoretical discharge into a coefficient whose numerical value depends upon the size of the aperture and head of the fluid. Moreover, all other circumstances being the same, it is ascertained that this coefficient remains constant, whether the aper- ture be circular, square, or oblong, which embrace all coefficient of cases of practice, provided that in comparing rectangular dli with circular orifices, we compare the smallest dimension of the former with the diameter of the latter. The value of this coefficient depends, therefore, when other circum-depends upon; stances are the same, upon the smallest dimension of the rectangular orifice, and upon the diameter of the circle, in the case of circular orifices. But should other cir- cumstances, such as the head of fluid, and the place of 618 NATURAL PHILOSOPHY. discharge through orifices in thin plates; coefficient deduced from experiments; the orifice, in respect to the sides *%• 356- and bottom of the vessel, vary, then will the coefficient also vary. When the flow takes place through thin plates, or through orifices whose lips are bevelled external- ly, the coefficient corresponding to given heads and orifices, may be found in the following table, pro- vided the orifices be remote from the lateral faces of the vessel. This table is deduced from the experi- ments of Captain Lesbros, of the French engineers, and agrees with the previous experiments of Bossut, Miche- lotti, and others. TABLE. Coefficient values, for the discharge of fluids through thin plates, the orifices being remote from the lateral faces of the vessel. table of coefficients; 1 ■———------------------------------------------—----------------------------------------------------------*---------------------------------------------------------------______________________________________________________________________________________________________ Values of the coefficients for orifices whose smallest dimensions or Head of fluid diameters are— above the centre of the orifice, in feet. ft- ft- ft- ft- ft- ft- 0.66 o.33 0.16 0.98 0.07 o.o3 o.o5 0.700 0.07 0.627 0.660 0.696 O. i3 0.618 o.632 0.657 o.685 O.20 0.592 0.620 o.64o o.656 0.677 0.26 0.602 0.625 o.638 o.655 0.672 o.33 0.593 0.608 o.63o 0.637 o.655 0.667 0.66 0.596 o.6i3 o.63i o.634 o.654 o.655 1.00 0.601 0.617 o.63o o.632 o.644 o.65o • 1.64 0.602 0.617 0.628 o.63o 0.640 o.644 3.28 o.6o5 o.6i5 0.626 0.628 o.633 o.632 5.oo o.6o3 0.612 0.620 0.620 0.621 0.618 6.65 0.602 0.610 o.6i5 o.6i5 0.610 0.610 32.75 0.600 0.600 0.600 0.600 0.600 0.600 coefficients for In the instance of gas, the generating head is always greater than 6 65 ft and the gas; and for coefficient 0.6, or 0.61, is taken in all cases. ' orifices not in the . Jor orUicfs larger than 0.66 ft., the coefficients are taken as for this dimenson • for table; «*»» "f" th- *» «* the coefficients are the same as for this latttTZn;, o orifices between those of the table, we take coefficients whose values are a mea_ between the latter, corresponding to the given head. MECHANICS OF FLUIDS. 619 As the orifice approaches one of the lateral faces of the reservoir, the contraction on that side becomes less and less, and will ultimately be- come nothing, and the coefficient will be greater than those of the table. If the orifice be near two of these faces, the contraction be- comes nothing on two sides, and the coefficient will be still greater. Under these circumstances, we have the following rules: Denote by C the tabular, and by C the true coefficient corresponding to a given aperture and head, then, if the contraction be nothing on one side, will C = 1.03 C; if nothing on two sides, C = 1.06 C; Fig. 357. ii!Im! Bl Fig. 358. h'llll'il1 III J I Hi when orifice is near one lateral face; near two lateral faces; coefficient in the first case; coefficient in the second; coefficient for no contraction on three sides. if nothing on three sides, C = 1.12 C; and it must be borne in mind, that these results and those of the table are applicable only when the fluid issues through holes in thin plates, or through apertures so bevelled externally that the particles may not be drawn aside by molecular action along their tubular contour. g 296.__When the orifice is rectangular, and has no Discharge upper limit, or is open at the top, it is called a sluice-way. *h'n0cuegwava. It is usually a cut made in the edge of a reservoir, through 620 NATURAL PHILOSOPHY. estimate of the expense through a sluice-way; which the fluid may flow when it rises above a certain level. The expense is esti- mated in this wise. Denote by I the length of the horizontal side of the sluice-way; by h the head or distance Bl, of the centre of gravity of a transverse section of the flowing fluid below the upper surface of the latter in the reservoir; by Hthe height of the fluid above the sill C, of the sluice- way ; and by Y the mean velocity: then, supposing the sluice-way filled to the upper level of the fluid in the reservoir, will h = \H, Y2 = 2gh = 2g X \H= ^(2gH); whence value of mean velocity; Y = 0.707 VJg~LT; theoretical expense; and the theoretical expense will be Fx IX H= 0.707 • VYg~Hx IX H. But this is too great, and experience shows that it should be multiplied by the coefficient 0.57 for all ordinary cases of practice; that is to say, the true expense, denoted by E, will be given by the equation, practical expense; i? = 0.57 X 0.707 XlXHX V^W= 0.403 I. H. y/TfW. . . (262). The experiments of Dubuat, Bidone, Eytelwein, and Lesbros, show that the coefficient 0.403 should be re- MECHANICS OF FLUIDS. 621 duced about 0.39 when H becomes equal to or greater than 0.66 of a foot, and increased to 0.415 when H be- comes less than 0.07 of a foot; but that it remains variation in the sensibly the same, whatever be the total contraction orval"eof'he , , coefficient; position of the sluice-way in regard to the vertical sides of the reservoir, provided H be measured from the level of the upper surface of the sill to that of a point, as A, in the surface of the fluid in the reservoir which has no sensible velocity. When the sill is on a level with the bottom of the reservoir, the velocity of the upper surface is everywhere sensible, and the coefficient increases to about 0.45. On the contrary, 0.403 is already too large when the sluice-way is prolonged into a trough-like duct, of slight inclination, wherein the fluid may have impressed upon it a whirling or irregular motion by the roughness of the surface. The foregoing conclusions suppose that the fluid is discharge discharged through orifices in thin plates, and that, du- JJ™^h thick ring the flow, the fluid particles are not drawn aside from the converging paths, along which they tend to approach the orifice, by the action of any extraneous cause. When the discharge is through thick plates without bevel, or through cylindrical tubes whose lengths are from two to three times the smaller dimension of the orifice, the expense is increased, the mean coefficient, in such cases, augmenting, acpording to experiment, to about 0.815 for values of the orifices of which the smaller dimension varies from 0.33 coefflc,ent8; to 0.66 of a foot, under heads which give a coefficient 0.619 in the case of thin plates. The cause of this in- crease is obvious. It is within the observation of every one, that water will wet most surfaces not highly polished or covered with an unctuous coating—in other words, that there exists between the particles of the fluid and explanation; those of solids an affinity which will cause the former to spread themselves over the latter and adhere with con- siderable pertinacity. This affinity becoming effective between the inner surface of the tube and those particles 622 NATURAL PHILOSOPHY. effects of of the fluid which enter the orifice near its edge, the latter molecular action. wiU not ^ be drawn ^^ from tneir converging direc- tions, but will take with them, by the force of viscosity, the other particles, with which they are in sensible con- tact. The fluid filaments leading through the tube will, therefore, be more nearly parallel than in the case of orifices through-thin plates, the contraction of the vein will be less, and the discharge consequently greater. XVII DISCHARGE OF FLUIDS THROUGH PIPES. Discharge of We have considered the discharge of fluids through Quids through tkm and thick plates. It remains to discuss the discharge pipes; x <-J through pipes. When the flow is through pipes whose length does not exceed two or three times their diameter, the quantity discharged in a given time is, as we have seen, greater than through bevelled orifices of the same size ; but when the length is increased much beyond this limit, the reverse is the case and, all other things being less than through equal, the discharge will be less as the pipe is longer. The same pipe may be of variable bore, that is to say, it may have a greater cross-section at one point than at another; in which case, the living force of any given portion of the moving fluid cannot be constant throughout. When of considerable length, pipes are rarely perfectly smooth, the fluid particles cannot, therefore, flow through them in par- allel filaments, but must be incessantly deflected from their onward course into partial eddies formed by the small ir- causes which regularities pf surface. Moreover, as the pipes increase in motion; ' length, will the surface exposed to fluid pressure increase, and as the extent of surface, all other things being equal, MECHANICS OF FLUIDS. 623 determines the amount of pressure, the friction, which de- friction. pends upon the pressure, augments so as greatly to impede the motion. We shall proceed to estimate the value of these influences. § 297.—But first of all let us compute the amount of living force resulting from the shock of fluids, flowing with different velocities. For this purpose, let the fluid in the pipe LK flow with the velocity Y, and denote by M the mass which flows into the vessel B C in a unit of time; also let the velocity of the fluid in the vessel B C be Y', and its mass M'; then will the correspond- ing living force be MY2 AM' Y'2; Fig. 360. Loss of living force arising from the impact fluids; living force before the impact; and supposing the fluid to be water, which we have re- garded as unelastic, the common velocity after impact will be obtained from either of the Eqs. (194) or (195), by making e = 0 ; hence the common velocity denoted by v, will be given by v = M. Y A M' . Y' MAM' common velocity after the impact; and the corresponding living force, ,ir_i_ rim • - (M-VAM>V>\> m^J£I±J£Il. corresponding (Af-r-ATK-^ M+M> ) *(M+M)- MAM' ' living force; and the loss of living force in a unit of time, denoted by A 624 NATURAL PHILOSOPHY. loss of living force l = m\ +m v -v- M . M,---—-Knrw>—' M A M'~ and, dividing by M', T _M{Y- Y')2 ^ M' (263); same when a small mass flows into a large mass. Loss of living force from contraction of or when the mass M' is very great as compared to M, L = M(Y- YJ . . . (264). § 298.—It will be an easy matter now to estimate the loss of living force, arising from a contraction of the vessel or pipe through which the fluid may be flowing. Let A B CD he a vessel containing a heavy fluid, cross-section of a of which A B is the upper level, pipe; hypothesis; notation; and issuing through an opening a b in the bottom CD; and suppose Al B' to be a diaphragm, pierced by an opening a' b'. Denote by A" the area of the section at A" B", by a the area of the contraction at a b, and by a' that of the contraction at a' b'. The fluid, in passing through the contraction a' b', im- pinges against that below the diaphragm A' B', and if the opening a b is beyond the reach of the eddies formed by this conflict, the velocity at either contraction may be computed from that at the other. Denote by Y the velocity of the fluid as it passes the contraction at a b, by V' that at the contraction a' b', and by Y" that at the section A" B", supposed beyond the region of eddies; and let m represent the coefficient of the expense at ab, and m' that at a'b': these coefficients MECHANICS OF FLUIDS. 625 the different sections; may be found from the table. The expense at a b will be maY, that through the section A"B" will be A" Y", and that through the contraction at a' b' will hem' a' V ; expense through but as the same quantity of fluid must pass through the sections of ab, A"B", and a'b', in the same time, we have maY = A" V", maY = m'a'Y; whence y„ __ ma T A" ' velocities; _-, maY and the velocity with which the fluid through a' b' impinges against that below the diaphragm, will be V' - Y" = mat-^-r----Ij.) Y relative velocity \ m a A I of the impact; Denoting by w the weight of fluid that passes a' b' in any small portion of time, its loss of living force will be -(Y' - Y'J = - • m2a2(—L- - -LY • T3; g v 9 \m'a A J and denoting the factor ra a (——- — -rn) by K, the quantity of work, lost will be —— K2 Y2. work lost; The work of the weight, during the same time, will be w h, and the quantity of work remaining will be 626 NATURAL PHILOSOPHY. the work remaining; W wh - -^-K2Y2; 2g but this must be equal to half of the living force, hence which is equal to half the living force; ™v2 W wh - -AL-K2Y2 2# whence we find velocity of egress through a b. V -^ 2gh A K2 (265); Loss of living force in short pipes; hypothesis and notation; Fig. 362. and from which we see that the velocity will be less than that due to the height A C, equal to h. § 299.—Let us apply this to the discharge of a fluid through a short pipe, inserted into the orifice in the side of a vessel. The fluid hav- ing contracted to its mini- mum dimensions at n, again dilates, and fills the tube at a' b'. Let Y be the mean velocity at a' b', where the ~ area of the cross-section of ;: the pipe is a. The fluid particles moving in parallel paths at a' b', the expense will be a X Y; while that through a section at a b, where the velocity is Y', and cross-section a', will be ma' Y', in which m is the co- efficient corresponding to the area a'; and, as these must be equal, we have whence velocity at the entrance of pipe; a Y = ma'Y'; Y' = ~ V; m a MECHANICS OF FLUIDS. 627 and the loss of living force, w w X (Y' - Yf = — x F 9 9 Vraa / loss of living force; The quantity of work of the weight, in the same time, is w X h, and this, diminished by half the loss above, must be equal to half the actual living force; and, therefore, w 2"? r«__A-f'.F,.(-i,-iJ,i 2 a \ma I or ma ■, — 1 = K, we find ->/_ 2gh velocity of egress from the pipe; AX2' When the tube is cylindrical a = a', and m when the contraction is complete in n, and the head varies from 3 to 7 feet, it is found that ra is equal to value of m; 0.62 very nearly ; whence K = 0.62 — 1 = 0.613 very nearly, and whence VT+ K2 =r = 0.85; F=0.85 VTgh. Experiments give the coefficient 0.82, but, in com value of the constant; final value for velocity of 628 NATURAL PHILOSOPHY. coefficient given puting the foregoing value, no account was taken of by experiment a f*ricti0T1 which is an additional cause to diminish the little less. ' work of the weight w h. § 300.—When the velocity of a fluid is considerable, Flow of fluids and the length of the pipe through which it flows is great, anyfengthT8° friction, which has thus far been neglected, becomes an effective cause of obstruction, and can never be neglected in estimating the circumstances which determine the quantity discharged. The amount of friction depends, as we have seen in the case of fluids, upon the pressure, and this latter is determined by the extent of surface, and the head which impresses the velocity, so that the length of pipe and the velocity of flow, are the elements from which friction is to be estimated. Let abb' a' he case stated; notation; Fig. 363. B a' a pipe of uniform bore throughout, con- necting two reser- voirs AC DB and A'C'D'B', partly filled with fluid, the former to the level A B, and the latter to the level A'B'. De- note by II the differ- ence of level between AB and A'B'; by a the area of a cross-sec- tion of the bore of the pipe; by C the contour of this section; by L the length of the pipe; and by Fthe constant velocity of the fluid flow- ing through it. Experience shows, and the computations of Coulomb, de Mest, Prony, Eytelwein, and Navier, loss of work from teach us, that the loss of work occasioned bv friction of pipes, in the time during which a weight of the fluid denoted by w is discharged, is proportional to the value friction in pipes; C.4B MECHANICS OF FLUIDS. 629 of the expression 2 W L X C X Y proportional to ~ * ~ ~ J this function; and that this loss is a certain fraction n of this function, or is equal to W L X C X Y the loss of work yi . ___ .___________---------— q a, from friction; If, therefore, there be neither contractions in the pipe, nor sudden turns giving rise to. shocks, the only loss of work will be that measured by the above expression, and by that due to a diminution at the orifice a b, measured by the expression work lost from W T-2 / 1 1 \ _ ^ 17- J?"2 diminution at tht ~2~Q '\m~)'"Tq ' entrance of the pipe; in which I - 1 = _T2; m and, because of the principle of fluid level, H is the only distance through which w can act to produce work, we have ^LY2 = wH- ^--Y2 ■ K2 - ».!_-. ^---Z-Z! . .(266); 2g 2g 9 * whence --------^—g- - - (267>'S::Tof 1 + K2 A 2n-L--- a from which the velocity may be found. The expense, denoted by Q, will be given by Q = a Y.....(268). expense; 630 NATURAL PHILOSOPHY. Taking the value of m equal to 0.60, (see table,) we find vaiueofthe 2 4- ^2 == 1.4444. constant; Experiment shows that, for water, value of the coefficient n, for n = 0.0035 J water; and for air or gas, and for gas; n = 0.00324 J modification in and it is important to remark that, when the question the formula for i _ . _i j • i o i gas; relates to the discharge of gas, we must make rr ,D„h-K'(. 8hh' 7\ as indicated by Eqs. (254), (257), in the latter of which h and h' denote the mercurial altitudes corresponding to the interior and exterior pressures. Denote by D the internal diameter of the pipe, then will C=ifD, and a = —7—, so that 4 C _ 4tZ> _ 4 a == TW ~ D' Substituting these different values and that of gravity, Eq. (22), in the expression for the velocity, we have, after dividing both terms of the fraction by 8n, velocity in case - __ I 7) TT of water; «» water, ... Y = 47.94 \/_ + 5i57~g ' ' <269)' Incase of air; for air,.....Y = 49.83 v/— ' L D . H A 55.72. D ' • (270); MECHANICS OF FLUIDS. 631 in which all linear dimensions are expressed in English either formula feet. The first formula may be employed even for gas, ™yg^ emplo: because of the small difference between the values of n for the two fluids, provided we employ the proper value for H. Finally, if the aperture a' b' of final egress be small- er than ab, or of less section =p^ than a, Y being the velocity __£_ within the pipe, the expense may still be deduced from a slight modification of the value of the velocity, as given by Eq. (267). For let Y' denote the velocity of egress, a' the area of the section at a'b', and m' its coefficient of contraction, then will Fig. 364. _^__aj(5~^ when the aperture of final egress is smaller than section of pipe; aY = m'a'Y; condition of permanent flow; whence Y = aY m' a' and the living force of the fluid as it issues through a'b', will be __ Y'2 = - x a' m'2 a'2 X Y2; living force of the discharging fluid; which, being placed equal to the second member of Eq. (266), will give Y = 2gH °>2 . z^2 r o T 0 —5—5 +K A2nL- — m'J a'* a . . (271). its velocity; When a' is very small as compared with a, the value of m' is about 0.60. If the values of a and a' differ but 632 NATURAL PHILOSOPHY. values of the slightly, or if the pipe term- coefficient in'; • . 7 » • -i mates at ab m a conical tube, then will the value of m' vary from 0.82 to 0.96. Example. Let the height example; data; of the reservoir above. the point of delivery be 70 feet, the diameter of the pipe 0.5 of a foot, and its length 1200 feet: required the quantity of water discharged in 24 hours. In this case, ft. ft. ft. D = 0.5; H = 70; L = 1200; which, in Eq. (269), give velocity; V = 47.94 ^ 0.5 x 70 = 8.102. area of the section of pipe; expense. 1200 + 51.57 X 0.5 The value of a, in Eq. (268), will be given by a =_ * ±L = 3.1416 X ^P- = 0.196; 4 4 which, in Eq. (268), gives Q = aY = 0.196 X 8.102 = 1.6 nearly; and this multiplied by the number of seconds in 24 hours, equal to 86400, gives 138240 for the number of cubic feet discharged in the given time. END OF MECHANICS.