The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031364320 Cornell University Library arW3892 Elements of mechanics: 3 1924 031 364 320 olin.anx ELEMENTS MECHANICS: FOK THE USE OF COLLEGES, ACADEMIES, AND HIGH SCHOOLS. BT WILLIAM G.'^ECK, M. A., ADJinrCT PBOFBSSOK OF UATHEMATIOS, COLUMBIA COLLEGE. NEW TOEK : A. S. BARNES & BURR, 61 & S8 JOHN STREET. 1859. Lk \aiBRARY -y Entered, according to Act of Congress^ in the year Eighteen Hnndred and Eifty-nine, By WILLIAM G. PECK, In the Clerk's Office of the District Court of the Southern SiBtrict of New York. C£ ■William Bkntse, Stereotyper and Eleotrotyper, 183 WiHiam Street, N. T. PREFACE. The following work was undertaken to supply a want felt by the author, when engaged in teaching Natural Philosophy to College classes. In selecting a text-hook on the subject of MECHAiacs, there was no want of material from which to choose ; but to find one of the exact grade for College instruction, was a matter of much diiBculty. The higher treatises were found too difficult to be read with profit, except by a few in each class, in addition to which they were too extensive to be studied, even by the few, in the limited time allotted to this branch of education. The simpler treatises were found too elementary for advanced classes, and on account of their non-mathematical character, not adapted to prepare the student for subsequent investi- gations in Science. The present volume was intended to occupy the middle ground between these two classes of works, and to form a connecting link between the Elementary and the Higher Treatises. It was designed to embrace all of the important propositions of Elementary Mechanics, arranged in logical order, and each rigidly demonstrated. If these designs IV PEEFACE. have been accomplished, this volume can be read with facility and advantage, not only by College classes, but by the higher classes in Academies and High Schools ; it will be foimd to contain a suiBcient amount of information for those who want either the leisure or the desire to make the mathematical sciences a specialty ; and finally, it will serve as a suitable introduction to those higher treatises on Mechanical Philosophy, which all must study who would appreciate and keep pace with the wonderful discoveries that are daily being made in Science. Columbia College, February 22, 1859. CONTENTS. CHAPTER I. FJlOS. DErmmoNS — Rest and Motion 13 Forces 14 Gravity 16 Weight— Mass 16 Momentum — Fropei^ties of Bodies IT Definition of Mechanics — Measure of Forces 21 Representation of Forces 23 CHAPTER II. Composition of Forces whose Directions coincide 25 Parallelogram of Forces 26 Farallelopipedon of Forces 27 Geometrical Composition and Resolution of Forces 28 Components in the Direction of two Axes 30 Components in the Direction of three Axes 32 Projection of Forces 84 Composition of a Group of Forces in a Plane 35 Composition of a Group of Forces in Space 36 Expression for the Resultant of two Forces 37 Principle of Moments : 40 Principle of Virtual Moments 43 V VI CONTENTS. PAGE. Resultant of Parallel Forces 45 Composition and Resolution — Parallel Forces 48 Lever arm of the Resultant 51 Centre of Parallel Forces 52 Resultant of a Group in a Plane 53 Tendency to Rotation — Equilibrium in a Plane 58 Equilibrium of Forces in Space 69 Equilibrium of a Revolving Body 60 CHAPTER III. Weight — Centre of Gravity 62 Centre of Gravity of Straight Line 64 Of Symmetrical Lines and Areas 64 Of a Triangle 65 Of a Parallelogram — Of a Trapezoid 66 Of a Polygon .' 67 Of a Pyramid 68 Of Prisms, Cylinders, and Polyhedrons VO Centre of Gravity Experimentally 71 Centre of Gravity by means of the Calculus 72 Centre of Gravity of an Arc of a Circle 73 Of a Parabolic Area 74 Of a Semi-Ellipsoid 75 Pressure and Stability 80 Problems in Construction '. 85 CHAPTER IV. Definition of a Machine 94 Elementary Machines — Cord 96 The Lever. 98 The Compound Lever 101 The Elbow-joint Press 102 The Balance, 103 CONTENTS. TU The Steelyard 105 The Bent Lever Balance — Compound Balances 106 The Inclined Plane 110 The Pulley 112 Kngle Pulley 118 Combinations of Pulleys US The Wheel and Axle 117 Combinations of Wheels and Axles 118 The Wmdlass 119 The Capstan— The Differential Windlass 1 20 Wheel-work 121 The Screw 123 The Differential Screw 125 Endless Scisew 126 The Wedge 1 27 General It^marks on Machines 129 Friction 130 Limiting Angle of Resistance 133 Kolling Friction — Adhesion 135 Stifliiess of Cords 136 Atmospheric Resistance — Friction on Inclined Planes 137 Line of least Fraction , 140 Friction on Axle ' 141 CHAPTER V. Uniform Motion 143 Varied Motion 144 Uniformly Varied Motion 146 Application to Falling Bodies 148 Bodies Projected Upwards 150 Restrained Vertical Motion 153 Atwood's Machines 156 Motion on Inclined Planes 158 Motion down a Succession of Inclined Planes 161 Periodic Motion 163 Vlll CONTENTS. FAOE. Angular Velocity 165 The Simple Pendulum 166 The Compound Eendulum 169 Practical Applications of the Pendulum 175 Graham's and garrison's Pendulums 176 Basis of a System of Weights and Measures 177 Centre of Percussion 179 Moment of Inertia 180 Application of Calculus to Moment of Inertia 182 Centre of Gyration ' .- 186 CHAPTER VI. Motion of Projectiles 188 Centripetal and Centrifugal Forces.. 197 Measure of Centrifugal Force 197 Centrifugal Force of Extended Masses 203 Principal Axes 206 Experimental Illustrations 207 Elevation of the outer rail of a Curved tracii 209 The Conical Pendulum , 210 . The Governor 212 Worli 215 Work, when the Power acts obliquely 217 Work, when the Body moves on a Curve 219 Rotation — Quantity of Work 22S Accumulation of Work 225 Living Force of Revolving Bodies 227 Fly Wheels 228 Composition of Rotations 280 Application to Gyroscope 232 CHAPTER VII. Classification of Fluids , 236 Principle of Equal Pressures 236 COKTENTS. IX pAoa Pressure due to Weight .*. 238 Centre of Pressure on a Plane Surface 243 Buoyant Effect of Fluids 2i9 Floating Bodies -. 249 Specific Gravity 251 Hydrostatic Balance 2S3 Specific Gravity of an Insoluble Body 253 Specific Gravity of Liquids 254 Specific Gravity of Soluble Bodies 255 Specific Gravity of Air and Gases 256 Hydrometers — Nicholson's Hydrometer 257 Scale Areometer 268 Volumeter 259 Densimeter 260 Centesimal Alcoholometer of Gay Lussac 261 Thermometer 263 Velocity of a Liquid through an Orifice 265 Spouting of Liquids on Horizontal Planes 268 Modifications due to Pressures. 269 Coefficients of Efflux and Velocity 270 Efflux through short Tubes 2*72 Motion of Wates in open Channels 274 Motion of Water in Pipes 2'?'? General Eemarks 279 Capillary Phenomena 280 Elevation and Depression between Plates 281 Attraction and Repulsion of Floating Bodies 282 Applications of the principle of Capillarity 283 ^ndosmose and E^osmose 284 CHAPTER VIIL Gases and Vapors 285 Atmospheric Air 285 Atmospheric Pressure 286 ^lariotte's Law 287 X CONTENTS. PAGE. Gay Lussac's Law 290 Manometers — The open Manometer 291 The closed Manometer 292 The Siphon Guage -. 294 The Barometer — Siphon Barometer 295 The Cistern Barometer 296 Uses of the Barometer 297 Difference of Level 298 Work of Expanding Gas or Vapor 304 Efflux of a Gas or Vapor 306 Steam 308 Work of Steam 310 Experimental Formulas 311 CHAPTER IX. Pumps — Sucking and Lifting Pumps 313 Sucking and Forcing Pump 318 Fire Engine , 321 The Rotary Pump 322 Hydrostatic Press 324 the Siphon t 326 Wurtemburg and Intermitting -Siphon 328 Intermitting Springs 328 Sphon of Constant Flow — Hydraulic Ram 329 Archimedes' Screw 331 The Chain Pump — The Air Pump ; 332 Artificial Fountains — Hero's Ball 386 Hero's Fountain ; 8317 Wine-Taster and Dropping Bottle 338 The Atmospheric Inkstand 338 MECHANICS. OHAPTEK I. DEFINITIONS AND INTEODUCTOBT EEMAEKS. Definition of Natural Philosophy. 1. Natueal Philosophy is that branch of Science which treats of the laws of the material universe. These laws are called laws of nature ; and it is assumed that they are constant^ that is, that like causes always pro- duce like effects. This principle, which is the basis of all Science, is an inductive truth founded upon universal experi- ence. Definition of a Body. 2. - A Body is a collection of material particles. When the dimensions of a body are exceedingly snlaU, it is called a material point. Rest and Motion. 3. A body is at rest when it retains the same absolute position in space ; it is in motion when it continually changes its position. A body is at rest with respect to surrounding objects, when it retains the same relative position with respect to them ; it is in motion with respect to them, when it con- tinually changes this relative position. These states are called relative rest and relative motion, to distinguish them from absolute rest and absolute motion. It is highly prob- able that no object in the universe is in a state of absolute rest. 14 MECHANICS. Trajectory. 4. The path traced out, or described by a moving point, is called its trajectory. When this trajectory is a straight line, the motion is rectilinear ; when it is a curved line, the motion is curvilinear. Translation and Rotation. 5. When all of the points of a body move in parallel straight lines, the motion is called motion of translation / when the points of a body describe arcs of circles about a straight line, the motion is called motion of rotation. Other varieties of motion result from a combination of these two. Unifonn and Varied IVTotion. 6. The velocity of a moving point, is its rate of motion. When the point moves over equal spaces in any arbitrary equal portions of time, the motion is uniform., and the velocity is constant ; when it moves over unequal spaces in equal portions of time, the motion is varied, and the velocity is variable. If the velocity continually increases, the motion is accelerated ; if it continually decreases, the motion is retarded. Forces. y. A FoECE is anything which tends to change the state of a body with respect to rest or motion. If a body is at rest, anything which tends to put it in motion is a force ; if it is in motion, anything which tends to make it move faster, or slower, is a force. The power with which a force acts, is called its intensity. Forces are of two kinds : extraneous, those which act upon a body from without ; molecular, those which are exerted between adjacent particles of bodies. An extraneous force may act for an instant and then cease, in which case it is called an impulse, or an impulsive force / or it may act continuously, in which case it is called an incessant force. An incessant force may be regarded as made up of a succession of impulses acting at equal but exceedingly small intervals of time. When these successive DEFINITIONS AND INTRODUCTORY REMARKS. 15 impulses are equal, the force is constant; when they are unequal, the force is variable. The force of gravity at any given place, is an example of a constant force ; the eifort of expanding steam, is an example of a variable force. Molecular forces are of two kinds ; attractive^ those which tend to draw particles together ; repellent, those which tend to separate them. These forces also exert an arranging ipower hy virtue of which the particles of bodies are grouped [into definite shapes. The phenomena of crystalization pre- sent examples of this action. Molecular forcest)f both kmds are continually exerted between the particles of all bodies, and upon their variation, in intensity and direction, depend the conditions of bodies, whether solid, liquid, or gaseous. Classification of Bodies. ^ S. Bodies are divided into two classes, solids Stadjhcids. A solid is a body which has a tendency to retain a perma- nent form. The particles of a solid adhere to each other so as to require the action of an extraneous force of greater or less intensity to separate them. A fluid is a body whose particles move freely amongst each other, each particle yield- ing to the slightest force. Fluids are divided into liquids and gases, liquids being sensibly incompressible, whilst gases are highly compressible. Many bodies are capable of exist- ing in either of these states according to their temperatui'e. Thus ice, water, and steam, are simply three different states of the same body. Gravity. 9. Experiment and observation-have shown that the earth exercises a force of attraction upon all bodies, tending to draw them towards its centre. This force, which is exerted upon every particle of every body, is called the force of gravity. When a body is supported, the force of gravity produces pressure or weight; when it is unsupported, the force pro- duces motion. Experiment and observation have shown that the entire force of attraction exerted by the earth upon any body, varies directly as the quantity of matter in the body. 16 ' MECHASrCS. and inversely as the square of its distance from the centre of the earth. This force of attraction is mutual, so that the body attracts the earth according to the same law. Obser- Tation has shown that this law of mutual attraction extends throughout the universe, and for this reason it has received the name of universal gravitation. Weight. 10. The WEIGHT of a body is the resultant action of the force of gravity upon all of its particles. If the body there- fore remain the same, its weight at different places will vary directly as the force of gravity, or inversely as the square of its distance from the centre of the earth. Mass. H. The MASS of a body is the quantity of matter which it contains. Were the force of gravity the same at eveiy point of the earth's surface, the weight of a body might be taken as the measure of its mass. But it is foimd that the force of gravity increases slightly in passing from the equa- tor towards either pole, and consequently the weight of the same body increases as it is moved from the, equator towards either pole ; its mass., however, remains the same. K we take the weight of a body at the equator as the measure' of its mass, it follows from what has just been said, that the mass wUl be equal to the weight at any place, divided by the force of gravity at that place, the force of gravity at the equator being regarded as the unit ; or, denoting the mass of any body by M, its weight at any place by T^ and the force of gravity at that place by g., we shall have W M = — ; ^vhence, W ==: Mg. The expression for the mass of a body is constant, as it should be, smce the quantity of matter r^Hiains the same. The UNTT OF MASS is any definite mass assumed as a stand- ard of comparison. It may be one pound, one ounce, or any DEFINITIONS AND INTEODUCTOET EEMAKKS. 17 other unit of weight, taken at the equator. The pound is generally assumed as the unit of mass. The terms weight and mass may be regarded as synonymous, provided we un- derstand that the weight is taken at the equator. Density. 12. The DENsrrr of a hody is the quantity of matter contained in a unit of volume of the body, or it is the mass of a unit of volume. At the same place the densities of two bodies are propor- tional to the weights of equal volumes. The mass of any body is therefore equal to its volume multiplied by its den- sity, or denoting the volume by V, and the density by D, we have M =^ VB. "We have also, M W B = ~ = -yr; whence, W - VJ>g, Momentum. 13. The MOMENTUM of a moving body, or its. quantity of MOTION, is the product obtained by multiplying the mass moved, by the velocity with which it is moved ; that is, we multiply the number of units in the mass moved by the num- ber of units in the velocity with which it is moved and the product is the number of units ia the momentum. This will be explained more in detail hereafter. Properties of Bodies. 14. All bodies are endowed with certain attributes, or properties, the most important of which are, magnitude and form,; impenetrability ; inohility ; inertia; divisibility, uni porosity ; compressibility, dilatibility and elasticity ; at- traction, repulsion, smA polarity. IVEagnitude and Form. 15. Magnitude is that property of a body by virtue of which it occupies a definite portion of space ; every body 18 MECHANICS. possesses the three attributes of extension, length, breadth, and height. The form of a body is its figure or shape. Impenetrability. 16. Impenetrability is that property by virtue of which no two bodies can occupy the same space at the same time. The particles of one body may be thrust aside by those of another, as when a nail is driven into wood ; but where one body is, no other body can be. Mobility. IT. Mobility is .that property by virtue of which a body may be made to occupy different positions at different in- stants of time. Since a body cannot occupy two positions at the same instant, a certain interval must elapse whilst the body is passing from one position to another. Hence motion requires time, the idea of time being very closely connected with that of motion. Inertia. 1§. Inertia is that property by virtue of which a body tends to continue in the state of rest or motion iu which it may be placed, until acted upon by some force. A body at rest cannot set itself in motion, nor can a body in motion in- crease or diminish its rate, or change the direction of its mo- tion. Hence, if a tody is at rest, it will remain at rest, or if it is in motion, it wUl continue to move uniformly in a straight line, until acted upon by some force. This princi- ple is called the law of inertia. It follows immediately from this law, that if a force act upon a body in motion, it will impart the same velocity, and in the same general di- rection as though the body were at rest. It also follows that if a body, free to move, be acted upon simultaneously by two or more forces in the same, or in different directions, it will move in the general direction of each force, as though the other did not exist. When a force acts upon a body at rest to produce motion, or upon a body in motion to change that motion, a resistance is developed equal and directly opposed to the effective force DEFINITIONS AND INTRODUCTOEY EEMAEKS. 19 exerted. This resistance, due to inertia, is called the force of inertia. The effect of this resistance is called re-action, and the principle just explained may be expressed by saying that action and re-action are equal and directly opposed. This principle is called the law of action and re-action. These two laws are deduced from observation and experi- menti and upon them depends the mathematical theory of mechanics. Divisibility and Porosity. 19. Divisibility is that property by virtue of which a body may be separated into parts. All bodies may be di- vided, and by successive divisions the fragments may be ren- dered very small. It is probable that aU bodies are composed of ultimate atoms which are indivisible and indestructible ; if so, they must be exceedingly minute. There are micro- scopic beings so small that millions of them do not equal in bulk a single grain of sand, and yet these animalcules possess organs, blood, and the like. How inconceivably minute, then, must be the atoms of which these various parts are composed. Porosity is that property by virtue of which the particles of a body are more or less separated. The intermediate spaces are called pores. When the pores are small, the body is said to be ^ense ; when they are large, it is said to be rare. Gold is a dense body, air or steam a rare one. Compressibility, Dilatability, and Elasticity. 20. Compressibility, or contractility, is that property by virtue of which the particles of a body are susceptible of being brought nearer together, and dilatability is that prop- erty by virtue of which they may be separated to a greater distance. All bodies contract and expand when their tem- peratures are changed. Atmospheric air is an example of a body which readily contracts and expands. Elasticity is that property by virtue of which a body tends to resume its original form after compiression, or^ extension. Steel and India rubber are instances of elastic bodies. E"o bodies are perfectly elastic, nor are any perfectly inelastic. The force which a body exerts in endeavoring to resume its 20 MECHANICS. form after distortion, is called the force of restitution. If ■we denote tlie force of distortion by d, the force of restitu- tion by r, and theu- ratio by e, we shall have r in which e is called • the modulus of elasticity. Those bodies are most elastic which give the greatest value for e. Glass is highly elastic, clay is very inelastic. Attraction, Repulsion, and Polarity. 2 1 . Attraction is that property by virtue of which one par- ticle has a tendency to pull others towards it. Repulsion is that property by virtue of which one particle tends to push others from it. The dissimilar poles of two magnets attract each other, whilst similar poles repel each other. It is sup- posed that forces of attraction and repulsion are continually exerted between the neighboring particles of bodies, and that the positions of these particles are continually changing, as these forces vary. Polarity is that property by virtue of which the attractive and repellent forces between the particles exert an arranging power, so as to give definite forms to masses. The phenom- ena of crystalization already referred to, depend upon this property. It is to polarity that many of the most interest- ing phenomena of physics are to be attributed. Equilibrium. 22. A system of forces is said to be in equilibrium when they mutually counteract each other's effects. If a system oi forces in equilibrium be applied to a body, they will not change its state with respect to rest or motion ; if the body be at rest it will remain so, or if it be in motion, it wUl con- tinue to move uniformly, so far as these forces are concerned. Th6 idea of an equilibrium of forces does not imply either rest or motion, but simply a continuance in the previous state, with respect to rest or motion. Hence two kinds of equilibrium are recognized ; the equilibrium of rest, called DEFINITIONS AND INTEODUCTOET EKMAEKS. 21 Statical equilibrium, and the equilibrium of motion, called dytiamical equilibrium. If we observe that a body remains a,t rest, we infer that all the forces acting upon it are in equi- librium ; if we observe that a body moves unifoi-mly, we in like manner infer that all the forces acting upon it are in equilibrium. Definition of Mechanics. ~T~23. Mechanics is that science which treats of the laws d^ equilibrium and motion. That branch of it which treats of the laws of equilibrium is called statics ; that branch which treats of the laws of motion is called dynamics. When the bodies considered are liquids, of which water is a type, these two branches are called hydrostatics and hydrodynarn- ics. When the bodies, considered are gases, of which air is a type, these branches are called cerostatics and cerody- nam,ics. Measure of Forces. 24. We know nothing of the' absolute nature of forces, and can only judge of them by their effects. We may, how- ever, compare these effects, and in so doing, we virtually compare the forces themselves. Forces may act to produce pressure, or to produce motion. In the foiTuer case, they are called forces of pressure ; ia the latter case, momng forces. There are two corresponding methods of measuring forces, _/irs^, by the pressure they can exert, secondly, by the quantities of motion which they can communicate. A force of pressure may be expressed in pounds ; thus, a pressure of one pound is a force which, if du-ected vertically upwards, would just sustain a weight of one pound ; a pres- sure of two pounds is a force which would sustain a weight of two pounds, and so on. A moving force may be a single impulse, or it may be made up of a succession of impulses. The unit of an impulsive force, is an impulse which can cause a unit of mass to move over a xmit of space ia a unit of time.- A force which can cause two units of mass to move over a unit of space in a unit of time, or which can cause a 22 MECHANICS. unit of mass to move over two units of space in a unit of time, is called a double force. A force which can cause three units of mass to move over a unit of space in a unit of time, or -which can cause a unit of mass to move over three units of space in a unit of tisne, is called a triple force, and so on. If we represent a ilnit of force by 1, a double force will be represented by 2, a triple force by 3, and so on. In general, a force which can cause m units of mass to move over n units of space in a unit of time, wiU be repre- sented by m X n. Hence, forces may be compared with each other as readily as numbers, and by the same general rules. The unit of mass,, the unit of space, and the unit of time, are altogether arbitrary, but having been once assumed they must remain the same throughout the same discussion. We shall assume a mass weighing one pound at the equator as the unit of mass, one foot, as the unit of space, and one second, as the unit of time. Let us denote any iaipulsive force, by f, the mass moved, by m, and the velocity whidh the impulse can impart to it by V. Then, since the velocity is the space passed over in one second, we shall have, from what precedes, f = mv. If we suppose m to be equal to 1, we shall have, f=v. That is, the measure of an impulse is the velocity which it can impart to a unit of mass. An incessant force is made of a succession of impulses. It has been agreed to take, as the measure of an incessant force, the quantity of motion that it can generate in one second, or the unit of time. If we denote an incessant force by f, the mass moved by m,, and the velocity generated in one second by v, we shall have, f z= mv. DEFINITIONS AND INTKODUCTOET REMARKS. 23 If we suppose m to be equal to 1, we shall have, /= V. That is, the measure of an incessant force is the velocity which it can generate in a unit of mass in a unit of time. If the force is of such a nature as to act equally upon every particle of a body, as gravity,, for instance, the vel- ocity generated wUl be entirely independent of the mass. In these cases, the velocity that a force can generate in a unit of time, is called the acceleration due to the force. If we denote the acceleration by/", the mass acted upon by w, and the entire moving force by/*, we shall have, f -— mf = mv. Since an incessant force is made up of a succession of im- pulses, its measure may be assimilated to that of an impul- sive force, so that both may be represented and treated in the same manner. Forces of pressure, if not counteracted, would produce motion ; and, as they differ in no other respect from the forces already considered, they also may be assimilated to impulsive forces, and treated in the same manner. Representation of Forces. /^ 25. It has been found convenient in Mechanics to repre- sent forces by straight lines; this is readily effected by ' taking lines proportional to the forces which they repre- sent. Having assumed some definite straight line to repre- sent a unit of force, a double force will be represented by a line twice as long, a triple force by a line three times as long, and so on. A force is completely given when we have its intensity, its point of application, and the direction in which it acts. When a force is represented by a straight Une, the length of the line represents the intensity, one extremity of the line represents the joom* ^ ^ of application, and the direction of the Fig. i. line represents the direction of the force. Thus, in figure \, P represents the intensity, O ih.e point 24 MECHANICS. of application^ and the direction from O to jP is the directiou of the force. This direction is gen- erally indicated by an arrow head. g p It is to be observed that the point of rig. i. application of a force may be taken at any point of its line of direction, and it is often found convenient to transfer it from one point to another on this line. The intensity of a force may be represented analytically by a letter, which letter is usually the one placed at the ar- row head ; thus, in the example just given, we should desig- nate the force OP by the single letter P. If forces acting in any direction are regarded as positive, those acting in a contrary direction must be regarded as nega^ tive. This convention enables us to apply the ordinary rules of analysis to the investigations of Mechanics. - Forces sitxiated in the same plane are generally referred to two rectangular axes, 0^ and O Y, which are called co-ordinate axes. The direction from towards JTis that of positive abscissas ; that fi-om O towards JP is that of negative ab- _ scissas. The directions from to- -^ wards Y and T', respectively, are those of positive and negative ordi- nates. Forces acting in the direc- , tions of positive abscissas and posi- tive ordinates are positive ; those acting in contrary directions, are • negative. -Q T' Fig. 2. Forces in space are referred to three rectangular co-ordinate axes, OX, Y, and OZ. Forces acting from O towards JT, Y, or Z, are positive, those acting in conti'ary directions, are negative. X TCf Fig. 8. COMPOSITION AJSD EESOLtTTION OF FORCES. 25 CHAPTER II. COMPOSITION, EESOLUTIOIT, AND EQUILIBEIUil OF FOEGES. Composition of Forces whose directions coincide. 26. Composition of forces, is the operation of finding a single force whose effect is equivalent to that of two or more given forces. This single force is called the resultant of the • given forces. Resolution of forces, is the operation of find- ing two or more forces whose united efiect is equivalent to that of a given force. These forces are calLed components of the given force. K two forces are applied at the same point, and act in the same direction, their resultant is equal to the sum of the two forces. If they act in contrary directions, their resultant is equal to their difference, and acts in the direction of the greater one. In general, if any number of forces are ap- plied at the same point, some of which act in one direction, and the others in a contrary direction, their resultant is equal to the sum of those which act in one direction, dimin- ished by that of those which act in the conti'ary direction ; or, if we regard the rule for signs, the resultant is equal to the algebraic sum of the components / the sign of this alge^ braic sum makes known the direction in which the resultant acts. This .principle follows immediately from the rule adopted for measuring forces. Thus, if the forces P, P', &c., applied at any point, act in the direction of positive abscissas, whilst the forces P", P"\ &c., applied to the same point, act in the direction Of nega- tive abscissas, then will their resultant, denoted by jB, be given by the equation, R-^{P -V P' ^. &c.,) - [P" + P'" -t- Ac.) 2 26 MKCHAifiaS. If the first terra of the second member of this equation is numerically greater than the second, H is positive, which shows that the resultant acts in the direction of positive ab- scissas. If the first term is numerically less than the second, JR is negative, which shows that the resultant acts in the direction of negative abscissas. If the two terms of the second member are numerically equal, H will reduce to 0. In this case, the forces will exact- ly counterbalance each other, and, consequently, wiU be in equilibrium. Whenever a system of forces is in equilibrium, their re- sultant must necessarily be equal to 0. ' When all of the forces of the system are applied at the same point, this sin- gle condition will be sufficient to determine an equilibrium. All of the forces of a system which act in the general di- rection of the same straight line, are called homologous, and their algebraic sum may be expressed by writing the ex- pression for a single force, prefixing the symbol 2, a sym- bol which indicates the algebraic sum of several hoonologous quantities. We might, for example, write the preceding equation under the form, ■ -B = 2(P) .- . . . . (1.) This equation expresses the fact, that the resultant of a sys- tem of forces, acting in the same direction, is equal to the algebraic sum of the forces. Parallelogram of Forces. "a'?. Let P and Q be two forces applied to the material point O, taken as a unit of mass, and acting in the directions OP and OQ. q JJ Let OP represent the velocity genex-- / ^^^/ ated by the force P, and OQ the ve- ,/ ^^ f locity generated by the force Q. Draw L^'^^^ PR parallel to OQ, and QB parallel o ? to OP ; di-aw also the diagonal OP. ^^- ■*• f'rom the law of inertia (Art 18), it follows that a mass acted upon by two simultaneous forces moves in the general COMPOSITION AND RESOLUTION OF FOECES. 27 direction of each, as thotigh the other did not exist. Now, if we suppose the material point 0, to be acted upon simul- taneously by the two forces P and Q, it will, by virtue of the first, be found at the end of one second somewhere on the line PB,\ and by virtue of the second somewhere on the line QR ; hence, it will be at their point , of intersection. But had the point O been acted upon by a single force, rep- resented in direction and intensity by Oi2, it would have moved fronl to i? in the same time. Hence, the single force B, is equivalent, in effect, to the aggregate of the two forces P and J|["Pif is. Therefore, theiFresultant. Hence, If two forces be represented in direction and intensity by. the adjacent sides of a parallelogram, their resultant will be represented in direction and intensity by that diagonal of the parallelogram which passes through their point of in- tersection. This principle is called the parallelogram of forces. In the preceding demonstration we have only considered moving forces, but the principle is equally true for forces of pressure ; for, if we suppose a force equal and directly op- posed to the resultant Ji, this force will be in equilibrium with the forces P and Q, which will then become forces of pressure. The relation between the forces wiU not be changed by this hypothesis, and we may therefore enunciate the principle as follows : If two pressures be represented in direction and ititensity by the adjacent sides of a parallelogram, their residtant will be represented in direction and intensity by that diago- nal of the parallelogram, which passes through their com men point. This principle is called \he parallelogram of pressures. Hence, we sfee that moving forces and pressures may be compounded and resolved a«!eerdiBg te the same principles, and by the same general laws. Parallelopipedon of Forces. 28. Let P, Q, and S represent three forces applied to the same point, and not in the same plane. Upon these lines. 28 MECHANICS. Fig. 5. as edges, constrnct the parallelopipedon OH, and draw OM, and SR. From the preceding article, Oil represents the resultant of P and §, and from the same article, OH rep- resents the resultant of OM and S. Hence, OM is the resultant of the three forces P, §, and S. That is, if three forces be represented m direc- tion and intensity by three adjacent edges of a parallelopipedon, their resultant will be repre- sented by that diagonal of the 2^arallelopipedon which passes through their point of intersection. \ This principle is known as the parallelopipedon of forces, and is equally true for moving forces and pressures. Geometrical Composition and Resolution of Foroes. 29. The following constructions depend upon the prin- ciple of the parallelogram of forces. 1. Having given the directions and intensities of two forces applied at the same point, to find the direction and in- tensity of their resultant. Let OP and OQ represent the given forces, and their point of ap- plication; draw Pi? parallel to OQ, and QP parallel to OP, and draw the diagonal OR ; it will be the re- sultant sought. 2. Having given the direction and intensity of the result- ant of two forces, and the direction and intensity of one of its components, to find the direction and intensity of the other component. Let P be the given resultant, P the given component, and their point of application ; drawJ?P, and through draw OQ parallel to MP, also through M draw MQ parallel to P ; then will OQ'be the component sought. 3. Having given the direction and intensity of the results ant of two forces, and the directions of the two components, to find the intensities of the components. Fig. 6. COMPOSITION AND EESOLUTION OF FOECES. 29 Fig. 7. Let H be the given resultant, OP and OQ the directions of the compo- nents, and O their point of applica- tion. Through Ji draw HJP ancl BQ respectively, parallel to QO and P 0, then will OF and OQ represent the intensities of the com-" ponents. From this construction it is evident that any force may be resolved into two components having any direction what- ever ; these, again may each be resolved into new compo- nents, and so on ; hence it follows that a single force may be resolved into any number of components having any as- sumed directions whatever. 4. Having given the direction and intensity of the re- sultant of two forces, and the intensities of the components, to find their directions. Let H be the given resultant, and O its point of application. With li as a centre, and one of the compo- nents as a radius, describe an arc of a circle ; with as a centre, and the other component as a radius, describe a second arc cutting the first at P ; di'aw JPIi and P 0, and complete the parallelogram PQ, then will OP and OQ he the directions sought. 5. To find the resultant of any number of forces, P, Q, S, T, &c., lying in the same plane, and applied at the same point. Construct the resultant P' - of P and Q, then construct the re- sultant P" of P' and S, then the resultant P of P" and T, and so on : the final resultant will be the result- ant of the system. By inspecting the preceding fig- ure, we see that in the polygon OQ P'P"PT, the side QP' is equal and parallel to the force P, the side ^'^' ®- P'P" to the force /S,-and the side P"P to the force T, ,-7>S" /^„->», 30 MECHANICS. and so on. Hence, we may construct the resultant of such a system of forces by drawing through the second extremity of the first force, a line parallel and equal to the second force, through the second extremity of this line, a line par- allel and equal to the third force, and so on to the last. The - line drawn from the starting point to the last extremity of the last line drawn, will represent the resultant sought. If the last extremity of the last force fall at the starting point, the resultant will be 0, and the system will be in equili- brium. This principle is called the polygon offerees ; its simplest case is the triangle of forces. / Components of a Force in the direction of two axes. 3©. To find expressions for the components of a force which act in directions parallel to two rectangular axes. Let O-STand 3^ be two such axes, and R any force lying- in their j)lane; construct the compo- nents parallel to OX and 1^ as be- fore explained, and denote the angle LAH, which the force makes with the axis of X^ by a. From the figure, we have, AL = JR cos a, and RL . — A3f = i? sin a ; or, making AL = X, and ^ Jtf = J^ we have, X = JR cos a, and ]F = i2 sin u. . . (2.) The angle a is estimated from the direction of positive abscissas around to the left through 860°. For all values of «. from 0° to 90°, and from 2*70° to 360', the cosine of a, will be positive, and, consequently, the com- ponent AIj will be positive ; that is, it will act in the direction of positive abscissas. For all values of a from 90° to 270°, the cosine of a, will be negative, and the component AZi will act in the direction of negative abscissas. Fig. 10. COMPOSITION AND RESOLUTION OF rOECES. 31 For all values of a from 0° to 180°, the sine of a will be positive, and the component AM wiU be positive ; that is, it wiU act in the direction of positive ordinates. For all values of a from 180° to 360°, the sine of u. will be negative, and the compo- nent AM will act in the direction of negative ordinates. For a = 90°, or a = 270°, we shall have Ajy = 0. For a. — 0, or a AM= 0. If we regard AJO and AM as two given forces, H will be leir result the figure. Fig. 10. 180°, we shall have their resultant ; and since J^ = AM, we shall have from B =^VX' + Y' (3.) Hence, tf/,e resultant of any two forces, at right-angles to each other, is equal to the square root of the sum of tlie squares of the two forces. From the figure, we also have, cos a =: — , and sin a = — • li Ji Hence, the resultant is completely determined. T^ PEACTICAL EXAMPLES. 1. Two pressures of 9 and 12 pounds, respectively, act upon a point, and at right-angles to each other. Required, the direction and intensity of the resultant pressure. SOLUTION. We have, X= 9, and Y -. 12; H Also, 9. COS a = ^ = .6 ; -v/81 + 144 = 15. 53° 1' 32." That is, the resultant pressure is 15 lbs., and it makes an angle of 53° 7' 32" with the direction of the first force. 2, Two forces arc to each other as 3 is to 4, and their 32 MECHANICS. resultant is 20 lbs. What are the intensities of the compo- nents ? SOLUTION. We have, 3^ = 4^ oi' #=3*^ ^^^^ i? = 20; .-. 20 = VX' + J/X' = fX; Hence, '.^= 12, and'^= ,lV. 3. A boat fastened by a rope (o a jfoint on the shore, is urged by the wind perpendicular to the current, with a force of 18 pounds, and down the current by a force of 22 pounds. What is the tension, or strain, upon the rope, and what angle does it make with the current ? SOLUTION. We have X=: 22, andF: Also, 22 cos a 28.425 iZSy/sOS = 28.425 ; a — 39° 11' 28". Hence the tension is 28.425 lbs., and the angle 39° 17' 28". Components of a l^orce in the direction of three axes. 31. To find expressions for the components of a force in the directions of three rectangu- lar axes. Let OB represent the force, and OX, OF, and OZ, three rectangular axes drawn through its point of application, 0. Construct a parallelopipedon on OJi as a diagonal, having three of its edges coinciding with the axes. Then will the lines OX, 03f, and OJST, represent the required components. Denote these components, re-- • spectively, by X, Y, and Z. Draw lines from H, to i, Mfaad ■?r-^ Fig. 11. COMPOSITION AND RESOLUTION OF F0ECE8. 33 iVj respectively ; these will be perpendicular to the axes, and with them, and the force H, will form thyee right-angled triangles. Denote the" angle between H and the axis of vX" by a, that between Ji and the axjs of I^by /S, and that between H and the axis of Z by y ; we shall harve from the right-angled triangles referred to, the follow- ing equations-: !*-^ Fig. U. X — M' cos a, IT = ^vcos /3, and Z — M cos 7. The angles a, /3, and 7, are estimated from the directions of the positive co-ordinates, through 360°. The components above found will be positive when they act in the direction of positive co-ordinates, and negative when they act in a contrary direction. If we regard JT, Y, and Z, as three forces, JR will be their resultant, and we shall have, from a known property of the rectangular paraUelopipedon, B = yX' + Y' + Z' (4.) That is, the resultant of three forces at right angles to each other, is equal to the square root of the sum of tlie squares of tJie components. We also have from the figure, cos a = X cos , j3 = -^, and cosy ,= -^, Hence, the position of the resultant is completely determined. EXAMPLES. 1. Eequired the intensity and dkection of the resultant of three forces at right angles to each other, having the in- tensities 4, 5, and 6 pounds, respectively. 2* 34 MECHANICS. SOLUTION. We have, X= 4, r= 6, and^= 6. .-. B = ^77= 8.115. Also, cosa =. g^, cos^ = ^,andcosr = ^g; whence, a = 62°53', /3 = 55°15'32", and 7 = 46°51'31". Hence, the resultant pressure is 8.775 lbs., and it makes, with the components taken in order, angles equal to 62° 53', 55° 15' 32",. and 46° 51' 31". 2. Three forces at right angles are to each other as the numbers 2, 3, and 4, and their resultant is 60 lbs. What are the intensities of the forces ? '. SOLUTION. We have T-iX, Z= 2X, and J2 = 60 ; Hence, 60 = VX^ + IX' + 4X= = \X^/2^ = 2.6925X: .-. X = 22.284. The components are, therefore, 22.284 lbs., 33.426 lbs., and 44.568 lbs. ■T-^ Projection of Forces. 32. If planes be passed through the extremities of a ' force, perpendicular to the direction of any straight line, that portion of the'line intercepted between them is the pro- jectioii of the force upon the line. The operation of resolv- ing forces into components in the direction of rectangular axes, is nothing more than that of finding their projections upon these axes. If two straight hnes be drawn through the extremities of a force, perpendicular to any plane, and the points in whjih they meet the plane be joined by a straight line, this line is the projection of the force upon the plane. COMPOSITION AND KESOLUTION OF FOEOES. 35 If we denote any force by P, and the angle whicli it makes Tvith any line or plane by a, -P cos a wiU refpresent the projection of the force on the line or plane. In botl^ cases the projection of the force is its effective component in the direction of the line or plane upon which it is projected. Composlton of a Group of Forces in a Plane. /'' 33. Let P, P', P", &c., denote any number of forces ( lying in the same plane, and applied at a common point, and ■ represent the angles which they make with the axis of ^ by a, a', a", &c. Their components in the direction o£^;he axis of ^ are P cos a, P' cos a', P" cos a", &c., and their com- ponents in the direction of the axis of IT, are P sin a, P' sin a', P" sin a", &c. If we ' denote tlie resultant .of the group of components which are parallel to the axis of X. by X, and the resultant of the group parallel to the axis of I^ by yi we shall have, (Alt. 26), X = ,2 (Pcosa), and F = 2 (P sin a) . . (5.) The resultant of Xand IT is the same as the resultant of the given forces. Denoting this resultant by P~, and recollecting that X smdi 3^ are perpendicular to each other, we have, as in Article 30, ' R = -v/Jr2 + Y^ . . . . ( 6.) If we denote the angle which the resultant makes with the axis of X by a, we shall have, as in Article 30, X y cos « = -=-, and sin a = -:=■• EX,AMPLES. 1. Three forces, whose intensities are respectively equal to 50, 40, and 70, lie in the same plane, and are applied at the same point, and make with an axis through that point, angles equal to 15°,^ 30°, and 45°, respectively. Required the intensity and direction of the resultant. 36 MECHANICS. SOLTJTION. "We have, X = 50 cos 15° + 40 cos 30° + '70 cos 45° = 132.435, and 7"= 50 sin 15° + 40 sin 30° +1© sin 45° = 82.45 ; whence, H - ^6198 + 17539 = 156. and cos a = 132.435 _._ ^^31054/24". 156 ' The resultant is 156, and the angle which it makes with the axis is equal to 31° 54' 24". 2. Three forces 4, 5, and 6, lie in the same plane, making equal angles with each other. Required the intensity of their resultant and the angle which it makes with the least force. SOLUTION. Take the least force as the axis of X. Then the angle between it and the second force is 120°, and that between it and the third foi'ce is 240°. We have X = 4 + 5 cos 120° + 6 cos 240° — .— 1.5 ; F= 5 sin 120° + 6 sin 240° = — .866; „ r- 1.5 . 8.66 ••• ^ = VS, cos « = - -■-, sma = - j^; .-. a = 210°. 3. Two forces, one of 5 lbs. and the other of 7 lbs., are applied at the same point, and make with each otlier an angle of 126°. What is the intensity of their resultant? Ans. 6.24 lbs. Composition of a Group of Forces in Space. 34. Let the forces he represented by JP, I", P", &c. The angles which they make with the axis of .X", by a, a', a",' &c., the angles which they make with the axis of T] by (S, j3', (3", &c., and the angles which they make with the axis COMPOSITION AND EESOLUTION OF FORCES. 37 of Z by y, y', y", &c. Resolving each force into compo- nents, respectively parallel to tte three co-ordinate axes, and denoting the resultants of the groups in the directions of the respective axes by X^ Y, and Z, we shall have, as in the preceding article, X = 2 (P cos a), Y—S. (P cos /3), Z = 2 (P cos 7.) If we denote the resultant of the system by J?, and the angles which it makes with the axes by «, 6, and c, we shall have, as in Article 31, R = v^X" + Y' + Z\ JC Y Z cos a = -= , cos 6 = ^, and cos c r= -^ • The application of these formulas is entirely analogous to that of the formulas in the preceding article. Expression for the Resultant of two Forces. 35. Let us consider two forces, P and P', situated ia the same plane. Since the position of the co-ordinate axes is perfectly arbitrary, let the axis of X be so taken as to coincide with the force P/ a will then be equal to 0, arid we shall have sin = 0, and cos a = 1.- " Fig.'i2. The valtie of X (Equation S), will become P -V P' cos a' and the value of Y will be- come P' sin a'. Squaring these values, substitutmg them in Equation ( 6 ), and reducing by the relation sm" a' -f cos'' a.' = 1, we have. a = -^P' + P" + 2PP' cos a' . ( 7.) The angle a' is the angle included between the given forces. Hence, The resultant of any tioo forces, applied at the same point, is equal to the square root of the sum of the squares 38 MECHANICS. of the two forces, plus twice the product of the forces into the cosine of their included angle. If we make a' greater than 90°, and less than 270°, its cosine will be negative, and we shall have. B -y/P^ + P"^ - IPP' COS a'. If we make a' == 0, its cosine wiU be 1, and we shall have, It = P + P'. If we make a' = 90°, its cosine will be equal to 0, and we shall have, B = -yJP' + P'\ If we make a' shall have, 180°, its cosine wUl be B^P - P'. 1, and we The last three results conform to principles already de- duced. Let P and § be two forces, and B their resultant. The figure QP being a parallelogram, the side PB is equal -to Q. From the triangle OBP we have, in accor- dance with the principles of trigo- nometry. Hg. 18. P: Q:B::wy OBP : sin BOP : sin OPB. ( 8.) If we apply a force B' equal and directly opposed to i2, the forces P, g,and B\ wUl be in equilibiium. The angles OBP., and QOB', being opposite exte- rior and interior angles, are sup- plements of each other; hence, sin OBP = sin QOB'. The angles BOP, and POB', are ad- jacent, and, consequently, supple- ^'^ ^*' mentary; hence, sin BOP = sin POB'. The angles COMPOSITION AND EE80LUTI0N OF FOEOES. 39 OP a,, and POQ, are interior angles on the same side, and, con- sequently, supplementary; hence, sin OPB = sin POQ. We have ^alsoi2 = P'. Making these sub- stitutions in the preceding propor- tion, we have, Fig 14 P : Q : P' :: sm QOP' : sin POP' : sin POQ. Hence, if three forces are in equilibrium, each is propor- tional to the sine of the angle between the other two. EXAMPLES. 1. Two forces, P and §, are equal in intensity to 24 and 30, respectively, and the angle between them is 105°. What is the intensity of their resultant ? P = ■\/24:' 4- 30' + 2 X 24 X 30 cos 105° — 33.21. 2. Two forces, P and Q, whose intensities are respec- tively equal to 5 and 12, have a resultant whose intensity is 13. Required the angle between them. 13 — ^25 -f- 144 -j- 2 X 5 X iJt^COS a. .". cos a = 0, or a = 90°. Ans. 3. A boat is impelled by the current at the rate of 4 miles per hour, and by the wind at the rate of 7 miles per hour. What wiU be her rate per hour when the direction of the wind makes an angle of 45° with that of the current? P = -v/16 -f 49 -f 2 X 4 X 7 cos 45° = 10.2m. Ans. 4. A Aveight of 50 lbs., suspended by a string, is drawn aside by a horizontal force until the string makes an angle of 30° with the vertical. Required the value of the hori- zontal force, and the tension of the string. Ans. 28.8675 lbs., and 57.735 lbs. 40 MECHANICS. 5. Two forces, and their resultant, are all equal. What is the value of the angle between the two forces ? 120°. 6. A point is kept at rest by three forces of 6, 8, and 11 lbs., respectively. Required the angles which they make Avith each other. SOLUTION'. We have P = 8, Q := 6, and li' ■— 11. Since the forces are in equilibrium, we shall have H' = H =11; hence from the preceding article. 11 = v'64 + 36 + 96 .cos QOF; .: cos QOP = fi; or, QOP = 11° 21' 52". From the last proportion we have, siXi. FOB' 6 . „/,„, ^„„„, -^-nriW = ^^'^ ■'■ smPOiJ' = .53224; sm QOF 11 or, FOB' = 147° 50' 34". or, Q0i2'.= 134°47'34" '' Principle of Moments. 36. The moment of a force, with respect to a point, is the product obtained by multiplying the intensity of the force by the perpendicular distaiice from the point to the line of direction of the force. The fixed point is called the centre of moments ; the per- pendicular distance is called the lever arm of the. force ; and the moment itself measures the tendency of the force to produce rotation about the centre of moments. COMPOSmON AND RESOLUTION OF FORCES. 41 Let P and Q be any %y,-o forces, and It their resultant ; assume any point (7, in their plane, as the centre of moments, and from it, let fall upon the di- rections of the forces, the per- pendiculars, Cp^ Cq, sjad.Gr; denote these perpendiculars resf pectively by p, q, and r. Thefi will Pp, Qq, and Bi; be the moments of the forces P, Q, and B. Draw CO, and from P let fall the perpendicular PS, upon OP. Denote the angle POP, by a, the angla POQ, or its equal, OPP, by /3, and the g.ngle POGhj cp.. Since PP = Q, we have from the right-angled triangles OPiS and PP8, the equations, P — -Q cos ^ + P cos a. := Q sin ^ — P sin a. Multiplying both members of the first equation by sin 9, and both members of the second by 00s (p, then adding the resulting equations, we find, -K sin 9 =3 Q (sin 9 cos ^ "+ sin ^ cos M and N the given J)oints of application. Through M and JV draw lines parallel to B. Make MA = B, and draw AN, cutting Bin B; make MP = SB and NQ = BB; they will be the required components. 8 3119 At-' — 7?ir — 7B A ri&8B. 60 MECHANICS. For, from the similar triangles AMN and B8N^ BS : AM : : SJSf: MW; or, BS : B :. : SJST : MN. But, from Proportion (8), we hare, P : B:': BN: MN; .: BS z= P, and BB = Q. M y^S ;*B k / Fig. 25. Re- EXAMPLB. Given i2 = 24 lbs., SM = 7 in., and SN = 5 in. quired P and Q. From Proportion (8), we have, 12 :. 7 : : 24 : ^ ; .-. Q = U lbs. 12 : 5 : : 24 : P ; .-. P - 10 lbs. 4. To resolve a given force into parallel components lying in opposite directions, and applied at given points.. Both points of application must lie on the game side of the given force. Let B be the given force, M and JST the given points of application. Through M and iV draw lines parallel to B ; make JSTB = B, and draw BM; through S, draw SA parallel to MB; then, will JVA and BA be equal to the in- tensities of the components. Make MP = AJV, and JSTQ = AB, and they 'wiU. be the components. For, from the triangles ASN, and BMJV, we have, AJV : BJSr : : SN: MJST; or, AN : B : : SN : MN. But, from Proportion (8), we have, P : B : : SN: MN; .: AN = P, and AB = Q. P. :nr-'i! r-g. 26. COMPOSITION AND EESOLtTTION OF FORCES. 51 '^ w :1 EXAMPLE. Given i? = 24 lbs., 81^= 18 in,, and S3I —Qin. Re- quii-ed P and Q. From Proportion (8), we have, -P : 24 : : 18 : 9 ; .-. P = 48 lbs. Q i 2i : : 9:9; .-. $ = 24 lbs. i? = P - § = 24 lbs. 5. To find the resultant of any number of parallel forces. Let P, P', P", P'", be such a system of forces. Find the resultant of P and P', by the rule already given, it will be P' = P' -|^ P' ; find the resultant of P' and JP", it will be B" = JP+ T' + P" ; find the resultant of H" and P", it will be P = P+P+P' + P'". K there is a gi-eater number of forces, the operation of composition may be con- tinued ; the final result wiU be the re- sultant of the system. If some of the .forces act in contrary directions, combine all which act in one direction, as just explained, and call their resultant H' ; then combine all those which act in a contrary direction, and call their resultant H" ; finally, combine P' and JR" by a preceding rule ; their resultant H wUl be the resultant of the system. K P' = P", the resultant will be 0, and its point of ap- pUcation wiU be at an infinite distance. In this case, the forces reduce to -a couple, the effect of which is simply to pro- duce rotation,^ Ziever Arm of the Resultant. 48. Let P, P', P", &o., denote any number of parallel forces, a,nd p,p',p",&a., their lever arms with respect to an axis of moments, taken perpendicular to the common direc- tion of the forces ; denote the lever arm of the resultant of Fig. 27. 52 MECHANICa. the system, taken with respect to the same axis, by r From the prinoii^le of moments (Art. 28), (P + P' + P' + &c.)r = Pp + Pp' + &c- ; . = 1^ . . ■ (-) Hence, the lever arm of a system of parallel forces, taken with respect to an axis at right-angles to- their direction, is equal to the algebraic sum of the moments of the forces divided by the cdgebraic sum of the forces. Centre of Parallel Forces. 43. Let there be any number of forces, P, P, P", &o., applied at points invariably connected, together, and -vvhose co-ordinates are x, y, z; x', y', z ; a;", y", z" ; &c. Let It denote their resultant, and represent the co-ordinates of its point of application, by ajj, y^, and Zj ; denote the angles made by the common direction of the forces with the axes of JE", Y, and Z, by a, ^, and y. Suppose each force resolved into three components, re- spectively parallel to the co-ordinate axes, the points of application being unchanged : The components parallel to the axis of JC are, . Pcosa, P'cosa, P"cosx, &c., -Scosa ; those parallel to the axis of Y" are, Pcos;S, P'cosS, P"cos/3, &c., -Bcos/3 ; and those parallel to the axis of Z are, Poos/, P'cos/, P'cosy, &c., Hoosy. If we take the moments of the components parallel to the axis of Z, with respect to the axis of T, as an axis Sf mo- ments, we shall have, for the lever arms of the components, X, x', x", &c. ; and from the principle of moments (Art. 36), /JcosyiBj = Poosyx+ P'cos/ as' -f &e. COMPOSITION AND KKSOLUTION OF FOEOES. .53 Striking out the common factor cos 7, and substituting for B, its value, we have, whence, In like manner, if ye take the moments of the same com- ponents, with respect to the axis of ^ we shall have, And, if we take the moments of the components parallel to the axis of Y, with respect to the axis of ^, we shall have, 3i = 2(P)- Hence we have for the co-ordinates of the point of appli- cation of the resultant, a^i - 2(p) ' yi - 2(P) ' a°* ^i - 2(P) ■ v^^-; These co-ordinates are entirely independent of the direc- tion of the parallel forces, and will remain the same so long as their intensities and points of application remain un- changed. The point whose co-ordinates we have just found, is called the centre of parallel forces. Resultant of a Group of Forces in a Plane, and applied at points invariably connected. 44. Let P, P', P", &c., be any number of forces lying in the same plane, and applied at points invariably connected together ; that is, at points of the same solid body. "^-Icosa X Kg. 28. 54r MECHANICS, ThroTigli any point in the plane of the forces, draw any two straight lines, OX and Y, at right angles to each other, and "^ lying in the plane of the forces ; assume these as co-ordinate axes. ' Denote the angles which the forces P, P^, P",&c., make with — the axis OJC, by a, a,', a", &o., and the angles which tliey make with the axis OY, by /3, /3', /3", &c. j denote, also, the co- ordinates of the points of application of the forces, by a;, y ; «', y' ; ««", y" ; <&«. Let each force be resolved ihta components parallel to the co-ordinate axes ; we shall have for the group parallel to th& axis of X, Pcosa, P'costt', P"cosa"5 &c. ; ' and, for the group parallel to the- axis of Y, PcosS, P'cos,S', P"cos;3''',.&c.^ The resultant of the first group is equal to the algebraie sum of the components (Art. 32) ; denoting this by X, we shall have, jr=2(Pcosa.) .... (14..) In like manner, denoting the resultant of the secoad group by Y, we shall have, F=: 2(PC08,S) .... (15.) The forces X and Y intersect in a point, which is the point of application of the system of forces. Denoting the resultant by H, we shall have (Art. 33), Ji = V X' +■ Y\ To find the point of application of J?, let O be taken as a centre of moments, and denote the lever arms of X and Y COMPOSITION AND RESOLUTION OF FOE0E8. 55 by yi and cb,, respectively. From the principle of Article 42, we shall have, S{Fcosl3x) ^^ ~ 2(Pcos^) • ■ . . (16.) ^(Pcosay) ^' ~ 2(Pcosa) ■ • • • (17.) K we denote the angles which the resultant makes with the axes of JST and ]F by a and b respectively, we shall have, as in Article 33, COS a — -^i cos 6 = -= . . . ( 18.) Equations (16) and (IV) make known the point of applica- tion, and Equations (18) make known its direction ; hence, the resultant is completely determined. ...v . To find the moment of H, with respect to O as a centre of moments, let us denote its lever arm by r, and the lever arms of -P, -P, P", &c., with respect to 0, hy p,p',p", &c.- The moment of the force Pcosa, is Pcosa y, and that of the force jPcos.S, is — JPcosfS x. The negative sign is given to the last result, because the forces Pcosa and PcoSjS tend to turn the system in contrary directions. From the principle of moments (Art. 36), the moment of F is equal to the algebraic sum of the moments of its com- ponents. Hence, Pp — Pcosa y — PC0S;8 X. In like manner, the moments of the other component forces may be found. Because the moment -df the resultant is equal to the algebraic sum of the moments of all its com- ponents (Art. 36), we have, Br = ^{Pp) = 2(Pcosa y — Pcos/S x) . (19.) 56 MECHANICS. »cl Fig. 29. Resultant of a Group of Forces situated in Space, and applied at 1 ^ points invariably connected. 45. Let P, P', P", &c., be any number of forces situated in any manner in space, and applied at points of the same solid body. Assume any point in space, and through it draw any three lines perpen- dicular to each other. Assume these lines as axes. Denote the angles which the forces P, P', P', &c., make with the axis of JE", by «, a', a", &c. ; the angles which they make with the axis of T, by /3, ^', ^", &c. ; the angles which they make with the axis of Z, by /, 7', /", &c., and denote the co-ordi- nates of their points of application by x, y, z; x', y\ z' ; x", y", s"; &c. Let each force be resolved into components respectively parallel to the co-ordinate axes. We shall have for the group parallel to the axis of X^ Pcosx, Pcosx', P'cosx",-. &c. ; for the group parallel to the axis of J^ Pcos^, Pcos^', P'cos,S", &c. ; and for the group parallel to the axis of Z, Pcos/, Pcos/', P'cos/", &c. Denoting the resultants of these several groups by J^ P; and Z, we shall have, X=:2(Pcosa,) F=i(Pcos,S,) andZ = 2(Pcosy) . (20.) These three forces must intersect at a point, which point is the point of application of the resultant of the entii-e sys- COMPOSITION AND EESOLUTION OF FORCES. 67 tern. Denote this resultant by H; then, since the forces ^X', Y, and Z are perpendicular to each other, we shall have, S- V^" + T' + Z' (21.) To find the Co-ordinates of the point of application of R. Consider each of the forces, X, T, and Z, with respect to the axis whose name comes next in order, and denote the lever arm of JT, with respect to the axis of Y, by gj ; that of Y, with respect to the axis of Z, by x^ ; and that of Z, with respect to the axis of JT, by y^ We shall have as in the last article, "'' ~ X{jPcosIB) yi = «i = 2(Pcosa s) 2{Pcosa) , (22.) in which Xi, j/^, and Sj, are the co-ordinates of the point of application of J2. Denoting the angles which H makes with the axes by a, b, and c, respectively, we have, as in the preceding article, i2' cos a = -v;, cos 5 = ^, cos c = Y Z (23.) The values of ^, Y, andZ, maybe computed by means of Equations (20), and these being substituted in (21), make known the value of the resultant. The co-ordinates of its point of application result from Equations (22), and its line of direction is shown by Equations (23). The iutensity, dii-ection, and point of application being known, the resul- tant is completely determined. 3* 58 MECHANICS. .yy Fig. 89. Measure of the tendency to Rotation about the Axes. 46. Let X, Y", and Z denote the components of the resultant of the system, as in the last artiele, and denote, as g before, the co-ordinates of the point of application of the re- sultant by «!, -y-i, and Zy To find the resultant moment, with re- spect to the axis of Z, it may be observed that the component Z, can produce no rotary effect, since it is parallel to the axis of Z; the moment of the compo- nent Y, with respect to the axis of Z, is Yx-^ ; the moment of the component -X", with respect to the same axis, is — -Zj/j, the negative sign being taken because the force X tends to produce rotation in a negative direction. Hence, the resultant moment of the system, with respect to the axis of Z, is, Tx^ — Xyi ; or, substituting for ^and I^ their values, we have, Tajj - Xyi = 2{Pcos^a; — Pcosay) . (24.) In like manner for the resultant moment of the system, with respect to the axis -2", Zy^ — Fs, - 2(Pcos7 y — Pcos^ s) . ( 25.) And' for the resultant moment, with respect to the axis ofT,' •■• Xsi — Zxj = 2(Pcosas — JPcosyx) . (26.) Equilibrium of Forces in a Plane. ^7. In order that a system of forces lying in the same plane, and applied at points of a free solid, may be in equilibrium, two conditions must be fulfilled : Krst, the resultant of the system must have no tendency to produce EQUILIBRITTM OF FOEOES. 59 motion of translation; and, secondly, it must have no tendency to produce motion of rotation. Conversely, if these conditions are satisfied, the system will be in equi- lihrium. The first condition will he fulfilled, and will only he ful- filled, when the resultant is equal to ; but from Art. 45, we have, R = ■s/'X' + T\ The value of li can only be equal to when -X" = 0, and y = ; or, what is the same thing, 2(Pcosa) = 0, and 2(Pcos/3) = . (27.) The second condition will be fulfilled, and will only be fulfilled, when the moment of the resultant, with respect to any point of the plane, is equal to 0, whence, Br - Q; or, ^Pp) = . . . ( 28.) Hence, from Equations (27) and (28), in order that a system of forces, lying in the same plane, and applied at points of a free solid body, may be in equilibrium, we must have, 1st. The algebraic sum of the components of the forces in the direction of any two rectangular axes separately equal to 0. 2d. 77ie algebraic sum of the moments of the forces, with respect to any point in the plane, equal to 0. £? — ■ C~""----A/r--: :"rh'A ^ the centre of gravity of the triangle -p-^'i lies upon the line DJEJ, drawn from the vertex D to the middle point of the opposite side AG ; it is, therefore, at G-, their point of intersection. 66 MECHANICS. Draw ^D ; then, since ^J) bisects A and J5 C, it is parallel to AJ3, and the triangles ^GD and AGB are similar. The ^ side JSD is equal to one-half of its . / / \ homologous side AS, consequently y^, and the vertical pressure on the pivot A. soiunoN. Denote the weight of the beam, together with its load, by W, and suppose its pomt of application to be at O. Denote 6 A hj p, and the perpendicular distance AI*] from ■ A to D£!, by/)'. Denote also the tension of the cord by t. If we regard A as the centre of moments, we shall have, in the case of an equilibrium, Fig. 51. Wp = tp'; t^W^.^ Or, denoting' the angles EDA by a, and the distance AD by 5, we shall have, t = W-, P fisina To find the vertical pressure on the pivot A, resolve the force t into two components, respectively parallel and per- 86 MKCHANICS. pendioular to AJ3. We shall have for the latter component, denoted by t', t' — t sina — W^ • The vertical pressure upon A, plus the weight W, must he equal to this value of t'. Denoting this pressure by P, we shall have, P + TT^ Ff ; or,P= Tr(f - l) = W^^); or. ^ - ^ AD When 2>C = ; or, when D and C coincide, the vertical pressure becomes 0. 2. A rope AB, supports a pole, D 0, of uniform thick- ness, one end of which rests upon a horizontal plane, and from the other D end is suspended a weight W. Re- ^^^ quired the tension of the rope, and , y^/yP^ the thrust, or pressure, on the pole, j^y^ /Y'' ^'^ the weight of the pole being neg- ■ ^^ " lected. Fig. 52. SOLTJTIOIir. Denote the tension of the rope by t, the pressure on the pole by p, the angle ADO by a, and the angle ODW by/3. There are threeforces acting at D, which hold each other in equilibrium ; the weight W, acting downwards, the ten- sion of the rope acting from D, towards A, and the thrust of the pole acting from towards D. Lay off Dd, to re"present the weight, and complete the parallelogram of forces doaD ; then wiU Da represent the tension of the rope, and Do the thrust on the pole. From Art. 35, we have, t : W: : sm 13 : sina; .-. t^W^^- STABILITY. 87 We have, also, from the same principle, p : W: : sin(a + /3) : sina ; .: p = j^ s^K" + /^) . ■* sm a. If the rope is horizontal, we shall have a =: 90° — /3, which gives, W t = TF tan/3, anAp = COS;S 3. A beam A£, is suspended by two ropes attached at its extremities, and fastened to pins A and IT. Required the tensions upon the ropes. soLtmoii. Denote the weight of the beam and its load by W, and suppose -that C is the point of application of this force. Denote the tension of the „■ „« rope £ir, by t, and that of the rope FA, by i'. The forces acting to produce an equilibrium, are W, t, and t'. The plane of these forces must be verti- cal, and further, the directions of the forces must intersect in a point. Produce AF, and Hff, till they intersect in IT, and draw ^0; lay off JTC, to represent the weight of the beam and its load, and complete the parallelogram of forces, Kb Cf ; then will Kh represent f, and Kf will represent t'. Denote the angle GKB by a, and the angle CKF by ^. We shall have, as in the last problem, W:t : : sin(a + /3) : sin/3 ; .: t= TF^-±^- sm /d And, W:t' :: sin(a + /3) : sina ; .-. t' = W 5^ ±D. ^ ' ' > sm a 4. A gate AH, is supported at upon a pivot, and at J. by a hinge, attached to a post AB. Hequired the pressure on the pivot, and also the tension of the hinge. 88 MECHANICS. A1_ -f S 12 Fig. 54. SOLUTION. Denote the weight of the gate and its load, by W. Produce the vertical through the point of application C, of the force W, till it intersects the hori- zontal through A in D, and draw the line DO. Then will DA and DO represent the directions of the requir- ed components of W. Lay off Dc, to represent the value of W, and complete the parallelogram of forces, Dcoa ; then will Do represent the pressure on the pivot 0, and Da the pres- sure on the hinge, A. Denoting the angle oDc by a, the pressure on the pivot by/;), and on the hinge by^', we shall have, W ^ , W p — , and p — —. cosa. -' sma If we denote the distance 0£J by b, and the distance D^ by h, we shall have, h , . b cos a = -/Fh- h' and sin a Hence, ^r * P = ^ , and^ = ^ 6. Having given the two rafters ^C and DC of a roof, abutting in notches of a tie-beam AD, it is required to find the pressure, or thrust, upon the rafters, and the di- rection and intensity of the pressure upon the joints at the tie-beam. SOLUTION. Denote the weight of the rafters and their load by 2w ; we may regard this weight as made up of three parts — a Kg. 55. STABILITY. 89 ■weight w, applied at O, and two eqiial weights ^w, applied at A and H respectively. Let us denote the half span AZi by s, the rise OJO by 7i, and the length of the rafter ^ C or CIS by I. Denote, also, the pitch of the roof 0J3Xi by «, the thrust on the rafter by t, and the resultant pressure at each of the joints A and J3 by p. Lay off Co to represent the weight w, and complete the parallelogram of forces Cboa ; then will Ca and Cb repre- sent the thrust upon the rafters ; and, since the figure Cboa is a rhombus, we shall have, w wl t sma = iw) .". t = ——. — = —Tf • 2 sma 2/i Conceive the force t to be applied at A, and resolve it into two components respectively parallel to CL and LA ; we shall have for these components, , ws t sma = iw), and t cosa — — =- • ^ ' 2A The latter component gives the strain on the tie-beam, AB. To find the pressure on the joint, we have, acting down- wards, the forces \w and iio, or the single force w, and, act- ing from L towards A, the force —r ; hence. If we denote the angle DAE by (8, we shall have from the right-angled triangle DAE, DE s tan/3=-j^ = ^^. The direction of the joint should be perpendicular to that of the force />, that is, it should make with the horizon an s angle whose tangent equals —j ' 90 MKCHANICS. 6. In the last problem suppose the rafters to abut against the wall. Reqiiired the least thickness that must be given to the wall to prevent it from being overturned. SOLUTION. Denote the entire weight thrown upon the wall by w, the length of that portion of the wall which sustains the pressure p by r, its height by A', its thickness by x, and the weight of each cubic foot of the material of the wall by w' ; then- will the weight of this part of the wall be equal to lo'h'l'x. The force —r- acts with an arm of lever h' to overturn the 2/t wall about its lower and outer edge ; this force is resisted by the weight lo + w'h'l'x, acting through the centre of gravity of the wall with a lever arm equal to ix. If there is an equilibrium, the moments of these two forces must be equal, . WS -, , ,,,,, ,83 wsh' , ,,,„ . that is, —r X h' = {w + whix) -, or —7— = wx + io h'l as • 2,lh U th Reducing, we have, ^ + ,,,-,, «! = ,,,, ; *" ' whl will w 2w w fws w' 7. A sustaining wall has a cross section in the form of a trapezoid, the face upon which the pressure is thrown being vertical, and ^^ the opposite face having a slope of \ six perpendicular to one horizontal. / ,\ ,( Required the least thickness that must / i ; j be given to the wall at the top, that — / J.|. J. _ , J} J*J-C It \j it may not be overturned by a nori- jj. gg zontal pressure, whose point of appli- cation is at a distance from the bottom of the wall equal to one-third of its height. SOLUTION. Pass a plane through the edge A parallel to the face £C, and consider a portion of the wall whose length is one BTABILITT. 91 foot. Denote the pressure upou this portion by P, the height of the wall by 6A, its thickness at the top by a;, and the weight of a cubic foot of the material by w. Let fall from the centres of gravity and 0' of the two portions, the perpendiculars OG and O'JS, and take the edge D as an axis of moments. The weight of the portion AJB GF is equal to QwJkc, and its lever arm, DG, is equal to h + \x. The weight of the portion ^Z'i^is Sio/i", and its lever arm, DjE, is f A. In case of an equilibrium, the sum of the mo- ments of their weights must be equal to the moment of P, whost lever arm is 2/i. Hence, &whx{h + \x) + 3wh' X f A = F x2h; 6whx + Siox' + 210^" = 2P. or. Whence, x' + 2hx 2(P-W0 3w X = — h± 4' 2(P - wK') Sw + h\ 8. Kequired the conditions of stability of a square pillar acted upon by a force oblique to the axis of the pillar, and applied at the centre of gravity of the pillar's upper base. SOLTJTIOIT. Denote the intensity of the oblique force by P, its inclination to the vertical by a, the length or breadth of the pillar by 2a, its height by x, and the weight of the pillar by W. Through the centre ' of gravity of the piQar draw the vertical A (7, and lay off J. C equal to w ; prolong PA and lay off J.P equal to P ; complete the parallelogram of forces ABB G, ' and prolong the diagonal till it intersects HG or IIG produced. If the point F falls between II and G, the pillar will be stable ; if it falls at H, it will be indifferent ; if it falls with- out il, it will be unstable. To find an expression for the .92 MECHANICS. distance FG, draw DE perpendicular to A Q. From the similar triangles ADE and AFG, we have, AGy. DE AE : AG : : DE : F.G; .: FG = But AG = hence we have, AE But AG = X, DE = Psina, and AE = W+ Pcosa, j?T^ _ ^^ sina W + Pcosa ' And, since ITG equals a, we have the following condi- tions for stability, indifference and instability, respectively. a > «<. Fx sina W + Pcosa ' Fx sina W + Pcosa ' J'a; sina TF + Pcosa ' If we denote the distance FG by y, and the weight of a cubie foot of the material of the pillar by W, we shall have, since TF= Aa'xw, FsinoL X il =: . 4:a\ox + Pcosa • - If, now, we suppose the intensity and direction of the force P to remain the same, whilst x is made to assume every possible value from up to any assumed limit, the value of y will undergo corresponding changes. The suc- cessive points thus determined make up a line which is called the line of resistance, arid whose equation is that just deduced. If the pillar is made up of uncemented blookfe, it will re- mam in equilibrium so long as each joint is pierced by the line of resistance, provided that the tangent to the line of resistance makes with the normal to the joint an angle less than that of least resistance (Art. 88). STABILITY. 93 The highest degree of stability will be attaiaed when the line of resistance is normal to every joint, and when it passes through the centre of gravity of each. 9. To determine the conditions' of equilibrium and sta- bility of an arch of imcemented stones. SOLUTION. Let MNLK represent half of an arch sustained in equi- librium by a horizontal force i-*, and by the weight of the arch- .It stones. Through the centre of ^'SkJ-lt "* gravity of the first arch-stone draw /„ yy \ a vertical line, and on it lay off a /~-^v distance to represent the weight i^( I of that stone. Prolong the direc- K^I • ^ tion of P, and lay off a distance ^'S- 53. equal to the horizontal pressure. Complete the parallelogram of forces, oabB, and draw the diagonal oJi. This will be the resultant of the forces com- bined. Combine this resultant with the weight of the second arch-stone, and this with the weight of the third, and so on, till the last inclusive. The polygon oBCDE, thus found, is the line of resistance, and if this lies wholly within the solid part of the arch, the arch will be stable ; but, if it does not lie within it, the arch will be unstable. A rupture will take place at the joint where the line of re- sistance passes withou.t the solid part of the arch. This problem may be solved analytically, in accordance with the principles already illustrated. It is only intended to indicate the general method of proceeding. 94 MECHANICS. CHAPTEE lY. ELEMENTARY MACHINES. Definitions and General Principles. TS. A MACHINE is a contrivance by means of which a force applied at one point is made to produce an effect at some other point. The force applied is called- the power, and the point at which it is applied, is called tJie point of application. The force to be overcome is called the resistance, and the pomt at which it is to be overcome is called the worleing point. The workiQg of any machme requires a contmued applica- tion of power. The source of this power is called the motoi: Motors are exceedingly various. Some of the most im- portant are muscular effort, as exhibited by man and beast in various kinds of work ; the weight and living force of water, as exhibited in the various kinds of water-mills ; the expansive force of vapors and gases, as displayed in steam and caloric engines; the ./brce of air in motion, as exhi bited in the wmdmill, and in the propulsion of sailing vessels ; the force of magnetic attraction and repulsion, sp- shown in the magnetic telegraph and various magnetic machines ; the elastic force of springs, as shown in watches and various other machines. Of these motors, the most important ones are steam, air, and water power. To work, is to exert a certain pressure through a certain distance. The measure of the quantity of work performed by any force, is the product obtained by multiplying the effective pressure exerted, by the distance through which it is exerted. Machines serve simply to transmit aud modify the action of forces. They add nothing to the work of the motor ; on ELEMENTAEY MA0HINK8. 95 the contrary, they absorb and render inefficient much of the woi-k that is impressed upon them. For example, in the case of a -water-mill, only a small portion of the worlc ex- pended by the motor is transmitted to the machine, on account of the imperfect manner of applying it, and of this portion a very large fraction is absorbed and rendered prac- tically useless by the various resistances, so that, in reality, only a small fractional portion of the work expended by the motor becomes effective. Of the applied worlc, a part is expended in overcoming friction, stiffness of cords, hands, or chains, resistance of the air, adhesion of the parts, &o. This goes to wear out the machine. A second portion is expended in overcoming sudden impulses, or shocks, arising from the nature of the work to be accompUshed, as well as from the imperfect con- nection of the parts, and from the want of hardness and elasticity in the connecting pieces. This also goes to strain and loear out the machine, and also to increase the sources of waste already mentioned. There is often a waste of work arising from a greater supply of motive power than is required to attain the desired result. Thus, in the move- ment of a train of cars on a railroad, the excess of the work of the steam, above what is just necessary to bring the train to the station, is wasted, and has to be consumed by the apphcation of brakes, an operation which not only wears out the brakes, but also, by creating shocks, injures and ulti- mately destroys the cars themselves. Such are some of the sources of the loss of work. A part of these may, by judicious combinations and apphances, be greatly diminished; but, under the most favorable cir- cumstances, there must be a continued loss of work, which i-equires a continued supply of power from the motor. In any machine, the quotient obtained by dividing the quantity of useful, or effective work, by the quantity of applied wnrle, is called the modulus of the machine. As the resistances are diminished, the modulus increases, and the machine becomes more perfect. Could the modulus ever 96 MECHANICS. become equal to 1, the machine would be absolutely ^er/ec<. Once set in motion, it would continue to move forever, realizing the solution of the problem of perpetual motion. It is needless to state that, until the laws of nature are changed, no such realization need be looked for. In studying the principles of machines, we proceed by approximation. For a first result, it is usual to neglect the effect of hurtful resistances, such as friction, adhesion, stiff- ness of cords, &c. Having found the relations between the power and resistance under this hypothesis, these relations are afterwards modified, so as take into account the various resistances. We shall, therefore, in the first instance, regard cords as destitute of weight and thickness, perfectly flexible, and inextensible. We shall also regard bars and connecting pieces as destitute of weight and inertia, and perfectly rigid ; that is, incapable of compression or extension by the forces to which they may be subjected. Elementary Machines. 'J6. The elementary machines are seven in number — viz., the cord ^ the lever ; the inclined plane ; the pidley, a combination of the cord and lever; the wheel and axle, also a combination of the cord and lever ; the screw, a combination of two inclined planes twisted about an axis ; and the wedge, a simple combination of two inclined planes. It may easily be seen that there are in reality but three elementary machines — the cord, the lever, and the inclined plane. It is, however, more convenient to consider the seven above- named as elementary. By a suitable combination of these seven elements, the most complicated pieces of mechanism are produced. The Cord. '!"!'. Let AS represent a cord solicited by two forces, P and JR, applied at its extremi- ties, A and £. In order that ^"^ 2~ B ^^ the cord may be in equilibrium, ^. g^ it is evident, in the first place, that two forces must act in the direction of tie cord, and in ELEMENTARY MACHINES. 97 such a manner as to stretch itj otherwise the cord would bend under the action of the forces. In the second place, the intensities of the forces must be equal, otherwise the greater force would prevail, and motion would ensue. Hence, in order that two forces applied at the extremities of a cord may be in equilibrium, the forces must he equal and directly opposed. The measure of the tension of the cord, or the force by which any two of its adjacent particles are urged to sepa- rate, is the intensity of one of the equal forces, for it is evident that the middle point of the cord might be fixed and either force withdrawn, without diminishing or increasing the tension. When a cord is solicited in opposite directions by unequal forces directed along the cord, the tension will be measured by the intensity of the lesser force. Let AJ3 represent a cord solicited by two groups of forces applied at its two extrem- ities. In order that these \ forces may be in equilibrium, "^~~^ the resultant of the group ap- ■^ ^ ^^ plied at ^ and the resultant of the group at JB must be equal and directly opposed. Hence, if we suppose all of the forces at each point to be resolved into components respectively coinciding with, and at right angles to A£, the normal 'components at each of the points must be such as to maintain each other in equilibrium, and the resultants of the remaining components at each of the points A, and JB must be equal and directly opposed. ^ Let ABCD represent a cord, at the different points A, B, G, D, of which are applied groups of forces. If these forces are in equili- brium through the interven- tion of the cord, there njust necessarily be an equili- "^ rig. ei. brium at each point of ap- plication. Denote the tension of AB,JBC, CD, by t, t', t", 5 yo MKOHANIOS. and the forces applied hj P, I", P", &o., as shown in the figure. The forces in equihbrium about the point A are P, I", J"', and t, directed from ^ to -S ; the forces in equili- brium about JB are P'", P'^, t, directed from P to A, and t', directed from P to O. The tension t is the same at all points of the branch AJi, and, since it acts at A in the direc- tion AH, and at P in the direction~P4, it follo\ys that these two forces exactly counterbalance each other. If, therefore, the forces P' and P" were transferred from A to P, unchanged in direction and intensity, the equilibrium at that point would be undisturbed. In like manner, it may be shown that, if all the forces now applied at P be trans- ferred to C, without change of direction or intensity, the equilibrium at G would be undisturbed, and so on to the last point of the cord. Hence we conclude, that a system of forces applied in any manner at different points of a cord will be in equilibrium, when, if applied at a single point without change of intensity or direction, they will maintain each other in equilibrium. Hence, we see that cords in naachinery sunply serve to transmit the action of forces, without in any other manner modifying their effects. The Lever. '78<, A lever is an inflexible bar, free to turn about an axis. This axis is called the fulcrum. Levers are divided into three classes, according to the relative positions of the points of application of the power and resistance. In the frst class, the resistance is- beyond both the power and fulcrum, ^'' ''''^*°- and on the side of the fulcrum. The i i ^ common weighing-scale is an example 1 of this class of levers. The matter to p j/ be weighed is the resistance, the -I counterpoising weight is the power, and the axis of suspension is the fulcrum. ELEMENTARY MACHmES. 99 inn Class. f Fig. 63. Sud Class. F In the second class, the resistance is between the power and the ful- crum. The oar used in rowing a boat is an example of this class of levers. The end of the oar in the water is the fulcrum, the point at which the oar is fastened to the boat is the point of application of the resist- ance, and the remaining end of the oar is the point of application of the power. In the t/iird class, the resistance is beyond both the fulcrum and the power, and on the side of the power. The treadle of a lathe is an example of a lever of this kind. The point at which it is fastened to the floor is the fulcrum, the point at which the foot is applied is the point of application of the power, and the pouit where it is attached to the crank is the point of appUcation of the resistance. Levers may be either curved or straight, and the direc- tions of the power and resistance may be either parallel or oblique to each other. We shall suppose the power and resistance to be situated in planes at right angles to the ful- crum ; for, if they were not so situated, we might conceive each to be resolved into two components — one at right angles, and the other parallel to the axis. The latter com- ponent would be exerted to bend the lever laterally, or to make it slide along the axis, developing only hurtful resist- ance, whilst the former only would tend to turn the lever about the fulcrum. The perpendicular distances from the fulcrum to the Unes of direction of the power and resistance, are called the lever arms of these forces. In the bent lever MJ^JV, the perpen- Flg. 64. 100 MECHANICS. dicular distances JFIA and J^Ji are, respectively, tlie lever arms of JP and H. To determine the conditions of equilibrium of the lever, let us denote the power by jP, the re- sistance by a, and their respec- tive lever arms by p and r. We have the case of a body restrained " j,. gg by an axis, and if we take this as the axis of moments, we shall have for the condition of equilibrium (Art. 49), Pp = Br; or, P : R : : r : p . . ( 36.) That is,, the power is to the resistaru^, as the lever arm of the resistance is to the lever arm of the power. This relation holds good for every kind of lever. The ratio of the power to the resistance when in equili brium, either statical or dynamical, is called the leverage, oi mechanical advantage. When the power is less than the resistance, there is said to be a gain of power, hut a loss of velocity ; that is, the space passed over by the power in performing any work, is as many times greater than that passed over . by the resis- tance, as the resistance is greater than the power. When the power is greater than the resistance, there is said to be a loss of power, but a gain of velocity. When the power and resistance are equal, there is neither gain nor loss of power, but simply a change of direction. In levers of. the first class, there may be either a gain or a loss of power ; in those of the second class, there is always a gain of power ; in those of the third class, there is always a loss of power. A gain of power is always attended with a corresponding loss of velocity, and the reverse. If several forces act upon a lever at different points, all being pei-pendicular to the direction of the fulcrum, they will be in equilibrium, when the algebraic sum of their moments, with respect to the fulcrum, is equal to 0. bd P 4 ■^jP' - 1- p i- i ELEMENTARY MACHINES. 101 This principle enables us to take into account the weight of the lever, which may be regarded as a vertical force applied at the centre of gravity. The pressure on the fulcrum is equal to the resultant of the power and resistance, together with the weight of the lever, when that is considered, and it may be found by the rule for finding the resultant of forces applied at points of a rigid body. The Compovmd Iiever. 79. A compound lever consists of a combination of simple levers A£, B G, CD, so arranged that the resis- 3. -g^n tance in one acts as a power ^ ^, ^ in the next, throughout the combination. Thus, a power P produces at ^ a resis- tance Jt'j which, in turn, produces at C a resistance Kg. 66. M", and so on. Let us as- sume the notation of the figure. From the principle of the simple lever, we shall have the relations, Pp = R'r", R'p' = i2'V, R"p" = Rr. Multiplying these equations together, member by member, and striking out the common factors, we have, Ppp'p" = Rrr'r" ; or, P : i2 : : rr'r" : pp'p". ( 37.) We might proceed in a similar manner, were there any number of levers in the combination. Hence, in the compound lever, the power is to the resis- tance as the continued product of the alternate arms of lever, commencing at the resistance, is to the continued pro- duct of the alternate arms of lever, commencing at the power. By suitably adjusting the simple levers, any amount of mechanical advantage may be obtained. 102 . MECHANICS. The following combination is used where a great pressure is to be exerted through a very small distance : The Elbow-joint Press. SO. Let CA, SD, and DJE represent bars, with hinge- joints at JB and D. The bar CA, has a fulcrum at -^ (7, and the bar DE works /^^^^^>^ B ,,--\^ through a guide between / J^^^Sss-^ '^ D and M When ^ is II^^-^^^^^^^C depressed, DJE is forced | against the upright F, so Fig. 67. as to compress, with great force, any body placed between E and F. This machine is called the elbow-joint press, and is used in printing, in moulding bullets, in striking coins and medals, in punching holes, riveting steam boilers, &c. Let P denote the force applied at A, perpendicular to A C, Q the resistance in the direction I)JB, and It the com- ponent of Q, in the direction ET>. Let G be taken as an axis of moments, and then, because P and Q are in equili- brium, we shall have, Px AG^ Qx FG, or, Q=P x ~^- If we draw PJT perpendicular to DJS, we shall have, cos PDJT = -jyn ; Ibut we have, for the component JR, TiTT B = Qoo8£I>JI= q X ^- JJ-tS Substituting for Q its value, and reducing, R _ AG BH P'' FG^ DB' When B is depressed, DH and BB approach equality, and FG continually diminishes ; that is, the mechanical ad- vantage increases, and finally, when B reaches ER, it becomes infinite. There is no limit to the pressure exerted at F, except that fixed by the strength of the machine. ELEMENTARY MACHINES. 103 Fig. 68. The Balance. SI. A Balance is a machine for -weighing bodies : it consists of a lever AJB, called the beam, a knife-edge fulcrum JF] and two scale-pans D and £J, suspended by knife-edges from the extremities of the lever arms J^£ and ^A. These arms should be symmetrical, and of equal length; the knife- edges A, JB, and F, should all lie in the same plane, and be perpen- dicular to a plane through their middle points and the centre of gravity of the beam ; they are, therefore, parallel to each other. This condition of parallelism in the same plane, is of essential importance. In addition to this, the middle points of the knife-edges A, £, and J^, should be on the same straight line, perpendicular to the plane through the fulcrum It] and the centre of gravity of the beam. The knife-edges should be of hardened steel, and their supports should either be of polished agate, or, Tvhait is stUl better, of hardened steel, so as to diminish the eifect of friction along the lines of contact. The fulcrum may be made horizontal, by leveling-screws passing' through the foot-plate i. A needle W, projects upwards, or some- times downwards, vrhich, playing in front of a graduated arc GJT, serves to show the deflection of the line of knife- edges from the horizontal. When the instrument is not in use, the fulcrum may be raised from its bearings by a pinion JS] working into a rack in the interior of the standard .Kffl The knife-edges A and Ji ma^, by a similar arrangement, be raised from their bearings also. The ordinary balances of the shops are similar in their general plan ; but many of the preceding arrangements are omitted. The scale-pans being exactly alike, the balance should remain in equilibrium, with the line A£ horizontal, not only when the balance is without a load, but also when the pans are loaded with equal weights ; and when AJB is 104 MECHANICS. deflected from the horizontal, it should return to this posi- tion. This result is attained by throwing the centre of . gravity sUghtly below the line AB. To test a balance, let two weights be placed in the pans that will exactly counter- balance each other, then change the weights to the opposite pans ; if the equilibrium is still maintained, the balance is said to be true. The sensibility of a balance is its capability of indicating small diiferences of weight. The sensibility will be greater, as the le?iffths of the arms increase, as the centre of gravity of the beam approaches the fulcrum, as the mass of the load decreases, and as the length of the needle increases. The centre of gravity of the beam being below the fulcrum, it may be made to approach to or recede from it, by a solid ball of metal attached to the beam by means of a screw, by which it may be raised or depressed at ple'asure. The remaining conditions of sensibility will be limited by the strength of the material, and the use to which it is to be apphed. Should it be found that a balance is not true, it may stiU be employed, with but slight error, as indicated below. Denote the length of the lever arms, by r and r', and the weight of the body, by W. When the weight W is applied at the extremity of the arm r, denote the counterpoising weights employed, by W ; and when it is applied at the extremity of the arm »•', denote the counterpoising weights employed, by W". We shall have, from the principle of the lever, Wr = W'r', and Wr' = Wr. Multiplying these equations, member by member, we have, Wrr' = W" Wrr.' ; .-. W~ y^' W" ; that is, the true weight is equal to the square root of the pro- duct of the apparent weights. A still better method, and one that is more free from the eflfects of errors in construction, is to place the body to be HinMnillirTTTTTfl uiiH Fig. 69. ELEMENTAKY MACHINES. 105 weighed in one scale and add counterpoising weights till the beam is horizontal ; then remove the body to be weighed and replace it by known weights till the beam is again hori- zontal ; the sum of the replacing weights will be the weight required. If, in changing the loads, the positions of the knife-edges are not moved, this method is almost exact, but this is a condition difficult" to fulfill in manipulation. The Steelyard. 83. The steelyard is an instrument used for weighing bodies. It consists of a lever AJi, called the beam; a ful- crum F; a scale-pan D, attached at the extremity of one arm; and a known weight -EJ movable along the other arm. We shall / \ "" E suppose the weight of JiJ to be 1 lb. This instrument is sometimes more conve- nient than the balance,- but it is more inaccurate. The con- ditions of sensibility are essentially the same as for the balance. To graduate the instrument, place a pound-weight in the pan D, and move the counterpoise £J till the beam rests horizontal — ^let that point be marked 1 ; next place a 10 lb. weight in the pan, and move the counterpoise £1 till the beam is again horizontal, and let that point be marked 10 ; divide the intermediate space into nine equal parts, and mark the points of division as shown in ,the figure. These spaces may be subdivided at pleasure, and the scale ex- tended to any desirable hmits. We have supposed that the centre of gravity coincides with the Mcrum ; when this is not the case, the weight of the instrument must be taken into account as a force appUed at its centre of gravity. We may then graduate the beam by experiment, or we may compute the lever arms, corresponding to the difierent weights, by the general principle of moments. To weigh any body with the steelyard, place it in the scale-pan and move the comiterpoise £J along the beam till 5* 106 MECHANICS. an equilibrium is established bet-ween the two ; the cor- responding mal-k on the beam will indicate the weight. The bent Lever Balance. 83. This balance consists of a bent lever AC£; fulcrum G ; a scale-pan D ; and a graduated arc £JF, whose centre co- incides with the centre of motion C. When a weight is placed in the scale-pan, the pan is de- pressed and the lever- arm of the weight is diminished ; the weight S is raised, and its lever-arm increased. When the moments of the two forces become equal, the instrument will come to a state of rest, and the weight will be indicated by a needle projecting from .B, and playing in front of the arc JFTJ. The zero of the arc jEI<^ is at the point indicated by the needle when there is no load in the pan D. The instrument may be gTaduated experimentally by~ placing weights of 1, 2, 3, &o., pounds in the pan, and mark- ing the points at which the needle comes to rest, or it may be graduated by means of the general principle of moments. We need not explain this method of graduation. To weigh a body with the bent lever balance, place it in the scale-pan, and note the point at which the needle comes to rest ; the reading will make known the weight sought. Compound Balances. 84. Compound balances are much used in weighing heavy articles, as merchandise, coal, freight for shipping, &c. A great variety of combinations have been employed, one of which is annexed. A£ is a platform, on which the object to be weighed is ELKMENTAET MACHINEB. 107 Fig. n. placed ; £G is a guard firmly attached to the platform ; the platform is supported upon the knife-edge fulcrum £J, and the piece D, through the medium of a brace CJ) ; GJFis a lever turn- ing about the fulcrum F, and suspended by a rod from the point ^ ; Z]V is a lever having its fulcrum at M, and sustaining the piece ^ by a rod JKJI; is a scale-pan suspended from the end iVof the lever iiV! The instrument is so constructed, that ^JJP: GF:: KM: LM; and the distance KM is generally made equal to J^ ofMJST. The parts are so arranged that the beam XiV shall rest horizontally in equilibrium when no weight is placed on the platform. If, now, a body Q be placed upon the platform, a part of its weight will be thrown upon the piece J), and, acting downwards, will produce an equal pressure at K. The remaining part will be thrown upon K,- and, acting upon the lever KG, will produce a downward pressure at G, which will be transmitted to X ; but, on account of the relation given by the above proportion, the effect of this pressure upon the lever i2V"wiU be the same as though the pressure thrown upon K had been applied directly at K. The final effect is, therefore, the same as though the weight of Q had been applied at K, and, to counterbalance it, a weight equal to j\ of Q must be placed in the scale-pan 0. To weigh a body, then, by means of this scale, place it on the platform, and add weights to the scale-pan tiU the lever ZiJV ia horizontal, then 10 times the sum of the weight added will be equal to the weight required. By making other combmations of levers, or by combining the princi- 108 MECHANICS. pie of tte steelyard with this balance, objects may be weighed by usiag a constant counterpoise. EXAMPLES. 1. In a lever of the first class, the lever arm of the resistance is 2| inches, that of the power, 33J, and the resistance 100 lbs. What is the power necessary to hold the resistance in equilibrium ? Ans. 8 lbs. 2. Four weights of 1, 3, 5, and Y lbs. respectively, are suspended from points of a straight lever, eight inches apart. How far from the point of apphcation of the first weight must the fulcrum be situated, that the weights may be in equilibrium ? SOLtTTIOlT. Let X denote the required distance. Then, from Art. (36) 1 Xx+3{x—8) + B{x — 16) + 1{x - 24) = ; x — 17 in. Ahs. 3. A lever, of uniform thickness, and 12 feet long, is kept horizontal by a weight of 100 lbs. applied at one extremity, and a force -P applied at the other extremity, so as to make an angle of 30° with the horizon. The fulcrum is 20 inches from the point of application of the weight, and the weight of the lever is 10 lbs. What is the value of P, and what is the pressure upon the fulcrum ? SOIXSTION. The lever arm of P is equal to 124 in. x sin 30° = 62 in., and the lever arm of the weight of the lever is 52 in. Hence, 20 X 100 = 10 X 52 + P X 62 ; .-. P = 24 lbs. nearly. We have, also, H = V'X" + T' = -v/{110 + 24 sin 30°)» 4- (24 cos BOS')'. .-. i2 = 123.8 lbs. ; ELEMENTAET MACHINES. 109 , X 20.V85 and, cos« = -^= ^^^ = .16789; .-. a = 80° 28' 10". 4. A heavy lever rests on a fulcrum wMcli is 2 feet from one end, 8 feet from the other, and is kept horizontal by a weight of 100 lbs., applied at the first end, and a weight of 18 lbs., apphed at the other end. What is the weight of the lever, supposed of uniform thickness throughout ? SOt-UTION. Denote the required weight by x ; its arm of lever is 3 feet. We have, from the principle of the lever, 100 X2=:£BX3 + 18X8; .-. x = 18|-lbs. Ans. 5. Two weights keep a horizontal lever at rest ; the pressure on the fulcrum is 10 lbs., the difference of the \ weights is 3 lbs., and the difference of lever arms is 9 inches. \ [ ^ What are the weights, and then- lever arms ? ; Ans. The weights are 1 lbs. and 10 lbs. ; their lever arms , JL are 15| in., and 6f in. 6. The apparent weight of a body weighed in one pan of a false balance is 5^- lbs., and in the other pan it is 6j\ lbs. What is the true weight ? W= -/y X If =: 6 lbs. '7. In the preceding example, what is the ratio of the lever arms of the balance ? SOLUTION. Denote the shorter arm by I, and the longer ai-m by nl. We shall have, from the principle of moments, 61 = 51 X nl, or, 6nl = GyV ; .-. n =: lyV- That is, the longer arm equals lyV times the shorter arm. 110 MECHANICS. The Inclined Plane. S5. An inclined plane is a plane inclined to the horizon. In this machine, let the power be a force applied to a body either to prevent motion down the plane, or to produce -motion up the plane, and let the resistance be the. weight of the body acting vertically downwards. The power may be applied in any direction whatever ; but we shall, for sim- pUcity's sake, suppose it to be in a vertical plane, taken per- pendicular to the inclined plane. Let A£ represent the inclined plane, O a body resting on it, R the weight of the body, and P the force applied to hold it in equilibrium. In order that these two forces may keep the body at rest, friction being neglected, their resultant must be perpendicular to AJB{An.n). ;p.g^2_ When the direction of the force P is given, its intensity may be found geometrically, as fol- lows : draw OR to represent the weight, and Q perpen- dicular to ^.S ; through P draw P, Q parallel to OP, and through Q draw QP parallel to OP ; then will OP repre- sent the required intensity, and OQ the preffl3ure on the plane. . ■"-. When the intensity of P is given, its direction may be found as follows : draw OP and § as before ; with B as a centre, and the given intensity as a radius, describe an arc cutting .OQm Q; draw PQ, and through draw OP parallel, and equal to PQ', it will represent the direction of the force P. If we denote the angle between P and P by (p, aad the inclination of the plane by a, we shall have the angle POQ equal to a, since OQ is perpendicular to AB, and OP to A 0, and, consequently, the angle Q OP = (p — a. From the principle of Art. 36, we have, P : P : : sxaa : sin((p — a) ... (38,) ELEMENTAET MACHINES. Ill From which, if either P or (p he given, the other can he found. If we suppose the power to he applied parallel to the plane, we shall have,

<^ 2. The distance between the threads of a screw is ^ of an inch. What resistance can be supported by a power of 60 lbs., acting at the extremity of a lever 15 inches long ? ^'^ ■ Ans. 16,964 lbs. 3. The distance from the axis of the tr unions of a gun weighing 2,016 lbs. to the elevating screw is 3 feet, and the distance of the centre of gravity of the gun from the same axis is four inches. If the distance between the threads of the screw be f of an inch, and the length of the lever 5 inches, what power must be applied to sustain the gun in a horizon- tal position? Ans. 4.754 lbs. The Wedge. lOO. The wedge is a solid, bounded by a rectangle BD, called the back; two equal rect- angles, AJF and DF, called faces; and two equal isosceles triangles, called 1» ends. The Une -EZ^ in which the faces meet, is called the edge. The power is applied at the back, to which its direction should be normal, and the resistance -is applied to the faces, and in directions normal ^ to them. One half of the resistance ^ig.89. 128 MEOHANICS. is applied normally to one face, and the other half normally to the other face. Let AJBG be a section of a wedge made by a plane at right angles to the edge. . Denote the power by P, and the resistance opposed to each face by \Ii ; denote ^ the angle J3AC of the wedge by 2gj. Produce, the directions of the resistances till they intersect in O. This point will be on the line of direc- tion of the power. Lay oif OF to represent the power, and complete the parallelogram JED; then will 0J3 and OD repre- sent the resistances developed by the power. Let each of the forces \R be resolved into two components, one per- pendicular to OF, and the other coinciding with it. The two former will be equal and directly opposed to each other, whilst the two latter will hold the force P in equilibrium. Since DF is perpendicular to FO, and DO perpen- dicular to GA, the angle ODE is equal to the angle OAO, or ip. The component of ^H in the direction of OF, is -^TJsin^ ; hence, twibe this, or i2sin(p = P. But sins = -=-r = ^, in which b denotes the breadth of the CA I back P C, and I the length of the face GA. Substituting this expression for sinp, and reducing, we have, Bx^h = Pl,orP:P::\b:l . ( 49.) That is, the power is to the resistance as one-half of the breadth of the back is to the length of the face of the loedge. The mechanical ^vantage of the wedge may be increased by diminishing the breadth of the back, or, in other words, by making the edge sharper. The principle of the wedge finds an important application in all cutting instruments, as knives, razors, and the like. By diminishing the thickness of the back, the instrument is rendered liable to break, hence the necessity of forming cutting instruments of the hardest and most tenacious niaterials. ELEMENTARY MACHINES. 129 General remarks on Elementary Machines. 101. We have thus far supposed the power and resist- ance to be in equilibrium, through the intervention of the machine, their points of application being at rest. If we now suppose the point of application to be moved through any distance, by the action of an extraneous force, the point of application of the power will move through a correspond- ing space. These spaces will be described in conformity with the design of the machine ; and it will be found, in each instance, that they are inversely proportional to the forces. If we suppose these spaces to be infinitely small, they may, in all cases, be regarded as straight lines, which will also be the virtual velocities of the forces. If the point of application moves in a direction contrary to the direction of the resistance, the point of application of the power wiU move in the direction of the power. If we denote the paths described by those points respectively, by 8r, and Sp, we shall have,' PSp — Mr = ; or PSp = MSr . . ( 50.) That is, the algebraic sum of the virtual moments is equal to 0, Or, we might enunciate the principle in another man- ner, by saying, that m all cases, the quantity of work of the power is equal to the quantity of work of the resistance. We shall illustrate this principle, by considering a single case, that of the single movable pulley, leaving its further application to the remaining machines, as exercises for the student. In the figure, suppose that an extraneous force acts to raise the resistance R, through the infinitely small space DJS, denoted by Sr ; the point of application of P must be raised through the infinitely small space FG, denoted 7 by Sp, in order that the equilibrium may be ^l' preserved In order that the resistance may be raised YD E through the distance DE, both branches of the rope enveloping the jDulley must be shortened by the same amount; or, what is the same Fig. 91. 130 MKCHANICS. thing, the fi-ee end of the rope must ascend through twice the distance DE. Hence, hp = 25r. But, from the conditions of equilibrium, Multiplying these equations, member by member, we have, P^p - JRSr. Hence, the principle is proved for this particular case. In like manner, it may be shown to hold good for all of the elementary machines. The principle of equality of work of the power and resist- ance being true for any infinitely short time, it must neces- sarily hold good for any time whatever. Hence, we con- clude, that the quantity of work of the power, in overcoming any resistance, is equal to quantity of work of the resist- ance. Although, by the application of a very small J)ower, we are able to overcome a very great resistance, the space passed over by the point of application of the power must be as much greater than that passed over by the point of application of the resistance, as the resistance is greater than the power. This is generally expressed by saying, that what is gained in power is lost in velocity. We see, therefore, that no power is, or can be, gained ; the only function of a machine being to enable a smaller force to accomplish in a longer time, what a larger force would be required to perform in a shorter time. Friction. 102. Feiction is the resistance which one body experi- ences in moving upon another, the two being pressed together by some force. This resistance arises from inequalities in the two surfaces, the projections of one sur- face sinking into the depressions of the other. In order to overcome this resistance, a sufficient force must be applied HURTFUL HESISTANCES. 131 to break off, or bend down, the projecting points, or else to lift the moving body clear of the inequalities. The force thus applied, is equal, and directly opposed to the force of friction, which is tangential to the two surfaces. The force which presses th« surfaces together, is normal to them both at the point of contact. Friction is distinguished as sliding and rolling. The for- mer arises when one body is drawn upon another ; the lat- ter when one body is rolled upon another. In the case of rolling friction, the motion is such as to Uft the projecting points out of the depressions ; the resistance is, therefore, much less than in sliding friction. Between certain bodies, the friction is somewhat different when motion is just beginning, &om what it is when motion has been established. The friction developed when a body is passing from a state of rest to a state of motion, is called friction of quiescence '; that which exists between bodies in motion, is called friction of motion. The following laws of friction have been established by numerous experiments, viz. : First, ther friction of quiescence between tJie same bodies, is proportional to the normal pressure, and independent of the extent of the surfaces in contact. Secondly, the friction of motion between the same bodies, is proportional to the normal pressure, und independent, both of the extent of the surfaces in contact, and of the velocity of the moving body. Thirdly, /or" compressible bodies, the friction of quiescence is greater than the friction of m,otion / for bodies which are sensibly incompressible, the difference is scarcely appre- ciable. Fourthly, friction may be greatly diminished, by inter- posing unguents between the rubbing surfaces. Unguents serve to fiU up the cavities of surfaces, and thus to diminish the resistances arising from their roughness. For slow motions and great pressures, the more consistent unguents are used, as lard, tallow, and various mixtures ; SlC 132 JIKCHANICS. for rapid motions, and light pressures, oils are generally em- ployed. The ratio obtained by dividing the entire force of friction by the normal pressure, is called the coefficient of friction ; the value of the coefficient of friction for any two substances, may be determined experimentally as follows : Let AB be a horizontal plane formed of one of the substances, and let be a cubical block of the other substance resting upon it. Attach a string OC, l^ip to the block, so that its direc- Kg. 92. tion shall pass through its cen- tre of gravity, and be parallel to AB ; let the string pass over a fixed pulley C, and let a weight F, be attached to its extremity. Increase the weight F till the body just begins to slide along the plane, thei/will this weight measure the whole force of friction. Denote this weight by F, that of the body, or the normal pressure, by P, and the coefficient of friction, hj f. Then, from the definition, we shall have, f=- J p In this manner, values for f corresponding to different substances, may be found, and an-anged in tables. This experiment gives the friction of quiescence. If the weight F is such as to keep the body in uniform motion, the resulting value of/ will correspond to friction of motion. The value of/", for any substance, is called the unit^ or coefficient of friction. Hence, we may define the unit, or coefficient of friction, to be the friction due to a normal lyressure of one pound. Having given the normal pressure in pounds, and the unit of friction, the entire friction will be found by multi- plying these quantities together. HUHTFUL RESISTANCES. 133 Kg. 93. There is a second method of finding the value of/ ex- ' perimentally, as follows : Let AJB be an inclined plane, formed of one of the sub- stances, and a cubical block, formed of the other substance, and resting upon it. Elevate the plane tUl the block just begins to sUde down the plane by its own weight. Denote the angle of in- clination, at this instant, by a, and the weight of 0, by W. Resolve the force W into two components, one normal to the sur- face of the plane, and the other one parallel to it. Denote the former component by P, and the latter by Q. Since O TP" is perpendicular to A C, and OP to A£, the angle WOP is equal to a. Hence, P z= TFcosa, and Q — TFsina. The normal pressure being equal to TFcosa, and the force of friction being Tf^ina, we shall have, from the principles already explained, _ TTsina BG TFcosa tana =: AG The angle a is called the angle of friction. "^ Iiimiting Angle of Resistance. 103. Let AB be any plane surface, and a body rest- ing upon it. Let B, be the resultant of all the forces acting upon it, in- cluding the weight applied at the centre of gravity. Denote the angle between B and the normal to AB, ■ by a, and suppose ^ to be resolved into two components P and Q, the former parallel to AJB, and the latter perpendicular to it ; we shall have, P = iJsina, and Q = ^cosa. 134 MECHANICS. The friction due to the noTmsd^ressure will be equal to yiJcosa. Now, when the tangeriiial component 7?sina is less than /LRcosa., the body will remain at rest ; when it is greater than fJicosm., the body will shde along the plane ; and when the two are equal, the body will be in a state bordering on motion along the plane. Placing the two equal, we have, fUcosx = ^sina ; .•._/"= tana. The value of a is called the limiting angle of resistance, and is equal to the inclination of the plane, when the body is about to slide down by its own weight. If, now, the V-— U*^^ line OH be revolved about the normal, it \ i W will generate a conical surface, within /A/ 7 which, if any force whatever, including / o / the weight, be applied at the centre of Fig. 95. gravity, the body will remain at rest, and without which, if a sufficient force be applied, the body wiU slide along the plane. This cone is called the limiting cone of resistance. The values of/, or the coefficient of friction, in some of the most common cases, as determined by MoEisr, is appended : TABLE. £o(U6s between wMch fricUon takes place. CoeffiGWid offri(M(m. Iron on oak, .62 Cast iron on oak, .49 Oak on oak, fibres parallel, .... .4S Do., do., greased, .10 Cast iron on cast iron, .15 Wrought iron on wrought iron, . . .14 Brass on iron, ,16 Brass on brass, .20 Wrought iron on cast iron, . . . .19 Cast iron on elm, .19 Soft limestone on the same, .... .64 Hard limestone on the same, ... .38 HURTFUL RESISTANCES. 135 BpMea ietwmi which friction takes place. Ooefficimt offHcHm. Leather belts on wooden pulleys, . .47 Leather belts on cast iron pulleys, . .28 Cast iron on cast iron, greased, . . .10 Pivots or axes of wrought or cast iron, on brass or cast iron pillojvs : 1st, when constantly supplied with oil, .05 2nd, when greased from time to time, .08 3rd, without any apphcation, ... .15 Rolling Prictiou. 104. RolHng friction is the resistance which one body offers to another when roUing along its surface, the two being pressed together by some force. This resistance, like that in sliding friction, arises from the inequalities of the two surfaces. The coefficient, or unit, of rolling friction is equal to the quotient obtained by dividing the entu-e force of friction by the normal pressure. This coefficient is much less than the coefficient of sliding friction. The following laws of friction have been established, when a cylindrical body or wheel rolls upon a plane : First, the coeffidient of rolling friction is proportioned to the normal pressure : Secondly, it is inversely proportional to the diameter of the cylinder or loheel: Thirdly, it increases as the surface of contact and velocity increase. In many cases there is a combination of both sliding and roUing friction in the same machine. Thus, in a car upon a railroad-track, the friction at the axle is sliding, and that between the circumference of the wheel and the track is rolling. Adhesion. 105. Adhesion is the resistance which one body ex- periences in moving upon another in consequence of the cohesion existing between the molecules of the surfaces in contact. This resistance increases when the surfaces are 136 MECHANICS. allowed to remain for some time in contact, and is very slight when motion has been established. Both theory and experiment show that adhesion between the same sm-faces is proportional to the extent of the surface of contact. The coefficient of adhesion is the quotient obtained by dividhig the entire adhesion by the area of the surface of contact. Or, denoting the entire adhesion by A, the area of the surface of contact by /S, and the coefficient of adhesion by a, we have, A A = aS. To find the entire adhesion, we multiply the unit of adhesion by the area of the surface of contact. Stiffness of Cords. 106. Let represent a pulley, with a cord AjB, wrapped around its circumference, and suppose a force P, applied at J5, to over- come the resistance It^ and impart motion to the pulley. As the rope winds upon the puUey, at C, its rigidity acts to increase the arm of lever of R, and to overcome this resistance to flexure an additional force is required. For the same pulley, this addi- tional force may be represented by the j,; 95 algebraic expression, 05+ hB, in which a and b are constants dependent upon the nature and construction of the rope, and 72 is the resistance to be overcome, or the tension of the cord^ C. The values of a and b for different ropes have been ascertained by experi- ment, and tabulated. Finally, if the same rope be wound upon pulleys of different diameters, the additional force is found to vary inversely as their diameters. If the diameter of the pulley be denoted by D, and the resistance due to stiffness of cordage be denoted by 8, we shall have. HUETFUL EKSIST^U^CES. 137 „ a + bH In the case of the pulley, if we neglect friction, we shall have, when the motion is uniform, P = i2 + ^^, for the algebraic expression of the conditions of equilibrium. The .values of a and b have been determined experi- mentally for all values of i? and D, and tabulated. Atmospheric Resistance. 107. The atmosphere exercises a powerful resistance to the motion of bodies passing through it. This resistance is due to the inertia of the particles of air, which must be overcome by the force of a moving body. It is evident, in the first place, other things being equal, that the resistance will depend upon the amount of surface of the moving body which is exposed to the air in the direction of the motion. In the second place, the resistance must increase with the square of the velocity of the moving body ; for, if we sup- pose the velocity to be doubled, there wUl be twice as many particles met with in a second, and each particle will collide against the moving body with twice the force, hence ; if the velocity be doubled, the resistance will be quadrupled. By a similar course of reasoning, it may be shown that, if the velocity be tripled, the retardation will become nine times as great, and so on. If, therefore, the retardation corresi^ond- ing to a square foot of surface, at any given velocity, be determined, the retardation corresponding to any surface and any velocity whatever may be computed. Influence of Friction on the Inclined Plane. 10§. Let it be required to determine the relation between the power and resistance, when the power is just on the point of imparting motion to a body up an inclined plane, friction being taken into account. 138 mkchVnics. Let AB represent the plane) the body, OP the power on the point of imparting motion up the plane, and OR the weight of the body. Denote the power by P, the weight by JR, the in- clination of the plane by a, and the angle between the direction of the power and the normal to the plane by ^. Let P and R be resolved into components re- j,. ^ spectively parallel and perpendi- dicular to the plane. We shall have, for the parallel com- ponents, Rsa\.a and J*sin/3, and for the perpendicular com-^ ponents, i?cosa and Paos^. The resultant of the normal components will be equal to -Rcosa — Pcos/3 ; and, if we denote the coefficient of friction by /, we shall have for the entire force of friction (Art. 102), /■(-Kcoso. — Jfeos^). When we consider the body on the eve of motion up the plane, the component Psm^ must be equal and directly opposed to the resultant of the force of friction and the component Rsma. ; hence, we must have, ^in^ — ^sinc, +_/" (iZcosx — Pcos/3). Performing the multiplications indicated, and reducing, we have, p = b\'^^^^±^A . . . (51.) ( sm/3 + /coSjS j If we suppose an equilibrium to exist, the body being on on the eve of motion down the plane, we shall have. PsinjS -f /(iJcosa — Pcos/3) = iJsina. Whence, by reduction, P^i?i^-^l . . . (52.) ( sm^ — /cos/3 j ^ ' HURTFUL RESISTANCES. 139 Pi-om these expressions, two values of P may be found, when a, /3, /, and B are given. It is evident that any value of P greater than the first will cause the body to slide up the plane, that any value less than the second will permit it to slide down the plane, and that for any inter- mediate value the body will remain at rest on the plane. If we suppose P to be parallel to the plane, we shall have sin/3 = 1, cos/3 = 0, and the two values of P reduce to P = J2(sina + /cosa) . . . ( 53.) and, P = i?(sina — /cosa) , . . ( 54.) If friction be neglected, we have /" = 0; whence, by substitution, „ „. P BG P=Bsmu, or ^ = ^^-; a result which agrees with that deduced in a preceding article. To find the quantity of work of the power whilst drawing a body up the ■ entire length of the inclined plane, it may be observed that the value of P, in Equation (53), is equal to that required to maintain the body in uniform motion after motion has commenced. Multiplying both members of that equation by AB, we have, P X AB = B X AB sin a + /B x AB cosa = B X BO + fB X AG. But B X BG is the quantity of work necessary to raise the body through the vertical height B G ; and fB x A G, is the quantity of work necessary to draw the body horizon- tally through the distance A G (Art. 75). Hence, the quan- tity of work required to draw a body up an inclined plane, when the power is parallel to the plane, is equal to the quan- tity of work -necessary to draw it horizontally across the base of the plane, plus the quantity of work necessary to raise it vertically through the height of the plane. 140 MECHANICS. A curve situated in a vertical plane may be regarded as made up of an infinite num^ber of inclined planes. . We infer, therefore, that the quantity of work necessary to draw a body up a curve, the power acting always parallel to the direction of the cuiwe, is equal to the quantity of work ne- cessary to draw the body over the horizontal projection of the curve, plus the quantity of work necessary to raise the body through a height equal to the difference of altitude of the two extremities of the curve. The last two principles enable us to compare the quanti- ties of work necessary to draw a train of ears over a hori- zontal track, and up an inclined track, or a succession of inclined tracks. "We may, therefore, compute the length of a horizontal track which will consume the same amount of work, furnished by the motor, as is actually consumed in consequence of the undulation of the track. We are thus enabled to compare the relative advantages of different proposed routes of railroad, with respect to the motive power required for working them. Iiine of Least Traction. 109. The force employed to draw a body with uniform motion along an inclined plane, is called the force of trac- tion / and the line of direction of this force is the line of traction. In Equation (51), JP represents the force of trac- tion required to keep a body in uniform motion up an inclined plane, and /3 is the angle which the line of traction makes ivith the plane. It is plain, that when 13 varies, other things being the same, the value of P will vary ; there will evidently be some value of |8, which will render P the least possible ; the direction of I' in this case, is called the line of least traction ; and it is along this line that a force can be applied with greatest advantage, to draw a body up an inclined plane. If we examine the expression for P, in Equation (51), we see that the numerator remains constant; therefore, the expression for JP will be least possible when the denominator is the greatest possible. By a simple pro- HURTFUL EESISTANCES. 141 cess of the Differfintial Calculus, it may be shown that the denominator will be the greatest possible, or a maximum, when, / = cot /3, or / = tan(90° — /3). That is, the power wiU be applied most advantageously, when it makes an angle with the inclined plane equal to the angle of friction. From the second value of P, it may be shown, in like manner, that a force wUl be most advantageously appUed, to prevent a body from sliding down the plane, when its direc- tion makes an angle with the plane equal to the supplement of the angle of friction, the angle being estimated as before from that part of the plane lying above the body. Friction on an Axle. HO. Let it be required to determine the position of equilibrium of a horizontal axle, resting in a cyliadrical box, when the power is just ^ . on the point of overcoming the friction if \ between the axle and box. \{i^^W ) Let 0' be the centre of a cross section ^^^m J of the axle, the centre of the cross sec- \ — j-'^^ tion of the boi, and N their point of con- \ ■ tact, when the power is on the point of j,. gg overcoming the friction between the axle and box. The element through iVwUl be the line of con- tact of the axle and' box. When the axle is only acted upon by its own weight, the element of contact will be the lowest element of the box. If, now, a power be, appUed to turn the axle in the direction indicated by the arrow-head, the axle will roll up the inside of the box until the resultant of all the forces acting upon it becomes normal to the surface of the axle at some point of the element through N. This normal force pressing the axle against the box, wiU give rise to a force of friction act- ing tangentially upon the axle, which will be exactly equal to the tangential force applied at the circumference of the 142 MECHANICS. axle to produce rotation. If the axle he rolled further up the side of the box, it -will slide back to iV; if it be moved down the box, it will roll back to W, under the action of the force. In this position of the axle, it is in the condition of a body resting upon an inclined plane, just on the point of sliding down the plane, but restrained hj the force of fric- tion. Hence, if a plane be passed tangent to the surface of the box, along the element ]V, it wiU make with the horizon an angle equal to the angle of friction. The rela- tion between the power and resistance may then be found, as in Art. 108. EECTItlNEAE MOTION. 143 CHAPTEE Y. EEOTILllIEAB AND PEEIODIO MOTION. Motion. 111. A material point is in motion when it continually changes its position in space. When the path of tha moving point is a straight line, the motion is rectilinear ; when it is a curved line, the motion is cwrvilinear. When the motion is curvilinear, we may regard the path as made up. of infi- nitely short straight lines ; that is, we may consider it as a polygon, whose sides are infinitely small. If any side of this polygon be prolonged in the direction of the motion, it will he a tangent to the curve. Hence, we say, that a point always moves in the direction of a tangent to its path. Unlfonn Motion. 112. Unieoem motion is that in which the moving point describes equal spaces in any arbitrary equal portions of time. If we denote the space described in one second by V, and the space described in t seconds by s, we shall have, from the definition, s = vt; .: V = - . . . (55.) V From the first of these equations, we see that the space de^ciribed in any time is equal to the product of velocity and the time ; and, from the second, we see that the velo- city is equal to the space described in any time, divided by that time. These laws hold true for all cases of uniform motion. If we denote by ds the space described in the infinitely short time dt, we shall have, from the last principle, v = % (56.) dt 144 SJECHANICS. ■which is the differential equation of uniform motion, v being constant. Clearing this equation of fractions, and integ- rating, we have, s -vt+ G . . . . (57.) which is the most general equation of uniform motion. If, in {51), we make t = 0, we shall have, s = a Hence, we see that the constant of integration represents the space passed over by the point, from the origin of spaces up to the beginning of the time t. This space is called the initial space. Denoting it by s', we have, s = vt+s' .... (58.) , If s' = 0, the origin of spaces corresponds to the origin of times, and we have, s = vt, the same as the first of Equations ( 55.) Varied Motion. ff 113. Varied motion is that in which the velocity is continually changing. It can only result from the action of an incessant force. To find the differential equations of varied motion, let us denote the velocity at . the time t, by v, and the space passed over up to that time, by s. In the succeeding instant dt, the space described will be ds, and the velocity gener- ■ ated vstU be dv. Now, the space ds, which is described in the infinitely small time dt, may be regarded as having been described with the uniform velocity v. Hence, from Equar tion (55), we have, ^ = 1 (^^•) Let us denote the acceleration due to the incessant force at the time t, by (p. We have seen (Art. 24), that the meas- EECTTLINEAR, MOTION. 145 ure of the acceleration due to a force, is the velocity that it can impart in a unit of time, on the hypothesis that it acts uniformly during that time. Now, it is plain that, so long as the force acts uniformly, the velocity generated will be proportional to the time, and, consequently, the measure of the acceleration will be, the quotient obtained by dividing the velocity generated in any time, by that time. The quan- tity (p is, in general, variable ; but it may be regarded as constant during the instant dt ; and from what has just been said, we shaU have, ''=dt ^^°-) / Differentiating Equatioij^ (59), we have, , d's which, being substituted in Equation (60) gives, ^=^ («^-) Equations (59), (60), and (61) are the differential equa- tions required. The acceleration q> , is the measure of the force exerted when the mass moved is the unit of mass (Art. 24) ; in any other case, it must be miiltiplied by the mass. Denoting the entire moving force applied to the mass m by jF^ we shall have, d^s F =mjtsf = m-^ . . . . ( 62.) This value of F is the measure of the effective moving force in the direction of the body's motion. When a body moves upon any curve in space, the motion may be regard- ed as taking place in the direction of three rectangular axes. If we denote the effective components of the moving force in the direction of these -axes, by ^, Y, and Z, the spaces 7 14:6 MECHANICS. described being denoted by x, y, and z, we shall have, from (62), ^=='^-^' ^^'^-di^^ ^='^W^- Unifomily Varied Motion. 114. Unifoemlt vaeibd motion is that in which the velocity increases or diminishes tmiformly. In the former case, the motion is accelerated ; in the latter case, it is re- tarded. In both cases, the moving force is constant. De- noting the acceleration due to this constant force, by/, we shall have, from Equation (61), *=/ (•'■) Multiplying by dt, and integrating, we have, %^ft+C . . . (64.) ds or, since -=- is equal to «, Equation (59), at v=ft-V G ... (65.) Multiplying both members of (64) by dt, and integratmg, we have, s^lff+Gt^-C ... (66.) Equations (65) and (66) express the relations between the velocity, space, and time, in the most general case of uniformly varied motion. These equations involve the two constants of integration G and C", which serve to make them conform to the different cases that may arise. To de- termine the value of these constants, make ^ = in the two equations, and denote the corresponding values of « and s, by v' and s'. We shall have, G =v'. G' = «'. . EECTILINEAE MOTION. 147 That is, C is equal to the velocity at the beginning of the time t, and C is equal to space passed over up to the same time. These values of the velocity and space are called, respectively, the initial velocity, and the initial space. Substitutmg for C and C" these values in (65) and (66), they become, • V = v' +ft^ (67.) s^s' + v't + \ff . . (68.) From these equations, we see that the velocity at any time t, is made up of two parts, the initial velocity, and the velocity generated during the time t ; we also see, that the space is made up of three parts, the initial space, the space due to the initial velocity for the time t, and the space due to the action of the incessant force during the same time. By giving suitable values to v' and s'. Equations (67) and (68) may be made to express every phenomenon of varied motion. If we suppose both v' and s' equal to 0, the body wiU move from a state of rest at the origin of times, anJ Equations (67) and (68) will become, ■o = ft (69.) s = iff (70.) From the first of these equations, we see that, in imiformly varied motion, the velocity varies as the time ; and, from the second one, we see that the space described varies as the square of the time. If, iu Equation (70), we make < = 1, we have, s = if; or, / = 2s. That is, when a body moves from a state of rest, under the action of a constant force, the acceleration is equal to twice the space passed over in the_first second of time. If, in the preceding equations, we suppose f to be essen- tially positive, the motion wiU be uniformly accelerated ; if we suppose it to be negative, the motion will be uniformly 148 MECHANICS. retarded. In the latter case, Equations (67) and (68) become, v = v'-ft (VI.) s ^s' + v't-^ff . . . (72.) Application to Falling Bodies. 115. The E'Oeob of geavity is the force exerted by the earth upon all bodies exterior to it, tending to draw them towards it. It is found by observation, that this force is directed towards the centre of the earth, and that its intensity varies inversely, as the square of the distance from tJie centre. Since the centre of the earth is so far distant from the surface, the variation in intensity for small elevations above the surface wiU be inappreciable. Hence, we may re- gard the force of gravity at any place on the earth's sur- face, and for small elevations at that place, as constant, in which case, the equations of the preceding article' become immediately applicable. The force of gravity acts equally upon all the particles of a body, and were there no resistance offered, it would impart the same velocity, in the same time, to any two bodies whatever. The atm^osphere is a cause of resistance, tending to retard the motion of all bodies faUing through it ; and of two bodies of equal mass, it retards that one the most, which offers the greatest surface to the direc- tion of the motion. In discussing the laws of faUing bodies, it will, therefore, be found convenient, in the first place, to regard them as being situated in vacuum, after which, a method will be pointed out, by means of which the veloci- ties can be so diminished, that atmospheric resistance may be neglected. Let us denote the acceleration due to gravity, at any point on the earth's surface, by g, and the space fallen through in the time t, by h. Then, if the body moves from a state of rest at the origin of times, Equations (69) and (70) will give, V — gt ( 73.) h = \gt ( 74.) EECTJLIIJEAR MOTION. 149 From these equations, we see that the velocities at two different times are proportional to the times, and the spaces to the squares of the times. It has been found by experiment that the velocity im- parted to a body ia one second of time by the action of the force of gravity in the latitude of New York, is about 32-^ feet. Making ^ = 32i ft., and giving to t the successive values 1% 2% 3% &c., in Equations (73) and (74), we shall have the results indicated in the following TABLE. TIME ELAPSED. TELOOITIES ACQUIRED. SPACES DISCKIBBD. SECONDS. FEET. FEET. 1 32i 16A 2 64J ^ 3 961 144| 4 128| 257^ 5 150f 402Jj &C. &c. &o. Solving Equation (74) with respect to t, we have, t = 9 (75.) That is, tTie tim,e required for a body to fall through any height is equal to the square root of the quotient obtained by dividing twice the height in feet by 32tV- Substituting this value of t in Equation ( 73), we have, f = 9\ or v' = igh; 150 MECHANICS. ■whence, by solving with reference to v and h respectively, V = ■\/'2gh, and A = — ■ • ( 76.) These equations are of frequent use ia dynamical investiga- tions. In them the quantity v is called the velocity due to the height h, and the quantity A, the height due to the velocity v. If we suppose the body to be projected downwards with a velocity «', the circumstances of motion will be made known by the Equations, V ~ v' + gt, h — v't + Igf. In these equations we have supposed the origin of spaces to be at the point at which the body is projected down- wards. Motion of Bodies projected vertically upwards. 116. Suppose a body to be projected vertically upwards from the origin of ^spaces with a velocity «', and afterwards to be acted upon by the force of gravity. Li this case, the force of gravity acts to retard the motion. Making in (VI) and (72), s' = o, f ^= g, and s = h, they become,' V = v'.— gt {^'^■) h = v't-lgt' .... (18.) In these equations, h is positive when estimated upwards from the origin of spaces, and consequently negative, when estimated downwards from the same point. From Equation (11), we see that the velocity diminishes as the time increases. The velocity will be 0, when, v' v' — gt = 0, or when t = —■ g v' If t contiaues to increase beyond the value — , v wiU 9 RECTILINBAE MOTION. 151 become negative, and the body will retrace its path. Hence, the time required for the body to reach its highest elevation, is equal to the initial velocity divided by the force of gravity. * Eliminating t from Equations {^1) and CZS), we have, ^ ( 79.) 2g ^ ' Making « = 0, in the last equation, we have, h= ^ (80.) Hence, the greatest height to which the body will ascend, is equal to the square of the initial velocity, divided by twice the force of gravity. This height is that due to the initial velocity (Art. 115). v' If, in Equation {11), we make « = t', we find, v = gt' (81.) v' y ^ K, in the same equation, we make t — -^y^ t, we find, v= -gf (82.) Hence, the velocities at equal times before and after reaching tJie highest points, are equal. The difference of signs shows that the body is moving in opposite directions at the times considered. If we substitute these values of v successively, in Equa- tion {19), we shall, in both cases, find _ v'^ — gH'^ . 2g which shows that the points at which the velocities are equal, both in ascending and descending, are equally distant from the highest point ; that is, they are coincident. Hence, 152 MECHANICS. if a tody he projected vertically upwards, it will ascend to a certain point, and then return upon its path, in such a man- ner, that the velocities ifi ascending and descending will be equal at the same points. EXAMPLES. 1. Through what distance will a body fall from a state of rest ia vacuum, in 10 seconds, and through what space will it faU during the last second ? Ans. 1608J ft., and 305^ ft. 2. In what time will a body fall from a state of rest through a distance of 1200 feet ? Ans. 8.63 sec. 3. A body was observed to fall through a height of 100 feet in the last second. How long was the body falling, and through what distance did it descend ? SOLUTION. If we denote the distance by h, and the time by t, we shall have, h = igt\ and A - 100 = ig{t — 1)" ; .-. t = 3.6 sec, and h = 208.44 ft. Ans. 4. A body falls through a height of 300 feet. Through what distance does it fall in the last two seconds ? The entire time occupied, is 4.32 sec. The distance fallen through in 2.32 sec, is 86.57 ft. Hence, the distance re- quired is 300 ft. — 86.57 ft. = 213.43 ft. Ans. 5. A body is projected vertically upwards, with a veloci- ty of 60 feet. To what height will it rise ? Ans. 55.9 ft. 6. A body is projected vertically upwards, with a veloci- ty of 483 ft. -In what time will it rise to a height of 1610 feet? We have, from Equation (78), 1610 = 483* -. le^^f ; .-. t =' V/# ± \W> or, t = 26.2 sec, and t = 3.82 sec. The smaller value of t gives the time required ; the larger EECTILINEAE MOTION. 153 value of t gives the time occupied in rising to its greatest height, and returning to the point which is 1610 feet from the starting point. 7. A body is projected vertically upwards, with a veloci- ty of 161 feet, from a point 214f feet above the earth. In what time will it reach the surface- of the earth, and with what velocity will it strike ? SOLUTION. The body will rise from the starting point 402.9 ft. The time of rising will be 5 sec. ; the time of fallmg from the highest point to the earth wiU be 6.2 sec. Hence, the re- quired time is 11.2 sec. The required velocity is 199 ft. 8. Suppose a body to have fallen through 50 feet, when a second begins to fall just 100 feet below it. How far 'tvill the latter body fall before it is overtaken by the former ? Ans. 50 feet. Restrained Vertical Motion. liy. We have seen that the entire force exerted in moving a body is equal to the acceleration, multiplied by the mass (Art. 24). Hence, the acceleration is equal to the moving force, divided by the mass. In the case of a falling body, the moving force varies directly as the mass moved ; and, consequently, the acceleration is independent of the mass. If, by any combination, the moving force can be diminished whUst the mass remains unchanged, there will be a corresponding diminution in the acceleration. This object may be obtained by the combination represented in the figure. A represents a fixed pulley, mounted ,on a horizontal axis, in such a manner that the /"TN fi-iction shall be as small as possible ; W and W are unequal weights, attached to a flexible cord passing over the pulley. If we suppose the weight TF greater than W, the former will ^^^, descend and draw -the latter up. If the dif- |1|W ference is very small, the motion will be very Fig. 99. slow, and if the instrument is nicely constructed, \^ 154 MECHANICS. ■we may neglect all hurtful resistances as inap- preciable. Denote the masses of the weights ^_^^ W and W, by m and m', and the force of /^ -f- gravity, by g. The weight W is urged down- V_y wards by the moving force mg, and this mo- tion is resisted by the moving force m'g. Hence, the entire moving force is equal to ^VT' ^^ mg — m'g, or, (m — m')g, and the entire mass p;^, gp moved, is m + m\ since the cord joining, the weights is supposed inextensible. If we denote the accd- eration by g', we shall have, from what was said at the beginning of this article, m — m' / „„ X a' = -,g (83.) By diminishing the difference between m and m', we may make the acceleration as small as we please. It is plain that g' is constant; hence, the motion of TF"is uniformly varied. /yyj lyY) If we replace ff by -, -,g, in Equations (73) and (74), they will make known the circumstances of motion of the body W. This principle is employed to illustrate the laws of falling bodies by means of Atwood's machine. Had the two weights under consideration been attached to the extremities of cords passing around a wheel and its axle, and in different directions, it might have been shown that the motion would be uniformly varied, when the mo- ment of either weight exceeded that of the other. The same principle holds good in the more complex combinations of pulleys, wheels and axles, &c. In practice, however, the hurtful resistances increase so rapidly, that even when the moving force remains constant, the velocity soon attains a maximum limit, after which the motion will be sensibly uniform. EXAMPLES. 1. Two weights of .5 lbs. and 4 lbs., respectively, are suspended from the extremities of a cord passing over a EEOTILINEAE MOTION. 155 fixed pulley. What distance will each weight describe in the first second of time, what velocity will be generated in one second, and what will be the tension of the connecting cord ? SOLUTION. Since the masses are proportional to the weights, we shall have, ^' =1^^ = -9 X32ift. = 3.574ft. Hence, the velocity generated is 3.574 ft., and the space passed over is 1.787 .ft. To find the tension of the string, denote it by x. The moving force acting upon the heavier body, is (5 — x)g, and the acceleration due to this force, — r — jff; the moving force acting upon the lighter body, — I — )^' But since the two bodies move together, these accelerations must be equal. Hence, /5 —x\ /x — 4\ .-. a; = 4f lbs., the required tension. 2. A weight of 1 lb., hanging on a pulley, descends and drags a second weight of 6 lbs. along a horizontal plane. Neglecting hurtful resistances, to what will the accelerating force be equal, and through what space will the descending body move in the first second ? SOLUTION. The moving force is equal to 1 x g', and the mass moved is equal to 6. Hence, the acceleration is equal to ^ = 5.1944 ft., and the space described will be equal to 2.5972 ft. 3. Two bodies, each weighing 5 lbs., are attached to a string passing over a fixed pulley. What distance will each 156 MECHANICS. body move in 10 seconds, when a pound weight is added to one of them, and what velocity will have been generated at the end of that time ? SOLTITTOIT. The acceleration will be equal to yVP" = 2.924 ft. = g'. But, s = \g't^-, "0 = g't. Hence, the space described in 10 seconds is 146.2 ft., and the velocity generated is 29.24 ft. 4. Two weights, of 16 oz. €ach, are attached to the ends of a string passing over a fixed pulley. What weight must be added to one of them, that it may descend through a foot in two seconds ? SOLUTION. Denote the required weight by x ; the acceleration wUl be equal to and 32 + SB ^ = 2, we have, 2;8 g — g'. But s = i^g'f : making s = 1 32 + a; X 321 = (}7*&7 oz. Ans. Atwood's Machine. 118. Atwood's machine is a contrivance to illustrate the laws of falling bodies. It consists of a vertical post AB, about 1 2 feet in height, supporting, at its upper extremity, a fixed pulley A. To obviate, as much as possible, the resistance of friction, the axle is made to turn upon friction rollers. A fine silk string passes over the pulley, and at its two extremities are fastened two equal weights G and D. In order to impart motion to the weights, a small weight G^ in the form of a bar, is laid upon the weight C, and by diminishing its mass, the acceleration may be rendered as small as desirable. The^ vertical rod AB, graduated to feet and decimals, is provided with two sliding stages JE and F; the upper one is in the form of a ring, which will permif the Kg. 100. KECTILINEAK MOTION. 157 weight (7, to pass, but not the bar G ; the lower one is in the form of a plate, which is intended to intercept the weight C. There is also connected with the instrument a seconds pendulum for measuring tune. Let us suppose that the weights of G and 7), are each equal to 181 grains, and that the weight of the bar 6r, is 24 grains. Then will the acceleration be 94. g' = 7 = 2 ft. : ^ 362 + 24 -^ ' and smce h = W^"^-: ^'^^ '^ = 9'^ {-^-rt. 116), we shall have, for the case in question, h = f , and v — 2t. If, in these equations, we make t = 1 sec, we shall have h — 1, and v = 2. If we make t = 2 sec, we shall, in like manner, have A = 4, and « = 4. If we make t =z 3 sec, we shall have h = 9, and w = 6, and so on. To verify these results experimentally, commencing with the first. The weight G is drawn up till it comes opposite the of the graduated scale, and the bar G is placed upon it. The weight thus set is held in its place by a spring. The ring JEJ is set at 1 foot from the 0, and the stage li] is set at 3 feet from the 0. When the pendulum reaches one of its extreme limits, the spring is pressed back, the weight G, G descends, and as the pendulum completes its vibratioij, the bar G strikes the ring, and is retained. The acceleration then becomes 0, and the weight G moves on uniformly, with the velocity that it had acquired, in the first second ; and it will be obsei-ved that the weight G strikes the second stage just as the pendulum completes its second vibration. Had the stage JT been set at 5 feet from the 0, the weight G would have reached it at the end of the third vibration of the pendulum. Had it been 7 feet from the 0, it would have reached it at the end of the fourth vibration, and so on. To verify the next result, we set the ring JS at four feet 158 mp:chanics. from the 0, and the stage i?' at 8 feet from the 0, and pro- ceed as before. The ring will intercept the bar at the end of the first vibration, and the weight will strike the stage at the end of the second vibration, and so on. By making the weight of the bar less than 24 grains, the acceleration is diminished, and, consequently, the spaces and velocities correspondingly diminished. The results may be verified as before. Motion of Bodies on Inclined Planes. 119. If a body be placed on an inclined plane, and abandoned to the action of its own weight, it will' either slide or roU down the plane, provided there be no friction between it and the plane. If the body is spherical, it will roU, and in this case the friction may be disregarded. Let the weight of the body be resolved into two components ; one perpendicular to the plane, and the other parallel to it. The plane of these components will be vertical, and it will also be perpendicular to the given plane. The efiect of the first component wiU be counteracted by the resistance of the plane, whilst the second component wiU act as a constant force, continually urging the body down the plane. The force being constant, the body will have a uniformly varied motion, and Equations (67) and (68) wiU be immediately applicsible. The acceleration will be found by projecting the accelei'ation due to gravity upon the inclined plane. Let AJB represent a section of the inclined plane made by a vertical plane taken perpendicular to the given plane, and let P be the centre of gravity of a body renting on the given plane. Let J'Q repre- sent the acceleration due to gravity, denoted by g, , and let Pi? be the , j,; ^^j , component of g, which is parallel to A£, denoted by, g\ I^S being the normal component. Denote the angle that AJ3 makes Vith the horizontal plane by a. Then, since PQia perpendicular to £0, and QH to EBOTILIHEAE MOTION. 169 A£, the angle JR^P is equal to ABC, or to a. Hence we have, from the right-angled triangle PQR, g' = gsma.. But the triangle AB G is right-angled, and, if we denote its height AC ^yJ h, and its length AB by I, we shall have sina = y , which, being substituted above, gives, 9'='-^ (84.) This value of ^' is the value of the acceleration due to the moving force. Substituting it for f in Equations (67) and -, (68), we have, y If the body starts from rest at A, taken as the origin of spaces, then will v' = and s' = 0, giving, v = ^t (85.) V ^ = fl'' (««•) To find the time required for a body to move from the top to the bottom of the plane, make s = ;, in (86) ; there will result. 21 t; .: . = ?y^.(87.) Hence the time varies directly as the length, and inversely; as the square root of the height. For two planes having the same height, but different lengths, the radical factor of the value of t will remain con- 160 SEECHANICS. stant. Hence, the times required for a body to move dmon any two planes having the same height, are to each other as their lengths. To determine the velocity ^vitl^ which a body reaches the hottom of the plane, suhstitute for t, in Equation (85) its value taken from Equation (8^). We shall have, after reduction, / V = '\/lgh. But this is the velocity due to the height h (Art. 115). Hence, the velocity generated in a body whilst moving down any inclined plane, is equal to that generated in falling freely through the lieight of the plane. • EXAMPLES. 1. An inclined plane is 10 feet long and 1 foot high. How long will it take for a body to move from the top to the bottom, and what velocity will it acquire in the descent ? SOLUTION. We have, from Equation (SV), T '--' gh^ substituting for I its value 10, and for h its value 1, we have, t ^ 2^ seconds nearly. From the formula v = s/lgh, Ave have, by making A = 1, V = v^64.33 = 8.02 ft. 2. How far will a body descend from rest in 4 seconds, on an inclined plane whose length is 400 feet, and whose height is 300 feet ? Ans. 193 ft. 3. How long .will it take for a body to descend 100 feet on a plane whose length is 150 feet, and whose height is 60 feef? Ans. 3.9 sec. EECTIUNEAR MOTION. 161 4. There is an inclined railroad track, 2^ miles in length, whose inclination is 1 in 35. What velocity will a car attain, in running the whole length of the road, by its o^vvn weight, hurtful resistances being neglected ? Ans. 155. "75 ft., or, 106.2 m. per hour. 5. A railway train, having a velocity of 45 miles per hour, is detached from the locomotive on an ascending grade of 1 in 200. How far, and for what time, will the train continue to ascend the inclined plane ? SOLUTION. "We find the velocity to be 66 ft. per second. Hence, 66 =; V^gh ; or, h = 67.7 ft. for the vertical height. Hence, 67.7 X 200 = 13,540 ft., or, 2.5644 m., the dista,nce which the train will proceed. We have, /2" t — I \ —T = 410.3 sec, or, 6 min. 50.3 sec, V gh for the time required to come to rest. 6. A body weighing 5 lbs. descends vertically, and draws a weight of 6 lbs, up an inclined plane of 45°. How far wiU. the first body descend in 10 seconds ? SOLUTION. The moving force is equal to 5 — 6 sin 45° ; and, conse. sequently, the acceleration, 5^6_dn^4_5: ^^ ^6 + 5 11 ' .-. s = ig't' — 3.4409 ft. A)is. Motion of a Body down a succession of Inclined Planes. 120. If a body start from the top of an inclined plane, with an initial velocity v', it will reach the bottom with a velocity equal to the initial velocity, increased by that due to the height of the plane. This velocity, called the terminal velocity, will, therefore, be equal to that which the body 162 MECHANICS. would have acquired by falling freely through a height equal to that due to the initial velocity, increased by that of the plane. Hence, if a body start from a state of rest at A, and, after having passed over one inclined plane AH, enters upon a second plane £G, without loss of velocity, it will reach ^ j,. ^^^ the bottom of the second plane with the same velocity that it would have acquired by falling freely through DC, the sum of the heights of the two planes. Were there a succession of inclined planes, so ar- ranged that there would be no loss of velocity in passing from one to another, it might be shown, by a similar course of reasoning, that the terminal velocity would be equal to that due to the vertical distance of the terminal point below the point of starting. By a course of reasoning entirely analagous to that em- ployed in discussing the laws of motion of bodies projected vertically upwards, it might be shown that, if a body were projected upwards, in the direction of the lower plane, with the terminal velocity, it would ascend along the several planes to the top of the highest one, where the velocity would be reduced to 0. The body would then, under the action of its own weight, retrace its path in such a manner that the velocity at every point in descending would be the same as in ascending, but in a contrary direction. The time occupied by the body in passing over any part of its path in descending, w^ould be exactly equal to that occupied in passing over the same portion in ascending. In the preceding discussion, we have supposed that there is no loss of velocity in passing from one plane to another. To ascertain under what circumstances this condition will be fulfilled, let us take the two planes A£ and B G. Prolong J3 C upwards, and denote the angle ABE, by cp. Denote the velocity of the body on reaching _B, by v'. Let v' be resolved into two components, one in the direction of B G, and the other at right angles to it. The effect of the latter PERIODIC MOTION. 163 ■will be destroyed by the resistance of the plane, and the former will be the effective velocity in the direction of the plane B G. From the rale for decomposition of velocities, we have, for the effective component of '«', the value v' cosp. Hence, the loss of velocity due to change of direction, is v' — «' cos(p ; or, v'(\ — cos:p), which is equal to v' ver-sin!p. But when (p is infinitely small, its versed-sine is 0, and there will be no loss of velocity. Hence, the loss of velocity due to change of direction wiU always be 0, when the path of the body is a curved line. This principle is general, and may be enunciated as follows : When a body is constrained to describe a cwvilinear path, there will be no loss of velo- city in consequence of the change in direction of the body'' a motion. Periodic Motion. 121. Periodic motion is a kind of variable motion, in which the spaces described in certain equal periods of time are equal. This kind of motion is exempUfied in the pheno- mena of vibration, of which there are two cases. 1st. Heotilinear vibration. Theory indicates, and experi- ment confirms the fact, that if a particle of an elastic fluid be slightly disturbed from its place of rest, and then aban- doned, it will be urged back by a force, varying directly as its distance from the position of equilibrium ; on reaching this position, the particle will, by vii-tue of its inertia, pass to the other side, again to be urged back, and so on. To determine the time required for the particle to pass from one extreme position to the opposite one and back, let us denote the displacement at any time t by s, and the accelera- tion due to the restoring force by (p ; then, from the law of the force, we shaU have 9 == n^s, in which n is constant for the same fluid at the same temperature. Substituting for

A is called the amplitude of vi- bration; half of this angle ADB, denoted by a, is called the angle of deviation y and I is called the length of the pendulum. If the point starts from rest, &t A, it will, on reaching any point S, of its path, have a velocity «, due to the height EK, denoted by h. Hence, V — s/^gh (92.) If we denote the variable angle HDJB by ^, we shall Fig. 104. PERIODIC MOTION. 167 have DK = Icos^ ; we shall also have DJEJ — Icom ; and since h is equal to D£^ — DJE, we shall have, h = 1 (cos^ — cosa). Which, being substituted in the preceding formula, gives, V — ■\/lgl(c,09A — cosa). From the preceding article, we have, at Equating these two values of «, we have, i-jf= ■\/2gl{- mation, 2, to that of integration, /, and to replace m by dM, and r by x. This gives, Kz=fx'dM (106.) Example 1. To find the moment of inertia of a rod or bar of uniform thickness with respect to an axis through its centre of gravity and perpendicular to the length of the rod. Let AB represent the rod, G its centre of gravity, and E any element contained by planes at right angles to the length of the rod and infinitely near each other. Denote the mass of the rod by Jf, its length, by 2Z, the distance GE, by x, and j,, ^^^ the thickness of the element E, by dx. Then will the mass of the element E be equal to G MOMBITT OF INERTIA. 183 M -J dx. Substituting this for dM, in Equation (106), and integrating between the limits — I and + I, we have, — I For any parallel axis whose distance from G is d, we shall have. IC' = ilf (I + d''j ( 107.) These two formulas are entirely independent of the breadth of the filament in the direction of the axis DO. They will, therefore, hold good when the filament AB is replaced by the rectangle ITM In this case, M becomes the mass of the rectangle, 21 the length of the rectangle, and d the distance of the centre of gravity of the rectangle from the axis parallel to one of its ends. Mcample 2. To find the moment of inertia of a thin circular plate about one of its diameters. Let ACB represent the plate, AB the axis, and CD' any element parallel to AB. Denote the radius C, by r, the distance OE, by a;, the breadth of the element EF, by t&, and its length D C, by 2y. IS we denote the entire mass of the plate, by Jf, the mass of the element CD will be equal to Jf ^ , ; or. •gr smce y = -y/r' — £B% we have, ,?jf = jfVrEZ dx. Substituting in Equation (106), we have, 184 MECHANICS. J ■ar' ^ Integrating by the aid of Formulas A and JB (Integral Calculus), and taking the integral between the limits X = — r, and x = + r, we find, and for a parallel axis at a distance from AB equal to d, ^' = Jf(^ + c?'') (108.) Mcample 3. To find the moment of inertia of a circular plate with respect to an axis through its centre perpendicular to the face of the plate. Let the dimensions and mass of the plate be the same as before. Let J5!X be an elemetary ring whose radius is x, and whose breadth dx. Then will the mass of the elementary ring be equal ,^2'Kxdx ,,_ iMxdx to M ^ , or dM = — -r— • Substituting this in Equation (106), and taldng the integral between the limits x = 0, and x = r, we have, ^ ''p2Mx^dx Mr'' ^= J—r~ = -T'- For a parallel axis at a distance d from the primitive axis, M ii+^) (109.) MOMENT OF INERTIA. 185 Example 4. To find the moment of inertia of a circular ring, such as may be generated by revolving a rectangle about a line parallel to one of its sides, taken with respect to an axis thi'ough the centre of gravity and perpendi- cular to the face of the ring. This case differs but little from the preceding. Denote the inner radius by r, the outer radius by r\ and the mass of the ring by M. If we take, as before, an elementary ring whose radius is CB, and whose breadth is dx,. we shall have for its mass. Fig. U3. dM = i!f-5 5 Substituting in Equation (106), and integrating between the limits r, and r', we have, J r^ _ r' 2{r — r") 2 For a parallel axis at a distance from the primitive axis equal to d, we have. .^' = Jf(— tr^ + 4Ay - 4A% / the point (a;', y'), will be without this parabola, the values of tana will both be imaginary, and there will be no angle imder which the point can be struck. B' 2Ji Ji. ZU. Fig. 117. Let the parabola B'LB represent the curve whose equa- tion is k" = ^hy' — 4A^ Conceive it to be revolved about AL^ as an axis generat- ing a paraboloid of revolution. Then, from what has preced- ed, we conclude, first, that every point lying within the surface may be reached from A^ with a given initial velocity, under two different angles of projection ; second, that every point lying on the surface can be reached, but only by a sin- gle angle of projection ; thirdly, that no point lying without the surface can be reached at all, If we suppose a body to be projected horizontally from an elevated point A, the trajectory wiU be made known by Equation (115) by sim- ply making a; = ; whence, sina = 0, and cosa = 1. Substituting and reduc- ing, we have, y= -?^~ (119.) rig. 118. 194 MECHANICS. , For every value of x, if is negative, whicli shows that every point of the trajectory lies helow the horizontal line through the point of projection. Kwe suppose ordinates to be estimated positively downwards, we shall have, y = S ■ ■ ■ ■ (^^°-) To find the point at which the trajectory will reach any horizontal plane J5 C, whose distance below the point A is h', we make y = h'., in (120), whence, /oh' X=:JBG= v\/^ . . . (121.) On account of the resistance of the ah', the results of the preceding discussion will be greatly modified. They will, however, approach more nearly to the observed phenomena, as the velocity is diminished and the density of the projec- tile increased. The atmospheric resistance increases as the square of the velocity, and as the cross section of the pro- jectile exposed to the action of the resistance. In the air, it is found that, under ordinary circumstances, the maximum range is obtained by an angle of projection not far from 34°. EXAMPLES. 1. What is the time of flight of a projectile, when the angle of projection is 45°, and the range 6000 feet? SOLUTION. When the angle of projection is 45°, the range is equal to twice the height due to the velocity of projection. Denot- ing this velocity by v, we shall have, «' = 2gh r= 2 X 32^ X 3000 = 193000. CUEVILINEAK AND EOTAEY MOTION. 195 Whence, we find, , V = 439.3 ft. From Equation (11^), we have, »• 6000 ,„„ . t — =; TiTTr;; rrr = 19.3 see. Ans. vcosa 439.3 cos45° 2. What is the range of a projectile, when the angle of projection is 30°, and the initial velocity 200 feet ? Ans. 1016.Q ft. 3. The angle of projection under which a shell is thrown is 32°, and the range 3250 feet. What is the time of flight? Ans. 11.25 sec, nearly. 4. Find the angle of projection and velocity of projec- tion of a shell, so that its trajectory shall pass through two points, the co-ordinates of the first being x = 1700 ft., y = 10 ft., and of the second, x = 1800 ft., y = 10 ft. SOLUTION. Substituting for x and y, in Equation (115), (1700, 10), and (1800, 10), we have, 10 = I700tana - \ ,'/ ; 2«''cosV and, (1800)V 10 = ISOOtana 2«''cos°a Finding the value of ^ , from each of these equa- Zl) COS ot tions, and placing the two equal to each other, we have, after reduction, (18)''(l-l70tana) = (17)'(l-180tana). 196 MECHANICS. Whence, by solution, tana = ^^5 = 0.01144, nearly ; .-. a = 39' 19". We have, from trigonometry, ^ Substituting for tana and cosa in tbe first equation their values as just deduced, we find, for v', (llOOYff 9296166 ,„„,,,„ «' = , y — '-^ r = = 4925442. 2cosV(lY00tana— 10) 18.89 Whence, V = 2219.3 ft. The required angle of projection is, therefore, 3^'19", and the required initial velocity, 2219.3 ft. 4. At what elevation must a shell be projected with a velocity of 400 feet, that it may range 7500 feet on a plane which descends at an angle of 30 ? soLrTioiir. The co-ordinates of the point at which the shell strikes, are cb' = '7500cos30° = 6495 ; and y' = — '7500sin30° = — 3750. And denoting the height due to the velocity 400 ft., by h, we have, h = — = 2486 ft. Substituting these values in the formula, tana = i 2 -^ , CUBVILINEAE AJSTD KOTAET MOTION. 197 and reducing, we have, 4972 ± 4453 tana = 6495 Hence, a = 4° 34' 10", and 85° 25' 50". Am. Centripetal and Centiifugal Forces. 134. Curvilinear motion can only result from the action of an incessant force, whose direction differs from that of the original impulse. This force is called the deflecting force, and may arise from one or more active forces, or it may result from the resistance offered by a rigid body, as when a baU is compelled to run ia a curved groove. What- ever may be the nature of the deflecting forces, we can always conceive them to be replaced by a single incessant force acting transversely to the path of the body. Let the deflecting force be resolved into two components, one nor- mal to the path of the body, and the other tangential to it. The latter force will act to accelerate or retard the motion of the body, according to the direction of the deflecting force ; the former alone is effective in changing the direction of the motion. The normal component is always directed towards tie concave side of the curve, and is called the centripetal force. The body resists this force, by virtue of its inertia, and, from the law of inertia, the resistance must be equal and directly opposed to the centripetal force. This force of resistance is called the centrifugal force. Hence, we may define the centrifugal force to be tJie resistance which a body offers to a force which tends to deflect it from a rectilineal path. The centripetal and centrifugal forces taken together, are called central forces. Measure of the Centrifugal Force. 135. To deduce an expression for the measure of the centrifugal force, let us first consider the case of a single material point, which is constrained to move in a circular T98 MECHANICS. path by a force constantly directed towards tiie centre, as when a soHd body is confined by a string and whirled around a fixed point. In this case, the tangential component of the deflecting force is always 0. There will be no loss of velo- city in consequence of a change of direction in the motion (Art. 120). Hence, the motion of the point will be uniform. Let ABD represent the path of the body, and Y its centre. Suppose the circumference of the circle to be a regular polygon, having an infinite number of sides, of which AH is one; and denote each of these sides by ds. When the body reaches A^ it tends, by virtue of its inertia, to move in the direction of the tangent AT; but, in consequence of the action of the centripetal force di- rected towards F", it is constrained to describe the side ds in the time dt. If we draw B G parallel to A T^ it wiU. be perpendicular to the diameter AD, and AG will represent the space through which the body has been drawn from the tangent, in the time dt. If we denote the acceleration due to the centripetal force by/, and suppose it to be constant during the t^me dt, we shall have, from Art. 114, AG =^fde ( 122.) From a property of right-angled triangles, we have, since AB = ds. Whence, = AG X AD ; or, ds'' z= AG X 1r. AG = 2r Substituting this value of ^C in (122), and solving with respect to f, -' ~ de^ r CUKVILINEAE AND KOTARY MOTION. 199 But — = v' (Art. 113), in which v denotes the velocity of the moving point. Substituting in the preceding equa- tion, we have, /=- (123.) Here /" is the acceleration due to the deflecting force ; and, since this is exactly equal to the centrifugal force, we have the acceleration due to the centrifugal force equal to the square of the velocity, divided hy the radius of the circle. If the mass of the body be denoted by il/j and the entire centrifugal force by F, we shall have (Art. 24), F=^ (124.) r ^ ' K we supposethe body to be-moving on any curve what- ever, we may, whilst it is passing over any two consecutive elements, regard it as moving on the arc of the osculatory circle to the curve which contains these elements ; and, fur- ther, we may regard th^ velocity as uniform during the infinitely smaU time required to describe these elements. The direction of the centrifugal force being normal to the curve, must pass through the centre of the osculatoiy circle. Hence, all the circunistaiices of motion are the same as before, and Equations (123) and (124) will be applicable, provided r be taken as the radius of the curvature. Hence, we may enunciate the law of the centrifugal foi-ce as follows : The acceleration due to the centrifugal force is equal to the square of the veloc^y of the body divided by the radius of curvature. ■ • The entire centrifugal force is equal to the acceleration, multiplied by the mass of the body. In the case of a body whirled around a centre, and re- sti'ained by a string, the tension of the string, or the force 200 MECHANICS. exerted to break it, -will he measured by the centrifugal force. The radius remaining constant, the tension will increase as the square of the velocity. Centrifugal Force at points of the Earth's Surface. 136. Let it be required to determine the centrifugal force at diiferent points of the earth's surface, due to its rotation on its axis. Suppose the earth spherical. Let A be any point on the surface, PQP' a meridian section through A, PP' the axis, FQ the equator, and AB perpendicular to PP', the radius of the parallel of latitude through A. Denote the radius of the earth by r, the radius of the parallel through A by r', and the latitude of A, or the angle ACQ, by I. The tune of revolution being the same for every point on the earth's surface, the velocities of Q and A will be to each other as their distances from the axis. Denoting these velocities by V and v', we have, ,-^ V : V whence, vr r But, from the right-angled triangle GAB, since the angle at A is equal to I, we have, >•' = rcosZ. r Substituting this value of t' in the value of v\ and re- ducing, we have, «' = vcos7. CUEVILINEAB AND EOTAEY MOTION. 201 If we denote the acceleration due to the centrifugal force at the equator by/, we shall have, Equation (123), f =} (125.) In like manner, if we denote the acceleration due to the centrifugal force at A, hjf, we shall have, Substituting for v' and r' their values, previously deduced, we get, -, «° OQsl , f'= -J- (126.) Comparing Equations (125) and (126), we find, /:/':: 1 : cos?, .-. f =fcoal . (127.) That is, the centrifugal force at any point on the earth^s surface is equal to the centrifugal force at the equator, multiplied hy tlie cosine of the latitude of the place. Let AjE, perpendicular to PP', represent the value of f, and resolve it into two components, one tangential, and the other normal to the meridian section. Prolong GA, and draw AD perpendicular to it at A. Complete the rectangle FD on AE as a diagonal. Then wiU AD represent the tangential, and AF the normal component of f. In the right-angled triangle AFF, the angle at A is equal to I. Hence, FF ^AD = /'sin? = faoslsml = ^^^ . ( 128.) AF = f'cosl = foos'l . . . . ( 129.) From (128), we conclude that the tangential component is 9* 202 MECHANICS. at the equator, goes on increasing till I = 45°, wKere it is a maximum ; then goes on decreasing till the latitude is 90° when it again becomes 0. The eflFect of the tangential component is to heap up the particles of the earth about the equator, and, were the earth in a fluid state, this process would go on till the effect of the tangential component was exactly counterbalanced by component of gravity acting down the inclined plane thus found, when the particles would be in a state of equili- brium. The higher analysis has shown that the form of equilibrium is that of an oblate spheroid, differing but slightly from that which our globe is found to possess by actual measurement. From Equation (129), we see that the normal component of the centrifugal force is equal to the centrifugal force at the equator multiplied by the square of the cosine of the latitude of the place. This component is directly opposed to gravity, and, con* sequently, tends to diminish the weight of all bodies on the surface of the earth. The value of this component is greatest at the equator, and diminishes towards the poles, where it becomes equal to 0. From the action of the normal component of the centrifugal force, and fi'om the flattened form of the earth due to the tangential component bringing the polar regions nearer the centre of the earth, the measured force of gravity ought to increase in passing from the equator towards the poles. This is found, by observation, to be the case. The radius of the earth at the equator is found, by measurement, to be about 3962.8 miles, which, multiplied by 251-, will give the entire circumference of the equator. If this be divided by the number of seconds iti a day, 86400, we find the value of v. Substituting this value of v and that of r just given, in Equation (125), we should find, / = 0.1112 ft., for the measure of the centrifugal force at the equator. If CUKVILINEAE AND ROTARY MOTION. 203 this be multiplied by the square of the cosine of the latitude of any place, we shall have the value of the normal com- ponent of the centrifugal force at that place. Centrifugal Force of Extended Masses. O 136. We have supposed, in what precedes, the dimen- sions of the body under consideration to be extremely small ; let us next examine the case of a body, of any dimensions whatever, constrained to revolve about a fixed axis, with which it is invariably connected. If we suppose this body to be divided into infinitely small elements, whose directions are parallel to the axis, the centrifugal force of each element will, from what has preceded, be equal to the mass of the element into the square of its velocity, divided by its i dis- tance from the axis. If a plane be passed through the cen- tre of gravity of the body, perpendicular to the axis, we gjiay, without impairing the generality of the result, suppose the mass of each element to be concentrated at the point in which this plane cuts the line of direction of the element. Let J^GY be the plane through the centre of gravity of the body perpendicular to the axis of revolution, AB the section cut out of the body, or the projection of the body on the plane, and G the point in which it cuts the axis. Take G as the origin of a system of rectangular axes, and let C-X" be the axis of ^, c X G Y the axis of Y, and let m be the pig. 121 point at which the mass of one of these 'filaments is concentrated, and denote that mass by m. De- note the co-ordinates of »i by sc and y, its distance from G by r, and its velocity by v. The centrifiigal force of the mass m . will be equal to Jf we denote the angular velocity of the body by F', the 204 MECHANICS. velocity of the point m will Ibe equal to r V, which, being substituted in the expression for the centrifugal force just deduced, gives mrV\ Let this force be resolved into two components, respec- tively parallel to the axes CX and GY. We shall have, for these components, the expressions, mr V'coam C-X", and mr "F'^sinm CX. But from the figure, we have, cosmCJT =. -, and sinmC^= -• r r Substituting these values in the preceding expressions, and reducing, we have, for the two components, mxV", and myV". In like manner, if we denote the masses of the remaining filaments by m', m", &o., the co-ordinates of the points at which they are cut by the plane ^Cy, by x', y' ; a;", y", &c., their distances from the axis by r\ r", &c., and resolve the centrifugal forces into components, respectively parallel to the axes, we shall have, since V remains the same, m' x' F", to' y' V" ; m"x"V", m"y"V"; &o., &o. If we denote the sum of the components in the direction of the axis of X' by JC, and in the direction of the axis of ]r by Y, we shall have, X=S{mx)V", and Y=:s{my)V'\ CUEVILINEAE ANB KOTAET MOTION. 205 If, no-w, we denote the entire mass of the hody, by M, and suppose it concentrated at its centre of gravity O, whose co-ordinates are designated by x^, and 2/1, and whose distance from O is equal to r^, we shall have, from the principle of the centre of gravity (Art. 51), S(mx) = J/ajj, and 2(/wy) = Mi/^. • Substituting above, we have, X=JfF"a!i, and Tz=MV"i/i. If we denote the resultant of all the centrifugal forces, which wUl be the centrifugal force of the body, by H, we shall have, B =-^X' + Y' = JlfFV^i' + 1/1 = MVr^. But if the velocity of the centre of gravity be denoted by V, we shall have, V = Vr,; or, F" = -^ ; which, substituted in the preceding result, gives, for the resultant, Ii = -^-^ (130.) The line of direction of M is made known by the equa- tions, cosa = — , and cos6 = — ; it, therefore, passes through the centre of gravity 0. Hence, we conclude, that the centrifugal force of an ex- tended mass, constrained to revolve about a fvxed axis, with which it is invariably connected, is the same as though the ' entire mass were concentrated at its centre of gravity. 206 MECHANICS. Pressure on the Axis. 137. The centiifugal force, passing through the centre of gravity and intersecting the axis, will exert its entire effect in creating a pressure upon the axis of revolution. By inspecting the equation, we see that this pressure will increase with the mass, the angular .velocity, and the distance of the centre of gravity from the axis. When the last distance is 0, that is, when the axis of revolution passes through the centre of gravity, there will be no pressure on the axis arising from the centri- fugal force, no matter what may be the mass of the body or its angular velocity. Such is the case of the earth revolving on its axis. Principal Axes. 138. Suppose the axis about which a body revolves to become free, so that the body can move in any direction. If that axis be not one of symmetry, it will be pressed un- equally in different directions by the centrifugal force, and will immediately alter its position. The body will for an instant rotate about some other line, which will immediately change its position, giving place to a new axis of rotation, which will instantly change its position, and so on, until an axis is reached which is pressed equally in all directions by the centrifugal forces of the elements. The body wiU then continue to revolve about this line, by virtue of its inertia, until the revolution is destroyed by the action of some extraneous force. Such an axis is called a principal axis of rotation. Every body has at least one such axis, and may have more. The axis of a cone or cylinder is a prin- cipal axis ; axij ^axaetev of a, sphere is a principal axis / in short, any axis of symmetry of a homogeneous solid is a principal axis. The shortest axis of an oblate spheroid is a principal axis ; and it is foimd by observation that all of the planets of the solar system, which are oblate spheroids, n E . «^ L OtrBVILINEAE "AND EOTAKY MOTION. 207 revolve about their shorter axes, whatever may be the incli- nation of these axes to the planes of theii- orbits. "Were the earth, by the action of any extraneous force, constrained to revolve about some other axis than that about which it is found to revolve, it would, as soon as the force ceased to act, return to its present axis of rotation. Experimental Illustrations. 139. The principles relating to the centrifugal forces admit, of experimental illustration. The instrument repre- sented in the figure, may be employed to show the value of the centrifugal force. ^ repre- sents a vertical axle upon which ^n is niounted a wheel Jf] commu- nicating with a train of wheel- work, by means of which the axle may be made to revolve ' |{{] ' with any angular velocity. At Fig.i2z. the upper end of the axle is a forked branch B 0, sustaining a stretched wire. D and JS are two balls which are pierced by the wire, and are free to move along it. Between Ji and ^ is a spiral spring, whose axis coincides with the ^ire. Immediately below the spring, on the horizontal part of the fork, is a scale for determining the distance of the ball £J, from the axis, and for measuring the degree of compres- sionjaf the spring. Before using the instrument, the force required to produce any degree of compression of the spring is determined experimentally, and marked on the scale. If now a motion of rotation be communicated to the axis, the ball D wiU at once recede to C, but the ball £1 will be restrained by the spiral spring. As the velocity of rotation is increased, the spring wUl be compressed more and more, and the ball £1, will approach JB. By a suitable arrange- ment of the wheelwork, the angular velocity of the axis corresponding to any degree of compression may be ascer- 208 MECHANICS. tained. "We have thus all the data necessary to a verifica- tion of the law of the centrifugal force. If a vessel of water be made to revolve about a vertical axis, the interior particles will recede from the axis on account of the centrifugal force, and will be heaped up about the sides of the vessel, imparting a concave form to the upper surface. The concavity will become greater as the angular velocity is increased. If a circular hoop of flexible metal be fastened so that one of its diameters shall coincide with the axis of a whirling machine, its lower point being fastened to the horizontal beam, and a motion of rotation be imparted, the portions of the hoop farthest from the axis will be most affected by the centrifugal force, and the hoop will be observed to assume an elliptical form. If a sponge, filled with water, be attached to one of the- arms of a whirling machine, and a motion of rotation be imparted, the water wUl be thrown from the sponge. This principle has been made use of in a machine for drying clothes. An annular trough of copper is' mounted upon an axis by means of radial arms, the axis being connected with a train of wheelwork, by means of which it may be put in motion. The outer wall is pierced with holes for the escape of the water, and a lid serves to confine the articles to be dried. To use this instrument, the linen, after being washed, is placed in the annular space, and a rapid motion . of rotation imparted to the'machine. The linen is thrown, by the centrifugal force, against the outer wall of the instru- ment, and the water, being partially squeezed out, and par- tially thrown off by the centrifugal force, escapes through the holes made for the purpose. Sometimes as many as 1,500 revolutions per minute are given to the drying machine, in which case, the drying process is very rapid and very perfect. If a body be whirled about an axis with sufiicient velo- city, it may happen that the centrifugal force generated will be greater than the force of cohesion which binds the CURVILINKAR AND EOTAET MOTION'. 209 particles together, in which case, the body will' be torn asunder. It is a common occurrence that large grindstones, when put into a state of rapid rotation, burst, the fragments being thrown with great velocity away from the axis, and often producing much destruction. When a wagon, or carriage, is driven rapidly around a corner, or is forced to tiirn about a circular track, the cen- trifugal force generated is often sufficient to throw out the loose articles from the vehicle, and even to overthrow the vehicle itself. When a car upon a railroad track is forced to turn around a shai;p curve, the centrifugal force generated, ' tends to throw the weight of the cars against the rail, pro- ducing a great amount of friction, and contributing to wear out both the track and the car. - To obviate this difficulty in a measure, it is customary to raise the outer rail, so that the resultant of the centrifugal force, and the force of grav- ity, shall be sensibly perpendicular to the plane of the two rails. Elevation of the outer rail of a curved track. 14®. To find the inclination of the track, that is, the elevation of the outer rail, so that the resultant of the weight and centrifugal force may be pei'pendicular to the line joining the two rails. Let G be the centre of gravity of the car, and let the figure represent a vertical section through the centre of gravity and the centre of the curved track. Let GA^ parallel to the hprizon, represent the ac- celeration due to the centrifugal force, and GB^ perpen- dicular to the horizon, the acceleration due to the weight of the car. Construct the resultant G (7, of these forces, then must the line DjE 'h<3 perpendicular to GO. Denote the velocity of the car, by v. and the radius of the curved track, by r. The aeoelera;tion due to the weight will be rig. 123. 210 MECHANICS. equal to g^ the force of gravity, and the acceleration due to the centrifugal force ■will be equal to — The tangent of the GS ^ angle GGB will be equal to ^tb ; 0^5 denoting the angle by a, we shall have, GB v' tana = y^^ = GJB gr But the angle BEF is equal to the angle GGB. Denot- ing the distance between the rails, by d, and the elevation of the outer rail above the inner one, by h, we shall have, tana = — , very nearly. Equating the two values of tana, we have, ■^ — —, .-. h — . . (131.) a gr gr ^ ' Hence, the elevation of the outer rail varies as the square of the velocity directly, and as the radius of the curve inversely. It is obvious that this connection would require to be different for different velocities, which, from the nature of the case, would be mamfestLy impossible. The correction is, therefore, made for some assumed velocity, aind then such a form is given to the tire of the wheels as wiU com- plete the correction for different velocities, ^ The Conical PendiUuin. 141. The conical pendulum consists of a solid ball at- tached to one end of a rod, the other end of which is con- nected, by means of a hinge-joint, 'with a vertical axle. When the axle is put in motion, the centrifugal force gene- rated in the ball causes it to recede from the axis, untU an equilibrium is established between the weight of the ball, the centrifugal force, and the tension of the connecting rod. CUKVILINBAB AND KOTAET MOTION. 211 When the velocity is constant, the centrifttgal force wili be constant, and the centre of the hall will describe a horizontal circle, whose radius will depend upon the velocity. Let it be required to determine the time of revolution. Let BD be the vertical axis, A the ball, B the hinge- joint, and AB the connecting rod, whose mass is so small, that it may be neglected, in comparison with that of the ball. Denote the required time of revolution, by t, the length of the arm, by I, the accele- ration due to the centrifugal force, by/, and the angle AB (7, by a?. Draw A G perpen- dicular to -B2>, and denote AC, by >■, and BCbjh. Z . From the triangle AB O, we have, r = hincp ; and since r is the radius of the circle described by A, we have the distance passed over by A, in the time t, equal to 2'n'r = 2*&in!p. Denoting the velocity of ^, by v, we have, from Equation (55), ■2*Mn!p But the centrifugal force is equal to the square of the velocity, divided by the radius ; hence, /=i!^ .... (132.) The forces which act upon A, are the centrifugal force in the direction AF, the force of gravity in the direction A G, and the tension of the connecting rod in the direction AB. Li order that the ball may remain at an invariable distance from the axis, these three forces must be in equilibrium. Hence (Art. 35), g :/ : : bidBAF : : siaBAG ; but, sinBAF = sin(90° -f -'Sl * perpendicular to the axis, inter- | //'','<.'-''' sects it. Let A and O be any I4v-'''' two points whatever, on the line fi<'.128. of direction of P. Suppose the force P to turn the system through an infinitely small angle, and let B and .D be the new positions of ^ and C. Draw OE, JBa, and Pc respectively perpendicular to PE ; draw 9lso,A0, BO, CO, and BO. Denote the distances OA, by r, 00, by ?•', OPJ, by p, and the path described by 224 MECHAJ ,-'k--!5 and CD = }•'<)'; and since the angular motion is infinitely small, ^^, these lines may be regarded as j,.^ straight lines, perpendicular re- spectively to OA and G. From the right-angled triangles A£a and CDc, we have, Aa = r^'cosBAa, and Og = r'^'cosDOc. In the right-angled triangles AJBa, and OAE, we have A£ perpendicular to OA, and Aa perpendicular to OJE; hence, the angles JBAa, and A OE, are equal, as are also their cosines ; hence, we have, cos^ J.a = cos J. OE = - ■ r In like manner, it may be shown, that cosZ) Cc = cosCOE=:^- ' r Substituting in the equations just deduced, we have, Aa = p6, and Gc = p& ; .: Aa = Cc ; whence, P .Aa = r. Gc = Tp6'. The first member of the equation is this quantity of work of JP, when its point of application is at ^ ; the second is the quantity of work of P, when its point of application is at G. Hence, we conclude, that the eLementary quantity of work of a force applied to produce rotation, is always tJie CUKVILmEAE AND ROTARY MOTION. 225 same, wherever its point of application may he taken, pro- vided its line of direction remains unchanged. We conclude, also, that the elementary quantity of work is equal to the intensity of the force multiplied by its lever arm into the elementary space described by a point at a unit's distance from the axis. If we suppose the force to act for a unit of time, the intensity and lever arm remaining the same, and denote the angular velocity, by &, we shall have, Q' = PpL For any number of forces similarly applied, we shall have, q = l.{Pp)(i .... (135.) If the forces are in equilibrium, we shall have (Art. 48), 2(Pp) = ; consequently, § = 0. Hence, if any number of forces tending to produce rotar tion about a fixed axis, are in equilibrium, the entire quan- tity of work of the system of forces wiU be equal to 0. Accumiilation of Wcirk. 148. When a body is put in motion by the action of a force, its iaertia .has to be overcome, and, in order to bring the body bacli again to a state of rest, a quantity of work has to be given out just equal to that required to put it in motion. This results from the nature of inertia. A body in motion may, therefore, be regarded as the representation of a quantity of work which can be reproduced xxpon any resistance opposed to its motion. Whilst the body is in motion, the work is said to be accumulated. In any given instance, the accum/ulated work depends, first, upon the mass in motion ; and, secondly, upon the velocity with which it moves. Take the case of a body projected vertically upwards in vacuum. The projecting force expends upon the body a quantity of work sufficient to raise it through a height equal 10* 226 MECHANICS. to that due to the velocity of projection. Denoting the weight of the body, by w, the height to which it rises, by h, and the accumulated work, by Q, we shall have, Q =. wh. But, A = i-, (Art. 116), hence, Denoting the mass of the body by m, we shall have, m = — (Art. 11), and, by substitution, we have, finally, '• Q — imv' ( 136.) If the body descends by its own weight, it wiU have impressed upon it by the force of gravity, during the descent, exactly the same quantity of work as it gave out in ascending. The amount of work accumulated in a body is evidently the same, whatever may have been the circumstances under which the velocity has been acquired ; and also, the amount of work which it is capable of giving out in overcoming any resistance is the same, whatever may be the nature of that resistance. Hence, the measure of the aacumulated worh of a moving mass is one-7ialf of the mass into the square of the velocity. The expression mv', is called the living force of the body. Hence, the living force of a body is equal to its mass, m,ultiplied by the square of its velocity. The living force of a body is the measure of twice the quantity of work expended in producing the velocity, or, it is the measure of twice the quantity of work which the body is capable of giving out. When the forces exerted tend to increase the velocity, CTJEVILINKAR AND EOTAKY MOTION. 227 their ■work is regarded as positive ; when they tend to dimin- ish it, their work is regarded as negative. It is the aggre- gate of all the work expended, both positive and negative, that is measured by the quantity, ^mv'. If, at any instant, a body whose mass is m, has a velocity V, and, at any subsequent instant, its velocity has become «', we shall have, for the accumulated worl^ at these two instants, Q = imw", Q' — Imv" ; and, for the aggregate quantity of work expended in the interval, Q" = im{v" -v'). . . . ( 137.) When the motive forces, during the interval, perform a greater quantity of work than the resistances, the value of v' will be greater than that of v, and there will be an accu- mulation of work in the interval. Wlen the work of the resistances exceeds that of the motive forces, the value of v will exceed that of v', Q" will be negative, and there will be a loss of living force, which is absorbed by the resistances. Iiiving Force of Revolving Bodies. 149. Denote the angular velocity of a body which is restrained by an axis, by ^ ; denote the masses of its ele- mentary particles by m, m', &c., and their distances from the axis of rotation, by r, r\ &c. Their velocities will be r6, r'6, &c., and their living forces will be mr^&', m'r"6', &c. Denoting the entire living force of the body, by Zi, we shall have, by summation, and recoUeoting that P is the same for all the terms, Z = S{mr')6'' .... (138.) But S(mr^) is the expression for the moment of inertia of the body, taken with respect to the axis of rotation. De- 228 MHICHANICS. noting the entire mass by M, its radius of gyration, with respect to the axis of rotation, by Ic, we shall have, L = MM&\ If, at any subsequent instant, the angular velocity has become ^', we shall, at that instant, have, and, for the loss or gain of living force in the interval, we shall have, L" = Mh\li"' — H") . . . (139.) If we make 6'^ — 6^ — 1, we shall have, Z'" - Mk' - 2{mr') . . ( 140.) which shows that the moment of inertia of a body, with respect to an axis, is equal to the living force lost or gained whilst the body is experiencing a change in the square of its angular velocity equal to 1. The principle of living forces is extensively applied in discussing the circumstances of motion of machines. When the motive power performs a quantity of work greater than that necessary to overcome the resistances, the velocities of the parts become accelerated, a quantity of work is stored up, to be again given out when the resistances offered require a greater quantity of work to overcome them than is furnished by the motor. In many machines, pieces are expressly introduced to equalize the motion, and this is particularly the case when either the motive power or the resistance to be overcome, is, in its nature, variable. Such pieces are called fly-wheels. Fly-Wheels. 150. A fly-wheel is a heavy wheel, usually of iron, mounted upon an axis, near the point of application of the CURVILINEAE AND EOTAEY MOTION. 239 Kg. 129. force which it is destined to regulate. It is generally com- posed of a heavy rim, connected with the axis by means of radial arms. Sometimes it consists of radiating bars, carrying heavy spheres of metal at their outer extremity. In either case, we see, from Equation 139, that, for a given quantity of work absorbed, the value of 6" — &' will be less as M and h are greater ; that is, the change of angular velocity wUl be less, as the mass of the fly-wheel and its radius of gyration increase. It is for this reason that the peculiar form of fly-wheel indicated above, is adopted, it being the form that most nearly realizes the conditions pointed out. The principal objection to large fly-wheels in machinery, is the great amount of hurtful resistance which they create, such as fric- tion on the axle, &c. Thus, a fly-wheel of 42000 lbs. would create a force of friction of 4200 lbs., the coefficient of fric- tion being but -^-^ ; and, if the diameter of the axle were 8 mches, and the number of revolutions 30 per minute, this resistance alone would be equal to 8 horse powfers. EXAMPLES. 1. The weight of the ram of a pile-driver is 400 lbs., and it strikes the head of a pUe with a velocity of 20 feet. What is the amount of work stored up in it ? SOLtTTION. The height due to the velocity, 20 feet, is equal to - = 6;22 ft., nearly. 64^ Hence, the stored up work is equal to dOtflbs. X 6.22 ft. = 2488 lbs. ft. ; 230 ivrEOiiANics. or, the stored up work, equal to half the living force, is equal to i2? X -^?^ = 2488 units. Ans. 32i 2 2. A train, weighing 60 tons, has a velocity of 40 miles per hour when the steam is shut off. How far will it travel, if no break be applied, before the velocity is reduced to 10 miles per hour, the resistance to motion being estimated at 10 lbs. per ton. Ans.l683iit. (j Composition of Rotations. 151. Let a body A CJ3D, that is free to move, be acted upon by a force which, of itself, would cause the body to revolve for the infinitely small time dt, about the line A£, with an angu- lar velocity v ; and at the same instant, let the body be acted upon by a second force, which would of itsejf cause the body to revolve about CD, for the time j,. ^^^ dt, with an angular velocity v'. Suppose the axes to intersect each other at 0, and let P be any point in the plane of the axes. Draw PF and PG res- pectively perpendicular to OP and C, denoting the for- mer, by X, and the latter, by y. Then mil the velocity of P due to the first force, be equal to vx, and its velocity due to the second force will be equal to v'y. Suppose the rota- tion to take place in such a manner, that the tendency of the rotation about one of the axes, shaE be to depress the point below the plane, whilst that about the other is to elevate it above the plane ; then will the effective velocity of P be equal to vx — v'y. If this effective velocity is 0, the point P will remain at rest. Placing the expression just deduced equal to 0, and transposing, we have. *■ vx = vy. CUEVILINEAE AND EOTAEY MOTION. 231 To determine the position of P, lay oif OH equal to w, 01 equal to u', and regard these lines as the representatives of two forces ; we have, from the equation, the moment of w, with respect to the point P, equal to the moment of «', with respect to the same point. Hence, the point P must be somewhere upon the diagonal OK^ of the parallelogram described on «, and v' . But P may be anywhere on this liae ; hence, every point of the diagonal OK^ remains at rest during the time dt, and is, consequently, the resultant axis of rotation. We have, therefore, the following principles : If a body be acted upon simultaneously by two forces, each tending to impart a motion of rotation about a sepa- rate axis, the residtant motion will be one of rotation about a third axis lying in the plane of tJie other two, and passing through their common point of intersection. The direction of the resultant axis coincides with the dia-gonal of a parallelogram, whose adjacent sides are the component axes, and lohose lengths are proportional to tlie impressed angular velocities. Let OH and 01 represent, as before, the angular veloci- ties V and v', and OK\h& diagonal of the parallelogram constructed on these lines I ^ as sides. Take any point I, on the second /^^'C''''/ axis, and let fall a perpendicular on OH and /^^^ / OK; denote the fomaer by r, and the H latter, by r" ; denote, also, the resultant -^'s- ^^'■ angular velocity, by v". Smce the actual space passed over by -Z", during the time t, depends only upon the first force, it will be the same whether we regard the revolution as taking place about the axis OH, or about the axis OK. If we suppose the rotation to take place about OH, the space passed over in the time dt, will be equal to rvdt ; if we sup- pose the rotation to take place about OK, the space passed over in the satne time will be equal to r"v"dt. Placing these expressions equal to each other, we have, after reduc- tion, v" = —V. r' 233 MECHANICS. \ But regarding Z as a centre of moments, we shall have, from the principle of moments, OK X r" = vr ; or, OJT — -„ v. By comparing the last two equations, we have, v" = OK. That is,JAe resultant angular velocity wiU he equal to the diagonal of the parallelogram described on the component angular velocities as sides. By a course of reasoning entirely similar to that employed in demonstrating the parallelopipedon of forces, we might show, that, If a body be acted upon by three simultaneous forces, each tending to produce rotation about separate axes inter- secting each other, the resultant motion will be one of rota- tion about the diagonal of the parallelopipedon whose adjor cent edges are the component angular velocities, and the resultant angular velocity will be represented by the length of this diagonal. The principles just deduced are called, respectively, the pa/rallelogram and tlie parallelopipedon of rotations. Application to the Gyroscope. 152. The gyroscope is an instrument used to illustrate the laAvs of rotary motion. It consists essentially of a heavy wheel A, mounted upon an axle BQ. This axle is attached, by means of pivots, to the inner edge of a cu-cular hoop DE, within which the wheel A can turn freely. On ig-i82. one side of the hoop, and in the prolongation of the axle B C, is a bar EF, having a conical hole drilled on its lower CUKVILINEAE AND EOTAET MOTION. 233 face to receive the pointed summit of a vertical standard G. If a string be wrapped several times around the axle B C, and then rapidly unwound, so as to impart a rapid motion of rotation to the wheel A, in the direction indicated by the arrow-head, it is observed that the machine, instead of sinking downwards under the action of gravity, takes up a retrogade orbital motion about the pivot G, as indicated by the arrow-head H. For a time, the orbital motion in- creases, and, under certain circumstances, the bar JSF is observed to rise upwards in a retrograde spiral direction; and, if the cavity for receiving the pivot is pretty shallow, the bar may even be thrown off the vertical standard. Instead of a bar EF, the instrument may simply have an ear at E, and be suspended from a point above by means of a string attached to the ear. The phenomena observed are the same as before. Before explaining these phenomena, it will be necessary to point out the conventional rules for attributing proper signs to the different rotations. Let OX, OY, and OZ, be three rectangular axes. It has been agreed to call all dis- tances, estimated from 0, to- wards either ^ I^ or Z, posi- tive; consequently, all distances q estimated in a contrary direction "S, ^ M must be regarded as negative. y V' If a body revolve about either Fig. iss. axis, or about any line through the origm, in such a manner as to appear to an eye beyond it, in the axis and looking towards the origin, to move in the same direction as the hands of a watch, that rotation is considered positive. If rotation takes, place in an opposite direction, it is negative. The arrow-head A, indicates the direction of positive rotation about the axis of JC. To an eye situated beyond the body, as at X, and looking towards the origin, the motion appears to be in the same direction as the motion of the hands of a watch. The arrowhead B, 23i MECHANICS, indicates the direction of positive rotation about the axis of Y, and the arrow-head G, the direction of positive rotar tion about the axis of Z. Suppose the axis of the wheel of the gyroscope to coincide with the axis of JC, taken horizontal ; let the standard be taken to coincide with the axis of Z, the axis of Y being perpendicular to them both. Let a positive rotation be communicated to the wheel by means of a string. For a very short time dt, the angular velocity may be regarded as constant. In the same time dt, the force of gravity acts to impart a motion of positive rotation to the whole instru- ment about the axis of Y, which may, for ■ an instant, be regarded as constant. Denote the former angular velocity by V, and the latter by «'. Lay off in a positive direction on the axis of JC, the distance OD equal to v, and, on the positive direction of the axis of Y, the distance OF equal to v', and complete the parallelogram 01*1 Then (Art. 151) will OF represent the direction of the resultant axis of revo- lution, and the distance . OF will represent the resultant angular velocity, which denote by v". In moving from OD to OF, the axis takes up a positive, or retrograde orbital motion about the axis of Z. To construct the position of the resultant axis for the second instant dt, we must com- pound three angular velocities. Lay off on a perpendicwlar to OF and OZ, the angular velocity OG due to the action of gravity during the time dt, and on OZ the angular velo^ city in the orbit ; construct a parallelopipedon on these lines, and draw its diagonal through 0. This diagonal will coincide in direction with the resultant axis for the second instant, and its length will represent the resultant angular velocity (Art. 151). For the next instant, we may proceed as before, and so on continually. Since, in each case, the diagonal is greater than either edge of the paral- lelopipedon, it follows that the angular velocity will contin- ually increase, and, were there no hurtful resistances, this increase would go on indefinitely. The effect of gravity is continually exerted to depress the centre of gravity of the CURVILINEAR AJSTD ROTARY MOTION. 236 instrument, whilst the effect of the orbital rotation is to elevate it. When the latter effect prevails, the axis of the gyroscope will continually rise ; when the former prevails, the- gyroscope will continually descend. Whether the one or the other of these conditions will be fulfilled, depends upon the angular velocity of the wheel of the gyroscope, and upon the position of the centre of gravity of the instru- ment. Were the instrument counterpoised so that the centre of gravity would lie exactly over the pivot, there would be no orbital motion, neither would the instrument rise or fall. Were the centre of gravity thrown on the opposite side of the pivot from the wheel, the rotation due to gravity would be negative, that is, the orbital motion would be direct, instead of retrograde. 236 MECHANICS. CHAPTER YII. MECHANICS OF LIQUIDS. Classification of Fluids. 153. A FLUID is a body whose particles move freely amongst eacli other, each particle yielding to the slightest force. Fluids are of two classes : liquids,- ot which water is a type, and gases, or vapors, of which air and steam are types. The distinctive property of the first class is, that they are sensibly incompressible; thus, water, on being pressed by a force of 15 lbs. on each square inch of surface, only suffers a diminution of about y&woS's *^f i^^ bulk. The second class comprises those which are readily compressible ; thus, air and steam are easily compressed into smaller vol- umes, and when the pressure is removed, they expand, so as to occupy larger volumes. Most liquids are imperfect ; that is, there is more or less adherence between their particles, giving rise to viscosity. In what follows, they will be regarded as destitute of vis- cosity, and homogeneous. For certain purposes, fluids may also be regarded as destitute of weight, without impairing the validity of the conclusions. Principle of Equal Pressures. 154. From the nature and constitution of a fluid, it fol- lows, that each of its particles is perfectly movable in all directions. From this fact, we deduce the following funda- mental law, viz. : If a fluid is in equilibrium under the action of any forces whatever, each particle of the mass is equally pressed in all directions ; for, if any particle were more strongly pressed in one direction than in the others, MECHANICS OF LIQUIDS. 237 it would yield in that direction, and motion would ensue, which is contraiy to the hypothesis. This is called the principle of equal pressures. It follows, from the principle of equal pressures, that if any point of a fluid in equilibrium, be pressed by any force, that pressure wiU be transmitted without change of intensity to every other point of the fluid mass. This may be Ulustrated experimentally, as follows : Let AH represent a vessel filled 'with a fluid in equili- brium. Let and D represent two openings, furnished with tightly-fit- ting pistons. Suppose that forces are applied to the pistons just sufiicient to maintain the fluid mass in equilibrium. If, now, any additional force be appli- ed to the piston P, the piston Q will be forced outwards : and in order to . . Fig 184. prevent this, and restore the equili- brium, it will be found necessary to apply a force to the piston Q, which shall have the same ratio to the force ap- plied at P that the area of the piston Q has to the area of the piston P. This principle will be found to hold true, whatever may be the sizes of the two pistons, or in what- ever portions of the surface they may be inserted. If the area of JP be taken as a unit, then will the pressure upon Q be equal to the pressure on JP, multiplied by the area of Q. The pressure transmitted through a fluid in equilibrium, to the surface of the containing vessel, is normal to that sur- face ; for if it were not, we might resolve it into two compo- nents, one normal to the surface, and the other tangential ; the effect of the former would be .destroyed by the resistance of the vessel, whilst the latter would impart motion to the fluid, which is contrary to the supposition of equilibrium. In Uke manner, it may be shown, that the resultant of all the pressures, acting at any point of the free surface of a fluid, is normal to the surface at that point. When the only force acting is the force of gravity, the surface is level. For 238 MECHANICS. small areas, a level surface coincides sensibly with a horizon- tal plane. For larger areas, as lakes and oceans, a level sur- face coincides with the general surface of the earth. Were the earth at rest, the level surface of lakes and oceans would be spherical ; but, on account of the centrifugal force aris- ing from the rotation of the earth, it is sensibly an ellip- soidal surface, whose axis of revolution is the axis of the earth. Pressure due to Weight. 155. If an incompressible fluid be in a state of eqviUi- brium, the pressure at any point of the mass arising from the weight of the fluid, is proportional to the depth of the point below the free surface. Take an infinitely small surface, supposed horizontal, and conceive it to be the base of a vertical prism whose altitude is equal to its distance below the free surface. Conceive this filament to be divided by horizontal planes into infi- nitely smaU, or elementary prisms. It is evident, from the 'principle of equal pressures, that the pressure upon the lower face of any one of these elementary prisms is greater than that upon its upper face, by the weight of the element, whilst the lateral pressures are such as to counteract each other's efiects. The pressure upon the lower face of the first prism, counting from the top, is, then, just equal to its weight ; that upon the lower face of the second is equal to the weight of the first, plus the weight of the second, and so on to the bottom. Hence, the pressure upon the assumed surface is equal to the weight of the entire column of fluid above it. Had the assumed elementary surface been oblique to the horizon, or perpendicular to it, and at the same depth as before, the pressure upon it would have been the same, from the principle of 'equal pressures. "We have, therefore, the foUffwing law : The pressure upon any elementary portion of the surface of a vessel containing a heavy fluid is equal to the weight of a prism of the fltvid whose base is equal to that surface., MECHANICS OF LIQUIDS. 239 and whose altitude is equal to its depth below the free surface. Denoting the area of the elementary surface, by s, its depth below the free surface, by z, the weight of a unit 'of the volume of the fluid, by w, and the pressure, by p, We shall have, p = wzs (141.) We have seen that the pressure upon any element of a surface is normal to the surface. Denote the angle which this normal makes mth the vertical, estimated from above, down- wards, by (p, and resolve the pressure into two components, one vertical and the other horizontal, denoting the vertical Fig. 135. component byjs', we shall have, p' = MSSCOS(p ( 142.) But scosip is equal to the horizontal projection of the elementary surface s, or, in other words, it is equal to a horizontal section of a vertical prism, of which that surface is the base. Hence, the vertical component of the pressure on any element of the surface is equal to the weight of a column of the fluid., whose base is equal to the horizontal projection of the element, and whose altitude is equal to the distance of the element from the upper surface of the fluid. The distance s has been estimated as positive from the surface of the fluid downwards. If (p < 90°, we have cos^ positive ; hence, p' will be positive, which shows that the vertical pressure is exerted downwards. If (p > 90°, we have cos(p negative ; hence, p' is negative, which shows that the vertical- pressure is exerted upwards (see Fig. 135). Suppose the interior surface of a vessel containing a heavy fluid to be divided into elementary portions, whose areas are denoted by s, s', s", &c. ; denote the distances of these L-. 240 MECHANICS. elements below the upper surface, by «, z\ z", &c. From the princij)le just demonstrated, the pressures upon these surfaces will be denoted by wsz, ws'z', ws"z", &c., and the entire pressure upon the interior of the vessel will be equal to, w{sz + s'z' + s"z" + &c.) ; or, w x S{sz). Let Z denote the depth of a column of the fluid, whose base is equal to the entire surface pressed, and whose weight is equal to the entire pressure, then wUl this pressure be equal to tB{s + s' + s" + &o.)Z ; or, loZ.^s. Equating these values, we have. 10 . S(S2) = wZ. 2(s), .-. Z = ^^ (143.) The second member of (143), (Art. 51), expresses the distance of the centre of gravity of the surface pressed, below the free surface of the fluid. Hence, The entire pressure of a heavy fluid upon the interior of the containing vessel, is equal to the weight of a volume of the fluid, whose base is equal to the area of the surface pressed, and whose altitude is equal to the distance of the centre of gravity of the surface from the free surface of the EXAMPLES. 1. A hollow sphere is filled with a liquid. How does the entire pressure, on the interior surface, compare with the weight of the liquid ? SOLUTION. Denote the radius of the interior surface of the sphere, by r, and the weight of a unit of volume of the liquid, by w. The entire surface pressed is measured by 4*j'''; and, since the centre of gravity of the surface pressed is at a distance r below the surface of the liquid, the entire pres- MECHANICS OF LIQUIDS. 241 sure on the interior surface will be measured by the expression, w X 4*r'' X >• = 4*M>*'. But the weight of the liquid is equal to Hence, the entire pressure is equal to three times the weight of the liquid. 2. A hollow cylinder, with a circular base, is filled with a liquid. How does the pressure on the interior surface com- pare with the weight of the liquid? SOLUTION. Denote the radius of the base of the cylinder, by r, and the altitude, by h. The centre of gravity of the lateral surface is at a distance below the upper surface of the fluid equal to ^h. If we denote the weight of the unit of volume of the liquid, by w, we shall have, for the entire pressure on the interior surface, wh'sr^ + Iwsr . \h^ = wttrh{r + h). But the weight of the liquid is equal to vMr'h. Hence, the total -pressure is equal to times the weight of the liquid. If we suppose h = r, the pressure will be twice the weight. If we suppose r = 2A, we shall have the pressure equal to |- of the weight. If we suppose h = 1r, the pressure will be equal to three times the weight, and so on. 11 242 MECHANICS. In all cases, the total pressure will exceed the weight of the liquid. 3. A right cone, with a circular base, stands on its base, and is fiUed with a liquid. How does the pressure on the internal surface compare with the weight of the liquid ? SOLUTION. Denote the radius of the base, by r, and the altitude, by A, then will the slant height be equal to ■y/h"^ + r'. The centre of graAdty of the lateral surface, below the upper surface of the liquid is equal to f A. If we denote the weight of a unit of volume of the liquid, by w, we shall have, for the total pressure on the interior surface, vnr'h + f w*7-A -v/A" + r' = 'wtrh{')+ f -/A" + r'). But the weight of the Uquid is equal to ^'ifr'h = loifrh X ^r. Hence, the total pressure is equal to ■ — times the weight. 4. Required the relation between the pressure and the weight in the preceding case, when the cone stands on its vertex. SOLUTION. The total pressure is equal to ^wirrhy/h' +7'; and, consequently, the pressure is equal to — — — — times the weight of the liquid. *" MECHANICS OF I.IQIIIDS. 243 5. What is the pressure on the lateral faces of a cubical vessel filled with water, the edges of the cube being 4 feet, and the weight of the water 62^ lbs. per cubic foot ? Ans. 8000 lbs. 6. A cylindrical vessel is filled with water. The height of the vessel is 4 feet, and the radius of the base 6 feet. What is the pressure on the lateral surface ? Ans. 18850 lbs;-, nearly. Centre of Pressure on a Plane Surface. 156. Let AH CD represent a plane, pressed by a fluid on its upper surface, AJ3 its intersec- tion with the free surface of the fluid, _, j^ G its centre of gravity, the centre s- "74^^^>-^ of pressure, and s the area of any ''*^/ i/s/ element of the surface at S. De- /^*^ / note the inclination of the plane to ^<^ / the level surface, by *, the perpendic- ^^^ idar distances from to AH, by x, Fig. ise. from G to AB., by p, and from S to AjB, by r. Denote, also, the entire area A (7, by A, and the weight of a unit of volume of the fluid, by w. The perpendicular distance from G to the free surface of the fluid, will be equal to p sina, and that of any element of the surface, will be r since. From the preceding article, it follows that the entire pressure exerted is equal to w^jogina, and its moment, with respect to AB as an axis of moments, is equal to WAp sina X X. The dementary pressure on s is, in like manner, equal to wsr sina, and its moment, with respect to AB., is wsr' sina, and the sum of all the elementary moments is equal to «osina2(fir°). 244 MECHANICS. But the resultant moment is equal to the algebraic sum of the elementary moments. Hence, wAp sina X X = w sina 2{sr') ; and, by reduction, Ap (144.) The numerator is the moment of inertia of the plane AH CD, with respect to AH, and the denominator is the moment of the area with respect to the same line. Hence, the distance from the centre of pressure to the intersection of the plane with the free surface, is equal to the moment of inertia of the plane, divided by the moment of the plane. If we take the straight line AD, perpendicular to AJ3, as an axis of moments, denoting the distance of from it, by y, and of s from it, by I, we shall, in a similar manner," have. wAp sinay = : wsina2(srZ); and. fey reduction, y = ^srl) Ap • ■ • • (145.) The values of x and y make known the position of the centre of pressure. EXAMPLES. 1. What is the position of the centre of pressure on a rectangular flood-gate, the upper line of the gate coinciding with the surface of the water ? SOLUTION. It is obvious that it will be somewhere on the line joming the middle points of the upper and lower edges of the gate. MKCHANICS OF LIQUIDS. ' 245 Denote its distance from the upper edge, by s, the depth of the gate, hy 21, and its mass, by M. The distance of the centre of gravity from the upper edge will be equal to l. From Example 1 (Art. 132), replacing d by I, and reducing, we have, for the moment of inertia of the rectangle, But the moment of the rectangle is equal to, Ml; hence, by division, we have, « = fz = m)- That is, the centre of pressure is at two-thirds of the distance from the upper to the lower edge of the gate. 2. Let it be required to find the pi-essure on a submerged rectangular flood-gate ABGD, the plane of the gate being vertical. Also, the distance E R f of the centre of pressure below the surface of the water. SOLUTION. Let JEF be the intersection of the plane iC -i — [B ifi.. D C Fig. 137. with the surface of the water, and suppose the rectangle AC to be prolonged tUl it reaches ^J?^ Let (7, C", and 0", be the centres of pressure of the rectangles JiJC, EB, and A G respectively. Denote the distance GG", by s, the distance ^i?, by a, and the distance JExi, by a'. Denote the breadth of the gate, by 5, and th6 weight, a imit of volume of the water, by w. The pressure on EG will be equal to ^a'bw, and the pres- sure on EJ3 wiU be equal to ^a'^bw ; hence, the pressure on A C wiU be equal to ibw(a' - a") ; which is the pressure required. 240 MK0HANIC8. From the principle of moments, the moment of the pres- sure on A G, is equal to the moment of the pressure on JE!0, minus the moment of the pressure on £!Ji. Hence, from the last problem, ^bw{a' — a") X z = ^bwa' x f a — ^bwa" x |a', a" a' + aa' + a'" s = *- a + a' which is the required distance from the surface of the water. 3. Let it he required to find the pressure on a rectangular flood-gate, when both sides are pressed, the water being at diiferent levels on the two sides. Also, to find the centre of pressure. soLTjnoiir. Denote the depth of water on one ^. ^^ side by a, and on the other side, by a', the other elements being the same as before. The total pressure will, as before, be equal to, ^bw{a' — a"). We shall also have, as before, __ 2 *' ~ '^" __ ^^ + '^'^' + <*" ^ - ^a'' -a" "^ a'+'a' 4. A sluice-gate, 10 feet square, is placed vertically, its upper edge coinciding with the surface of the watei-. What is the pressure on the upper and lower halves of the gate, respectively, the weight of a cubic foot of water being taken equal to 62} lbs. ? Ans. '7S12.5 lbs., and 2343V.6 lbs. 5. What must be the thickness of a rectangular dam of granite, that it may neither rotate about its outer angular MECHANICS OF LIQUIDS. 247 point nor slide along its base, the weight of a cubic foot of granite being 160 lbs., and the coefficient of friction between it and the soil being .6 ? SOLUTION. First, to find the thickness necessary to prevent rotation outwards. Denote the height of the wall, by A, and sup- pose the water to extend from the bottom to the top. De- note the thickness, by t, and the length of the wall, or dam, by I. The weight of the wall in pounds, will be equal to Iht X 160 ; and this being exerted through its centre of gravity, the moment of the weight with respect to the outer edge, as an axis, wiU be equal to UHh X 160 = BQlhf. The pressure of the water against the inner face, in pounds, is equal to I yM X62.5 = Ih? X 31.25. This pressure is applied at the centre of pressure, which is (Example 1) at a distance from the bottom of the wall equal to ^h ; hence, its moment with respect to the outer edge of the wall, is equal to W X 10.4166. The pressure of the water tends to produce rotation out- wards, find the weight of the wall acts to prevent this rota- tion. In order that these forces may be in equilibrium, their moments must be equal ; or SOlht" - Ih' X 10.4166. 248 MECHANICS. Whence, "we find, t = A-v/.1302 =: .36 X h. Next, to find the thickness necessary to prevent sliding along the base. The entire force of friction due to the weight of the wall, is equal to IQQlht X .6 = 96ZA< ; and in order that the wall may not slide, this must be equal to the pressure exerted horizontally against the wall. Hence, 96?A< = 31.25^A'. Whence, we find, t = .325A. If the wall is made thict enough to prevent rotation, it will be secure against sliding. 6. What must be the thickness of a rectangular dam 15 feet high, the weight of the material being 140 lbs. to the cubic foot, that, when the water rises to the top, the structure may be just on the point of overturning ? Ans. 5.7 ft. 7. The staves of a cylindrical cistern filled with water, are held together by a single hoop. Where must the hoop be situated ? Ans At a distance from the bottom equal to one-third of the height of the cistern. 8. Required the pressure of the sea on the cork of an empty bottle, when sunk to the depth of 600 feet, the diameter of the cork being f of ^n inch, "and a cubic foot of sea water being estimated to weigh 64 lbs. ? Ans. 134 lbs. MECHANICS OF LIQUIDS. 249 Buoyant Effort of Fluids. ISY. Let A represent any solid body suspended in a heavy fluid. Conceive this solid to be divided into vertical prisms, whose horizontal sections are infinitely small. Any one of these prisms will be pressed downward by a force eqvial to the weight i ® of a column of fluid, whose base (Art. 155) is -^ jgg~ equal to the horizontal section of the filament, . and whose altitude is the distance of its upper surface from the surface of the fluid ; it will be pressed upward by a force equal to the weight of a column of fluid having the same base and an altitude equal to the distance of the lower base of the filament from the surface of the fluid. The re- sultant of these two pressures is a force exerted vertically upwards, and is equal to the weight of a column of fluid, equal in bulk to that of the filament and having its point of application at the centre of gravity of the volume of the filament. This being ti-ue for each filament of the body, and the lateral pressures being such as to destroy each other's effects^, it follows, that the resultant of all the pres- sures upon the body will be a vertical force exerted upwards, whose intensity is equal to the weight of a portion of the fluid, whose volume is equal to that of the sohd, and the point of application of which is the centre of gravity of the volume of the displaced fluid. This upward pressure is call- ed the buoyant effort of the fluid, and its point of application is called tJie centre of buoyancy. The line of direction of the buoyant efibrt, in any position of the body, is called a line of support. That line of support which passes through the centre of gravity of a body, is called the line of rest. > Floating Bodies. 158. A body wholly or partially immersed in a heavy fluid, is urged downwards by its weight applied at its cen- tre of gravity, and upwards, by the buoyant effort of the fluid applied at the centre of buoyancy. 11* 250 MKCHANICS. ^ I H" The body can only be in equilibrium when the line through the centre of gravity of the body, and the centre of buoy- ancy, is vertical ; in other words, when the line of rest is ver- tical. When the weight of the body exceeds the buoyant effort, the body will sink to the bottom ; when they are just equal, it will remain in equilibrium, wherever placed in the fluid. When the buoyant effort is greater than the weight, it will rise to the surface, and after a few oscillations, will come to a state of rest, in such a position, that the weight of the displaced fluid is equal to that of the body, when it is said to float. The upper surface of the fluid is then called the plane of floatation, and its intersection with the surface of the body, the line of floatation. If a floating body be slightly disturbed from its position of equilibrium, the centres of grav- ity and buoyancy will no longer be in the same vertical line. Let DE represent the plane of floata- tion, G the centre of gravity of the body (Fig. 141), 6?^ its line of rest, and G the centre of buoyancy in the disturbed, position of the body. If the line of support GB, in- tersects the line of rest in M, above G, as in Fig. 141, the buoy- ant effort and the weight will conspire to restore the body to its position of equilibrium ; in this case, the equilibrium must be stable. If the point M falls below i times heavier, than its volume of water, whence the rule. The auxiliary liquid, in some cases, might be a saturated solu- tion of the given body in water ; the rule remains unchanged. To determine the Specific Gravity of the Air. 164. Take a hollow globe, fitted with a stop-cock, to shut ofE" communication with the external air, and, by means of the air-pump or condensing syringe, pump in as much air as is convenient, close the stop-coc]s, and weigh the globe thus filled. Provide a glass tube, graduated so as to show cubic inches and decimals of a cubic inch, and, having filled it with mer- cury, invert it over a mercury bath. Open the stopcock, and allow the com- pressed air to escape into the inverted tube, taking care to bring the tube into such a position that the mercury without the tube is at the same level Fig. i46. as within. The reading on the tube will give the volume of the escaped air. "Weigh the globe again, and subtract the weight thus found from the first weight; this diiference will indicate the weight of the escaped air. Having reduced the measured volume of au- to what it would have occupied at a standard temperature and barometric pressure, by means of rules yet to be deduced, compute the weight of an equivalent volume of water ; divide the weight of the corrected volume of air by that of an equivalent volume of distilled water, and the quotient wUl be the specific gravity sought. To determine the Specific Gravity of a Gas. 165. Take a glass globe of suitable dimensions, fitted with a stop-cock for shutting ofi" communication with the atmosphere. Fill the globe with air, and determine the weight of the globe thus filled referred to a vacuum, as already explained. From the known volume of the globe MECHANICS OF LIQUIDS. 257 and the specific gravity of air, the weight of the contained air can be computed; subtract this from the previous weight, and we shall have the true weight of the globe alone; determine in succession the weights of the globe filled with water a,nd with the gas in vacuum, and from each subtract the weight of the globe ; divide the latter result by the former ; the quotient will be the specific gravity required. Hydrometers. 166. A hydrometer is a floating body, used for the pur- pose of determining specific gravities. Its construction de- pends upon the principle of floatation. Hydrometers are of two kinds. 1. Those in which the submerged volume is constant. 2. Those in which the weight of the instrument remains constant. Nicholson's Hydrometer. 167. This instrument consists of a hollow brass cylinder^ A, at the lower extremity of which is fastened a basket JB, and at the upper extremity a wire, bearing a scale-pan C. At the bottom of the basket is a ball of glass E, containing mer- cury, the object of which is, to cause the in- strument to float in an upright position. By means of this ballast, the instrument is ad- justed so that a weight of 500 grains, placed in the pan C, wUl sink it in distilled water to a notch Z>, filed in the neck. To determine the specific gravity of a solid which weighs less than 500 grains. Place the body in the pan C, and add weights tUl the instrument sinks, in distilled water, to the notch D. The added weights, substracted from 500 grains, will give the weight of the body in air. Place the body in the basket JB, which generally has a reticulated cover, to prevent the body from floating away, and add other weights to the pan, until the instrument again sinks to the notch D. The weights last added give the weight of the water displaced by the body. Eig. 147. 258 MECHANICS. Divide the first of these weights by the second, and the quotient will be the specific gravity required. To find the specific gravity of a liquid. Having carefully weighed the instrument, place it in the liquid, and add weights to the scale-pan tiU it sinks to D. The weight of the instrument, plus the sum of the weights added, will be the weight of the liquid displaced by the instrument. Next, place the instrument in distilled water, and add weights till it sinks" to 1>. The weight of the instrument, plus the added weights, gives the weight of the displaced water. Divide the first result by the second, and the quotient wUl be the specific gravity required. The reason for this rule is evident. A modification of this instrument, in which the basket S^ is omitted, is sometimes constructed for determining specific gravities of liquids only. This kind" of hydrometer is generally made of glass, that it may not be acted upon chemically, by the liquids into which it is plunged. The hydrometer just described, is generally known as Fahren- heit's hydrometer, or Fahrenheit's areometer. Scale Areometer. 16§. The scale areometer is a hydrometer whose weight remains constant ; the specific gravity of a liquid is made known by the depth to which it sinks in it. The instrument consists of a hollow glass cylinder jL, o with a stem (7, of uniform diameter. At the bottom of the cylinder is a bulb S, containing mercury, to make the instrument float upright. By introducing a suitable quantity of mercury, the instrument may be adjusted so as to float at any desired point of the stem. When it is de- signed to determine the specific gravities of liquids, both heavier and lighter than water, it is bal- lasted so that in distilled water, it will sink to the rig. us. middle of the stem. This point is marked on the stem with a file, and since the specific gravity of water is 1, it is numbered 1 on the scale. A liquid is then formed by dissolving common salt in water whose specific gravity is MECHANICS OF LIQUIDS. 259 1.1, and the instrument is allowed to float freely in it; the point E^ to which it then sinks, is marked on the stem, and the intermediate part of the scale, HE, is divided into 10 equal parts, and the graduation continued above and below throughout the stem. The scale thus constructed is marked on a piece of paper placed within the hollow stem. To use this hydrometer, we have simply to put it into the liquid and allow it to come to rest ; the division of the scale which cor- responds to the surface of floatation, makes known the spe- cific gravity of the liquid. The hypothesis on which this instrument is graduated, is, that the increments of specific gravity are proportional to the increments of the submerged portion of the stem. This hypothesis is only approximately true, but it approaches more nearly to the truth as the dia- meter of the stem diminishes. When it is only desired to use the instrument for liquids heavier than water, the instrument is ballasted so that the division 1 shall come near the top of the stem. If it is to be used for liquids lighter than water, it is ballasted so that the division 1 shall fall near the bottom of the stem. In this case we determine the point 0.9 by using a mixture of alcohol and water, the principle of graduation being the same as in the first instance. Volumeter. 169. The volumeter is a modification of the scale areo- meter, difiering from it only in the method of graduation. The graduation is efifected as follows : The instru- ment is placed in distilled water, and allowed to come to a state of rest, and the point on the stem where the surface cuts it, is marked with a file. The submerged volume is then accurately deter- mined, and the stem is graduated in such a man- i/lp ner that each division indicates a volume equal to a hundredth part of the volume originally sub- mei'ged. The divisions are then numbered from the tirst mark in both directions, as indicated in ^ the figure. To use the instrument, place it in the '^' **'' iquid, and note tlie division to which it sinks ; 260 MECHANICS. divide 100 by the number indicated, and the quotient will be the specific gravity sought. The principle employed is, that the specific gravities of liquids are inversely as the vol- umes of equal weights. Suppose that the instrument indi- cates X parts ; then the weight of the instrument displaces X parts of the liquid, whilst it displaces 100 parts of water. Denoting the specific gravity of the liquid by S, and that of water by 1, we have, S : 1 :: 100 : X, -■. S = A table may be computed to save the necessity of per- forming the division. Densimeter. lyo. The densimeter is a modification of the volum- eter, and adniiits of use when only a small portion of the liquid can be had, as is often the case in examining animal secretions, such as bUe, chyle, &c. The construction of the densimeter differs from that of the volumeter, last described, in having a small cup at the upper extremity of the stem, destined to receive the fluid whose specific gravity is to be determined. The instrument is ballasted so that when the cup is empty, the densimeter wiU sink in distilled water to a point £, near the bottom of the stem. This point is the of the instrument. The cup is then filled with distilled water, and the point C, to Fig. iso. which it sinks, is marked ; the space JB C, is divi- ded into any number of equal parts, say 10, and the grad- uation is continued to the top of the tube. To use the instrument, place it in distilled water, and fill the cup with the liquid in question, and note the division to which it sinks. Divide 10 by the number of this division, and the quotient will be the specific gravity required. The principle of the densimeter is the same as that of the volu- meter. MECHANICS OF LIQUIDS. 261 Centesimal Alcoholometer of Oay Lussac. 171. This instrument is the same in construction as the scale areometer ; the graduation is, however, made on a diff- erent principle. Its object is, to determine the percentage of alcohol in a mixture of alcohol and Trater. The graduation is made as follows : the instrument is first placed in absolute alcohol, and ballasted so that it wiU sink nearly to the top of the stem. This point is marked 1 00. Next, a mixture of 95 parts of alcohol and 5 of water, is made, and the pomt to which the instrument sinks, is marked 95. The inter- mediate space is divided into 5 equal parts. Next, a mix- ture of 90 parts of alcohol and 10 of water is made ; the point to which the instrument sinks, is marked 90, and the space between this and 95, is divided into 5 equal parts. In this manner, the entire stem is graduated by successive operations. The spaces on the scale are not equal at differ- ent points, but, for a space of five fiarts, they may be re- garded as equal, without sensible error. To use the instrument, place it in the mixture of alcohol and water, and read the division to which it sinks ; this will indicate the percentage of alcohol in the mixture. In all of the instruments, the temperature has to be taken into account ; this is usually effected by means of correc- tions, which are tabulated to accompany the different instruments. x On the principle of the alcoholometer, are constructed a great variety of areometers, for the purpose of determining the degrees of saturation of wines, syrups, aind other liquids employed in the arts. In some nicely constructed hydrometers, the mercury used as ballast serves also to fill the bulb of a delicate ther- mometer, whose stem rises into the cylinder of the instru- ment, and thus enables us to note the temperature of the fluid in which it is immersed. EXAMPLES. 1. A cubic foot of water weighs 1000 ounces. Required 262 MECHANICS. the weight of a cuhical block of stone, one of whose edges is 4 feet, its specific gravity being 2.5. Ans. 10000 lbs. 2. Required the number of cubic /eet in a body whose weight is 1000 lbs., its specific gravity being 1.25. Ans. 12.8. 3. Two lumps of metal weigh respectively 3 lbs., and 1 lb., and their specific gravities are 5 and 9. What will be the specific gravity of an alloy formed by melting them together, supposing no contraction of volume to take place. g-^ / Ans. StS- 4. A body weighing 20 grains has a specific gravity of 2.5. Required its loss of weight in water. Ans. 8 grains. 5. A body weighs 25 grains in water, and 40 grains in a liquid whose specific gravity is .1. "What is the weight of the body in vacuum ? Ans. 15 grains. 6. A Nicholson's hydrometer weighs 250 graios, and it requires an additional weight of 726 grains to sink it to the notch in the stem, in a mixture of alcohol and water. What is the specific gravity of the mixture ? Ans. .781. 7. A block of wood is found to sink in distilled water tiU ■| of its volume is submerged. What is its specific gravity ? Ans. .8825. 8. The weight of a piece of cork in air, is -f oz.'^tne" weight of a piece of lead in water, is 6^ oz. ; the weight of the cork and l^d together in water, is 4^^ oz. What is the specific gravity of the cork ? ^ Ans. 0.24. 9. A solid, whose weight is 250 grains, weighs in water, 147 grains, and, in another fluid, 120 grains. What is the specific gravity of the latter fluid ? Ans. 1.262. 10. A solid weighs 60 grains in air, 40 in water, and 30 in an acid. What is the specific gravity of the acid ? Ans. 15. MECHANICS 01? LIQUIDS. 263 The following table of the specific gravity of some of the most important solid and fluid bodies, is compiled from a table given in the Ordnance Manual. TABLE OF SPECIFIC GKAVITIES OF SOLIDS AND LIQUIDS. Antimony, cast. ... Brass, cast Copper, cast Gold, hammered . . . Iron, bar Iron, oast Lead, cast Mercury at 32° F . . " at 60° Platina, rolled " hammered. Silver, hammered. . Tin, cast Zinc, cast Bricks Chalk Coal, bituminous.. . Diamond Earth, common. ... Gypsum iTory SPEC. QEAT. 6.712 8.396 8.788 19.361 7.788 7.207 11.352 13.598 13.680 22.069 20.337 10.511 7.291 6.861 1.900 2.784 1.270 3.521 1 500 2.168 1.822 Limestone Marble, common. Salt, common . . . Sand Slate Stone, common . . Tallow Boxwood Cedar Cherry Lignum vitae .... Mahogany Oak, heart Pine, yellow Nitric acid Sulphuric acid Alcohol, absolute . Ether, sulphuric . Sea water Olive oil Oil of Turpentine 8PE0. SEAT. 3.180 2.686 2.130 1.800 2.672 2.520 0.945 0.912 0.696 0.715 1.333 0.854 1.170 0.660 1.217 1.841 0.792 0.715 1.026 0.915 0.870 Thermometer. ITS. A thermometer is an instrument used for measur- ing the temperatures of bodies. It is found, by observation, that almost all bodies expand when heated, and contract when cooled, so that, other things being equal, they always occupy the same volumes at the same temperatures. It is also found that difierent bodies expand and contract in a different ratio for the same increments of temperature. As a general rule, liquids expand much more rapidly than solids, and gases much more rapidly than liquids. The construc- tion of the thermometer depends upon this principle of unequal expansibility of different bodies. A great variety of combinations have been used in the construction of ther- 9 Fig. 151. 264 MECHANICS. mometers, oniy one of which, the common mercurial ther- mometer, will be described. The mercurial thermometer consists of a cylindrical or spherical bulb A, at the upper extremity of which, is a narrow tube of uniform bose, hermetically sealed at its upper end. The bulb and tube are nearly filled with mercury, and the whole is attached to a frame, on which is a scale for deter- mining the temperature, which is indicated by the rise and fall of the mercury in the tube. The tube should be of uniform bore through- out, and, when this is the case, it is .found that the relative expansion of the mercury and glass is very nearly uniform for constant increments of temperature. A thermometer maybe constructed and graduated as follows : A tube of uniform bore is selected, and upon one extremity a bulb is blown, which may be cylindrical or spherical; the former shape is, on many accounts, the preferable one. At the other extremity, a conical-shaped funnel is blown open at the top. The funnel is filled with mercury, which should be of the purest quality, and the whole being held vertical, the heat of a spirit-lamp is applied to the bulb, which expand- ing the air contained in it, forces a portion in bubbles up through the mercury in the funnel. The instrument is next allowed to cool, when a portion of mercury is forced down the capillary tube into the bulb. By. a repetition of this process, the entire bulb may be filled with mercury, as well as the tube itself. Heat is then applied to the bulb, until the mercury is made to boU ; and, on being cooled down to a little above the highest temperature which it is desired to measure, the top of the tube is melted ofi' by means of a jet of flame, urged by a blow-pipe, and the whole is her- metically sealed. The instrument, thus prepared, is attached to a frame, and graduated as follows : The instrument is plunged into a bath of melting ice, and, after being allowed to remain a sufficient time for the MECHANICS OF LIQUIDS. 265 parts of the instrument to take the uniform temperature of the melting ice, the height of the mercury in the tube is m.arked on the scale. This gives the freezing point of the scale. The instrument is next plunged into a bath of boiling water, and allowed to remain long enough for all of the parts to acquire the temperature of the water and steam. The height of the mercury is then marked on the scale. This gives the boiling point of the scale. The freezing and boiling points having been determined, the intermediate space is divided into a certain number of equal parts, according to the scale adopted, and the graduation is then continued, both upwards ^and downwards, to any .desired extent. Three principal scales are used. Fahkenheit's scale, in which the space between the freezing and boiling point is divided into 180 equal parts, called degrees, the freezing point being marked 32°, and the boiling point 212°. In this scale, the point is 32 degrees below the freezing point. IVie Centigrade scale,\ n which the space between the fixed points is divided into 100 equal parts, called degrees. The of this scale is at the freezing point. Reaumur's scale, in which the same space is divided into 80 equal parts, called degrees. The of this scale also is at the freezing point. If we denote the number of degrees on the Fahrenheit, Centigrade, and Reaumur scales, by F, 0, and M respec- tively, the following formula will enable us to pass from any one of these scales to any other : i(J'°-32) = i(7° =iiJ°. The scale most in use in this country is Faheenheit's The other two are much used in Europe, particularly the Centigrade scale. Velocity of a liquid flowing through a small orifice. 1T3. Let ABJ> represent a vessel, having a very small orifice at its bottom, and filled with any liquid, 1? 266 MECHANICS. A- 13S Denote the area of the orifice, by a, and its depth below the upper surface, by h. Let D represent an infinitely smaU, layer of the liquid situated at the orifice, and denote its height, ^^^_ by h'. This layer is (Art. 155) urged down- 'V'^--'- ■wards by a force equal to the weight of a 'gisa. column of the liquid whose base is equal to the orifice, and whose height is h ; denoting this pressure, by p, and the weight of a unit of volume of the liquid, by w, we shall have, p = wah. If the element is pressed downwards by its own weight alone, this pressure being denoted by^', we have, p' = wah'. Dividing the former equation by the latter, member by member, we have, p _ h p' ~h''' that is, the pressures are to each other 'as the heights h and h' . "Were the element to fall through the small height A', imder the action of the pressure^', or its own weight, the velocity generated would (Art. 115) be given by the equation, v' = ^Jlgh'. Denoting the velocity actually generated whilst the ele- ment is falling throught the height A', by «, and recol- lecting that the velocities generated in falling through a MEOIIANICS OF LIQUIDS. 267 given height, are to each other as the square roots of the pressures, we shall have. V : v' : : '\fp : ^fp\ .; v = v' \/— , • y p' Substituting for v' its value, just deduced, and for ^ its value, Y7, we have P (150.) Hence, we conclude that a liquid will issue from a very small orifice at the iottom of the containing vessel, with a velocity equal to that acquired by a heavy body in falling freely through a height equal to the depth of the orifice below the surface of the fluid. We have seen that the pressure due to the weight of a fluid upon any poiat of the surface of the containing vessel, is normal to the surface, and is always proportional to the depth of the point below the level of the free surface. Hence, if the side of a vessel be thin, so as not to affect the flow of the liquid, and an orifice be made at any point, the liquid win flow out in a jet, normal to the surface at the. opening, and with a velocity due to a height equal to that of the orifice from the free surface of the fluid. If the orifice is on the vertical side of a vessel, the initial direction of the jet will be horizontal ; if it be made at a ( point where the tangent plane is oblique to the horizon, the initial direction of the jet wiU be oblique ; if the opening is made on the upper side of a por- tion of a vessel where the tangent is horizontal, the jet will be directed upwards, and will rise to a height due to the velocity ; that is, to the height of the ng. les. upper surface of the fluid. This T 1 •,B ■■,6 t- k \ 1 — ^0 268 MECHANICS. can be illustrated experimentally, by introducing a tube near the bottom of a vessel of water, and bending its outer extremity upwards, when the fluid will be observed to. rise to the level of the upper surface of the water in the vessel. Spouting of Iiiquids on a Horizontal Plane. 174. Let KL represent a vessel filled with water. Let D represent an orifice in its ver- tical side, and DM the path described by the spouting fluid. We may regard each drop of water as it issues from the orifice, as a projectile shot forth hori- zontally, and then acted upon by the force of gravity. Its path wUi, therefore, be a parabola, and the cii-cumstances of its motion will be made known by a discussion of Equations (115) and (120). Denote the distance MK, by A', and the distance -Z>X, by h. We have, from Equation (120), by making y equal to h\ and x = KE, Kg. 154. KjE = 2gh ; Jience, by substitu- But we have found that v = tion, we have, ITU = 2yT/i/. If we describe a semicircle on JTZi, as a diameter, and through D draw an ordinate DJS, we shall have, from a well-known property of the circle. Hence we have, by substitution, KE = 2BS. MECHANICS OF LIQTJIPS. 269 Since there are two points on KL at which the ordinates are equal, it follows that there are two orifices through which the fluid wUl spout to the same distance on the horizontal plane; one of these wiU be as far above the centre 0, as the other is below it. If the orifice be at 0, midway between K and i, the ordinate 08 will be the greatest possible, and the range KE' wUl be a maximum. The range in this case will be equal to the diameter of the circle LHK, or to the distance from the level of the water in the vessel to the horizontal plane. If a semi-parabola LE be described, having its axis ver- tical, its vertex at i, and focus at K^ then may every point jP, within the curve, be reached by tw;o separate jets issuing from the side of the vessel ; every point on the curve can be reached by on%, and only one ; whilst points lying without the curve cannot be reached by any jet whatever. K the jet is directed obliquely upwards by a short pipe A (Fig. 153), the path desci-ibed by each particle will still be the arc of a parabola AJBC. Since each particle of the liquid may be regarded as a body projected obUquely up- ward, the nature of the path and the circumstances of the motion will be given by Equation ( 115 ). In like manner, a discussion of the same equation will rdake known the nature of the path and the drcumstances' of motion, when the jet is directed obliquely downwards by means of a short tube. Modifications due to extraneous pressure. ITS. If we suppose the upper surface of the liquid, in any of the preceding cases, to be pressed by any force, as when it is urged downwards by a piston, we may denote the height of a column of fluid whose weight is equal to the ex- traneous pressure, by h'. The velocity of efflux will then be given by the equation, V = ^2g(h + h'). 370 TVIECHANICS. The pressure of the atmosphere acts equally on the upper surface and the surface of the opening ; hence, in ordinary cases, it may be neglected ; but were the water to flow into a vacuum, or into rarefied air, the pressure must be taken into account, and this may be done by means of the formula just given. Should the flow take place into condensed air, or into any medium which opposes a greater resistance than the atmos- pheric pressure, the extraneous pressure would act upwards, h' would be negative, and the preceding formula would become, V = ■\/2g{h - A'), Coefficients of Efflux and Velocity. 1 '!'6. When a vessel empties itself through a small orifice at its bottom, it is observed that the particles of fluid near the top descend in vertical lines ; when they approach the bottom they incline towards the orifice, the converging Unes of fluid particles tending to cross each other as they emerge from the vessel. The result is, that the streanj grows nar- rower, after leaving the vessel, until it reaches a point at a distance from the vessel equal to al^out the radius of the orifice, when the contraction becomes a minimum, and below that point the vein again spreads out. This phenomenon is called the contraction of the vein. The cross section at the most contracted part of the vein, is not far from /J^ of the area of the orifice, when the vessel is very thin. If we de- note the area of the orifice, by a, and the area of the least cross section of the vein, by «', we shall have, a' = ha, in which ^ is a number to be determined by experiment. This number is called the coefficient of contraction. To find the quantity of water discharged through an ori- fice at the bottom of the containing vessel, rt^ a second, we have only to multiply the area of the smallest cross section MKCHANIOS OF LIQUIDS. 271 of the vein, by the velocity. Denoting the quantity dis- charged in one second, hy Q\ we shall have, O! = hay/igh. This formula is only true on the supposition that the actual velocity is equal to the theoretical velocity, which is not the case, as has been shown by experiment. The theo- retical velocity has been shown to be equal to ■\/2gh, and if we denote the actual velocity, by v', we shall have, in which I is to be determined by experiment ; this value of I is slightly less than 1, and is called the coefficient ofveloo- ity. In order to get the actual discharge, we must replace ■ijigh by l-^/lgh, in the preceding equation. Doing so, and denoting the actual discharge per second, by §, we have, Q = hla-\/%gh. The product M, is called the coefficient of effiux. It has been shown by experiment, that this coefficient for orifices in thin plates, is not quite constant. It decreases slightly, as the area of the orifice and the velocity are increased ; and it is further found to be greater for circular orifices than for those of any other shape. If we denote the coefficient of efflux, by w, we have. In this equation, h is called the head of water. Hence, we may define the head of water to be the distance from the orific^tp the plane of the upper surface of the fluid. The meaa 'Value of m corresponding to orifices of from i to 6 inches in diametei', with from 4 to 20 feet head of 272 MECriANICS. water, has been found to be about .615. If we take the value of k = .64, we shall have, 1 = k ■615 ^640 - .96. That is, the actual velocity is only ^^ of the theoretical velocity. This diminution is due to friction, viscosity, &c. Fig. 155. Efflux through Short Tubes. 171'. It is found that the discharge from a given orifice is increased, when the thickness of the plate through which the flow takes place,is increased ; also, when a short tube is introduced. When a tube AB, is employed which is not more than four times as long as the diameter of the orifice, the value of m becomes, on an aver- age, equal to .813; that is, the discharge per second is 1.325 times greater when the tube is used, than without it. In using the cylindrical tube, the contraction takes place at the outlet of the vessel, and not at the outlet of the tube. Compound mouth-pieces are sometimes used formed of two conic frustrums, as shown in the figure, having the form of the vein. It has been sho^vn by Etelwein, that the most effec- tive tubes of~this form should have the diameter of the cross section CD, equal to .833 of the diameter AB. The angle made by the sides CJE and JDF, should be about 5° 9', and the length of this portion should be three times that of the other. EXAMPLES. 1 . With what theoretical velocity wiU water issue from a small orifice 16^'^ feet below the surface of the fluid ? Ans. 32|ft. MECHANICS OF LIQUIDS. 273 2. If the area of the orifice, in the last example, is -^^ of a square foot, and the coefficient of efflux .615, how many- cubic feet of water will lie discharged per minute ? • Ans. 118.695 ft. 3. A vessel, constantly filled with water, is 4 feet high, with a cross-section of one square foot ; an oi-ifice ia-the bottom has an area of one square inch. In what time will three-fourths of the water be drawn ofi", the coefficient of efflux being .6 ? Ans. 1 minute, nearly. 4. A vessel is kept constantly fuU of water. How many cubic feet of water will be discharged per minute from an orifice 9 feet below the upper surface, having an area of 1 square inch, the coefficient of efflux being .6 ? Ans. 6 cubic feet, about. 5. In the last example, what will be the discharge per minute, if we suppose each square foot of the upper surface to be pressed by a force of 645 lbs. ? Ans. 8f cubic feet, about. 6. The head of water in a vessel kept full of water is -5-^ of a square foot. What quantity of water will be discharged per second, when the orifice is through a thin plate ? SOLUTION. In this case, we have. Q =z .615 X .01/2 X 321 x 16 = .197 cubic feet. When a short cylindrical tube is used, we have, Q = .197 X 1.325 =: .261 cubic feet. In Etblwbin's compound mouthpiece, if we take the smallest cross-section as the orifice, and denote it by a, it is found that the discharge is 2^ times that through an orifice of the same size in a thin plate. In this case, we have, sup- posing a = y^(j- of a cubic foot, Q = .197 X-2^ — .49 cubic feet. 12* 274 MECHANICS. Motion of vrater in open channels. l'y§. When water flows through an open channel, as in A river, canal, or open aqueduct, the form of the channel being always the same, and the supply o# water being con- stant, it is a matter of observation that the flow becomes uniform ; that is, the quantity of water that flows through any cross-section, in a given time, is constant. On account of adhesion, friction, &c., the particles of water next the sides and bottom of the channel have their motion retarded. This retardation is imparted to the next layer of particles, but in a less degree, and so on, till a line of particles is reached whose velocity is greater than that of any other filament. This line, or filament of particles, is called the axis of the stream. In the case of cylindrical pipes, the axis coincides sensibly with the axis of the pipe ; in straight, open channels, it coincides with that line of the upper sur- face which is midway between the sides. A section at right-angles to the axis is called a cross-sec- tion, and, from what has been shown, the velocities of the fluid particles will be difierent at different points of the same cross-section. The mean velocity corresponding to any cross-section, is the average velocity of the particles at every point of that section. The mean velocity may be found by dividing the volume which flows through the sec- tion in one second, by the area of the cross-section. Since the same volume flows through each ci'oss-section per second, after the flow has become uniform, it follows that, in channels of varying width, the mean velocity, at any section, will be inversely as the area of the section-. The intersection of the plane of cross-section with the •sides and bottom of the channel, is called the perimeter of the section. In the case of a pipe which is constantly filled, the perimeter is the entire line of iutersection of the plane of cross-section, -with the interior surface of the pipe. The mean velocity of water in an open channel depends, in the first place, upon its incUnation to the horizon. As the inclination becomes greater, the component of gravity in the MECHANICS OF LIQtIIDS. 275 direction of the channel increases, and, consequently, the velocity becomes greater. Denoting the inclination by Z, and resolving the force of gravity into two components, one at right angles to the upper surface, and the other parallel to it, we shall have for the latter component, gsinl. This is the only force tfiat acts to increase the velocity. The velocity will be diminished by friction, adhesion, &c. The total eflfect of these resistances will depend upon the ratio of the perimeter to the area of the cross section, and also upon the velocity. The cross-section being the same, the resistances will increase as the perimeter ioereases ; con- sequently, for the same cross-section, the resistance of fric- tion win be the least possible when the perimeter is least possible. The retardation of the flow will also diminish as the area of the cross-section is increased, other things re- maining unchanged. If we denote the area of the cross-section by a, the perimeter, by P, and the velocity, by v, we shall have, in which / denotes some function of v. Since the LQcUnation is very small in all practical cases, we may place the inclination itself for the sine of the Laclin- ation, and doing so, it has been shown by Pbony, that the function of v may be expressed by two terms, one of which is of the first, and the other of the second degree, with re- spect to V ; or. Denoting -= by i2, — by A, and - by I, we have, finally, kv -f Iv"^ = EI, 276 MECHANICS. in which h and I are constants, to be determined by experi- ment. According to Etelwein, we have, k = .0000242651, and I — .0001114155. Substituting these values, and solving with respect to v, we have. V — ~ 0.1088941604 + v'.0118580490 + 8975.414285i2j; from which the velocity can be found when JR and I are known. The values of h and ?, and consequently that of w, were found by Peony to be somewhat diiferent from those given above. Those of Etelvstein are selected for the reason that they were based upon a much larger number of exper- iments than those of Peony. Having the mean velocity and the area of the cross-sep- tion, the quantity of water delivered in any time can be computed. Denoting the quantity delivered in n seconds, by §, and retaining the preceding notation, we have, Q, = nav. The quantity of water to be delivered is generally one of the data in all practical problems involving the distribution of water. The difference of level of the point of supply and delivery is also known. The preceding principles ena- ble us to give such a form to the cross-section of the canal, or aqueduct, as will ensure the requisite supply. Were it requii-ed to apply the results just deduced, to the case of irregular channels, or to those in which there were many curves, a considerable modification would be required. The theory of these modifications does not come within the limits assigned to this treatise. For a complete discussion of the whole subject of hydraulics in a popular .form, the reader is referred to the Traite d'Hydraulique D'Aubisson. MECHANICS OF LIQinDS. 277 • Motion of Tirater in pipes. ITO. The circumstances of the motion of water in, pipes, are clgsely analagous to those of its motion in open channels. The — ■ . forces which tend to impart motion ^ ' are dependent upon the weight of 1^^^^^' g ^^ c the water in the pipe, and upon the ^^^ ^^M height of the water in the upper rig. 15t. reservoir. Those which tend to prevent motion depend upon the depth of water in the lower reservoir, friction in the pipe, adhesion, and shocks arising from irregularities in the bore of the pipe. The re- tardation due to shocks will, for the present, be neiglected. Let AJS represent a straight cylindrical pipe, connecting two reservoirs H and H'. Suppose the water to maintain its level at JS, in the upper, and at C, in the lower reservoir. Denote AS, by h, and JB G, by h'. Denote the length of the ijipe, by I, its circumference, by c, its cross-section, by a, its incUnation, by (p, and the weight of a unit of volume of water, by w. Experience shows that, under the circumstances above indicated, the flow soon becomes uniform. We may then regard the entire mass of fluid in the pipe as a coherent solid, moving with a mean uniform velocity down the inclined plane AS. The weight of the water in the pipe will be equal to wal„ If we resolve this weight into two components, one perpen- dicular to, and the other coinciding with the axis of the tube, we shall have for the latter component, wal&imp. But Isinq) is equal to DJB. Denoting this distance by A", we shall have for the pressm-e in the direction of the axis, due to the weight of the water in the pipe, the expression wah". This pressure acts from A towards B. The pressure due to the weight- of the water in M, and acting in the same direction, is wah. The forces acting from B towards A, are, first, that due 278 MECHANICS. to the weight of the water in M', which is equal to wah' ; and, secondly, the resistance due to friction ^d adhesion. This resistance depends upon the length of the pipe, its circumference and the velocity. It has been shown, by experiment, that this force may be expressed by the term, cl{kv + k'v'). Since the velocity has been supposed uniform, the forces acting in the direction of the axis, must be in equUibrium. Hence, wah + wah" = wah' + cl{kv + k'v') ; whence, by reduction, k k' a/h + h"-h\ — v-\ w' = -( J ) • WW e\ I J The factor - is equal to one-fourth of the diameter of the c ^ pipe. Denoting this by c?, we shall have, — = i(? ; denot- h ^ *' , , h+h"-h' ^, mg — by m, — by n, and j — - by s, we have, mv + no' = \ds. The values of m and w, as determined experimentally by Peony, are, m = 0.00017, and n = 0.000106. Hence, by substitution, .OOOlVw + .OOOlOGw' = \ds. If V is not very small, the first term may be neglected, which wiU give, V = 48.56-v/^. MECHANICS OF LIQUIDS. 279 If we denote the quantity of water delivered in n sec- onds, by Q, we shall have, Q — nav — 48.56wa-v/S^ The velocity will be greatly diminished, if the tube is curved to any considerable extent, or if its diameter is not uniform throughout. It is not intended to enter into a discussion of these cases ; their complete development would require more space than has been allotted to this branch of Mechanics. General Remarks on the distribution and flow of water in pipes. 180. Whenever an obstacle occurs in the course of an open channel or pipe, a change of velocity must take place. In passing the obstacle, the velocity of the water mil increase, and then, impinging upon that which has already passed, a shock will take place. This shock consumes a certain amount of living force, and thus diminishes the velocity of the stream. All obstacles should be avoided ; or, if any are unavoidable, the stream should be diminished, and again enlarged gradually, so as to avoid, as much as possible, the necessary shock incident to sudden changes of velocity. For a like reason, when a branch enters the main channel, it should be made to enter as nearly in the dii-ection of the current as possible. All changes of direction give rise to mutual impacts amongst the particles, and the more, as the change is more abrupt. Hence, when a change of du-ection is necessary, the straight branches should be made tangential to the curved portion. The entrance to, and outlet from a pipe or channel, should be enlarged, in order to diminish, as much as possible, the coefficients of ingress and egress. When a pipe passes over uneven ground, sometimes as- cending, and sometimes descending, there is a tendency to a collection of bubbles of air, at the highest points, which 280 MECHANICS. may finally come to act as an impeding cause to the flow. There should, therefore, be suitable pipes inserted at the highest points, to permit the confined air to escape. Finally, attention should be given to the form of the cross- section of the channel. If the channel is a pipe, it should be made cylindrical. If it is a canal or open aqueduct, that form should be given to the perimeter which would give the greatest cross-section, and, at the same time, conform to the necessary conditions of the structure. The perimeter in open channels is generally trapezoidal, from the necessity of the case ; and it should be remembered, that the nearer the form approaches a semi-ckcle, the greater will be the flow. Capillary Phenomena. 181. When a liquid is in equilibrium, under the action of its own weight, it has been shown that its upper surface is level. It is observed, however, in the neighborhood of solid bodies, such as the walls of a containing vessel, that the surface is sometimes elevated, and sometimes depressed, according to the nature of the liquid and sohd in contact. These elevations and depressions result from the action of molecular forces, exerted between the particles of the liquid and solid which are in contact ; from the fact that they are more apparent in the case of small tubes, of the diameter of a hair, these phenomena have been called capillary phenom- ena, and the forces giving rise to them, capillary forces. These forces only produce sensible effects at extremely small distances. Claieaut has shown, that when the inten- sity of the force of attraction of the particles of the solid for those of the liquid, exceeds one-half th^t of the particles of the liquid for each other, the liquid wUl be elevated about the sohd ; when less, it will be depressed ; when equal, it will neither be elevated nor depressed. In the first case, the resultant of the capillary forces is a force of capillary attrac- tion ; in the second case, it is a force of capillary repulsion ; and in the third case, the capillary forces are in equilibrium. The following are some of the observed efi^ects-of capillary MECHANICS OF LIQUIDS.' 281 action : When a solid is plunged into a liquid -which is capable of moistening it, as when wood or glass is plunged into water, the surface of .the liquid is heaped up about the solid, taking a concave form, as shown in Fig. 158. When a solid is plunged ioto a liquid which is not capable of moistening it, as when glass is plunged into mercury, the surface of the liquid is depressed about the solid, taking a convex form, as shown in Fig. 159. rig. 158. The surface of the liquid in the neighbor- hood of the bounding surfaces of the con- taining vessel takes the form of concavity or convexity, according as the material of the vessel is capable of being moistened, , j,j jgg or not, by the liquid. These phenomena become more apparent when, instead of a solid body, we plunge a tube into a liquid, according as the material of the tube is, or is not, capable of being moistened by the liquid, the liquid will rise in the tube or be depressed in it. When the liquid rises in the tube, its upper surface takes a concave shape ; when it is depressed, it takes a con- vex form. The elevations or depressions increase as the dia- meter of the tube diminishes. Elevation and Depression between plates. 1§2. If two plates of any substance are placed parallel to each other, it is found that the laws of ascent and descent of the liquid into which they are plunged, are essentially the same as for tubes. For example: if two plates of glass parallel to each other, and pretty close together, are plunged into water, it is found that the water will rise between them to a height which is inversely proportional to their dist- ance apart ; and further, that this height is equal to half the height to which water would rise in a glass tube whose internal diameter is equal to the distance between the plates. 282 MECHANICS. If the same plates are plunged into mercury, there will be a depression according to an analagous law. If two ]3lates of glass, AB and A O, inclined to each other, as shown in Fig. 160, their line of junction being vertical, be plunged into any liquid which wiU moisten them, the liquid will rise between them. It will rise higher near the , • ••, ^i& ISO- junction, the surface takmg a curved form, such that any section made by a plane through A, will be an equilateral hyperbola. This form of the elevated fluid conforms to the laws above explaiaed. If the line of junction of the two plates is -=::^r^ horizontal, a small quantity of a liquid between them, which will moisten them, will assume -=^IlB the shape shown at A. If the liquid does Kg. lei. not moisten the plates, it wiU take the form shown at .B. Attraction and Repulsion of Floating Bodies. 183. If two small balls of wood, both of which can be moistened by water, or two small balls of wax, which cannot be moistened by water, be placed in a vessel of water, and brought so near each other that the surfaces of capillary elevation or depression interfere, the balls wUl attract each other and come together. If one ball of wood and one of wax be brought so near that the surfaces of capillary eleva- tion and depression interfere, the bodies wUl repel each other and separate. If two needles be carefully oiled and laid upon the surface of a vessel of water, they will repel the water from their neighborhood, and float. If, 'whilst floating, they are brought sufficiently near to each other to permit the surfaces of capillary depression to interfere, the needles will immediately rush together. The reason of the needles floating is, that they repel the water, heaping it up on each side, thus forming a cavity in the surface ; the needle is buoyed up by a force equal to the weight of the displaced fluid, and, when this exceeds the weight of the MECHANICS OF LIQUIDS. 283 needle, it will float. It is on this principle that certain insects move freely over the surface of a sheet of water; their feet are lubricated with an oily substance which repels the water from around them, producing a hollow around each foot, and giving rise to a buoyant effort greater than the weight of the insect. The principle of- mutual attraction between bodies, both of which repel water, or both of which attract it, accounts for the fact that small floating bodies have a tendency to collect in groups about the borders of the containing vessel. When the material of which the vessel is made, exercises a different capillary action from that of the floating particles, they will aggregate themselves at a distance from the sur- face of the vessel. Applications of the Principles of Capillarity. 1§4. It is in consequence of capillary action that water rises to fill the pores of a sponge, or of a lump of sugar. The same principle, causes the oil to rise in the wick of a lamp, which is but a bundle of fibres very nearly in contact, leaving capillary interstices between them. The siphon JUter differs but little in principle from the wick of a lamp. It consists of a bundle of fibres like a lamp-wick, one end of which dips into a vessel of the liquid to be filtered, whilst the other hangs over the edge of the vessel. The liquid ascends the fibrous mass by the principle of capillary attraction, and continues to advance till it reaches the overhanging end, when, if this is lower than the upper surface oi the liquid, the liquid will fall by drops from the end of the wick, the impurities being left behind. The principle of capillary attraction is used for splitting rocks and raising weights. To employ this principle in cleaving mill-stones, as is done in France, the stone is first dressed to the form of a cylinder of the required diameter for the mill-stone. Grooves are then cut around it where the divisions are to take place, and into these grooves thoroughly dried wedges of willow-wood are driven. On being exposed to the action of moisture, the cells of the 284 MECHANICS. ■U'ood absorb a large quantity of water, expand, and finally split the rock. To raise a weight, let a thoroughly dry cord be fastened to the weight, and then stretched to a point above. If, now, the cord be moistened, the fibres will absorb the moisture, " expanding laterally, the rope will be diminished in length, and the weight raised. The principle of capillary attraction is also very exten- sively employed in metallurgy, in a process of purifying metals, called cupellation. Sndosmose and SKOsmose. 1§5. The names endosmose and exosmose have been given to two currents flowing in a contrary direction between two liquids, when they are separated by a thin porous partition, either organic or inorganic. The discovery of this phenomena is due to M. Duteochet, who called the flowing in, endosmose, and the flowing out, exosmose. The existence of the currents was established by means of an instrument, to which he gave the name endosmomelre. ' This instrument consists of a long tube of glass, at one end of which is attached a membranous sack, secured by a tight ligature. If the sack is filled with gum water, a solution of sugar, albumen, or, in fact, with almost any solution denser than water, and then plunged into water, it is observed, after a time, that the fluid rises in the stem, and is depressed in the vessel, showing that water has entered the sack by passing through the pores. By applying suitable tests, it is also found, that a portion of the liquid in the sack has passed through the pores into the vessel. Two currents are thus established. If the operation be reversed, and the bladder and tube be fllled with pure water, the liquid in the vessel will rise, whilst that in the tube falls. The phenomena of endosmose and exosmose are extremely various, and serve to explain a great variety of interesting facts in animal and vegetable physiology. The cause of the currents is the action of molecular forces exerted between the particles of the bodies employed. MECHANICS OF GASES AND VAPOES. 286 CHAPTER YIII. MECHANICS OF GASES AND VAPOES. Gases and Vapors. 186. Gases and vapors are distinguished from otlier fluids, by their great compressibility, and correspondingly great elasticity. These fluids continually tend to occupy a greater space ; this expansion goes on till counteracted by some extraneous force, as that of gravity, or the resistance offered by a containing vessel. The force of expansion, which is common to all gases and vapors, is called their tension or elastic force. We shall take for the unit of this force at any point, the pressure which would be exerted upon a square inch of surface, were the pressure the same at every point of the square inch as at the point in question. If we denote this unit, by p, the area pressed, by a, and the entire pressure, by P, we shall have, P — ap ( 161.) Most of the priaciples already demonstrated for liquids hold good for gases and vapors, but there are certain pro- perties arising from elasticity which are peculiar to seriform fluids, some of which it is now proposed to investigate. Atmospheric Air. !§'}'. The gaseous fluid which envelops our globe, and extends on all sides to a distance of many miles, is called the atmosphere. It consists principally of nitrogen and oxygen, together with variable, but small portions of watery vapor and carbonic acid, all in a state of mixture. On an average, it is found by experiment that 1000 parts by volume of 286 MECHANICS. atmospheric air, taken near the surface of the earth, consists of about, 788 parts of nitrogen, 197 parts of oxygen, 14 parts of watery vapor, 1 part of carbonic acid. The atmosphere may, physically speaking, be taken as a type of gases, for it is found by experiment that the laws regulating the density, expansibility, and elasticity, are the same for aU gases and vapors, so long as they maintain a purely gaseous form. It is found, however, in the case of vapors, and of those gases which have been reduced to a liquid form, that the law changes just before actual lique- faction. This change appears to be somewhat analagous to that observed when water passes from the liquid to the solid form. Although water does not actually freeze tiU reduced to a temperature of 32° Fah., it is found that it reaches its maximum density at about 38°. 75, at which temperature the particles seem to commence arranging themselves according to some new laws, preparatory to talcing the soKd form. Atmospheric Pressure. 188. If a tube, 35 or 36 inches long, open at one end and closed at the other, be filled with pure mercury, and inverted in a basin of the same, it is observed that the mercury will fall in the tube until the vertical distance from the surface of the mer- cury in the tube to that in the basin is about 30 inches. This column of mercury is sustained by the pressure of the atmosphere exerted upon the surface of the mercury in the basin, and transmitted through the fluid, according to the general law of transmission of pressures. The column of mercury sustained by the elasticity of j,. ^^^ the atmosphere is called the barometric column, because it is generally measured by an instrument called a barometer. In fact, the instrument just described, when MECHANICS OF GASES AND VAPORS. 287 provided with a suitable scale for measuring the altitude of the column, is a complete barometer. The height of the barometric column fluctuates somewhat, even at the same place, on account of changes of temperature, and other causes yet to be considered. Observation has shown, that the average height of the Jjarometrio column at the level of the sea, is a trifle less than 30 inches. The weight of a column of mercury 30 inches in height, having a cross section of one square inch, is nearly 15 pounds. Hence, the unit of atmospheric pressure at the level of the sea, is 15 pounds. This unit is called an atmosphere, and is often employed in estimating the pressure of elastic fluids, particularly in the case of steam. Hence, to say that the pressure of steam in a boiler is two atmospheres, is equivalent to saying, that there is a pressure of 30 pounds upon each square inch of the interior of the boiler. In general, when we say that the tension of a gas or vapor is n atmospheres, we mean that each square inch is pressed by a force of n times 15 pounds. Maiiotte's Law. 1§9. When a given mass of any gas or vapor is com- pressed so as to occupy a smaller space, other things being equal, its elastic force is increased ; on the contrary, if its volume is increased, its elastic force is diminished. The law of mcrease and diminution of elastic force, first discovered by Makiottb, and bearing his name, may be enunciated as follows : The elastic force of a given mass of any gas, whose tem- perature remains the same, varies inversely as the volume which it occupies. As long as the mass remains the same, the density must vary iaversely as the volume occupied. Hence, from Maei- 'otte's Law, it follows, that. The elastic force of any gas, whose temperature remains the same, varies as its density, and conversely, the density varies as the elastic force. 288 MECHANICS. n\fp Fig. 168 Maeiotte's law may be verified in the case of atmospheric ail-, by the aid of an instrument called Mauiotte's Tube. This instrument consists of a tube AH CD, of uniform bore, bent so that its two branches are parallel to each other. The shorter branch AJB, is closed at its upper extremity, whUst the longer one remains open for the reception of mercury. Between the two branches of the tube, and attached to the same frame with it, is a scale of equal parts for measuring distances. To use the instrument, place it in a vertical position, and pour mercury into the tube, until it just cuts oS the communication between the two branches. The mercury will then stand at the same level £ G, in both branches, and the tension of the confined air in AH, will be exactly equal to that of the external atmosphere. If an additional quantity of mer- cury be poured into the longer branch, the confined air in the shorter branch will be compressed, and the mercury will rise m both branches, but higher in the longer, than in the shorter one. Suppose the mercury to have risen in the shorter branch, to IT, and in the longer one, to H. There will be an equilibrium in the mercury lying below the hori- zontal plane lOT; there will also be an equilibrium between the tension of the air in AIT, and the forces which give rise to that tension. These forces are the pressure of the exter- nal atmosphere transmitted through the mercury, and the weight of a column of mercury Avhose base is the cross-sec- tion of the tube, and whose altitude is PIT. If we denote the height of the column of mercury which wUl be sustained by the pressure of the external atmosphere, by h, the ten- sion of the air in AIT, will be measured by the weight of a column of mercury, whose base is the cross-section of the tube, and whose height is A + JPIT. Since the weight is proportional to the height, the tension of the confined air will be proportional to A + JPJT. Now, whatever may be the value of -PJi^, it is found that, MECHANICS OF GASES AND VAPOES. 289 AB .h AK = h + PK If PK = h, we shall have, AK = \AB ; if PK = 2h, we shall have, AK = ^A£ ; in general, if PK = nh, n being any positive number, either entire or fractional, we AJB shall have, AK — — Maeiotte's Law was verified in this manner by Dulong and Aeago for all values of n, up to n =. 21. The law may also he verified when the pres- sure is less than an atmosphere, by means of the following apparatus. AK represents a straight tube of uniform bore, closed at its upper and open at its lower extremity : 0J> is a long cistern of mercury. The tube AK is either graduated into equal parts, commencing at A, or it has attached to it a scale of brass or ivory. To use the instrument, pour mercury into the tube till it is nearly full ; place the finger over the open end, and invert it in the cistern of mer- cury, and depress it till the mercury stands at the same level without, as within the tube, and suppose the surface of the mercury in this case riTm to cut the tube at JB. Then will the tension of the confined air in AB, be equal to that of the external atmosphere. If now the tube be raised vertically, the air in AJB will expand, its tension wUl diminish and the mercury will fall in the tube, to maintain the equlibrium. Suppose the level of the mercury in the tube to have reached the point K. In this position of the instrument the tension of the air in AK, added to the weight of the column of mer- cury, KE will be equal to the tension of the external air. Now, it is found, whatever may be the value oi KE, that AK- ^-^ 13 • A B K L 290 MECHANICS. K T:K= \h, we have, AK = 2AB\ if EK = fA, we have, AK — ZAB; in general, '\i EK = -A, we have, AK=^. '"^ w + 1 Maeiotte's law has been verified in this manner, for all values of n, up to w = 111. It is a law of Physics that, when a gas is suddenly com- pressed, heat is evolved, and when a gas is suddenly ex- panded, heat is absorbed ; hence, in making the experiment, care must be taken to have the temperature kept uniform. Gay Iiussac's Iiavr. 190. If, whilst the volume of any gas or vapor remains the same, its temperature be increased, its tension is in- creased also. If the pressure remain the same, the volume of the gas will increase as the temperature is raised. The law of increase and diminution, as deduced by Gat LtrssAC, whose name it bears, may be enunciated as follows : In a given mass of any gas, or vapor, if the volume^ remains the same, the tension varies as the temperature ; if the tension remains the same, the volume varies as the tem- perature. According to Rbgnault, if a given mass of atmospheric air be heated from 32° Fahrenheit to 212°, the tension, or pressure remaining constant, its volume will be increased by the .3665tb part of the volume at 32°. Hence, the increase of volume for each degree of temperature is the .00204th part of the volume at 32°. If we denote the volume at 32° by v, and the volume at the temperature t', by v', we shall there- fore have, v' — w[l H- .00204(i!'— 32)] . . ( 152.) Solving with reference to v, we have, v' " =" 1 + .00204(«'- 32) • • • ( ^^^-^ Formula (153) enables us to compute the volume of any MECHANICS OF GASEB AND VAPORS. 291 mass of air at 32°, knowing its volume at the temperature t\ the pressure remaining constant. To find the volume at the temperature t'\ we have simply to substitute t" for t' in (152.) Denoting this volume by ■w", we have, v"— v\\ + .00204(<"- 32)]. Substituting for v its value from (153), we get, ,1 + .00204(i5"— 32) 5," — ^' — I ^ i . . / 154 ^ ~ l+.00204(«' -32) ^^^ -^ This formula enables us to compute the volume of any mass of air, at a temperature <", when we know its volume at the temperature t' ; and, since the density varies in- versely as the volume, we may also, by means of the same formula, find the density of any mass of aii-, at the temper- ature t", when we have given its density at the tempera- ture t'. Manometers. 191. A MANOMETEE is an iustrument used for measuring the tension of gases and vapors, and particularly of steam. Two principle varieties of manometers are used for measur- ing the tension of steam, the open manometer, and the closed m,anam,eter. The open Manometer. 192. The open manometer consists, essentially, of an open glass tube AJB, terminating below, nearly at the bottom of a cistern EF. -^^[j The cistern is of wrought iron, steam tight, and filled with mercury. Its dimen- sions are such, that the upper surface of _3j?n the mercury wiU not be materially lowered, when a portion of the mercury is forced up the tube. JEJD is a tube, by mean^ of which, steam may be admitted from the boiler to the surface of the mercury in the j. - B cisterr;. This tube is sometimes filled with rig. 166, 292 MECHANICS. water, through which the pressure of the steam is trans^ mitted to the mercury. To graduate the instrument. All communication with the boiler is cut ofl^, by closing the stop-cock U, and commu- nication with the external^ air is made by opening the stop- cock B. The point of the tube AB, to which the mercury rises, is noted, and a distance is laid off, upwards, from this point, equal to what the barometric column wants of 30 inches, and the point JZ'thus determined, is marked 1. This point will be very near the surface of the mercury in the cistern. From the point H, distances of 30, 60, 90, &c., inches are laid off upwards, and the corresponding points numbered 2, 3, 4, &c. These divisions correspond to atmospheres, and may be subdivided into tenths and hundredths. To use the instrument, the stop-cock D is closed, and a communication made with the boiler, by opening the stop- cock E. The height to which the mercury rises in the tube, will indicate the tension of the steam in the boiler, which may be read from the scale in terms of atmospheres and decimals of an atmosphere. If the pressure in pounds is wished, it may at once be found, by multiplying the reading of the instrument by 15. The principal objection to this kind of manometer, is its want of portability, and the great length of tube required, when high tensions are to be measured. The closed Manometer. >^ 193. The general construction of the closed manometer is the same as that of the open raanometei", with the excep- tion that the tube AB is closed at the top. The air which is confined in the tube, is then compressed in the same way as in Maeiotte's tube. To graduate this instrument. We determine the division M, as before. The remaining divisions are found by apply- ing Maeiotte's law. Denote the distance in inches, from JET to the top of the MECHANICS OP GASES AHD VAPOBS. 293 tube, by I; the pressure on the mercury, expressed in atmosphere, by n, and the distance Ln inches,,-from IT to the upper surface of the mercury in the tube, by x. The tension of the air in the tube will be equal to that on the mercury in the cistern, diminished by the weight of a column of mercury, whose altitude is x. Hence, in atmos- pheres, it is X ""-To- The bore of the tube being uniform, the volume occupied by the compressed air will be proportional to its height. When the pressure is 1 atmosphere, the height is I; when the pressure is ra atmospheres, the height is I — x. Hence, from Makiotte's law, I : n : : I — X : I . 30 Whence, by reduction, x' — (30w H- l)x = — 30l{n — 1). Solving, with respect to x, we have, son + 1 , / ZT. . , fSOn + iy The upper sign of the radical is not used, as it would give a value for x, greater than I. Taking the lower sign, and, aa a particular case, assuming ^ = 30 in., we have. X = 15n + 15 - -/ - 900(w - 1) + {15n + 15)'. Making w = 2, 3, 4, &c., in succession, we find for x, the corresponding values, 11.46 in., 11.58 in., 20.92 in., &c. These distances being set off from JT, upwards, and marked 2, 3, 4, &c., indicate atmospheres. The intermediate spaces are subdivided by means of the same formula. 294 MECHANICS. Fig. 166. The use of this instrument is the same as that of the manometer last described. In making the graduation, we have supposed the tem- perature to remain the same. If, however, it does not remain the same, the reading of the instrument must be corrected by means of a table computed for the purpose. The instruments already described, can only be used for measuring tensions greater than one atmosphere. The Siphon Guage. 194. The SIPHON GUAGE is an instrument employed to measure tensions of gases and vapors, when they are less than an atmosphere. It consists of a tube A£ 0, bent so that its two branches are parallel. The branch JBC is closed at the top, and filled with mercury, which is retained by the pres- sure of the atmosphere, whilst the branch AB is open at the top. If, now, the air be rarified in any manner, or if the mouth A of the tube, be exposed to the action of any gas whose tension is sufficiently small, the mercury will no longer be sup"J)orted in the branch BG, but will fall in that and rise in the other. The distance between the surfaces of the mer- cury in the two branches, as given by a scale placed between them, will indicate the tension of the gas. If this distance is expressed in inches, the tension can be found, in atmos- pheres, by dividing by 30, or, in pounds, by dividing by 2. The Diving-Bell. 195. The DIVING-BELL is a bell-shaped vessel, open at the bottom, used for descending below the surface of the water. The bell is placed so that its mouth shall continue horizontal, and is let down by means of a rope AB, and the whole apparatus is sunk by weights properly adjusted. The air con- tained in the bell before immersion, will be compressed by the weight of the Fig. 167. MECHANICS OF GASES AND VAPOE8. 295 water, but its increased elasticity will prevent the water from rising to the top of the bell, which is provided with seats for the accommodation of those wishing to descend. The air within is constantly contaminated by breathing, and is continually replaced by fresh air, pumped in through a tube FG. "Were there no additional air introduced, the volume of the compressed air, at any depth, might be com- puted by Maeiotte's law. The unit of the compressing force, in this c'ase, is the weight of a column of water whose cross-section is a square inch, and whose height is the distance from D C, to the surface of the water. The Barometer. 196. The BAEOMETEK is an instrument for measuring the pressure of the atmosphere. As already explained, it consists of a glass tube, hermetically sealed at one extre- mity, which is filled with mercury, and inverted in a basin of that fluid. The pressure of the air is indicated by the height of the column of mercury which it supports. A great variety of forms of the mercurial barometer have been devised, all involving the same mechanical principle. The two most important of these are the siphon and the cistern barometer. The Siphon Barometer. 19'J'. The siphon barometer consists essentially of a tube CDE^ bent so that its two branches, CD and DIE, shall be parallel to each other. A -c scale of equal parts is placed between them, ^|a and attached to the same frame with the tube. The longer branch CD, is about 32 or 33 inches in length, hermetically sealed at the top, and filled with mercury ; the shorter one is open to the action of the au-. When the instrument is placed vertically, the mercury sinks in the longer branch and rises in the ■''' shorter one. The distance between the sur- '^' face of the mercury in the two branches, as measured by the scale of equal parts, indicates the pressure of the atmos- phere at the particular time and place. I 296 MECHANICS. AK N iM. The Cistern Barometer. 198. The cistern barometer consists of a glass tube, filled and inverted in a cistern of mercury, as already explained. The tube is surrounded by a frame of metal, firmly attached to the cistern. Two opposite longitudinal openings, near the upper part of the frame, permit the upper surface of the mercury to be seen. A slide, moved up and down by means of a rack and pinion, may be brought exactly to the upper level of the mercury. The height of the column is then read from a, scale, so adjusted as to have its at the surface of the mercury in the cistern. The scale is graduated to inches and tenths, and the smaller divisions are read by means of a vernier. The figure shows the arrangement of parts in a complete cistern barometer. JJOT represents the frame of the barometer ; HIT that of the cistern, open at the upper part, that the level of the mercury in the cistern may be seen through the glass; Zi, an attached thermo- meter, to show the temperature of the mer- cury in the tube ; JV, a part of the sUding ring bearing the vernier, and moved up and down by the milled-headed screw JIf. The particular arrangement of the cistern is shown on an enlarged scale in Fig. 110. A represents the barometer tube, terminating in a small opening, to prevent too sudden shocks when the instrument is moved from place to place ; JI represents the frame of the cis- tern; S, the upper portion of the cistern, made of glass, that the surface of the mercury may be seen ; JEJ, a conical piece of ivory, pro- jecting from the upper surface of the cistern : when the surface of the mercury just touches the point of the ivory^ it is at the of the scale; CG represents the lower part of the cistern, and is made of leather, or some other „. „„ J! Ig. ItO* D Fig. 169. MECHANICS OF GASES AND VAPORS. 297 flexible substance, and firmly attached to the glass part; J? is a screw, working through the bottom of the frame, and against the bottom of the bag CC, through the medium of a plate F. The screw Z>, serves to bring the surface of the mercury to the point of the ivory piece E, and also to force the mercury up to the top of the tube, when it is desired to transport the barometer from place to place. To use this barometer, it should be suspended vertically, and the level of the mercury ia the cistern brought to the point of the ivory piece E, by means of the screw D ; 8 smart rap with a key upon the frame will detach the mer cury from the glass to which it sometimes tends to adhere. The sliding ring iV, is next run up or down by means of the screw ilf, till its lower edge appears tangent to the upper surface of the mercury in the tube, and the altitude is read from the scale. The height of the attached thermometer should also be noted. The requirements of a good barometer are, sufficient width of tube, perfect purity of the mercury, and a scale with a vernier accurately graduated and adjusted. The bore of the tube should be as large as practicable, to diminish the effect of capillary action. On account of the mutual repulsion between the particles of the glass and mer- cury, the mercury is depressed in the tube, and this depres- sion increases as the diameter of the tube diminishes. ■ In all cases, this depression Should be allowed for, and corrected by means of a table computed for the purpose. To secure purity of the mercury, it should be carefully distilled, and after the tube is filled, it should be boiled over a spirit-lamp, to drive off any bubbles of air that might ad- here to the walls of the tube. Uses of the Barometer. 199. The primary object of the barometer is, to meas- ure the pressure of the atmosphere at any time or place. It is used by mariners and others, as a weather-glass. It is also extensively employed for determining the heights of points on the earth's surface, above the level of the ocean. 13* 298 MECHANICS. The principle on which it is employed for the latter pur- pose is, that the pressure of the atmosphere at any place depends upon the weight of a column of air reaching from the place to the upper limit of the atmosphere. As we as- cend above the level of the ocean, the weight of the column diminishes ; consequently, the pressure becomes less, a fact which is shown by the mercury falling in the tube. We- shall investigate a formula for determining the difference of level between any two points. Difference of Level. 200. Let aJi represent a portion of a vertical prism of air, whose cross-section is one square inch. De- note the pressure on the lower base J3, by jo, and ^ ^a a' fi on the upper base aa', by p' ; denote the density of the air at -B, by d, and at aa', by d\ and sup- pose the temperature throughout the column to be 32° Fah. Pass a horizontal plane hh', infinitely near to aa', and denote the weight of the elementary j,; ^^^ volume of air ab, by w. Conceive the entire column to be divided" by horizontal planes into elementary prisms, such that the weights of each shaU be equal to w, and denote their heights, beginning at a, by s, s', s", &c. From Maeiotte's law, we shall have, p ~ d The air throughout each elementary prism may be re- garded as homogeneous ; hence, the density of the air in ab is equal to its weight, divided by its volume into gravity (Art. 12). But its volume is equal to lxlxs = s; hence, , d'='^. gs ■ Substituting this in the preceding equation, we have, whence, MECHANICS OF GASES AND VAPOES. 299 s =.§- X-, ■ • .^. (155.) From Daties' Bourdon, page 297, we have, by substitut- w ing for 2/ the fraction — , , the equation, to But — being infinitely small, all the terms in. the second member, after the first, may be neglected, giving, - = l{l + -y, or, _ = Z(^); or finally. I = l{p' + w)- Ip', in which I denotes the Napierian logarithm. In this equation, p' denotes the pressure on the prism ab ; hence, p' -{- w denotes the pressure on the next prism below, that is, on the prism be. . 10 If we substitute this value of -7 in Equation (155), we shall have, for the height of the prism ab, Substituting in succession for p', the values p'+ w,p'+ 2w, p' + 3w, &c., we shall find the heights of the elementary prisms be, cd, &o. We shall therefore have, 300 MECHANICS. s = §^U{f'+ ^)-¥l s'= ^[l{p'+2w)-l{p' + w)l s"= ^[lip'+3w)-l(p'+2w)l s«'z= ^ il{p'+ nw) - i{p' +{n- l)w)]. ag If n denote the number of elementary prisms in AB, the sum of the first members will be equal to AjB. Adding the equations member to member, and denoting the sum of the first members by z, we have, Because nw denotes the weight of the column of air AB, we shall have, p' + nw = p, hence, ^=-^1^, (156.) dg p' Denoting the modulus of the common system of loga- rithms by M, and designating common logarithms by the Symbol log, we shall have, Mz = ~ log ". , or a = ^=^ log — • dg ®y Mdg ^ p' Now, the pressures j9 and^' are measured by the heights of the columns of mercury which they will support ; denoting these heights by ^and IT', we have, p _ ^ p' ~ H'' MECHANICS OF GASES AND VAPOES. 301 whence, by substitution, ^=]4'°^5- • • • ^'"'-^ We have supposed the temperature, both of the air and mercury, to be 32°. In order to make the preceding for- mula general, let T represent the temperature of the mer- cury at jB, J", its temperature at a, and denote the cor- responding heights of the barometric column by h and h' ; also, let t denote the temperature of the air at £, and t' its temperature at a. The quantity ^ is the ratio of the density of the air at ^, to the corresponding pressure, the temperature being 32°.. According to Mabiotte's law, this ratio remains constant, whatever may be the altitude of -B above the level of the ocean. If we denote the latitude of the place by I, we have, (Art. 124), g = g'{l — 0.002695 cos20. It has been shown, by experiment, that, when a column of mercury is heated, it increases in length at the rate of ygL^ths of its length at 32°, for each degree that the tem- perature is elevated. Hence, , j^/, , T-32\ ^9990+2^-32 A =B:{1+ —^^) = H -g^ ; V ^ 9990 / 9990 Dividing the second equation by the first, member by member, A_ _ JT 9990 -f T— 32 h! ~ .^''9990+ 2"- 32* 302 MECHANICS. Dividing both terms of the fractional coeflScient of -™ by the denominator, and neglecting the quantity 7'— 32, in comparison with 9990, we have. Whence, by reduction, H h 1 H' ~ h' 1 + .0001(r- T') The quantity z denotes, not only the height, but also the volume of the column of air aB, at 32°. When the tem- perature is changed from 32°, the pressures remaining the same, this volume will vary, according to the law of Gat LuSSAC. If we suppose the temperature of the entire column to be a mean between the temperatures at B and a, which we may do without sensible error, the height of the coluron will become, Equation (153), s Fl + .00204 C-~- - 32 ^1 = s[l + .00102(«+<'-64)] Hence, to adapt Equation (157) to the conditions pro- posed, we must multiply the value of s by the factor, 1 + .00102(< + t' — 64). Substituting in Equation (157), for -= and g, the values shown above, and multiplying the resulting value of s, by the factor 1 + .00102(« + t' — 64), we have, _ p 1 + .00102 (i^-f t'—U) h ^ ~Mcl 1 - 0.002695COS2Z '^^ h'll+.OOOl^T-T')]' (158.) MECHANICS OF GASES AND VAPORS. 303 n The factor -=^ is constant, and may be determined as follows: select two points, one of which is considerahly higher than the other, and determine, by trigonometrical measurement, their difference of level. At the lower point, take the reading of t^e barometer, of its attached ther- mometer, and of a detached thermometer exposed to the air. Make similar observations at the upper station. These observations, together with the latitude of the place, will give all the quantities eptering Equation (158), except the factor in question. Hence, this factor may be deduced. It is found to be 60345.51 ft. Hence, we have, finally, the barometric formula, s = 60345.51 ft. X 1 + .00102 (^ +e'- 64) h ^ 1-0.002695COS2/ ^ A'[l + .0001(2' - J")] ^ ' To use this formula for determining the difference of level between two stations, observe, simultaneously, if possible, the heights of the barometer and of the attached and de- tached thermometers, at the two stations. Substitute these results for the corresponding quantities in the formula ; also substitut-e for I the latitude of the place, and the resulting value of z, wUl be the difference of level required. If the observations cannot be made simultaneously at the two stations, make a set of observations at the lower station ; after a certain interval, make a set at the upper station ; then, after an equal interval, make another set at the lower station. Take a mean of the results of observation at the lower station, as a single set, and proceed as before. For the more convenient application of the formula for the difference of level between two points, tables have been computed, by means of which the arithmetical operations are m.uch facilitated. 304 MECHANICS. Work due to the Expansion of a Gas or Vapor. 201. Let the gas or vapor be confined in a cylinder closed at its lower end, and having a piston working air-tight. When the gas occupies a portion of the cylinder whose height is /i, denote the pressure on each square inch of the piston by p ; when the gas expands, so that the altitude of the column be- comes X, denote the pressure on a square inch by y. Since the volumes of the gas, under these suppositions, are proportional to their altitudes, we shall have, from Maeiotte's laws, Fig. 172. whence, p : y : : X : h; xy = ph If we suppose p and h to be constant, and x and y to vary, the above equation will be that of an equilateral hyperbola referred to its asymptotes. Draw A C perpendicular to AM, and on these lines, as asymptotes, construct the curve NLH, from the equation, xy — ph. Make AG = h, and draw GS parallel to AO; it wUl represent the pressure p. Make AM = x, and draw MISF parallel %(> AG ; it will represent the pressure y. In like manner, the pressure at any elevation of the piston may be constructed. Let KL be drawn infinitely near to GS, and parallel with it. The elementary area GKLS will not differ sensibly from a rectangle whose base is jo, and altitude is GK. Hence, its area may be taken as the measure of the work whilst the piston is rising through the infinitely small space GK. In like manner, the area of any infinitely small element, bounded by lines parallel to A (7, may be taken to represent the work whilst the piston is rising through the MECHANICS OF GASES AND VAPOKS. 305 height of the element. If we take the sum of all the elements between the ordinates GH and MN, this sum, or the area GMNH^ will represent the total quantity of work of the force of expansion whilst the piston is rising from G to M. But the area included between an equilateral hyper- bola and one of its asymptotes, and limited by lines parallel to the other asymptote, is equal to the product of the co- ordinates of any point, multiplied by the Naperian logarithm of the quotient obtained by dividing one of the limiting ordinates by the other ; or, in this particular case, it is equal to pli x ^( )• Hence, if we designate the quantity of woi-k performed by the expansive force whilst the piston is moving over GM, by g', we shall have, This is the quantity of work exerted upon each square inch of the piston ; if we denote the area of the piston, by A, and the total quantity of work, by §, we shall have, q = Aph X ?(^) = Aph X Z(-J) . ( 160.) If we denote by c the number of cubic feet of gas, when the pressure is p, and suppose it to expand tUl the pressure is y, we shall have, Ah — c; or, if J. be expressed in square Ah feet, we shaU have, c = — — • Hence, by substitution. Q = .lUcpXl(^)- Finally, if we suppose the pressure at the highest point to be p', we shall have, Q=:UicpXl{^), 306 MECHANICS. an equation which gives the quantity of work of c cubic feet of gas, whilst expanding from a pressure p, to a pres- sure jo'. Efflux of. a Gas or Vapor. 202. Supi^ose the gas to escape from a small orifice, and denote its velocity by v. Denote the weight of a cubic foot of the gas, by w, and the number of cubic feet dis- charged in one second, by c, then will the mass escaping in cw one second, be equal to — ■ , and its living force will be cw equal to — ■v'. But, from Art. 148, the living force is double the accumulated quantity of work. If, therefore, we denote the accumulated work by Q, we shall have, But the accumulated work is due to the expansion of the gas, and if we denote the pressure within the orifice, by p, and without, by/>', we shall have, from Art. 201, Q = lucp X i(py Equating the second members, we have, f^v' = lUepxl{f:); Whence, U = 12. ■V¥^) Substituting for g, its value, 321 ft.^ ^q have, after reduction. ''=^Vs^\ times as much to convert all of the water into steam as it requires to raise it from 32° to 212°. Hence, the entire amount of heat which becomes latent is 5i X (212° — 32°) = 990°. That the heat applied becomes latent, may be shown experimentally as follows : Let a cubic inch of water be converted into steam at MECHANICS OF GASES AND TAJOES. 309 212°, and kept in a close vessel. Now, if 5J cubic inches of water at 32° be injected into the -vessel, the steam will aU ■ be converted into water, and the 6^ cubic inches of water will be found to have a temperature of 212°- The heat that was latent becomes sensible again. When water is converted into steam under any other pressure than that of the atmosphere, or 1 5 pounds to the square inch, it is found that, although the boiling point will be changed, the entire amount of heat required for convert- ing the water into steam will remain unchanged. If the evaporation takes place under such a pressure, that the boiling point is but 150°, the amount of heat which becomes latent is 1052°, so that the latent heat of the steam, plus its sensible heat, is 1252°. If the pressure under which vaporization takes place is such as to raise the boiling point to 500°, the amount of heat which becomes latent is "702°, the sum 702° + 500° being equal to 1252°, as before. Hence, we conclude that t/ie same amount of fuel is required to convert a given amount of water into steam,, no matter what m^ay he the pressure under which the evapora- tion takes place. When water is converted into steam under a pressure of one atmosphei'e, each cubic inch is expanded into about 1700 cubic inches of steam, of the temperature of 212° ; or, since a cubic foot contains 1728 cubic inches, we may say, in round numbers, that a cubic inch of water is converted into a cubic foot of steam. If water is converted into steam under a greater or less pressure than one atmosphere, the density wiU be increased or diminished, and, consequently, the volume will be dimin- ished or increased. The temperature beLug also increased or diminished, the increase of density or decrease of volume mil not be exactly proportional to the increase of pressure ; but, for purposes of approximation, we -may consider the densities as directly, and the volumes as inversely propor- tional to the pressures under which the steam is generated. Under this hypothesis, if a cubic inch of water be evapo- 310 MECHANICS. rated under a pressure of a half atmosphere, it will afford two cubic feet of steam ; if generated under a pressure of two atmospheres, it will only afford a half cubic foot of steam. Work of Steam. 205. When water is converted into steam, a certaia amount of work is generated, and, from what has been shown, this amount of work is very nearly the same, whatever may be the temperature at which the water is evaporated. Suppose a cylinder, whose cross-section is one square inch, to contain a cubic inch of water, above which is an air- tight piston, that may be loaded with weights at pleasure. In the first place, if the piston is pressed down by a weight of 15 pounds, and the inch of water converted into steam, the weight will be raised to the height of 1728 inches, or 144 feet. Hence, the quantity of work is 144 x 15, or, 2160 units. Again, if the piston be loaded with a weight of 30 pounds, the conversion of water into steam wUl give but 864 cubic inches, and the weight wiU be raised through 72 feet. In this case, the quantity of work wUl be 72 X 30, or 2160 units, as before. We conclude, therefore, that the quantity of work is the same, or nearly so, whatever may be the pressure under which the steam is generated. We also conclude, that the quantity of work is nearly proportional to the fuel consumed. Besides the quantity of work developed by simply con- verting an amount of water into steam, a further quantity of work is developed by allowing the steam to expand after entering the cylinder. This principle is made use of in steam engines working expansively. To find the quantity of work developed by steam acting ex- pansively. Let AS represent a cylinder, closed at A, and having an air-tight piston D. Suppose the steam to enter at the bottom of the cylinder, and to push the piston upward to <7, and then suppose the opening at which the steam enters, to be closed. If the piston is not too heavily loaded. the steam will continue to expand, and the piston _. ,.. MECHANICS OF GASES AND VAPORS. 311 will be raised to some position, S. The expansive force of the steam will obey Maeiotte's law, and the quantity of work due to expansion will be given by Equation ( 160). Denote the area of the piston in square inches, by A ; the pressure of the steam on each square inch, up to the moment when the communication is cut off, by p ; the distance A G, through which the piston moves before the steam is cut off, by h ; and the distance AD, by nh. If we denote the pressure on each square inch, when the piston arrives at D, by p\ we shall have, by Maeiotte's law, p : p' :: nh : h, . • . »' = — , n an expression which gives the limiting value of the load of the piston. The quantity of work due to expansion being denoted by q, we shall have, from Equation (160), q == Aph X I ( -y- ) = Aphl {n). If we denote the quantity of work of the steam, whUst the piston is rising to 0, by q", we shall have, q'' = Aph. Denoting the total quantity of work during the entire stroke of the piston, by Q, we shall have, Q = Aph[l+\n)] . . . (163.) Experimental Pormulas. 206. Numerous experiments have been made for the purpose of determining the relation existing between the elasticity and temperature of steam in contact with the water by which it is produced, and many foi-mulas, based 312 MECHANICS. upon these experiments, have been given, two of which are subjoined : The formula of Dulong and Aeago is, p = {I + .ooiissty, in which p represents the tension in atmospheres, and t the excess of the temperature above 100° Centigrade. Teedgold's formula is, t = 0.85-v/^- 75, in which t is the temperature, in degrees of the Centigrade thermometer, and p the pressure, expressed in centimeters of the mercurial column. HYDEAULIC AND PNEUMATIC MACHINES. 313 CHAPTEE IX. HYDRAULIC AND PNEUMATIC MACHINES. Definitions. 207. Hydraulic machines are those used in raising and distributing water, such, as pumps, siphons, hydraulic rams, &c. The name is also applied to those machines in which water power is the motor, or in which water is employed td transmit pressures, such as water-wheels, hydraulic presses, &c. Pneumatic machines are those employed to rarefy aiid condense air, or to impart motion to the air, such as air- pumps, ventilating-ilowers, &e,. The name is also applied to those machines in which currents of air furnish the motive power, such as windmills, &c. Water Pumps. 208. A water pump is a machine for raising water from a lower to a higher level, generally by the aid of atmospheric pressure. Three separate principles are employed in the working of pumps: the sucking, the lifting, and the forcing principle. Pumps are frequently named according as one or more of these principles are. employed. Sucking and Lifting Pump. 209. This pump consists of a cylindrical barrel A, at the lower extremity of which is attached a sucking-pipe B, leading to a reser- voir. An air-tight piston C is work- ed up and down in the barrel by means of a lever U, attached to a piston-rod D. P represents a valve opening upwards, which, when the 14 Fig. 174. 314 MECHANICS. pump is at rest, closes by its own -weight. This valve is. called, from its position, the piston-valve. A second valve G^ also opening upwards, is filaced at the junction of the pipe with the barrel. This is called the sleeping-valve. The space JLM, through which the piston can be moved up and down by the lever, is called the play of the piston. To explain the action of the pump, suppose the piston to be at the lowest limit of the play, and everything in a state of equilibrium. If the extremity of the lever E be depressed, and the piston consequently be raised, the air in the lower part of the barrel will be rarefied, and that in the pipe B will, by virtue of its greater tension, open the valve, and a portion of it will escape into the barrel. The air in the pipe, thus rarefied, will exert a less pressure upon the water in the reservoir than that of the external air, and, consequently, the water wiU rise in the pipe, until the tension of the internal air, plus the weight of the column of water raised, is equal to the tension of the external air ; the valve Q will then close by its own weight. If the piston be again depressed to the lowest limit, by means of the lever E, the air in the lower part of the barrel will be compressed, its tension will become greater than that of the external air, the valve F wiU be forced open, and a portion of the air will escape. If the piston be raised once more, the water will, for the same reason as before, rise still higher in the pipe, and after a few double strokes of the piston, the air will be completely exhausted from beneath the piston, the water will pass through the piston valve, and* finally escape at the spout P. The water is raised to the piston by the pressure of the air on the surface of the water in the reservoir ; hence, the piston should not be placed at a greater distance above the level of the water ita the reservoir, than the height to which the pressure of the air wiU sustain a column of water. In fact, it should be placed a little lower than this limit. The specific gravity of mercury being about 13.5, the height of a colunm of water which will exactly counterbalance the HYDRAULIC AND PNEUMATIC MACHINES. 315 pressure of the atmosphere, will he found hy multiplying the height of the harometrio column by 13^. At the level of the sea the average height of the baro- metric column is 2i feet ; hence, the theoretical height to which water can be raised by the principle of suction alone, is a little less than 34 feet. The water having passed through the piston valve, it may be raised to any height by the lifting principle, the only limitation being the strength of the pump and want of power. There are certain relations which must exist between the play of the piston and its height above the water in the reservoir, in order that the water may be raised to the piston ; for, if the play is too small, it will happen after a few strokes of the piston, that the air between the piston and the surface of the water will not be sufficiently compressed to open the piston valve ; when this state of affairs takes place, the water will cease to rise. To investigate the relation that must. exist between the play and the height of the piston above the water. Denote the play of the piston, by p, the distance from the upper surface of the water in the reservoir to the highest position of the piston, by a, and the height at which the water ceases to rise in the pump, by x. The distance from the surface of the water in the pump to the highest position of the piston wUl then be equal to a — x, and the distance to the lowest position of the piston, will be a — p — x. Denote the height at which the atmospheric pressure will sustain a column of water in vacuum, by 7», and the weight of a column of water, whose base is the cross-section of the pump, and whose altitude is 1, by w ; then will wh denote the pressure of the atmosphere exerted upwards through the water in the reservoir and pump. Now, when the piston is at its lowest position, in order that it may not thrust open the piston valve and escape, the pressure of the confined air must be exactly equal to that of the external atmosphere; that is, equal to wh. When the 316 MECHANICS. piston is at its highest position, the confined air will 'he rare- fied, the volume occupied being proportional to its height. Denoting the pressure of the rarefied air by wh', we shall have from Maeiotte's law, wh : wh' : :' a — X : a —p — X. .•. wh' =: w7i — • a — X If the water does not rise when the piston is at its highest position, the pressure of the rarefied air, plus the weight of the column already raised, will be equal to the pressure of the external atmosphere ; or , a —p ~x - wh + wx = wh. a — X Solving this equation with respect to x, we have, _ a ± '\/a? — 4ph X — - If we have^ iph > a" ; or, P>j^, the value of x wUl be imaginary, and there wUl be no point at which the water will cease to rise. Hence, the above inequality expresses the relation that must exist, in order that the pump may be effective. This condition expressed in words, gives the following rule : The pump will be effective, when tJie play of the piston is greater than the square of the distance from the surface of the water in the reservoir, to the highest position of the piston, divided by four times the height at which the pres- sure of the atmosphere will support a column of loater in a vacuum. Let it be required to find the least allowable play of the piston, when the highest position of the piston is 16 feet HTDEAULIC AND PNEUMATIC MACHINES. 3lt above the water in the reservoir, and when the barometer stands at 28 inches. In this case, a = 16 ft., and h = 28 in. X 13^- = Sl8 in. = 31^ ft. Hence, ^>f||ft.; or, p>2-i^.ft. To iind the quantity of work required to make a double stroke of the piston, after the water reaches the level of the spout. In depressing the piston, no force is required, except that necessary to overcome the inertia of the parts and the fric- tion. Neglecting these for the present, the quantity of work in the downward stroke, may be regarded as 0. In raising the piston, its upper surface will be pressed down- wards, by the pressure of the atmosphere wh, plus the weight of the column of water from the piston to the spout ; and it will be pressed upwards, by the pressure of the atmosphere, transmitted through the pump,, minus the weight of a column of water, whose cross-section is equal to that of the barrel, and whose altitude is the distance from the piston to the surface of the water in the reservoir. If we subtract the latter pi-essure from the former, the difference will be the resultant downward pressure. This difference will be equal to the weight of a column of water, whose base is the cross-section of the barrel, and whose height is the distance of the spout above the reservoir. Denoting the height by IT, the pressure will be equal to wIT. The path through which the pressure is exerted during the ascent of the piston, is equal to the play of the piston, or p. Denoting the quantity of work required, by Q, we shall have, Q = wpJI. But wp is the weight of a volume of water, whose base is the cross-section of the barrel, and whose, altitude is the play of the piston. Hence, the value of Q is equal to the 318 MECHANICS. quantity of work necessary to raise this volume of water from the level of the water in the reservoir to the spout. This volume is evidently equal to the volume actually delivered at each double stroke of the piston. Hence, the quantity of work expended in pumping with the sucking and lifting pump, all hurtful resistances being neglected, is equal to the quantity of work necessary to lift the amount of water, actually delivered, from the level of the water in the reservoir to the height of the spout. In addition to this work, a sufficient amount of power must be exerted, to overcome the hurtful resistances. The disadvantage of this pump, is the irregularity with which the force must act, being in depressing the piston, and a maximum in raising it. This is an important objection when machinery is em- ployed in pumping ; but it may be either partially or entirely overcome, by using two pumps, so arranged, that the piston of one shall ascend as that of the other descends. Another objection to the iise of this kind of pump, is the irregularity of iiow, the inertia of the column of water having to be overcome at each upward stroke. This, by creating shocks, consumes a portion of the force applied. Sucking and Forcing Pump. 210. This pump consists of a cylindrical barrel A, with its attached sucking-pipe £, and sleeping-valve G, as in the pump just discussed. The piston C is solid, and is worked up and down in the barrel by means of a lever £J, attached to the piston-rod D. At the bottom of the barrel, a branch-pipe leads into an air-vessel IT, through a second sleeping-valve ^, which opens upwards, and closes by its own weight. A delivery- pipe S, enters the air-vessel at its top, and terminates near its bottom. To explain the action of this E I Fig. ITS. HYDRAULIC AND PNEUMATIC MACHINES, 319 pump, suppose the piston C to be depressed to its lowest liiait. Now, if the piston be' raised to its highest position, the air in the barrel will be rarefied, its tension will be diminished, the air in the tube B, will thrust open the valve, and a portion of it will escape into the barrel. The pres- sure of the external air will then force a column of water up the pipe £, untU the tension of the rarefied air, phis the weight of the column of water raised, is equal to the tension of the external air. An equilibrium being produced, the valve G closes by its own weight. If, now, the piston be again depressed, the air in the barrel will be condensed, its tension wUl increase till it becomes greater than that of the external air, when the valve F will be thrust open, and a portion of it will escape through the delivery-pipe H. After a few double strokes of the piston, the water will rise through the valve G, and then, as the piston descends, it will be forced into the air-vessel, the air wiU be condensed in the upper part of the vessel, and, acting by its elastic force, will force a portion of the water up the delivery-pipe and out at the spout P. The object of the air-vessel is, to keep up a continued stream through the pipe S, otherwise it would be necessary to overcome the inertia of the entire column of water in the pipe at every double stroke. The flow having commenced, at each double stroke, a volume of water will be delivered from the spout, equal to that of a cylinder whose base is the area of the piston, and whose altitude is the play of the piston. The same relative conditions between the parts should exist as in the sucking and lifting pump. To find the quantity of work consumed at each double stroke, after the flow has become regular, hurtful resistances being neglected : When the piston is descending, it is pressed downwards by the tension of the air on its upper surface, and upwards by the tension of the atmosphere, transmitted through the delivery-pipe, plus the weight of a column of water whose base is the area of the piston, and whose altitude is the 320 MECHANICS. distance of the spout above the pistou. This distance is variable during the stroke, but its mean value is the distance of thfi middle of the play below the spout/ The difference between these pressures is exerted upwards, and is equal to the weight of a column of water whose base is the area of the piston, and whose altitude is the distance from the middle of the play to the spout. The distance through which the force is exerted, is equal to the play of the piston. Denoting the quantity of work during the descending stroke, by Q' ; the weight of a column of water, having a base equal to the area of the piston, and a unit in altitude, by w ; and the height of the spout above the middle of the the play, by A', we shall have, Q' — wh' X p. When the piston is ascending, it. is pressed downwards by the tension of the atmosphere on its upper surface, and upwards by the tension of the atmosphere, transmitted through the water in the reservoir and pump, minus the weight of a column of water whose base is the area of the piston, and whose altitude is the height of the piston above the reservoir. This height is variable, but its mean value is the height of the middle of the play above the water in the reservoir. The distance through which this force is exerted, is equal to the play of the piston. Denoting the quantity of work during the ascending stroke, by Q", and the height of the middle of the play above the reservoir, by A", we have, Q" =z wh" X p. Denoting the entire quantity of work during a double stroke, by Q, we have, Q = Q'+ Q" = wp{h' + h"). But wp is the_ weight of a volume of water, the area of whose base is that of the piston, and whose altitude is the HYDEAULIC AND PNEUMATIC MACHINES. 321 play of the piston ; that is, it is the weight of the volume delivered at the spout at each double stroke. The quantity h' + A", is the entire height of the spout above the level of the cistern. Hence, the quantity of work expended, is equal to that required to raise the entire volume delivered, from the level of the water in the reservoir to the height of the spout. To this must be added the work necessary to overcome the hurtful resistances, such as fric- tion, &c. If h' — h", we shall have, Q' = Q" ; that is, the quan- tity of work during the ascending stroke, wiU be equal to that during the descending stroke. Hence, the work of the motor win be more nearly uniform, when the middle of the play of the piston is at equal distances from the reservoir and spout. Fire Sngine. 211. The fire engine is essentially a double sucking and forcing pump, the two piston rods being so connected, that when one piston ascends the other descends. The sucking and delivery pipes are made of some flexible material, gen- erally of leather, and are attached to the machine by means of metallic screw joints. The figure exhibits a cross-section of the essential part of a Fire Engine. A A' are the two barrels, C C" the two pistons, con- nected by the rods, 2?2>', with the lever, E E' . B is the sucking pipe, termi- nating in a box from which the water may en- ter either barrel through the valves, G Q' . K is the air vessel, common to both pumps, and com- municating with them by the valves F F'. H is the delivery pipe. Fig. 176. 322 MECHANICS. The instrument is mounted on wheels for convenience of transportation. The lever JiJ JE' is worked by means of rods at right angles to the lever, so arranged that several men can apply their strength in working the pump. The action of the pump differs in no respect from, that of the forcing pump ; but when the instrument is worked vigor- ously, there is more water forced into the air vessel, the tension of the air is very much augmented, and its elastic force, thus brought into play, propels the water to a consider- able distance from the mouth of the delivery pijje. It is this capacity of throwing a jet of water to a great distance, that gives to the engine its value in extinguishing fires. A pvimp entirely similar to the fire engine in its construc- tion, is often used under the name of the double action forc- ing pump for raising water for other purposes. The Rotary Pump. 212. The rotary pump is a modification of the sucking and forcing pump. Its consti-uction will be best understood from the drawing, which represents a vertical section through the axis of the sucking-pipe, and at right angles to axis of the rotary portion of the pump. A A represents an annular ring of metal, which may be made to revolve about its axis 0. J) D is a second ring of metal, concentric with the first, and forming with it an inter- mediate annular space. This space communicates with the sucking-pipe IT, and the de- livery pipe X. Four radical paddles C, are disposed so as to slide backwards and for- wards through suitable open- ings, which are made in the ring A, and which are moved around with it. (? is a solid guide, firmly fastened to the end of the cylinder enclosing Fig. ITT. HYDRAULIC AND PNEUMATIC MACHINES. 323 the rotary apparatus, and cut as represented in the figure. JE E are two springs, attached to the ring i?, and acting by their elastic force, to press the paddles firmly against the guide. These springs are of such dimensions as not to impede the flow of the water yVom the pipe K, and into the pipe L. When the axis is made to revolve, each paddle, as it reaches and passes the partition H^ is pressed against the guide, but, as it moves on, it is forced, by the form of the guide, against the outer wall D. The paddle then drives the air in front of it, around, in the direction of the arrow- head, and finally expels it through the pipe L. The air behind the paddle is rarefied, and the pressure of the exter- nal air forces a column of water up the pipe. As the paddle approaches the opening to the pipe X, the paddle is pressed back by the spring E^ against the guide, and an outlet into the ascending pipe i, is thus provided. After a few revo- lutions, the air is entirely exhausted from the pipe K. The ■water enters the channel S J3, and is forced up the pipe I/, from which it escapes by a spout at the top. The quantity of work expended in raising a volume of water to the spout, by this pump, is equal to that required to lift it through the distance from the level of the water in the cis- tern to the spout. This may be shown in the same manner as was explained under the head of the sucking and forcing- pump. To this quantity of work, must be added the work necessary to overcome the hurtful resistances, as fric- tion, &o. This pump is well adapte.d to machine pumping, the work being very nearly uniform. A machine, entirely Snular to the rotary pump, might be constructed for exhausting foul air from mines ; or, by re- versing the direction of rotation, it might be made to force a supply of fresh air to the bottom of deep mines. Besides the pumps already described, a great variety of others have been invented and used. All, however, 324 SIECHANICS. r depend upon some modification of the principles that have just been discussed. The Hydrostatic Press. 213. The hydrostatic press is a machine for exerting great pressure, through small spaces. It is much used in compressing seeds to obtain oil, in packing hay and bales of goods, also in raising great weights. Its construction, though requiring the use of a sucking-pump, depends upon the prin- ciple of equal pressures (Art. 154). It consists essentially of two vertical cylinders, A and B, each provided with a solid pis- ton. The cylinders communi- cate by means of a pipe C, whose entrance to the larger cylinder is closed by a sleeping valve JE. The smaller cylinder communicates with the reser- voir of water ^ by a sucking- pipe a, whose upper extremity is closed by the sleeping-valve D. The smaller piston B, is worked up and down by the lever G. By working the lever G, up and down, the water is raised from the reservoir and forced into the larger cylinder A ; and when the space below the piston F is filled, a force of compression is exerted upwards, which is as many times greater than that applied to the piston B, as the area of Fis greater than 5 (Art. 154). This force may be util- ized in compressing a body L, placed between the piston and the fi-ame of the press. Denote the area of the larger piston by P, of the smaller, hjp, the pressure applied to B, by /, and that exerted at F, by F; we shall have, u m fig. 178. F:f:.P:p, P If we denote the longer arm of the lever G, by Z, and HYDBADLIO AND PNEUMATIC MACHINES. 325 the shorter arm, by I, and represent the force applied at the extremity of the longer arm, by JT, we shall have from the principle of the lever (Art. 18), Substituting this value of y above, we have, F^ pi To illustrate, let the area of the larger piston be 100 square inches, that of the smaller piston 1 square inch ; sup- pose the longer arm of the lever to be 30 inches, and the shorter arm to be 2 inches, and a force of 100 pounds to be applied at the end of the longer arm of the lever ; to find the pressure exerted upon F. From the conditions, P =. 100, !£ = 100, i =: 30, p = 1, and I = 2. Hence, „ 100 X 100 X 30 ,,„„„„,, F= — = 150000 lbs. 2 We have not taken into account the hurtful resistances, hence, the total pressure of 150000 pounds must be some- what diminished. The volume of water forced from the smaller to the larger piston, during a smgle descent of the piston F', will occupy in the two cylindei-s, spaces whose heights are inversely as the areas of the pistons. Hence, the path, over which / is exerted, is to the path over which F is exerted, as P is to p. Or, denoting these paths by s and S, we have, s : S :: P -.p; or, smce F : p :: F : f, we shall have, s: S:: F:f, .: fs = FS. 326 MECHANICS. That is, the quantities of work of the power and resistance are equal, a principle which holds good in all machines. EXAMPLES. 1. The cross-section of a sucking and forcing pump is 6 square feet, the play of the piston 3 feet, and the height of the spout, above the level of the reservoir, 50 feet. "What must be the effective horse power of an engine which can impart 30 double strokes per minute, hurtful resistances being neglected ? SOLUTION. The number of units of work required to be performed each minute, is equal to 6 X 3 X 60 X 62| = 56250. Hence, ™ 56250 1 93 Airia '' — ffS^B^O^ — ■*■ T3 a' • -"-'l^- 2. In a hydrostatic press, the areas of the two pistons are, respectively, 2 and 400 square inches, and the two arms of the lever are, respectively, 1 and 20 inches. Required the pressure on the larger piston for each pound of pressure applied to the longer arm of the lever ? A.ns. 4000 lbs. 3. The areas of the two j)istons of a hydrostatic press are, respectively, equal to 3 and 300 square inches, and the shorter arm of the lever is one inch. What must be the length of the longer arm, that a force of 1 lb. may produce a pressure of 1000 lbs. Ans. 10 inches. The Siphon. 214. The siphon is a bent tube, used for transferring a liquid from a higher to a lower level, over an in- termediate elevation. The siphon consists of two branches, AB and Ji C, of which the outer one is the longer. To use the instrument, the tube is filled with the liquid in any manner, the end of the longer branch being stopped with the finger or a stop-cock, in which case, the pressure of the atmosphere will prevent the liquid from escaping jig. ir9. HTDEAULIC AND PNEUMATIC MACHINES. 327 at the other end. The instrument is then inverted, the end G being submerged in the liquid, and the stop removed from A. The liquid will begin to flow through the tube, and the flow will continue till the level of J;he Uquid in the reservoir reaches that of the mouth of the tube G. To find the velocity with which water will issue from the , siphon, let us consider an infinitely small layer at the orifice A. This layer will be pressed downwards, by the tension of the atmosphere exerted on the surface of the reservoir, diminished by the weight of the water in the branch BD, and increased by the weight of the water in the branch BA. It will be pressed upwards by the tension of the atmosphere acting directly upon the layer. The diiference of these forces, is the weight of the water in the portion of the tube DA, and the velocity of the stratum will be due to that weight. Denoting the vertical height of DA, by A, we shall have, for the velocity (Art. 173), This is the theoretical velocity, but it is never qtdte realized in practice, on account of resistances, which have been neglected in the preceding investigation. The siphon may be filled by applying the mouth to the end A, and exhausting the air by suction. The tension of the atmosphere, on the upper surface of the reservoir, wiU press the water up the tube, and fiU it, after which the flow will go on as before. Sometimes, a sucking-tube AD, is in- serted near the opening A, and rising nearly to the bend of the siphon. In this case, the opening A, is closed, and the air exhausted through the sucking-tube AD, after which the flow goes on as before. The Wurtemburg Siphon. 215. In the Wurtemburg siphon, the ends of the tube are 328 MKCHANICS. m m bent twice, at right-angles, as shown in the figure. The advantage of this arrangement is, that the tube, once filled, remains so, as long as the plane of its axis is kept vertical. The siphon may be lifted out and replaced at pleasure, thereby stopping the flow at will. It is to be observed that the siphon is only effectual when the distance from the highest point of the tube to the level . of the water in the reservoir is less than the height at which the atmospheric pressure will sustain a column of water in a vacuum. This will, in general, be less than 34 feet. Fig. 181. The Intermitting Siphon. 216. The intermitting siphon is represented in the figure. AS is a curved tube issuing from the bottom of a reservoir. The reservoir is supplied with water by a tube S, having a smaller bore than that of the siphon. To explain its action, suppose the reservoir at first to be empty, and the tube £1 to be opened; as soon as the reservoir is filled to the level of CD, the water will begin to flow from the opening .B, and the flow once commenced, will continue till the level of the reservoir is again reduced to the level G'J)\ drawn through the opening A. The flow will then cease till the cistern is again filled to CD, and so on as before. Fig. 183. Intermitting Springs. aiY. Let A represent a subterranean cavity, communi- cating with the surface of the earth by a channel ABC, bent like a siphon. Suppose the reservoir to be fed by percolation through the crevices, or ' by a small channel D. When the rig. i83. HYDRAULIC AND- PNEUMATIC MACHINES/ 329 watei- in the reservoir rises to the height of the horizontal plane BD^ the flow will commence at C, and, if the chan- nel is sufficiently large, the flow will continue till the water is reduced to the level plane drawn through G. An inter- mission of flow will occur till the reservoir is again filled, and so on, intermittingly. This phenomena has been observed at various places. Siphon of Constant Floixr. 218. We have seen that the velocity of efflux depends upon the height of the water in the reservoir above the external opening of the siphon. When the water is drawn off from the reservoir, the upper surface sinks, this height diminishes, and, consequently, the velocity continually diminishes. K, however, the shorter branch CLZ?, of the tube, be inserted through a piece of cork large enough to float the siphon, the instrument wUl sink as the upper surface is' depressed, the height of DA will remain the same, and, consequently, the flow will be uniform till the bend of the siphon comes in contact with the upper edge of the reservoir. By suitably adjusting the siphon in the cork, the velocity of efflux can be increased or decreased within certain limits. In this manner, any desired quantity of the fluid can be drawn off in a given time. The siphon is used in the arts, for decanting liquids, when it is desirable not to stir the sediment at the bottom of a vessel. It is also employed to draw a j)ortion of a liquid from the interior of a vessel when that liquid is overlaid by one of less specific gravity. The Hydraulic Ram. 219. The hydraulic ram is a machine for raising water by means of shocks caused by the sudden stoppages of a stream of water. The instrument' consists of a reservoir J5, which is sup- plied with water by an inclined pipe A ; on the upper surface 330 MECHANICS. Fig. 184. of the reservoir, is an orifice which may be closed by a spherical valve D ; this valve, • when not pressed against the opening, rests in a metallic framework immediately below the orifice ; (? is an air-vessel ■ communicating with the reser- voir by an orifice I^, which is fitted with a spherical valve .£"; this valve closes the orifice F, except when forced upwards, in which case its motion is restrained by a metallic frame- work or cage; IT represents a delivery-pipe entering the air-vessel at its upper part, and terminating near the bot- tom. At P is a small valve, opening inwards, to supply the loss of air in the air-vessel, arising from absorption by the water in passing through the air vessel. To explain the action of the instrument, suppose, at first, that it is empty, and all the parts in equilibrium. If a cur-, rent of water be admitted to the reservoir, through the in- clined pipe A, the reservoir will soon be filled, and com- mence rushing out at the orifice C. The impulse of the water wiU force the spherical valve D, upwards, closing the opening ; the velocity of the water in the reservoir will be suddenly checked ; the reaction will force open the valve £J, and a portion of the water will enter the air-chamber G. The force of the shock having been expended, the spherical valves will both fall by their own weight ; a second shock wiU take place, as before ; an additional quantity of water will be forced into the air-vessel, and so on, indefinitely. As the water is forced up into the air-vessel, the air becomes compressed ; and acting by its elastic force, it urges a stream of water up the pipe II. The shocks occur in rapid succes- sion, and, at each shock, a quantity of water is forced into the air-chamber, and thus a constant stream is kept up. To explain the use of the valve .P, it maybe remarked that water absorbs more air under a great pressure, than under HYDEAXTLIC AND PNEUMATIC MACHINES. 331 a smaller one. Hence, as it passes through the air-chamber, a portion of the air contained is taken up by the water and carried out through the pipe H. But each time that the vaiye D falls, there is a tendency to produce a vacuum in the upper part of the reservoir, in consequence of the rush of the fluid to escape through the opening. The pres- sure of the external air then forces the valve P open, a small portion of air enters, and is afterwards forced up with the water into the vessel 6^, to keep up the supply. The hydraulic ram is only used where it is required to raise small quantities of water, such as for the supply of a house, or garden. Only a small fraction of the amount of fluid which enters the supply-pipe actually passes out through the delivery-piioe ; but, if the head of water is pretty large, the column may be raised to a great height. Water is often raised, in this manner, to the highest points of lofty buildings. Sometimes, an additional air-vessel is introduced over the valve E^ for the purpose of deadening the shock of the valve in its play up and down. Archimedes' Screvr. 220. This machine is intended for raising water through small heights, and consists, in its simplest form, of a tube wound spirally around a cylinder. This cylinder is mounted so that its axis is oblique to the horizon, the lower end dip- ping into the reservoir. When the cylinder is turned on its axis, by a crank attached to its upper extremity, the lower end of the tube describes a circumference of a circle, whose plane is perpendicular to the axis. When the mouth of the tube comes to the level of the axis and begins to ascend, there will be a certain quantity of water in the tube, which will flow so as to occupy the lowest part of the spire ; and, if the cylinder is properly inclined to the horizon, this flow will be towards the upper end of the tube. At each revolution, an additional quantity of water will enter the tube, and that already in the tube will be forced, or raised, higher and 332 MECHANICS. Fig. 185. higher, till, at last, it will flow from the orifice at the upper end of the spiral tube. The Chain Pump. 221. The chain pump is an-instrument for raising water through small elevations. It consists of an endless chain passing over two wheels, A and JB, having their axes horizontal, the one being below the surface of the water, and the other above the spout of the pump. At- tached to this chain, and at right angles to it, are a system of circular disks, just fitting the tube CD. If the cylinder A be turned in the di- rection of the arrow-head, the buckets or disks will rise through the tube OD, carrying the water in the tube before them, until it reaches the spout (7, and escapes. The buckets thus emptied returo through the air to the reservoir, and so on perpetually. One great objection to this machine is, the difficulty of making the buckets fit the tube of the pump. Hence there is a constant leakage, requiring a great additional expend- iture of force. Sometimes, instead of having the body of the pump ver- tical, it is inclined ; in which case it does not differ much in principle from the wheel with flat buckets, that has been used for raising water. The Air Pump. 222. The air pump is a machine for rarefying the air in a closed space. It consists of a cylindrical barrel A, in which a piston £, fitting air-tight, is work- ed up and down by a lever 0, attached to a piston-rod D. The barrel communi- cates with an air-tight ves- Fig. 1S6. HTDEAULIO AND PNEUMATIC MACHINES. 333 sel^, called a receiver, by means of a narrow pipe. The receiver, which is usually of glass, is ground so as to fit air- tight upon a smooth bed-plate KK. The joint between the receiver and plate may be rendered more perfectly air-tight by rubbing it with a little oil. A stop-cock H, of a peculiar construction, permits communication to be made at pleasure between the barrel and receiver, or between the barrel and the external air. When the stop-cock is turned in a partic- ular direction, the barrel and receiver are made to commu- nicate ; but on turning it through 90 degrees, the communi- cation with the receiver is cut off, and a communication is opened between the barrel and the external air. Instead of the stop-cock, valves are often used, which are either opened* and closed by the elastic force of the air, or by the force that works the pump. The communicating pipe should be exceedingly small, and the piston B should, when at its low- est point, fit accurately to the bottom of the barrel. To explain the action of the air pump, suppose the piston to be depiressed to its lowest position. The stop-cock li, is turned so as to open a communication between the barrel and receiver, and the piston is raised to its highest point by a force applied to the lever C. The air which before occu- pied the receiver and pipe, wUl expand so as to fill the bar- rel, receiver, a-nd pipe. The stop-cock is then turned so as to cut off communication between the barrel and receiver, and open the barrel to the external air, and the piston again de- pressed to its lowest position. The rarefied air in the barrel is expelled into the external air by the depression of the piston. The air in the receiver is now more rarefied than at the beginning, and by a continued repetition of the' process just described, any degree of rarefaction may be attained. To measure the degree of rarefaction of the air in the receiver, a siphon-gauge may be used, or a glass tube, 30 ' inches long, may be made to communicate at its upper extremity with the receiver, whilst its lower extremity dips into a cistern of mercury. As the air is rarefied in the receiver, the pressure on the mercury in the tube becomes 334 MECHANICS. less than that on the surface of the mercury in the cistern, and the mercury rises in the tube. The tension of the air in the receiver will be given by the diiference between the height of the barometric, column and that of the mercury in the tube. To investigate a formula for computing the tension of the air in the receiver, after any number of double strokes, let us denote the capacity of the receiver in cubic feet, by r, that of the connecting-pipe, by p, and the space between the bottom of the barrel and the highest position of the piston, by b. Denote the original tension of the air, by t ; its tension after the first upward stroke of the piston, by t' ; •after the second, third, ...n**, upward strokes, by t, t'\ . . . <»'. The air which originally occupied the receiver and pipe, fills the receiver, pipe, and barrel, after the first upward stroke ; according to Maeiotte's law, its tension in the two cases varies inversely as the volumes occupied ; hence, t : t' : : p + r+b : p + r, .: t' = t- ^'^^ p + r + b In like manner, we shall have, after the second upward stroke, t' : t" : : p + r + b : p + r, .: t" = t' —^^- - p + b + r Substituting for t' its value, deduced from the preceding equation, we have, \p+b + r) In like manner, we find, HTDRA,ULIC AND PNEUMATIC MACHINES. 335 and, in general, \p + + r / If the pipe is exceedingly small, its capacity may be neglected in comparison with that of the receiver, and we shall then have, b + n Let it be required, for example, to determine the tension of the air after 5 upward strokes, when the capacity of the barrel is one-third that of the receiver. T In this case, , = f, and w = 5, whence, fv * 243 . Hence, the tension is less than a fourth part of that the external air. Instead of the receiver, the pipe may be connected by a screw-joint with any closed vessel, as a hollow globe or glass flask; In this case,' by reversing the direction of the stop- cock, in the up and down motion of the piston, the in- strument may be used as a condenser. When so used, the tension, after n downward strokes of the piston, is given by the formula, ■b + r^ -■- (^) Taking the same case as that before considered, with the exception that the instrument is used as a condenser instead of a rarefier, we have, after 5 downward strokes. l — O 253^ That is, the tension is more than four times that of the external air. 336 MECHANICS. When the pump is used for condensing aii", it is called a condenser. Artificial Fountains. 223. An artificial fountain is an instrument by means of which a liquid is forced upwards in the form of a jet, by the tension of condensed air. The simplest form of an arti- ficial fountain is called Heeo's ball. Hero's Ball. 224. This instrument consists of a hollow globe A^ into the top of which is inserted a vertical tube ^, reaching nearly to the bottom of the globe. This tube is provided with a stop-cock C, by means of which it may be closed, or opened to the external air, at pleasure. A second tube _Z?, enters the globe near the top, which is also provided with a stop-cock E. To use the instrument, close the stop-cock C, Fig. ist. and fill the lower portion of the globe with water through- the tube D ; then attach the tube Z* to a condenser, and pump air into the upper part of the globe, and confine it there by closing the stop-co,ck E. If, now, the stop-cock C be opened, the pressure of the confined air on the surface of the water in the globe, wUl force a jet up through the tube S. This jet will rise to a greater or less height, according to the greater or less quantity of air that was forced into the globe. The water will continue to flow through the tube as long as the tension of the confined air is greater than that of the external atmosphere, or else till the level of the water in the globe reaches the lower end of the tube. Instead of using the condenser, air may be introduced by blowing with the mouth through the tube Z>, and then con- fined as before, by turning the stop-cock E. The principle of Heeo's ball is the same as that of the air- chamber in the forcing pump and fire-engine, already ex- plained. HYDRAULIC AND PNEUMATIC MACHINES. 337 ' m A 3 Fig. 188. Hero's Fovmtain. 825. Hero's fountain is constructed on the same prin- ciple as Hero's ball, except that the compression of the air is effected by the weight of a column of water, instead of by aid of a condenser. A represents a cistern, similar to Hero's ball, with a tube £, extending nearly to the bottom of the cis- tern. C is a second cistern placed at some .^ distance below A. This cistern is connected with a basin D, by a bent tube £!, and also with the upper part of the cistern A, by a tube F. When the fountain is to be used, the cistern A is nearly filled with water, the cistern G being empty. A quantity of water is then poured into the basin D, which, acting by its weight, sinks into the cistern C, compressing the air in the upper portion of it into a smaller space, thus increasing its tension. This increase of tension acting on the surface of the water in A, forces a jet through the tube .5, which rises to a greater or less height according to the greater or less increase of the atmospheric tension. The flow will con- tinue till the level of the water in A, reaches the bottom of the tube £. The measure of the compressing force on a unit of surface of the water in C, is the weight of a column of water, whose base is a square unit, and whose altitude is the difference of level between the water in D and C. If Hero's ball be partially filled with water and placed under the receiver of an air pump, the water will be ob- served to rise in the tube, forming a fountain, as the air in the receiver is exhausted. The principle is the same as before, an excess of pressure on the water Avithin the globe over that without. In both cases, the flow is resisted by the tension of the air without, and is urged on by the tension within. Wine-Taster and Dropping-Bottle. 826. The wine-taster is used to bring up a small por- 15 338 MECHAincs. tion of wine or • other liquid, from a cask. It consists of a tube, open at the top, and terminat- ing below in a very narrow tube, also open. When it is to be used, it is inserted to any depth in the liquid, which will rise in the tube to the level of the upper surface of that liquid. The finger is then placed so as to close the upper orifice of the tube, and the instrument is raised out of the '^' cask. A portion of the fluid escapes from the lower orifice, until the pressure of the rarefied air in the tube, plus the weight of a column of liquid, whose cross-section is that of the tube, and whose altitude is that of the column of fluid retained, is just equal to the pressure of the external air. If the tube be placed over a tumbler, and the finger re- moved from the upper orifice, the fluid brought up will escape into the tumbler. If the lower orifice is very small, a few drops may be allowed to escape, by taking oif the finger and irmnediately replabing it. The instrument then constitutes the dropping tube. The Atmospheric Inkstand. aay. The atmospheric inkstand consists of a cylinder A, which communicates by a tube with a second cylinder JB. A piston 0, is moved up and down in A, by means of a screw D. Suppose the spaces A and £, to be filled with ink. If the piston C is raised, the pressure of the. external air forces the ink to follow it, and the part -S is emptied. If the iigTiso" operation be reversed, and the piston depressed, the ink is again forced into the space -B. This operation may be repeated at pleasure. NATIONAL SERIES OF STANDARD SCHOOL-BOOKS. D AV I E S' Complete Course of Mathematics.. 5Blementavs ffioursc. RB.,vii Pri«e. DAYIES' PRIMARY AKITHMETIO AND TABLE-BOOK $0 15 DA VIES' FIRST LESSONS IN ARITHMETIC 20 UAVIES' INTELLECTUAL ARITHMETIC 25- IJAVIES' NEW SCHOOL ARITHMETIC 45 KEY TO DAVIES' NEW SCHOOL AKITHMETIO 46 DAVIES' NEW UNIVERSITY ARITHMETIC 75 KEY TO DAVIES' NEW UNIVERSITY ARITHMETIC 50 DAVIES' GRAMMAR OP ARITHMETIC 30 DAVIES' ELEMENTARY ALGEBRA 75 KEY TO DAVIES' ELEMENTARY ALGEBRA 50 DAVIES' ELEMENTARY GEOMETRY AND TRIGONOMETRY .... 1 00 DAVIES' PRACTICAL MATHEMATICS 1 00 31)tianceti bourse. 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