1 &*uszJ* ! I AN ELEMENTARY TREATISE ON DYNAMICS. FIFTH EDITION. AN ELEMENTARY TREATISE ON THE INTEGRAL CALCULUS, CONTAINING APPLICATIONS TO PLANE CURVES AND SURFACES. BY BENJAMIN WILLIAMSON, F. E. S. SIXTH EDITION. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL CALCULUS, CONTAINING THE THEORY OE PLANE CURVES. BY BENJAMIN WILLIAMSON, F.R.S. AN ELEMENTARY TREATISE DYNAMICS, CONTAINING APPLICATIONS TO THERMODYNAMICS, NUMEROUS EXAMPLES BY BENJAMIN WILLIAMSON, M.A., F.R.S., FELLOW OF TRINITY COLLEGE, AND PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF DUBLIN; ANT) FRANCIS A. TARLETON, LL.D., FELLOW AND TUTOR OF TRINITY COLLEGE. SECOND EDITION, REVISED AND ENLARGED. NEW YORK: D. APPLETON AND COMPANY.. 1889. DBPT . DUBLIN : PRINTED AT THE UNIVERSITY PRESS, BY PONSONBY AND WELDRICK. <&4 ZrVr W 7 ! k PHYSICS DEPf, PREFACE. Although in recent years several important works on Dynamics have been published in England, yet none have been issued which seem to fill the role contemplated in this book. In its composition we have started from the most ele- mentary conceptions, so that any Student who is acquainted with the conditions of Equilibrium and with the notation of the Calculus can commence the Treatise without requiring the previous study of any other work on the subject. The first half contains a tolerably full treatment of what is usually styled the Dynamics of a Particle. The latter half treats of the Kinematics and Kinetics of Eigid Bodies ; and throughout we have kept the practical nature of the subject in view, and have, in general, avoided purely fancy problems. In an early chapter we have introduced and elucidated the general principle of "Work or Energy, and have given subsequently a more complete treatment of this great principle, illustrating it by a brief application to the theory of Thermodynamics. In the latter part of the book we have borrowed largely from Thomson and Tait's Natural Philosophy; Routh's Rigid Dynamics; Schell's Theorie der 665465 vi Preface. Bewecjung und der Krtifte ; and Clausius' Mechanical Theory of Heat; our aim having been simply to enable the Student to acquire as easily as possible a knowledge of the subject of which we treat. In this Edition we have carefully revised and to a con- siderable extent rearranged the entire Work. In doing so we have developed, and in some cases rewritten, many por- tions of the subject, more especially that on generalized coordinates in connexion with Lagrange's and Hamilton's methods. We have also exhibited the general theory of small oscillations in a new form, and one which we hope will be easily comprehended by the Student. To those who desire to pursue the study of Dynamics to its highest development, the perusal of the great treatise of Thomson and Tait, as also that of Routh, will, we hope, be facilitated by using the present Work as an introduction. We may add that to the latter writer our obligations, as the reader will find, have been largely increased in this Edition. Teinity College, i%, 1889. TABLE OF CONTENTS CHAPTEE I. VELOCITY. PAGE Uniform Motion, . 2 Variable Motion, . 3 Kinematics, . 5 Composition of Velocities, 7 Relative Velocity, . . 9 Examples, . 11 CHAPTEE II. ACCELERATION. Uniform Acceleration, . . . .12 Variable Acceleration, 1» Accelerations Parallel to Fixed Axes, . . . ■ • ■ .16 Total Acceleration, . . . . 17 Tangential and Normal Accelerations, . . . . • • .17 Hodograph, 19 Angular Acceleration, . . . . - • • • • .19 Areal Acceleration, - 1 Accelerations Parallel to Moving Axes, . 22 Units of Time and Space, 23 viii Table of Contents. CHAPTER III. LAWS OF MOTION. Section I. — 'Rectilinear Motion. PAGE First Law of Motion, 25 Second Law of Motion, 26 Mass, 32 Motion on a Smooth Plane, 34 Line of Quickest Descent to a Curve, . . ■ 36 Section II. — Parabolic Motion. Construction of Path, 39 Eange and Time of Flight, 40 Morin's Apparatus, 46 Section III. — Friction. Elementary Laws of Friction, 50 Motion on a Rough Inclined Plane, 51 Section IV. — Momentum. Force measured by Momentum, 53 Absolute and Gravitation Units, 54 Impulses, 56 Equations of Motion, 56 Section V. — Action and Reaction. Third Law of Motion, 58 Forces of Inertia, 59 Atwood's Machine, 60 Examples, 64 CHAPTER IY. IMPACT AND COLLISION. Direct Collision of Homogeneous Spheres, ...... 66 Height of Rebound, 69 Oblique Collision, 70 Vis Viva of a System, 7;* Table of Contents. ix PAGE Momentum of a System, 74 Conservation of Momentum, ......... 75 Examples, .78 CHAPTER V. CIRCULAR MOTION. Section I. — Harmonic Motion. Uniform Circular Motion, 84 Elliptic Harmonic Motion, 86 Section II. — Centrifugal Force. Circular Orbits, 90 Centrifugal Force of Earth, .91 Verification of the Law of Attraction, 92 Centrifugal Force in Eotation of a Rigid Body, 94 Section III. — Motion in a Vertical Circle. Motion in a Vertical Curve, 98 Simple Pendulum, . . . . . . . • • .100 Time of a Small Oscillation, 101 Seconds' Pendulum, .......... 102 Effect of Change of Place, 104 Airy on Mean Density of Earth, 107 Time of Oscillation for any Amplitude, . . . - - - .108 Cycloidal Pendulum, . . . . . . . - - .111 Conical Pendulum, . . . . . . . • • .115 Eevolution in a Vertical Circle, • - 117 Examples, . . . . . • . • • • • .122 CHAPTER VI. WORK AND ENERGY. Gravitation Unit of Work, 125 Absolute Unit of Work, I' 26 Work by a Variable Force, 128 Potential of a Sphere, . . 130 X Table of Contents. PAGE Work by a Stress, 131 Energy, 133 Kinetic Energy, 133 Equation of Energy, 136 Energy of Eotation, 137 Work Done by an Impulse, 140 Compound Pendulum, . .141 Motion Round a Fixed Axis, . . 144 Examples, . 145 CHAPTER VII. CENTRAL FORCES. Section I. — Rectilinear Motion. Centre of Force, 147 Force varying as the Distance, 148 Force varying as Inverse Square of Distance, 149 Application to the Earth, 151 Application to Spheres, . . 152 Application to Elastic Strings, 155 Secion II. — Central Orbits. Differential Equations of Motion, 160 Law of Direct Distance, . . . 161 Equable Description of Areas, 164 ^ . dhi F Equation — — + u = rrr—z, 171 u dd- h~ u 2 Law of Inverse Square, 173 Kepler's Laws, 175 Law of Gravitation, 176 Velocity at any Point in Orbit, 177 Change of Absolute Force, 179 Application of Hodograph, 181 Lambert's Theorem, .......... 183 Mass of Sun, 186 Table of Contents. xi PAGE Mean Density of Sun, . . . - . . . • • .187 Planetary Perturbations, . . . . • • • • - 188 Tangential Disturbing Force, 189 Normal Disturbing Force, . . - • • • • • .189 Apsides, 190 Approximately Circular Orbits, 194 Movable Orbits, Newton, 196 Examples, .....-..•-•• 197 CHAPTER VIII. CONSTRAINED MOTION, RESISTING MEDIUM. Motion on a Fixed Curve, 206 Theorem of Ossian Bonnet, .....•••• -08 Motion on a Fixed Surface, . . . . • • • • .211 Motion on a Sphere, .......••• 212 Rectilinear Motion in a Resisting Medium, 219 Examples, 223 CHAPTER IX. THE GENERAL DYNAMICAL PRINCIPLES. D'Alembert's Principle, 227 Initial Motion, 230 Bertrand's Theorem, 23i Thomson's Theorem, 23! Equation of Vis Viva, 232 Effect of Impulses on Vis Viva, 23 '^ Equations of Motion, 24 ° Constraints and Partial Freedom, 24i Moments of Momentum, . . . . • • • • -243 Conservation of Moment of Momentum, ...-•• 2 ^8 Examples, 248 xii Table of Contents. CHAPTER X. MOTION OF A RIGID BODY PARALLEL TO A FIXED PLANE. Section I. — Kinematics. PAGE Degrees of Freedom, 254 Translation and Eotation, 255 Composition of Finite Displacements, ....... 256 Composition of Velocities, ......... 257 Space Centrode and Body Centrode, 261 Pure Rolling, 261 Geometrical Representation of Motion, ....... 262 Examples, 263 Section II. — Kinetics — Coyistrained Motion. Degrees of Freedom, 268 Motion Round a Fixed Axis, 270 Moments of Momentum, . . . . . . . . .273 Stresses on Axis of Rotation, 274 Stress Due to Impulses, 275 Centre of Percussion, 276 Examples, 279 Section III. — Kinetics of Free Motion Parallel to a Fixed Plane. Equations of Motion, 283 Equation of Vis Viva, 285 Moment of Momentum, relative to any Point, ..... 286 Impact, 288 Stress in Initial Motion, 292 Friction, 296 Tendency of a Rod to Break, 302 Impulsive Friction, 308 Roiling and Twisting Friction, 311 Examples 312 Table of Contents. xiii CHAPTER XI. MOTION OF A RIGID BODY IN GENERAL. Section I. — Kinematics. PA.GE Composition of Rotations, . . . . . . . . .317 Motion of a Body entirely Free, ........ 320 Analytical Treatment of Motion, 321 Velocity of any Point of a Body, ........ 325 Acceleration of Rotation, 327 Complete Determination of Motion of a Body, 329 Screws, 333 Composition of Twists, 334 Examples, 335 Section II. — Kinetics. Moments of Momentum round a Fixed Point, . . . . .315 Motion of a Body Round a Fixed Point under the Action of Impulse. . 346 Couple of Principal Moments, . . . . . . . .347 Motion of a Free Body under Impulse, ....... 350 Vis Viva of a Rigid Body, . . . . . . . . .351 Equations of Motion of a Body having a Fixed Point, .... 353 Equations of Motion of a Free Body, 354 Motion of a Body Round a Fixed Point under no External Force, . . 357 Conjugate Ellipsoid and Conjugate Line, ...... 363 Stress exerted by a Body on a Fixed Point, ...... 367 Centrifugal Couple, 369 Motion relative to Centre of Inertia, . . . . . . .371 Impact, ............ 379 Impulsive Friction, 380 Collision of Rough Spheres, 381 Equations of Motion Referred to Body-axes, 385 Motion consisting of Successive Rotations, ...... 385 Motion of a Solid of Revolution, 389 Examples, 390 xiv Table of Contents. CHAPTER XII. ENERGY AND THE GENEEAL EQUATIONS OF DYNAMICS. Section I. — Energy. PAGE Equation of Energy, 396 Conservation of Energy, 397 On the Ultimate Permanent Forces of Nature, 399 Forces which appear in the Equation of Energy, 400 General Form of Equation of Energy, 402 Equivalent Systems of Forces, ........ 404 Wrenches, ............ 405 Examples, ............ 405 Section II. — The General Equations of Dynamics. General Equations of Motion, 412 Equation of Energy when Conditions Involve the Time, . . .413 Similar Mechanical Systems, 414 Generalized Coordinates, . . .415 Kinetic Energy in Generalized Coordinates, 416 Generalized Equations of Motion under Impulse, . . . . .416 Generalized Expression for Kinetic Energy, . . . . . .419 Energy of Initial Motion, 420 Lagrange's Generalized Equations of Motion, ..... 420 Deduction of Equation of Energy, 423 Ignoration of Coordinates, 426 Components of Momentum and Velocity, ...... 429 Hamilton's Form of Equations of Motion, 431 Calculus of Variations, 433 Examples, 433 Principle of Least Action, 436 Hamilton's Characteristic Function, 439 Examples 442 Table of Contents. xv CHAPTEE XIII. SMALL OSCILLATIONS. PAGE Oscillation on a Plane Curve, 445 Oscillation on a Surface, ...... . . 446 Conditions for Stable Equilibrium, 451 Equations of Motion for an Oscillating System, 452 General Solution of these Equations, 454 Harmonic Determinant, . . . . . . . . .454 Lemma in Determinants, 455 Transformation of Harmonic Determinant, 457 Reality of Roots of Harmonic Determinant, ...... 457 Stability of the Motion, . .' .459 Case of Equal Roots, 462 General Solution in this Case, . 464 Principal Coordinates and Directions of Harmonic Vibration, . . 465 Effect of an Increase of Inertia, 469 Energy of an Oscillating System, 470 Examples, ............ 471 CHAPTER XIY. THERMODYNAMICS. Mechanical Equivalent of Heat, . . . . . . .477 Equation of Energy, .......... 478 Specific Heat, 479 Perfect Gas, 481 Reversibility and Cyclical Processes, 483 Isothermals and Adiabatics for a Perfect Gas, ..... 484 Fundamental Principles of Thermodynamics, ..... 485 Carnot's Cycle, 486 xvi Table of Contents. PAGE Extension of Camot's Cycle, 488 Entropy, 489 Energy and Entropy, 490 Elasticity and Expansion, 491 Examples, 492 Non-reversible Transformations, . 495 Examples, 496 Absolute Scale of Temperature, 497 Absolute Zero, 499 Change of State, 501 Examples, 504 Available Energy, 507 Dissipation of Energy, 508 Increase of Entropy, 509 Path of Least Heat, 511 Examples, 512 Miscellaneous Examples 514 DYNAMICS. CHAPTER I. VELOCITY. 1. Matter. — "We give the name of matter to that which exclusively occupies space, and which we regard as the permanent cause of any of our sensations. Portions of matter which are bounded in every direction are called bodies. Every body has necessarily a determinate volume, and an external form or surface ; and exists, or is conceived to exist, in space. A portion of matter indefinitely small in all its dimen- sions is called a material particle. Every body may be re- garded as consisting of an indefinitely great number of particles. The name of force is given to any cause which produces, or tends to produce, motion in matter. The branch of Mechanics which treats of motion produced in a body by the action of force is commonly called Dynamics. "We commence with the consideration of motion in itself, without any regard to its cause. 2. Motion, Velocity. — When a body continually changes its position in space, it is said to be in motion ; and the rate and the direction of the motion of any of its points at any instant is called the velocity of the point at that instant. The motion of a point is said to be rectilinear or curvilinear according as its path is a right line or curved. In the case of curvilinear motion, the direction of motion of a particle at any instant is that of the tangent to its path, drawn at the point occupied by the moving particle at the instant. 2 • ' '' 'Velocity. 3. Motion 0f TFr&n^lation. — If all the points of a rigid body move, at each "instant, in parallel directions, the body is said to have a motion of translation only ; and the motion of the body is completely determined when that of any one of its points is known. It is usual, in this case, to take its centre of mass as the point whose motion determines that of the hody. . . In our earlier chapters, whenever we speak of a rigid body moving, we suppose it to have a motion of transla- tion solely, and we consider its path as that of its centre of mass. 4. Uniform Motion, Velocity. — If a point move over equal lengths or spaces,* in equal intervals of time, however short the intervals be taken, its motion is said to be uniform ; and its velocity is measured by the space described in the un it of time : this is the same at every instant so long as the motion continues uniform. A second is usually adopted as the unit of time ; and, in this country, a foot as the unit of length. Thus, the velocity of a point which moves over five feet in each second is said to be a velocity of 5 feet per second, and is numerically denoted by 5 ; and similarly in other cases. If any other units of time and space be adopted, the number which represents the velocity of the moving point will have to be altered pro- portionally. Thus, we speak of a velocity of 10 miles an hour, or 100 yards a minute, &c. : each of these can be readily expressed in feet per second, when necessary. The space, or length of the path described during any time, is usually denoted by the letter s, the velocity by v, and the time estimated in seconds by t.f In the case of uniform motion, the relation connecting these quantities can be imme- diately obtained. For, if the space described in one second be represented by v 3 that described in two seconds is repre- sented by 2v, that in three seconds by 3r, and that in any number (t) of seconds by vt. * The -word space is employed in abbreviation for length of path described. f Unless the contrary be stated, we shall in all cases assume a foot and a second as our units of space and time, i.e. we shall regard t as representing a number of seconds or parts of a second, and s as a number of feet. Variable Motion. 3 Accordingly we have in the case of uniform motion the relation s = vt. (1) This formula evidently holds good whatever be the units of space and time, and introduces the unit of velocity as that of a unit of space described in a unit of time. It is true for uniform curvilinear, as well as rectilinear motion ; and also whether t represents a number of seconds, or any part of a second, however small. Again, if s denote the space described in the time t\ we have «' = vtf, and hence s'-s or the velocity, when uniform, is measured by the space de- scribed during any interval of time divided by the number by which that time is represented. This result equally holds good if we suppose the interval of time, denoted by if - t, to become indefinitely small ; in which case the limiting value of -, — - or — will still represent the v — L CIO velocity v. Examples. 1. If a body, moving uniformly, pass over 10 miles in an hour, find its ve- locity in feet per second. Ans. 14|, 2. If a body, moving uniformly with a velocity of 16 feet per second, pass V over 100 miles, find the time of its motion. Ans. 9 hrs. 10 min. 3. Assuming that light travels from the sun to the earth in 8 m 30 s , and that ^ its velocity is 180,000 miles per second, calculate the distance of the sun. Ans. 91,800,000 miles. 4. If a velocity of 20 miles an hour be the unit of velocity, and a mile the v" unit of space, find the number which represents a velocity of 32 feet per second. Ans. \r£' b. Find in metres the velocity of a point on the earth's equator arising from the rotation of the earth on its axis. Ans. 463. 5. Variable Motion. — If the spaces described in equal intervals of time be not equal, the motion is said to be ■variable, and the velocity can no longer be measured by the space actually described in one second. The movable has, however, at each instant a certain definite velocity which is b2 4 Velocity. measured by the space which it ivould describe during a second, if it were conceived to move uniformly during that time with the velocity which it has at the instant under consideration. For example, when we say that a railway train is moving at the rate of 40 miles an hour, we mean that it would pass over 40 miles in the hour if it continued to move during that time with the speed which it has at the instant referred to. Again, if we suppose that there are no sudden changes of velocity, the change in the velocity of a movable in any in- definitely small portion of time must be itself indefinitely small; as otherwise the velocity would not vary continuously. Accordingly, in such cases, we may suppose the motion as uniform during the indefinitely small time dt; and we shall have (as in the last Article) for the velocity v at any instant the equation s - s ds v m lm - 7=i = If (2) That is, in all cases the velocity of a point at any instant is measured by the limiting value of the space described in a small interval of time, divided by the number which repre- sents that interval of time. This method of expressing velo- city is sometimes concisely represented in the notation of Newton by the symbol s. 6. Mean Telocity. — If a body describe the space s in the time t, then its mean or average velocity during that time is represented by -, being the velocity with which a body, V moving uniformly, would describe the same space in the time t. The formula (2) can be immediately deduced from the consideration of mean or average velocity — for we may consider the velocity of a point at any instant as being its mean velocity during an infinitely small interval of time; ds whence we get, as before, the relation v = — . OjTi 7. Geometrical Representation of a Velocity. — Uniform rectilineal motion is completely determined when the direction and rate of motion are known. Hence the velocity of a point can be represented both in magnitude and direction by a right line. Kinematics. 5 Thus, if a point move uniformly in the line OP, so as to describe the space OA in the unit of time (one second suppose), the line OA may be taken to repre- sent the velocity of the point both in magnitude and direc- tion. The arrow head denotes the direction in which the motion takes place, namely from to A. This method of representation holds good also in the case of variable velocity, provided OA be the space which the body would describe in one second if its velocity remained unaltered in magnitude and direction (Art. 5). In accordance with the principles established in Geometry, if the velocity of a particle moving from to P be regarded as positive, velocity in the opposite direction, i. e. from P to 0, must be regarded as negative. 8. Kinematics. — As our ideas of motion and velocity depend solely on our conceptions of space and time, the whole subject of motion admits of being treated as a branch of pure Mathematics ; and, as such, has been discussed in many important treatises during recent years. This branch of Mathematics is called Kinematics* (from Kivr/jua, motion), and in it the motion of a body is discussed without any reference to the force or forces by which the motion is produced. Questions of the latter class, i. e. of motion with reference to force, belong to the science of Dy- namics, or what is now usually styled Kinetics. The foregoing distinction should be observed by the student, as much indistinctness of conception arises from its not being carefully kept in mind in the study of Dynamics. In the present treatise it is not proposed however to divide the treatment of the subject in the manner indicated, as to do so would require a complete discussion of motion (in- cluding rotation and kindred subjects) before entering on the most elementary problems in Dynamics. At the same time it will aid the student towards obtaining clear mechanical conceptions if he will consider what part of each problem * The name " Cinematique " was first given to this hranch of Mathematics by Ampere, in his " Essai sur la philosophie des Sciences," 1834 6 Velocity. discussed belongs properly to the science of Kinematics, and what to that of Dynamics or Kinetics. 9. Rest and Motion, Relative. — We have defined rest and motion with reference to space. Now of space in itself or absolute space our senses take no cognizance, all that we perceive being matter or body as occupying or existing in space ; but our senses give us no information as to whether any body occupies the same absolute position in space during successive intervals of time or not. Hence, of absolute rest we can have no perception or knowledge ; and when we say that a body is at rest we mean that it does not alter its posi- tion with relation to other bodies which are considered fixed. For instance, bodies on the earth's surface are said to be at rest when they do not alter their position relatively to the earth's surface ; we know however that the earth has at least two distinct motions, one of rotation relative to its axis ; the other around the sun, regarded as fixed. As our idea of rest is only relative, so also must be our idea of motion : thus, a body is said to be in motion when it alters its position with respect to other bodies regarded as being at rest. Hence all motions must be considered as relative : for in- stance, when we say that a body is moving at the rate of thirty miles an hour, we mean that such is its velocity relative to a place on the earth : its absolute velocity is immensely greater, and is obtained by combining this velocity with the absolute velocity of the earth itself. Again, we speak of the same body as at rest, or as in motion, according as we compare its position with that of one object or of another. For example, a person seated in a railway carriage is said to be at rest relatively to the carriage, and to be in motion relatively to the earth, &c. That a body may be regarded as having at the same in- stant two or more velocities is a matter of common experience : for instance, if a ball roll along the deck of a vessel, which is descending a river, we conceive the ball as having simul- taneously one velocity along the deck ; another, that of the vessel in the stream ; a third, that of the river relatively to its banks, &c. The velocity of the ball, relatively to the earth, is got by compounding these separate velocities. We proceed to show in what manner this can be done. Composition of Velocities. 7 10. Composition of Velocities. — Suppose a point to move uniformly, with a velocity v, along the line AB, while the line moves uniformly parallel to itself ; then the point may be regarded as having the two velocities simulta- neously. In order to find its position at the end of any time t, let AB be the space which it would describe in that time along AB considered as fixed; and let CD be the position of the moving line at the end of the same time; complete the parallelogram ABDC; then D will plainly be the position of the moving point at the end of the time t. Also, if v be the uniform velocity of the point along the line A C, we shall have AC = v't, and CD = vt. Hence AC jf CI)~ v' Again, as this is independent of t, the ratio of AC to CD will be constant during the entire motion ; and consequently the point will move from A to D along the diagonal AD. To find the velocity of the moving point, we make t = 1 (or the unit of time) in the last; then AB and AC represent in magnitude and direction the component velocities^ of the moving point, and AD represents the resultant velocity : in other words, if a body be animated by two velocities repre- sented in magnitude and direction by the sides of a parallel- ogram, the resultant velocity is represented in magnitude and direction by the diagonal of the parallelogram. Conversely, any velocity may be regarded as equivalent to two velocities in any two directions, and the magnitudes of the component velocities can be determined by the preceding construction. In like manner, if a body be animated simultaneously with three velocities, its resultant velocity is represented in magnitude and direction by the diagonal of the parallelepiped whose edges represent the component velocities. For we can compound two of these velocities by the method given above, and then compound their resultant with the third velocity. This principle can, plainly, be extended to the case of a point 8 Velocity. supposed to be animated by any number of velocities simul- taneously. 11. Polygon of Velocities. — It immediately follows that if a point be subjected to any number of simultaneous velocities its resultant velocity can be obtained by the fol- lowing geometrical construction : — From 0, the original position of the point, draw OA, representing one of the given velocities in magnitude and direction ; from A draw AB, parallel and equal to the line which represents a second velocity ; and so on for the remaining velocities ; then the line which connects with the extremity of the line drawn parallel and equal to the line representing the last velocity will represent the resultant velocity, both in magnitude and direction. This construction is called the polygon of velocity, and is in general a gauche polygon. The preceding result admits of being stated otherwise, thus : If a body be subjected to two or more uniform veloci- ties it will arrive at the same position at the end of any time as it would have arrived at if the several motions had taken place successively instead of simultaneously. This is adopted as an axiom by some writers on Mechanics, for it appears to be an immediate consequence of our ideas of motion. The student can easily see that the whole theory of the composi- tion of velocities can be deduced from this principle. 12. Component and Resultant Velocities. — The velocities represented by AB and AC, in Art. 10, are called the components of the velocity represented by AD. If a point describe a plane path, the usual method of representing its position is with reference to two fixed rect- angular axes lying in the plane. Then, if cc, y be the coordinates of the moving point at any instant, its component velocities parallel, respectively, to the coordinate axes, are evidently, by Art. 5, represented by dx , clu ■ — and — • dt dt Also, if a be the angle which the direction of motion at the instant makes with the axis of x, the component veloci- ties are represented by v cos a and v sin a, respectively;^', e. the velocity with which a point is moving in any fixed direc- Relative Velocity. 9 tion is equal to the component of its velocity in that direc- tion. TT i dx . du Hence we get v cos a = — , Psma = -J. (3) If we square and add, we get dt) + \it) ~\Jt) ; •'• v ~Jt ; i.e. the velocity in a curvilinear path is represented in the same matter as in a rectilinear ; this result might have been directly established from oflier considerations. More generally, if a?, i/, z be the coordinates of a moving point at any instant, with reference to any system of coordinate axes, its component velocities parallel to the coordinate axes are plainly represented by — , -~ and — , re- at at etc spectively. If the axes be rectangular, and if o, |3, 7 be the direction angles, and v the magnitude of the velocity of the point, then the component velocities parallel to the coordinate axes are represented by v cos a, v cos |3, v cos 7, respectively. Hence, in this case, we have dx _ du dz ... v cos a = — , v cos p = -£, v cos 7 = — . (4) dt ' dt ' dt x In Newton's notation, as in Art. 5, these component velocities are represented by the symbols, x t i/, z. 13. Relative Velocity. — If the point A be in motion along AB with a velocity represented by AB, and, at the same time, A! be in motion along A'B' with a velocity re- presented by A'B\ to find their relative velocity. Draw AD parallel and equal to A'B\ and construct the parallelogram ACBD; then the velocity AB may be regarded as equivalent to the velocities AD and AC; now the former velocity, being equal and in the same direction as that of the other point A', will not alter the relative 10 Velocity. position of the points (Art. 10) ; consequently the latter com- ponent AC represents the relative velocity of the moving points, i.e. the velocity with which A is moving relatively to A' 9 regarded as at rest. Hence, to get the velocity of one moving point relatively to another which is also in motion, we suppose equal and parallel motions given to both, each equal and opposite to the motion of the second point: by this means that point is brought to rest, and the velocity of the other, relative to it, is had by compounding the new velocity with its original velocity. 14. Components of Relative Telocity. — Suppose (x, y, z), (/, y', z) to be the coordinates of the two moving points (M, M'), respectively, with reference to any coordi- nate system of fixed axes. Then, to get the motion of M\ relatively to 31, we suppose three axes drawn through M parallel, respectively, to the coordinate axes ; and let £, rj, £ denote the coordinates of M, relative to these axes, and we have £ = %' - x, ri = y'-y, Z = s' - s ; and hence d% dx f dx dr) _ dy dy a% _ dz' dz ^ _ di = dt~It' It = dt " di' di'pdi'di' ^' i.e. dx' dx df dy dz dz dt ~ di' di "df Jt " 'di' or x - x, y - y, represent the components of the relative velocity of the two moving particles. Examples. 1 1 Examples. 1. Two points are moving in rectangular directions, with velocities of 300 and 400 yards per minute ; find their relative velocity in feet per second. Ans. 25. 2. Two particles start simultaneously from different points, in givendirec- tions, with uniform velocities. Show how, hy a geometrical construction, to determine the relative distance at the end of any time ; and find when this distance is a minimum. 3. The tide is running out of the mouth of a harbour at the rate of 2^ miles per hour ; in what direction must a man, who can row in still water at the rate of 5 miles per hour, point the head of the boat in order to make for a point directly across the harbour ? 4. A boat starts with a given velocity across a river ; find the direction in which she should steer, in order, without altering her course, to land at a given station at the opposite side of the river — the velocity of the stream, and also of the boat, being supposed known. 5. Two trains are moving, one due south, the other north-east. ^ If their velocities be 25 and 30 miles an hour, respectively, calculate their relative velocity. 6. A railway train is moving at the rate of 30 miles an hour, when it is struck by a stone, moving horizontally and at right angles to the train with the velocity of 33 feet per second. Find the magnitude and direction of the velo- city with which the stone appears to meet the train. Ans. Resultant velocity is 55 feet. Indian Civil Service Exam., 1876. 7. Two particles start simultaneously from A, JB, two of the angular points of a square ABC I), in the directions AB, BC; and describe the periphery with constant velocities V, v, respectively, where V is greater than v, until one par- y ticle overtakes the other. Prove that the minimum distances between the par- ticles occur at equal intervals of time, and that if V : v : : m + 1 : m, where m is an integer, the sum of all these minimum distances is m (m + 1) . , . ,, x a side of the square. 2y / wH(»»+ l) a Camb. Math. Trip., 1871. 12 Acceleration. CHAPTEE II. ACCELERATION. 15. Acceleration and Retardation of Motion. — The velocity of a point is said to be accelerated or retarded according as it increases or diminishes with the time. This acceleration, or rate of change of velocity in a fixed direction, may be either uniform or variable. Retardation of motion is to be regarded as a negative acceleration, i.e. as an accelera- tion in the opposite direction to that of the motion. 16. Uniform Acceleration. — The motion of a point moving in a straight line is said to be uniformly accelerated when it receives equal increments of velocity in equal times. In this case the acceleration is measured by the additional velocity received in each unit of time. As a second is usually taken as the unit of time, we may define the acceleration of velocity in this case to be measured by the additional velocity received by the movable in each second; this acceleration is usually denoted by the letter /. In the case of uniform acceleration in a right line we proceed to find expressions for the velocity at the end of any given time, and also for the space described. 17. Velocity at any Instant. — Let v denote the velo- city at the instant from which the time is reckoned; then, since the point receives in each second an additional velocity /, its velocity at the end of the first second is v + /; at the 'end of the next second, v + 2/; at the end of the third, v + 3/; and at the end of n seconds, v + nf. Or, if ^ t denote the number of seconds in question, and v the velocity at the end of that time, we have V = Vo +ft. (1) If the point be supposed to start from rest, we have v=ft; Space described in any Time. 13 that is, the velocity acquired at the end of t seconds is t times that acquired at the end of one second. In the case of a "uniformly retarded motion,/ denotes the Telocity lost in each second ; and, if v he the initial velocity, we shall have, as before, for the velocity at the end of t seconds, v = v -ft. (2) In this case the velocity becomes zero at the instant when r =ft, or at the end of the time -^ • If the retardation con- tinued afterwards, the velocity would become negative ; that is, the point should proceed to move back in a direction opposite to that of its former motion. It will be observed that the formula (1) and (2) differ only in the sign of/; they may accordingly be regarded as comprised in the same general formula, in which a retarda- tion, as stated before, is regarded as a negative acceleration. Examples. 1 . If a body start from rest with a uniform acceleration of 7 feet per second, find its velocity at the end of three minutes. Ans. 1260 feet. 2. In what time would a body acquire a velocity of 100 feet per second if it start from rest with a uniform acceleration of 32 feet per second ? Ans. Z\ seconds. 3. A body starts from rest with the velocity of 1000 feet per second, and its motion is uniformly retarded by a velocity of 16 feet each second ; find when it would be brought to rest. Ans. 1 m. 2 -J sec. 4. A velocity of one foot per second is changed uniformly in one minute to a velocity of one mile per hour. Express numerically the rate of change of velocity when a yard and a minute are taken as the units of space and time. Ans. - 3 *. 18. Space described in any Time. — To find the space described in any time in the case of uniform accelera- tion in a straight line. From equation (2) we get ds hence, by integration, s = v t+ I ft 2 ; \J 14 Acceleration. no constant being added since the space is measured from the position of the point when t = 0. If the point start from rest we have s = ift\ In the case of uniformly retarded motion we have s = v t - iff-. This and the preceding formula are represented by the single expression s = v t±ift\ (3) in which the upper or lower sign is given to /, according as the acceleration has place in the positive or negative direc- tion. Similarly, equations (1) and (2) are combined in the state- ment v = v ±ft. (4) The preceding result admits also of being established geo- metrically in the following manner, as given by Newton : — Suppose the point to start from rest, and on any right line AX take portions AD, AE, &c, proportional to the intervals of time from the commencement of the motion, and erect perpendiculars DB, EC, &c, representing the corresponding velocities ; then since the velocity at the end of any time (Art. 18) is proportional to that time, the ordinates BD, CE, &c, will be to one another in the same ratio as the times, i. e. as AD y AE, &c. ; and consequently the points A f B, C, &c, all lie on a right line. Again, let AD = t, DE = At, BD = v; then the space described in the infinitely small time At will be represented by vAt, i. e. by the area BDEC; and accordingly the whole space described in the time represented by AN will be repre- sented by the sum of the elementary areas, BDEC, &c, or by the whole area, APN, i. e. by I AN x PN, or by \vt ; therefore s = \ft", as before. If the point be supposed to start with an initial velocity Variable Acceleration. 15 r , the student will find no difficulty in supplying the corre- sponding construction. 19. Relation between Velocity and Space. — If we eliminate t between equations (3) and (4), we get v- = iv ± 2/s, (5) in which the upper or lower sign is taken according as the acceleration is in the direction of the motion or in the oppo- site direction. We shall resume the consideration of these equations when we come to the investigation of the motion of a body under the action of a constant force. 20. Algebraic Expression for an Acceleration. — In the case of a point moving with a uniform acceleration, let v represent the velocity at the end of the time t, and v that at the time tf; then by (1) we have v = v +ft, v'=v +ft', and hence / = j, — -. Moreover, since this result holds, however small the in- terval of time represented by f - t may be, we have, as in Art 4, civ J ~ dt' 21. Variable Acceleration. — In the case of the motion of a point in a right line, if the acceleration is not uniform, but varies continuously according to any law, we plainly (as in Art. 5) may suppose that the motion is uniformly accelerated during an infinitely small time dt ; or (which is the same thing) that the acceleration at any instant is measured by what the increase of velocity in a unit of time would hare been if its rate of increase had been uniform during that time, and the same as that at the instant in question. Hence the accelera- tion at any instant is defined as the rate of change of the velo- city at that instant, and is measured in all cases by the ratio of the increment of the velocity at the instant to the incre- ment of the time. 16 Acceleration. Accordingly we have, whether the acceleration be uniform or variable, the relations /=*?«*?. (6) J dt df- K j These are expressed in Newton's notation in the form / = b = s. All these results apply equally to the case of retardation of motion, which is always to be regarded as a negative acce- leration. 22. Geometrical Representation of an Accelera- tion. — From the preceding it appears that the acceleration of the motion of a point, whether it be uniform or variable, is in all cases measured by a velocity. Hence it can be re- presented, both in magnitude and direction, by a right line, in the same manner as velocity (Art. 7). Hence, also, we may regard a point as receiving two or more simultaneous accelerations of motion, and can deter- mine the resultant acceleration by a geometrical construction, as in Arts. 10 and 11. Consequently, accelerations are compounded and resolved according to the same laws as velocities. 23. Component Accelerations Parallel to Fixed Axes. — If x, y, z denote the coordinates relative to a fixed rectangular system of axes, of the position of a moving point at the end of the time t ; then, as in Art. 12, its com- ponent velocities parallel to the axes of coordinates are re- . _ . dx dy dz .. 1 presented by — , — , —, respectively. Hence, since the acceleration of motion in any direction is measured by the rate of change of the velocity in that direction, we have for the accelerations parallel to the axes of coordinates the expressions or (7) Total Acceleration. 17 where, in accordance with Newton's notation, x, y, z denote the accelerations parallel to the axes of x, y, z, respectively. The total acceleration of the motion of the point is the resultant of these accelerations. It is plain that this acceleration is independent of any previously existing velocity, which may or may not be in the same direction. The question of acceleration in curvilinear motion can also be treated in another manner, as follows : — 24. Curvilinear Motion, Change of Velocity, Total Acceleration. — Suppose a point to move in a curvilinear path, and from any point let the line OA be drawn, representing in magnitude and direction the velocity of the moving point at any c B instant. Let OB, in like manner, / ^^^^/ represent its velocity at the end / ^^-^^ of the interval of time At. Join ^^— ' AB, and complete the parallelo- gram OABC. Then the velocity represented by OB is equi- valent to the component velocities represented by OA and OC; but if the velocity of the point had not changed during the interval At, it would have been represented by OA ; hence OC, or AB, represents in magnitude and direction the change of velocity in the time At. Again, since the acceleration of the velocity of a mov- able, at any instant is, in all cases, measured by the rate of change of the velocity for that instant, it follows, as in (5), that if we regard the interval of time At as becoming infinitely small, the acceleration of the motion is represented by the AB limiting value of — — . This limiting value is called the total At acceleration of the motion of the particle at the instant. 25. Tangential and Normal Accelerations. — Again, suppose a to denote the position of the moving point at the end of the time t, and b its position after a small interval of time, At, and draw tangents to the path at the points a and b. Also, as before, from any point draw OA, OB parallel to these tangents, and representing the velocities 18 Acceleration. at a and b, respectively. Then, by the preceding Article, AB represents the total change in the velocity in the interval At. Draw AN perpendicular to OB, and suppose the velocity AB resolved into the two, AN and BN; then, the former re- presents the resulting change of velocity in the normal direction, and the latter in the tangential. The corresponding accelerations are represented by the AN BN limiting values of — — and — , respectively. Again, let the angle BOA, or the angle between the tangents at a and b, when indefinitely small, be denoted by d(p, and we have AN = OAdcj> = vd•' _ = constant ; from which we infer that the radius vector cit describes equal areas in equal times round the point 0. Equations (11) and (12) above can otherwise be obtained with great facility by a method analogous to that employed in Art. 25. 29. Areal Velocity, Areal Acceleration. — It is obvious, geometrically, that rdd represent double the area 22 Acceleration. described by the line OP in the time dt ; consequently —r— represents the rate of increase of double the area de- 2 fjfi scribed by the point P round the point 0. Hence -£- —r— ctt is called the areal velocity of the point P relative to the origin 0. Similarly \ — ( r 2 — ) represents the areal accele- dt \ dt t ration of P relative to the same origin. 30. Moving Axes. — In some cases it is necessary to- refer the motion of a point in a y plane to rectangular axes, which are themselves in motion. Thus let OX, OF be two fixed rect- angular axes in the plane, and OM, ON be two moving axes. Let P be any point in the plane ; then £ = OM, rj = ON, where t and r) are the coordinates of P, relative to the moviDg axes. rid Also, if 9 = L XOM, we have w = — , the angular velocity at of the moving axes. Then the motion of P is got by com- pounding the motions of M and N. dl Now, by Art. 27, the components of the velocity of M are — along OM, and w£ along MP. Likewise, the components CITi for N are — along ON, and - wy] along NP. at Hence, if u and v denote the components of the velocity of P, relative to the moving axes, we have dl dt I)bJ dx) (13) Again, by Art. 28, the acceleration of M along OM is- Units of Time and Space. 23 -~ - w% and that along MP is p — (w£ 2 ) ; with similar ex- at s wt pressions for the accelerations of N. Hence, finally, we get d 2 £ 1 d acceleration parallel to 0M= — - w 2 £ - - -r. (wrj 2 ) ; (14) acceleration parallel to ON = — - w 2 r] + ^-r (w£ 2 ) • 31. Units of Time and Space. — With respect to the units of time and space, as well as of all other quantities, it should he remarked that the units assumed must in all cases he finite magnitudes. For instance, the unit of time may be taken as a second, an hour, a day, or any other finite interval of time, but it should never be assumed to be an indefinitely small portion of time ; for if so, numbers which represent finite intervals of time become infinitely great, and accord- ingly arguments based on such an assumption become illusory and unmeaning when applied to finite intervals of time. This remark is requisite, as fallacious proofs are sometimes given in books on dynamics from overlooking this obvious principle. The unit of time most universally adopted is a second, as already stated. Different units of length prevail in different countries. Since in this country the foot is the standard of length, and areas and volumes are each referred to units of their own, we shall sometimes employ such units for the purpose of illustrating mechanical principles by familiar examples. But, when desirable, we shall avail ourselves of the metric system. In it the unit of length is a metre (3-2809 feet, or 39-37079 inches). From this, by the simple processes of squaring and cubing, units of area and volume are derived ; and decimal multiples and submultiples are respectively indicated by the use of Greek and Latin pre- fixes. For example, the centimetre is the hundredth part of the length of a metre. Again, one cubic decimetre is the measure of capacity called a litre, and is about 61 cubic inches, or 1-76 pints. We shall subsequently see that a cubic centimetre of distilled water at its greatest density 24 Acceleration. furnishes this system with another unit : to this the name gramme is applied. One thousand grammes are called a kilogramme, equivalent to about two and one-fifth pounds avoirdupois. It should also be observed that in the numerical expres- sion for an acceleration there is a double reference to the unit of time; so that, in strict accuracy, what we have called an acceleration of 7 feet per second should be called an acceleration of 7 feet per second per second. This mode of expression is, however, cumbrous, and quite unnecessary, since in ordinary language, as well as in mathematical de- ductions, it is assumed that velocities and their rates of change are referred to the same unit of time, unless the contrary be stated. First Law of Motion. 25 CHAPTEE III. LAWS OF MOTION. Section I. — Rectilinear Motion. 32. Motion in relation to Force. — In the preceding Chapters motion has been considered from a purely kine- matical point of view ; we now proceed to consider it in connexion with the force or forces by which it is produced. The science of Rational Dynamics is usually founded on three principles, or Laws of Motion, which have been stated in their simplest form by Newton, and are fully verified by their agreement with experience. In the present Chapter it is proposed to discuss and illustrate various cases of applica- tion of these Laws, chiefly when the forces supposed to act are constant both in direction and magnitude. The discus- sion of motion produced by varying force will be dealt with subsequently. We follow Newton's method, commencing with the statement of his First Law. 33. First law of Motion. — A body continues in its state of rest, or of straight uniform motion, except in so far as it is compelled to alter that state by impressed force. This law asserts that a body has no power or tendency in itself to alter either its velocity or the direction of its motion : this is usually called the Law of Inertia of Matter. Hence, if a body be conceived to be set in motion, and no external force act upon it afterwards, it should continue to move indefinitely in a right line with a uniform velocity. Conversely, if a body be in a state of uniform rectilinear motion, we infer that the forces which act on it are in equili- brium. For example, if a train be in a state of uniform motion on a horizontal railway, we infer that the force arising from the action of the steam is exactly equal, and opposite to, the entire resistance arising from friction and resistance of the air. 26 Rectilinear Motion. Hence, all questions of uniform rectilinear motion may be regarded as problems of equilibrium, and treated by the principles arrived at in Statics. In all applications of the Laws of Motion to a body of finite dimensions, the only motion considered in this Chapter is one of pure translation. Again, if the motion of a body be not uniform, or not rectilinear, we infer that it must be acted on by some ex- ternal force or forces. The connexion between the motion produced and the force which produces it is contained under the next Law. Example. A railway train is moving with constant velocity along a horizontal rail- road. The resistance from friction, &c, for each carnage is one-hundredth part of the pressure. Find the tension of the couplings of the last carriage, if its weight he four tons. In this case, since the motion is uniform, the tension of the couplings must be equal to the resistance to be overcome, or to the one-hundredth part of four tons, i.e. 89f lbs. 34. Second Iiaw of Motion. — Change of motion* is proportional to the impressed motive force, and takes place in the right line in which that force is impressed. As this statement is very comprehensive, it will be^ neces- sary to dwell on it with some detail, commencing with the case of a body under the influence of a force which acts uni- formly and in the same right line during the motion. The body is supposed, in the first instance, to start from rest, and the direction of the force to pass constantly through its centre of mass, in which case the motion is one of translationf solely. * For the present we shall consider that it is one and the same body which is acted on by forces passing through its centre of mass, in which case the force varies directly as the velocity generated in the unit of time. We shall subse- quently treat of the case where the mass acted on varies also. In that case, by the word "motus," here translated motion, we must understand quantity of motion. t A force applied at the centre of mass of a rigid body is equivalent to an indefinite number of equal and parallel forces applied to the several equal particles of which the body is conceived to be constituted ; but as the forces are equal, and the masses moved by each are equal, the velocities generated, in the same time, are also equal : hence the motion of the entire body is one of pure translation. The simplest case of this is that of bodies falling under the action of the force of gravity. Velocity Generated. 27 35. Velocity Generated. — Suppose a force to act uni- formly on a body, and let /denote the velocity generated at the end of the first second (taken as the unit of time) , then during the next second, in accordance with our law, the uni- form force will generate an additional velocity of the same amount/; and in each successive second the force generates the same additional velocity ; consequently the motion is in this case uniformly accelerated, and the velocity at the end of t seconds (Art. 17) is given by the equation v -A Again, if the body be supposed to start with the velocity «7 in the direction in which the force acts, we shall have for the velocity v, at the end of the time t, V = V +ft, (1) as in Art. 17. If the force act in a direction opposite to that of the motion it is called a retarding force ; which, if uniform, will diminish the velocity by the quantity / during each second, and we shall have, as before, the equation V = r -ft. The student should bear in mind that / in all cases is measured by the velocity generated or destroyed in the movable in each second during the motion ; /consequently may always be regarded as an acceleration — a retardation being considered as a negative acceleration. It may be observed that the entire reasoning in this Article depends on the following principle — contained in the Second Law of Motion — that the change of velocity produced by a force in any time is independent of the previous velocity of the movable. The Second Law of Motion equally applies to the case of a body acted on by any number of forces, in which case it may be stated as follows : — If any number of forces act simultaneously on a body, then, during any instant, each force produces the same change of motion in its own direction as if it had acted singly on the body. Prom this it follows that forces are compounded in the 28 Rectilinear Motion. same manner as velocities. The law of the composition of forces was thus establishad by Newton — Leges Mot us, Cor. 2. 36. Space described in any Time. — Since we have seen that in the case of a uniform force the velocity is uni- formly accelerated or retarded, we can at once apply the re- sults already arrived at in Arts. 18, 19. Hence, the space described from rest, in the time t, is given by the formula s = i/t\ (2) If the body start with an initial velocity v along the line in which the force acts, we shall have 8 = V t ± \ft\ (3) in which the upper or lower sign is taken according as the uniform force acts in the same or the opposite direction to that of the initial velocity. It is plain that the space described in the first second from rest is J/, or half the velocity acquired at the end of the second; and, in general, the space described in anytime from rest is half of that described by a body moving uniformly with the velocity acquired at the end of the time. 37. Relation between Velocity and Space de- scribed. — If the body start from rest, by eliminating t between the equations v =ft and s = ^ft 2 , we get v-=2fs; and, more generally, if v be the initial velocity, v*=v*±2fs. (4) From the preceding results it is seen that the question of rectilinear motion under the action of a constant force is com- pletely solved whenever the value of the acceleration /can be determined. In a subsequent Article we shall show how this can be done in elementary cases, but before doing so we pro- ceed to apply the preceding results to the important case of falling bodies. 38. Vertical Motion. — In order to get rid of the re- tardation caused by the resistance of the air, we shall sup- Vertical Motion. 29- pose the motion to take place in a vacuum. Under these circumstances it is found that all bodies, no matter what their density or chemical constitution may be, fall through the same vertical height and acquire the same velocity in the same time. That this is so is best established by means of pendulum experiments ; but it can also be tested by allowing different bodies to fall in an exhausted receiver. We hence infer that the attractive force of the Earth acts equally on all bodies. If g denote the acceleration due to the force of gravity, that is the increment of velocity per second acquired by a body falling in a 'vacuum, then, from what has been stated, the value of g is the same for all bodies at the same place on the Earth's surface. Again, since at any place the force of gravity may be assumed as a constant force (t. e. within moderate distances from the Earth's surface), we may apply to the case of falling bodies the results arrived at in the preceding Articles by sub- stituting g in place of/. Hence, if the body start from rest, we have • = 9t> 8= \gt\ e = 2gs. (5) Again, if it start downwards with a given vertical velo- city 0„, v = Vo + gt, s = v t + igt 2 , v 2 = ev 8 + 2gs. (6) If the body be projected vertically upwards with a velocity Vo, gravity becomes a uniformly retarding force, and we have v = Vo- gt, s = vf - igt 2 , v 2 = v 2 - 2gs. (7) To find in this case the height H to which the body would ascend, we make v = in the last equation, and we get *-t (8) The time T of ascent is given in like manner by the equation r=-°. (9) 9 The subsequent motion of the body is got from equations (5), in which we suppose the body to start from rest at the 30 Rectilinear Motion. height H. It immediately follows that the times of ascent and descent are equal, and that the body returns to its ori- ginal position with the velocity with which it was projected upwards. For this reason we say that the velocity v is due to the height H; and reciprocally, that the height H is due to the velocity v . We shall meet frequent applications of these expressions. As the motion is supposed to take place in a vacuum, the preceding results can only be regarded as approximate for motion in the air. 39. Variation of Gravity. — It is found that the value of g varies, within small limits, from place to place on the Earth's surface. It increases with the latitude, and when referred to feet and seconds, has its least value, 32*091, at the equator, and its greatest, 32*255, at the pole. It also diminishes as the body is raised above the Earth's surface, since the attraction of the Earth varies as the inverse square of the distance from its centre. The value of g at London, referred to the same units, is 32*19, and this may be em- ployed, in ordinary calculations, as an average value. It will be seen subsequently that the rotation of the Earth on its axis has the effect of diminishing the velocity of a falling body ; and, accordingly, the observed value of g is the difference between its value arising from the Earth's attraction and the component of the centrifugal acceleration in the vertical direction. As a rough approximation we may assume g = 32 ; and, when numerical results are required, this may be taken as its value in these and all subsequent examples, unless otherwise specified, inasmuch as they are given chiefly for the purpose of familiarizing the student with the application of mechani- cal principles. Examples. 1. Find the velocity acquired in 5 minutes by a falling body, assuming ff = 32-19. Arts. 9657 feet. 2. In what time will a falling body acquire a velocity of 400 feet per second if it start from rest ? Ans. 12-5 sec. 3. If a body move under the action of a constant force, its average velocity during anytime is an arithmetical mean between its velocities at the commence- ment and the end of that time ? Examples. 31 4. If one minute be taken as the unit of time, what should be taken as the value of g ? Ans. The velocity per minute acquired in one minute by a falling body, or 115,200 feet. 5. Two bodies start together from rest, and move in directions at right angles to each other. One moves uniformly with a velocity of 3 feet per second ; the other moves under the action of a constant force : determine the acceleration due to this force if the bodies at the end of 4 seconds be 20 feet apart. Ans. 2 feet per second. 6. If a uniform force generate in a body a velocity of 30 feet a second after •describing 25 yards, find the acceleration. Ans. f= 6. 7. A stone is let fall from a height into a well, and is heard to strike the water after t seconds ; find the depth of the well; assuming the velocity of sound to be V, and neglecting the resistance of the air. ^/S The required height h is got by solving the equation ~\ 9 h In applying this equation practically, it may be observed that — is, in all cases, small in comparison with t : accordingly, if we transpose and square, we get, neglecting — in comparison with — , 2(V+gt) 8. A person drops a stone into a well, and after three seconds hears it strike ^/ the water. If the velocity of sound be 1127 feet per second, find the depth of the water. Ans. 132-68 feet. 9. Prove that the spaces described by a falling body in successive equal y intervals of time are proportional to the series of odd numbers. 10. A body moves from rest under the action of a constant force during four seconds, when the force is supposed to cease ; in the next five seconds the body describes 200 feet; find the acceleration due to the constant force — (1) if one second ; (2) if one minute be taken as the unit of time. Ans. (1) 10 ; (2) 36000. 11. A body is projected upwards with any velocity, and t, t' denote the times in which it is respectively above and below the middle point of its path ; v ■< find the value of |. Ans ' ^ 2+1 ' 12. Assuming g to be represented by 32 when the units of space and time are one foot and one second; what number would represent its value if one mile and one day be taken as the units ? Ans. 45242181&. " « 13. A ball is dropped from the masthead of a ship sailing n miles an hour. Through how many feet must it have fallen when the direction of its motion is inclined at 45° to the horizon ? . 121 " 2 *"*' 3600* 32 Rectilinear Motion. 40. Acceleration Varies as Pressure. — If we sup- pose different forces to act uniformly during equal times on the same body, it follows from the Second Law of Motion that the forces will be to one another in the same ratio as the velocities generated in equal times. If we suppose the time of action to be one second, the velocities generated are represented by the corresponding accelerations / and f. Also, if F, F' denote the statical* measures of the forces, i. e. the total pressures which they are capable of producing, we have F\F'=f:f. (10) If one of the constant forces be the attraction of the Earth, since its statical measure is W, or the weight of the body moved ; and since g is the corresponding acceleration, we have F:W=f:g; (11) F hence ^ = W ' ^ This equation enables us to determine the velocity generated in one second by a constant force at any place whenever the pressure F which measures the force is known, and also the weight of the body. We suppose, as stated already, that the body is rigid, and that the force F acts through its centre of mass. When /has been determined by the foregoing equa- tion, and the force continues to act uniformly, we may apply the results arrived at in the preceding Articles to determine the subsequent motion (see Art. 37) . 41. Mass. — Our ordinary experience suggests to us that the amount of the acceleration produced in a body by a force depends not only on the magnitude of the force but also on the body which is moved. When exact experiments are carried out it is found that the same force acting on different * The magnitude of a force is estimated in Statics by the weight which it is just capable of supporting. Thus, a force which is capable of supporting a weight of 112 lbs. is called a force of 112 lbs., &c. Mass. 33 bodies produces different accelerations, and that different forces acting on the same body produce accelerations pro- portional to the forces. Hence we conclude that the accelera- tion produced in the motion of a body by a force is equal to that force multiplied by a factor which is invariable for the same body, but which varies for different bodies. Conversely, if F denote the magnitude of a force, and / that of the acceleration thereby produced, we have the equa- tion F= mf y (13) where m is always the same for the same body, but varies for different bodies. This quantity m is called the Mass of the body, and is estimated, like other quantities, by comparing it with a standard quantity of the same kind. It is found that at any fixed place on the Earth's surface the weight of a body (if permitted to accelerate its motion) produces an ac- celeration which is the same for all bodies (Art, 38). Now W being the weight and g the acceleration thereby produced, we have as above W-mg\ but g is the same for all bodies at the same place, hence W is proportional to m ; or, in other words, if there be two bodies whose weights are W, W, and whose masses are m. m', we have ==, = — . Hence, in order W m to find the ratio of the masses of two bodies, we have only to find the ratio of their weights at the same place. Examples. 1. A uniform pressure of 6 lbs. is applied in a horizontal direction to a body of 10 lbs. mass placed on a smooth horizontal table. Find — (1) the velocity gene- rated in one second ; (2) that acquired after describing 500 yards along the plane. and, if T be the time of descent for AB, we have, by (15), AB A Ti i a T 2 —— • T 2 -2 AC ' hence w/ (17) where a denotes the radius of the circle. Hence, the time down any chord such as AB of this circle is constant. It can at once be seen, in like manner, that the time of descent down BC has the same value. 45. Line of Quickest Descent to a Circle. — To find the right line down which a particle under the action of gravity would descend in the shortest time from a given point to a given vertical circle. d2 36 Rectilinear Motion. Draw AC, the vertical diameter of the circle, and join OC, meeting the circle in B, then OB is the line of quickest descent in question. For, join AB, and pro- duce it to meet the vertical drawn through in D. Then it is ohvious that the circle described on OD as diameter touches the given circle in B ; consequently the time of descent down OB being the same as that down any other chord of the circle OBD, drawn from 0, is less than the time down any other right line drawn from to meet the circle ABC. The preceding method of investigation applies equally if the point lie inside the given circle. 46. Iiine of Quickest or Slowest Descent to any Curve. — It is easily seen from the preceding Article that the determination of the right line of quickest or slowest descent to any given vertical curve from any point in its plane re- duces to the problem of drawing a circle, touching the given curve and having the given point for its highest point. The problem admits also of being treated by the ordinary method of maxima and minima, as follows : Suppose the curve referred to polar co- ordinates, the given point being taken as pole, and the vertical OD through it as prime vector ; then, if t be the time of de- scent down any radius vector OP, we have r-igfcosO, out ■h 2r cos 6 Accordingly, the time t is a maximum or a minimum when r . COS0 is a maximum or a minimum. u = To find the maximum or minimum values, assume „ ; then since — . = 0, we have cos cW cos — , + r sin = 0. dU (18) Line of Quickest Descent. 37 The solutions are obtained by combining this equation with that of the curve. To distinguish between the maximum and minimum solutions, we proceed to differentiate the equation cos 9 -TTj + r sin die ad eld cos 2 dr observing that, in this case, cos 9 —„ + r sin 9 = 0. Hence [Biff. Calc, Art. 138), t is a minimum or a maximum according as r + — is positive or negative. These results can be readily verified from geometrical considerations. Examples. 1. If the hypothenuse of a right-angled triangle be placed in a vertical position, prove that the times of descending from rest will be the same for each, of its sides. 2. Prove that the velocity acquired down any chord, terminated at the lowest point of a vertical circle, is proportional to the length of the chord. 3. If the length of an inclined plane be 150 yards, and its inclination 30*, what velocity would a body acquire in descending it ? Am. 40 yards per second. 4. A body slides down a smooth inclined plane of given height; prove that the time of descent varies as the length of the plane. 5. Find the inclination of a plane, of given length I, so that the velocity acquired in moving down it shall be of a given amount V. . . _ V 2 JX'tlS. Sill t — T La 6. Given the base a of an inclined plane, find its height so that the hori- zontal velocity acquired by descending it may be the greatest possible. Ans. h=a. 7. Find the gradient in a railway so that a carriage descending the plane by its own "weight may move through one quarter of a mile in the first minute ; and find how far the carriage will move in the next minute, friction being neglected. (1) sin t= 4^0; (2) f of a mile. 8. A body is attached by a string to a point in a smooth inclined plane, on which it rests : if it be projected from its position of rest up the plane with a velocity just sufficient to take it to the highest point to which the string allows it to go, find the time of its motion. I j Ans. t = 2 / — : — , the length of the string being /. yjff sin . y / 38 Rectilinear Motion. 9. A groove is cut in an inclined plane, making an angle a with the inter- section of the plane and the horizon. If a heavy particle he allowed to descend the groove (supposed smooth) , prove that its acceleration is g sin i sin o ; where i denotes the inclination of the plane. 10. If two vertical circles have a common highest point, then if any line he drawn from that point, the time of descending the portion intercepted between the circles is constant. 11. Find the right line of quickest descent from a point to a given right line lying in the same vertical plane as the point. 12. Find the right line of quickest descent from a given right line to a given vertical circle. 13. Find the lines of quickest and slowest descent between two vertical circles which lie in the same plane. 14. A parabola whose latus rectum is p is placed in a vertical plane, with its axis horizontal. Find the inclination of the normal terminated by the axis down which a particle would descend in the shortest time, and find the time of its descent. • j^Z Am. i = 45°, time ; 9 ■ 15. Find the latus rectum of a parabola, so that when it is placed in a ver- tical plane with its axis horizontal the least time in which a particle falls from rest down a normal from the curve to the axis may be one second. 16. Prove that the chords of quickest and slowest descent from the highest or to the lowest point of a vertical ellipse are at right angles to each other, and parallel to the axis of the curve. 17. Show immediately, from equation (18), that the right line of quickest descent from a given point to a given curve makes equal angles with the nor- mal at its extremity and the vertical ; and verify the result geometrically. 18. An ellipse is placed with its major axis vertical ; find the semi-diameter along which a particle will descend in the shortest time possible from the cir- cumference to the centre. Am. It makes with the axis major the angle sec-^V^), where e is the eccentricity. If e < — , the line of quickest descent is the axis major. 19. An ellipse is placed with its major axis vertical ; find the line of quickest descent from the upper focus to the curve. Am. It makes with the axis major the angle cos" 1 — . If e<|,the axis major is the required line. 20. AB is a quadrant of a circle whose centre is 0, the radius OB being horizontal; C is a point on the quadrant, and the angle BOG—d. Show that the time of falling from A to C is to that of falling from C to B as Jcos?toJ^f. Parabolic Motion. 39 Section II. — Parabolic Motion; 47. Path of a Projectile. — We have hitherto considered rectilinear motion ; we now proceed to the case of a body projected in any direction, and acted on only by the force of gravity, which is supposed to be uniform. In this case it is easily shown that the path* described by the projectile is a parabola. For, suppose a body projected from with the velocity V, in the direction OX, and draw OY xt ^x vertically downwards. Let ON he the space which the body, moving with the velocity V, would describe in t seconds ; then, if no force were to act on the body, m N would represent its position at the end of that time. Again, as the force of gravity acts in the direction OY, it will produce its effect in that direction, by the Second Law of Motion, independently of the previous velocity of the body : i. e. it will produce the same effect as if the body fell freely from rest. Measure off, accordingly, OM=\gf ; then OM represents the space moved through in the vertical direction in the time t. Complete the parallelogram OMPJSf, and by the combined effect of the two motions P will be the position of the projec- tile at the end of the time t. Let x y y be the co-ordinates of P referred to the axes OX and OY, and we have x = ON=Vt, y=OM=±gt\ If t be eliminated between these equations, the equation of the path described is Q ~ 2 *•-—*. (i) * As before, by the path described by a body we understand the path de- scribed by its centre of mass. 40 Parabolic Motion. This equation represents a parabola, touching OX and having its axis vertical. If H be the height due to the velocity V (Art. 38), the equation of the parabola becomes x* = ±Hy. (2) 48. Construction for Focus and Directrix. — From the preceding equation it follows (Salmon's Conic Sections, Art. 214) that H is the dis- tance of from the focus of the parabola, and also from its directrix. Hence, if OD be measured vertically upwards equal to IT, and DR drawn in Q a horizontal direction, the line DR will be the directrix of the parabolic trajectory. Also if OF be drawn through 0, making the angle XOF equal to the angle XOD, and if we take OF = OB ; then F will be the focus of the trajectory. Hence, as the focus and directrix of the curve are known, it is completely determined. Again, the velocity at any point in the trajectory is equal to that which the body would acquire in falling from the direc- trix. We have seen that this property holds good for the point of projection : moreover, after passing through any point the body will move in the same path as if it had been projected from that point, in the direction and with the velocity that it has at the instant; therefore the property is true for any point in the path. Hence, whenever the velocity at any point is given, the position of the directrix is completely determined. Definition. — The angle which the direction of projection makes with the horizontal line is called the angle of elevation of the projectile. 49. Horizontal Range and Time of Flight. — Let R be the point in which the projectile strikes the horizontal plane through 0; then OR is called the horizontal range, Range and Time of Flight for an Oblique Plane. 41 and the time Tof describing the corresponding path is called the time of flight. Through R draw RQ in the vertical direction. Let OR = R, L QOR = e ; then we have OQ = FT, QR = \gT\ But QR = OQ sin e ; hence we get 2 Fsin e T = (3) V 2 . Also R = OQ cos e = FT cos e = 2 — sin e cos e therefore R = 2Hsm2e. (4) If F be given, the horizontal range is the greatest when sin2 = Fcos e, and v sin

or AB . cot = PZ> cot A- AD cot P. Again, if a and ]3 be the angles e which CD makes with AC and BC respectively, we have AB cot 9 = AD cot a - PP cot j3. This follows at once by drawing AE parallel to BC, and applying the preceding result. 56. Being given the direction and the velocity of projection, to find the velocity with which a projectile would strike an oblique plane, and also the direction of its motion at the instant of impact. Let i be the inclination of the plane to the horizon ; then, by the preceding lemma (see figure on last page], cot a - cot a = 2 tan i. (11) Hence, the angle a is determined from the known angles a and i. 46 Parabolic Motion. v sin o Again, since v sin a = v sin a', we have which determines v'. sin a If the projectile impinge at right angles on the plane, we have a = 90° ; therefore cot a = 2 tan t, which determines a, or the corresponding angle of elevation. Also the velocity with which the projecticle strikes the plane is v sin a in this case. 57. motion on a Smooth Inclined Plane. — In our discussion of motion on an inclined plane in Art. 42 the movable was supposed to start from rest : in this case the motion is rectilinear. It is also rectilinear if the initial motion has place in the direction of the line of greatest slope in the plane. But when the body is projected along the plane in any other direction the problem is the same as that pre- viously discussed, namely, the motion of a projectile acted on by a constant force, parallel to a given direction. Its path along the plane is, accordingly, a parabola ; and its axis is in the direction of the line of greatest slope. 58. Morin's Apparatus. — "We conclude with a short description of the apparatus, designed by Poncelet, and con- structed by Morin, for experimentally exhibiting the laws of falling bodies. A cylinder is made by clock-work mechanism to revolve around a fixed vertical axis. A weight is suspended at the summit of the cylinder close to the outer surface and between two vertical guides. "When the rotation has become perfectly uniform, the weight is allowed to fall. A pencil, attached to the falling weight, is so arranged as to trace a line on a sheet of paper, which is wrapped tightly around the revolving cylinder. When the paper is taken off and unrolled on a plane surface, the curve traced on it by the pencil is found to be a parabola. That this curve is a parabola, may be shown in the follow- ing manner : — Let GP'P represent the curve traced out by the pencil. Monti's Apparatus. 47 Draw the tangent GL to the curve at the initial point G, and at any point P draw the tangent PL, and erect LF perpendicular to it at the point L. Make a corresponding construction for the other points on the path ; then the lines LF, L'F, &c, are all found to intersect in a common point F. This is a characteristic property of the parabola which has its focus at F, and its vertex at G. Having found the curve to be a para- bola, we can show that the motion of the weight has been uniformly accelerated. Let PM, FN be the coordinates of P, referred to the axes GL, GF, then if t denote the time in which the moving weight arrived at the position P, the line PM will be equal to the arc of the point on the circumference of the cylinder has rotated in the time t. Let V denote the constant velocity of any point on the circumference of the cylinder, and we get PM = Vt. Again, from the property of the parabola, circle through which a Accordingly, PM 2 =4:FGxMG t PM 2 V 2 MG= 4FG = ±FG t2 '> but MG is the space through which the weight has descended vertically in the time t ; hence the spaces described by the falling body vary as the squares of the times ; its motion consequently is uniformly accelerated. Comparing with the equation s = J gf, we get g = ; that is, the distance of the focus of the parabola from its summit is equal to the height due to the velocity of a point on the surface of the rotating cylinder. The student can easily prove that the parabola described is the same as that of a body projected horizontally from a point with the velocity V. 48 Parabolic Motion. 59. In the preceding investigations we have neglected the effects of the resistance of the air. When this is taken into account the problem becomes one of great uncertainty, arising from the law of resistance of fluids not being accurately known, and from the difficulties still remaining in the integration of the equations of motion, when the law of resistance is assumed. The most generally received theory is that the resistance of fluids is proportional to the square of the relative velocity of the fluid and the movable. When the resistance of the air is taken into account, it is easily shown that the preceding results are not even approximate in cases of high velocity ; such, for instance, as shot and shell projected by artillery. Examples. 1. Determine the elevation of a projectile, so that its horizontal range may be equal to the space to be fallen through to acquire the velocity of projection. Am. e = l5°. 2. If a number of particles be projected simultaneously from the same point with a common velocity, but in different directions, prove that at any subse- quent instant they will all be situated on the surface of a sphere. 3. Given the horizontal range and the time of flight of a projectile ; find its initial velocity and angle of elevation. 4. If a body be projected obliquely on a smooth inclined plane, the path in which it moves will be a parabola. Find the position of the focus and directrix of the parabola when the initial velocity and direction of motion are given. 5. Given the velocity with which a shot is projected from a certain point ; find the locus of the extremities of the maximum ranges on inclined planes pass- ing through that point. 6. If a body be projected with a velocity of 100 feet per second from a height of 66 feet above the ground, in a direction making an angle of 30° with the horizon ; find when and where it will strike the ground. Am. Time = 4£ sec. Range = 357*23 feet. 7. If A, B be two points on a parabolic trajectory, prove that the time of passage from one to the other is proportional to tan

the acceleration down the plane is^(sin» cos i). 7. A body slides down a rough roof and afterwards falls to the ground : find the whole time of motion. Momentum. 53 8. Several bodies start from the same point and slide down different inclined planes of the same roughness : find the locus of their positions after the lapse of .s a given time. Find also the locus of the positions arrived at with a common ^ velocity. "V" 1 t - , 9. A rough plane makes an angle of 45 3 with the horizon ; a groove is cut in the plane making an angle o with the intersection of this plane and the horizon- tal plane ; if a heavy particle he allowed to descend the groove from a given U^ height h find the velocity with which it reaches the horizontal plane. Ans. ffr*(«jn«-#Q. yf sin a 10. A body moves from rest down an inclined plane whose inclination is 30°, and limiting angle of resistance 15°: find the velocity acquired if the length of the plane be 200 feet. Here v 2 = 400# tan 15°; therefore v = 58-56 feet per second. 11. A railway train is moving up an incline of 1 in 120 with a uniform velocity. Find the tension of the couplings of the carriage which is attached to the engine ; assuming the weight of the train (exclusive of the engine) to be 80 tons, and the friction 8 lbs. per ton. Ans. 19cwt. 5^ lbs. 12. In the same case, if the acceleration of the train be 2 feet per second, find the tension of the couplings. / Here we must add to the preceding W -, i. e. 5 tons ; and the entire tension is nearly 6 tons. Section IV. — Moment urn. 63. Force measured by Quantity of motion gene- rated in Unit of Time. — The product of its mass and the velocity which a body has at any instant is called its quantity of motion or momentum at that instant. Accordingly we con- clude, from equation (13), Art. 41, that F varies as the quantity of motion it can generate in one second (taken as the unit of time), the force being supposed to act uniformly during that time. Again, since the velocity (g) which gravity can produce in one second is the same for all bodies, the quantity of motion gravity can generate in one second in a falling body of mass m is represented by mg ; hence, in this case, we have W = mg ; in which the units of mass and weight are connected in such a manner that when one is fixed the other is also determined. 54 Momentum. 64. Absolute Unit of Force. — In accordance with equation (13), Art. 41, the unit of force is defined as the force which, acting uniformly during the unit of time on a unit of mass, produces a unit of velocity. This is called by Gauss the absolute unit of force. The most convenient unit of mass in the British Isles is the mass contained in one standard pound avoirdupois. Hence, adopting as before a second as the unit of time, and a foot as the unit of length, the absolute unit of force is that which, acting during one second, would produce in a standard pound mass a velocity of one foot per second. This unit of force is sometimes called a poundal. Hence, if g = 32*19 with reference to the preceding units, the unit of force is 32 T 19 part of the attraction of the earth, at London, on a standard pound ; i. e. about half an ounce, approximately. In the metric system the force which in one second would generate a velocity of one centimetre per second in a gramme of matter is called a dyne. Hence, since 1 lb. = 453*6 grammes, and 1 foot = 30*48 centimetres, one poundal is approximately 13825 dynes. 65. Gravitation Units of Force and Mass. — In practical questions concerning bodies on the earth's surface, it is in general more convenient to measure forces by weights, and to speak of a force of so many pounds weight. In this system the unit of force is the weight at some definite place (London) of the pound mass ; or of a kilogramme when the metric system is taken. This is called the gravitation or statical measure of force ; and since the unit of force in this system, acting on one pound mass for one second, produces a velocity of 32*19 feet per second, we see that this unit is 32*19 times the absolute unit. Moreover, since the weight of a body varies, within certain small limits from place to place (Art. 38), when scientific accuracy is required we must correct for the change in the value of g due to any difference in altitude or latitude from those of the place to which the standard was originally referred. In practice this correction seldom requires to be taken into account, as the variation in the value of g is generally too small to aifect the result appreciably (Art. 39). Tuo Classes of Forces. 55 Examples. 1. An ounce being taken as the unit of mass, a second as the unit of time, and an inch as the unit of length, compare the unit of force with the weight of one pound. Here the unit of force is that which in one second would generate a velocity of one inch per second in an ounce mass ; and therefore is — — -— pait of the weight of one pound, or 1*25 grains. 2. Determine the unit of time in order that g may he expressed by unity when the foot is the unit of length. Ans. - V2 seconds. 8 3. Find the units of space and time in order that the acceleration of a body falling in vacuo, and the velocity it acquires in one minute, may respectively be the units of acceleration and of velocity. 66. Two Classes of Forces. — There are two classes of forces to be considered in Dynamics : one, such as gravity and those hitherto discussed, which require a finite time to pro- duce a finite change of velocity. Forces of this class, when uniform, are, as has been stated, measured by the change produced in one second (taken as unit of time) in the mo- mentum of the body acted on. There is another class, called ordinarily impulses, such as blows, sudden impacts. &c, which act only during a very short time, but are capable of pro- ducing a finite change of velocity in that time. These are sometimes called instantaneous forces ; it is ne- cessary, however, to observe that force in all cases requires some time to produce its effects, though that time may be exceedingly small. In fact, we cannot conceive that a force could produce any change in the velocity of a body if its time of action were absolutely nothing. Forces of the former class are frequently styled finite or continuous forces, to distinguish them from the other class, namely, impulsive forces. It should be observed that whenever both impulsive and finite forces act at the same time on a body, the latter may in general be neglected in determining the motion at the instant; since the effects produced by them, in the time during which the impulsive forces act, are so small that they may be neglected in comparison with the effects of the im- pulses. 56 Momentum. 67. Impulses. — The measure of an impulse, i.e. of the entire action of a force of great intensity, which acts during a very short time, and then ceases, is the whole change in the quantity of motion which it communicates to the body on which it acts. We may here observe that, if .Fbe the instantaneous value of an impulsive force, and r the time of action, the whole impulse is represented by ( Fdt, in which, as already observed, Jo t is a very small interval of time. 68. General Equations of Motion of a Particle. — Suppose that the force F acts as before in the line of motion of the mass acted on, but that it varies continuously, then we may consider that in an indefinitely small portion of time its intensity is unaltered. The variable acceleration/, caused by it, is determined by the equation F= mf: hence, as in Art. 21, we have at any instant dv d-s .. F = mf = m — = m -— = ms . (6) at dv Hitherto the motion has been supposed rectilinear. In the case of curvilinear motion the last equation expresses the tangential component of the force, and it can be similarly seen (Art. 25) that the normal component is expressed by v 2 m — . We now proceed to consider the motion of a particle P of mass m, under the action of any forces. If the particle be referred to a system of rectangular axes in space, and x, y, z, be the coordinates of its position at any instant, i. e. at the end of the time t, reckoned from any fixed instant, the com- ponents of its velocity parallel to the axes of coordinates are (Art. 12) represented by x, i/ } z. Resolve the whole force acting on the particle at the instant into three components, parallel to the axes of x, y, z, respectively ; and let these components be represented by X, Y, Z; then, since by the Second Law of Motion each of these forces produces its change of velocity in its own General Equations of Motion of a rartick. 57 direction, we deduce from what precedes (see Art. 24) the equations _ d fdx\ d 2 x These are called the differential equations of motion of the particle ; and the solution of the problem depends in each case on the integration of these equations. As already stated, the preceding equations hold for the motion of any rigid body, provided the direction of the force which acts on it always passes through its centre of mass. 69. In some problems the mass acted on constantly varies during the motion ; in this case equation (3) becomes F--(mv). (5) For instance, suppose a ball projected vertically upwards, a chain of indefinite length being attached to it, and drawn up gradually by it ; to investigate the motion. Here, if m be the mass of the ball, fj. that of a unit of length of the chain, and s the length of chain in motion at any instant, we have M = m + ps ; and if m = />>, our equa- tion gives ds\ dt\ K dt\ or dt nf (k + s)g, n \2 ds Hence {k + sy-(^j=c--y(k + sy. If V be the initial velocity, we have & 2 F 2 = c-yv. 58 Action and Reaction, Hence This determines the velocity at any height ; also H, the height of ascent, is given by the equation s =«f^%- k - (7) V 2 If k = oo , this is easily seen to become -jr— , which agrees with Art. 38. Section V. — Action and Reaction. 70. Third Law of Motion. — Reaction is always equal and opposite to action : that is, the mutual actions of two bodies are always equal and take place in opposite directions. On this law Newton remarks as follows : — " If any person press a stone with his ringer, his finger is pressed by the stone. If a horse draw a body by means of a rope, the horse also is drawn (so to speak) towards the body ; for the rope being strained equally in both directions, draws the horse towards the body as well as the body towards the horse, and impedes the progress of one as much as it promotes that of the other. Again, if any body impinge on another, whatever quantity of motion it communicates to that other it loses itself (on account of the equality of the mutual pressure)." • Newton verified this law experimentally in the case of the collision of spherical bodies. — See Scholium, Axiomata. He also showed that the law holds good in the case of the attraction of bodies, as follows : — Let A and B be two mutually attracting bodies, and con- ceive some obstacle interposed by which their approach to one another is prevented. If the body A be acted on towards B by a greater force than B is acted on towards A, then the obstacle will be more urged by the pressure of A than by the pressure of B. The stronger pressure should prevail, and cause the system consisting of the two bodies and the obstacle to move in directum towards B ; also, as the force is uniform the motion would be accelerated ad infinitum, which is absurd, and contrary to the first law of motion ; for, by that law, such Stress, Forces of Inertia. 59 a system, as it is not acted on by any external force, should continue in a state of rest or of uniform rectilinear motion. 71. Stress, Forces of Inertia.— The fact is that force is always exhibited as a mutual action between two bodies ; and this phenomenon, regarded as a whole, is described by the term stress, of which action and reaction are but different aspects. Thus to the action of a force producing an acceleration of motion in a body corresponds an equal and opposite reaction against acceleration; this is called the force of inertia of the body. It thus folldfe that the force of inertia of any material particle must be ^ual and opposite to the resultant of all the forces which act on the particle, whether arising from the action of the other parts of the system or from that of forces external to the system. Hence, inthe motion of any material system, since the actions and reactions of its different parts equilibrate in pairs, we infer that there is equilibrium between the external forces which act on the system and the several forces of inertia of the different par- ticles of which the system is composed. This is equivalent to the celebrated principle introduced by D'Alembert, and called by his name, but which is directly implied in Newton's Scholium on the Third Law of Motion. This has been observed by many writers on Mechanics, but the connexion of New- ton's Scholium with the modern theory of work and energy was first pointed out by Thompson and Tait : see their Treatise on Natural Philosophy, vol. i., pp. 247-8. 72. The laws of Motion, like every law of nature, must ultimately depend for their establishment on their agreement with experiment and observation. Accounts of the different apparatus that have been devised for the purpose of verify- ing these laws will be found in the books especially devoted to the purpose, such as Ball's Experimental Mechanics. The most complete proof of the laws of motion, however, is derived from Physical Astronomy. The Lunar motions, for instance, have been calculated from equations depending solely on these laws ; and the observed and calculated posi- tions are found to agree with a precision that could only arise from the perfect accuracy of the principles from which they were deduced. w' 60 Action and Reaction. One of the simplest contrivances for illustrating the laws of motion, in the case of falling bodies, is that devised by Atwood, which we shall now proceed to consider. 73. Atwood's Machine. — In its simplest form this machine may he regarded as consisting of two masses connected by a string which passes over a small fixed pulley. We shall neglect the weight of the pulley, and also that of the string, as well as the friction at the axle of the pulley. Suppose W and W |fc represent the w r eights of the bodies, of which flRis the greater. Let T denote the tendon of the string at any instant : then considering the pulley as perfectly smooth, this tension, by the law of action and reaction, must act equally, and in opposite directions, on the two masses. Accordingly, we may regard the body W as acted on by the pressure W downwards, and the tension T upwards ; i. e. by the single force W - T acting downwards — then, the corresponding acceleration /, from Art. 40, is given by the equation . W-T Similarly, the upward acceleration of the other body is repre- T - W sented by ^, g. Again, as the string is supposed inextensible, the velocities of the bodies at any instant are equal and opposite, and hence their accelerations also. Accordingly we have W- T T- W w w ' or w+ w" a) This determines the tension of the string. Again, we have W- 9 9 At wood's Machine. 61 therefore W-W=(W+ W) '-, W-W or /- w+W ' () This determines the acceleration. By aid of it the velo- city and the space described in any time can be readily deduced. The most important advantage of this apparatus is that, by taking bodies of nearly equal weights, we can make the acceleration — — — , g as small as we please. A complete account of Atwood's apparatus is beyond the scope of this treatise. In a subsequent place we shall consider the modification required when allowance is made for the mass of the pulley. Examples. 1. A mass of 488 grammes is fastened to one end of a chord which passes over a smooth pulley. What mass must he attached to the other end in order *f that the 488 grammes may rise through a height of 200 centimetres in 10 seconds, assuming^ = 980 centimetres? Ans. 492 grammes. 2. Two weights of 14 and 18 ozs. are suspended hy a fine thread which passes over a smooth pulley, if the system be free to move ; find how far the \r heavier weight will descend in the first three seconds of its motion, and also the tension of the string. Ans. 18 feet ; and 15f ozs. 74. Suppose that one of the bodies is placed on a smooth horizontal table, and that the string, by which the bodies are attached, passes over a smooth pulley placed at the edge of the table ; then, denoting the tension of the string by T, we have, as before, Again, since the motion of the body on the smooth table arises from the tension T, we have / T 62 Action and Reaction. W Eliminating T, we get /= ^ + w , g. (3) Again, equating the two values of/, T= WW ' . (4) It may be observed that the tension of the string in this case is half of that in Atwood's machine for the same masses. 75. Masses on Two Smooth Inclined Planes. — Suppose two bodies, of weights W and W\ placed on two planes, of inclinations i and %' to the horizon ; and suppose the connecting string to lie in a vertical plane at right angles to the line of intersection of the two inclined planes, and to pass over a small pulley placed at the common summit of the planes ; then, representing as before the tension of the string by T, since W sin i is the component of W acting parallel to the plane, we have W JFrint-jP-— /, 9 , W" and T-W'mii'=—f. 9 W sin i - W f sin x (KS Hence /= — — w+ w > 9- &) WW Also T = w+ w , (sin i + sin i) . (6) It is evident that W or W will descend according as W sin i or W sin i' has the greater value. The results of the two former Articles are particular cases of the preceding ; and are, accordingly, cases of the formulae (5) and (6) . We shall next consider the preceding problems for rough planes. 76. Motion on Uniformly Rough Planes.— Suppose two bodies connected as in Art. 74, and let fx denote the coefficient of friction for the horizontal plane. The friction acting against the motion of W is represented Motion on Uniformly Bough Planes. 63 hy [iW; hence the pressure producing motion is T- llW . We accordingly have the equation W j W and also W- T= — /, as before. W- llW Hence we get / = w+ w , g, (7) WW and T =W7W' {l+lx) - (8) There can be no motion unless W is greater than /.t W ; as is also evident from elementary considerations. Equation (7) may also be written in the form from which li can be determined when W and W are known, /having been obtained by observation. By this means the value of /i, the coefficient of dynamical friction was obtained for several substances by Coulomb. Again, let li, ll be the coefficients of dynamical friction for the inclined planes, in Art. 75. Since the pressures on the planes are represented by W cos i and W' cos i\ respectively, the corresponding fric- tions are ll W cos i and fx W cos i ; consequently the total pressure acting on W, down the plane, is represented by W (sin i - li cos i) - T ; W and we get TF(sin i - li cos i) - T= — /. W And, similarly, T - W (sin i + li cos i') = — /. Hence we have TF(sin i - li cos i) - W (sin % + u cos i) . n A . /= w^w —'' (10) WW and T = -== — = { sin i + sin i + li cos i - li cos i) . (11) 64 Action and Reaction. Examples. 1. If the two equal masses in Atwood's machine be each lib.; required the additional mass which, added to one of them, would generate a velocity of one foot in each mass at the end of the first second. ^ 2 ^s 2. In the same case find the tension of the string which connects the two masses - Ans. 9 -^± lbs. 9 3. Two smooth inclined planes are placed back to back : the inclination of one is 1 in 7, and of the other 1 in 10 ; a mass of 20 lbs. is placed on the first, and is connected by a string with a mass of 30 lbs. placed on the second plane. Find the acceleration of the descent, and the tension of the string. 4. A mass of 10 lbs., falling vertically, draws a mass of 15 lbs. up a smooth plane, of 30° inclination, by a string pa'ssing over a pulley at the top of the plane. Find the acceleration, the space fallen through in 10 seconds, and the tension of the string. ^ / = ff_ . $ = hg . T== 9 lbs> 5. A mass, descending vertically, draws an equal mass 25 feet in 2£ seconds up a smooth plane, inclined 30° to the horizon, by means of a string passing over a pulley at the top of the plane. Determine the corresponding value of g. * J * Ans. 32. 6. Given the height, h, of a smooth inclined plane, find its length so that a given weight F, descending vertically, shall draw another given weight Q up the plane in the least possible time. . g 2QA 7. A mass P, falling vertically, draws another, Q, by a string passing over a fixed pulley : if, at the end of t seconds, the connecting string be cut, find the height to which Q will ascend afterwards. ( P-Q \ 2 g&_ m ' \P+Ql 2 ' 8. A mass, hanging vertically, draws an equal mass along a rough horizontal plane. If at the end of one second the string be cut, find how far the mass will move along the plane before it is brought to rest by the friction. An, o ^ sec o n P, and P> Q. After the united masses Q and P have descended s feet from rest, R is detached : find how much further Q will move before being brought to rest. Let/be the acceleration in the first stage of the motion,/' that in the second, v the velocity at the instant P is detached, x the required distance ; then therefore 2/5 = p* = 2f'x ; / P+ Q - P P+ Q f P+ Q F + P' P- Q ( 66 CHAPTER IV. IMPACT AND COLLISION. 77. Collision of Homogeneous Spheres. — In this chap- ter it is proposed to consider some elementary cases of impact of solids, but principally the collision of homogeneous spherical bodies, moving without rotation, whose centres, at the instant of collision, move in right lines lying in the same plane (all friction being neglected). There are two cases to be considered, according as the centres of the spheres move in the same or in different right lines. The former is called direct, the latter oblique collision. We commence with the former case, and at first suppose the centres to move in the same direction along the line. 78. Direct Collision. — Let M and M represent the masses of the bodies, V and V their velocities before, v and v those after, collision. We also suppose M to impinge on M'. The whole impact may be divided into two stages. During the first, the bodies compress each other, and the impinging body M, moving with a greater velocity than the other, accele- rates its motion, until the exact instant at which their mutual compression is the greatest, when they are moving with a common velocity. During the second stage, the bodies tend to revert to their original shape, and the forces thus brought into play, called the forces of restitution, tend to cause the bodies to separate by still further diminishing the velocity of the impinging body, and increasing that of the other. Suppose u represents the common velocity at the instant of greatest compression ; then the quantity of motion lost by M during the first stage of the shock is M(V - u), and that gained by M' is M\u - V). Collision of Homogeneous Spheres. 67 These are the measures, by Art. 68, of the entire actions of the mutual forces during this stage of the collision ; and, since by the Third Law of Motion, the forces must be equal and opposite, so also are their actions in the same time. Hence, we have H{V-u) =M'(u- V), MV+M'W or u = M+M' (1) In the case of perfectly non-elastic bodies, in which no forces of restitution are brought into play, the bodies would proceed after collision to move with this common velocity, j There is probably no case in nature of a perfectly non- elastic solid. All solid bodies with which we are acquainted have a tendency to recover, in different degrees, their original forms after being compressed. This tendency arises from their elasticity. Bodies are said to be perfectly elastic when the forces of restitution, brought into play during the second stage, are exactly equal to the forces of compression, which act during the first. In this case the impinging body M would lose a further quantity of motion, M(V-u), equal to that which it lost in the first stage. Therefore its velocity v, at the end of the shock, will be equal to 2u - V. In like manner we have v = 2u - V '. Thus, in direct collision, we are enabled to determine the velocities, v, v, in the case of perfectly elastic bodies. Bodies are, however, in general imperfectly elastic ; that is, the whole force of restitution is less than that of compres- sion. The Law of restitution, as derived from experiment, is usually stated thus: — The ratio which the whole impulse of restitution bears to that of compression is constant while the imping- ing substances remain the same. This ratio is usually repre- sented by the letter e, having been by many writers called the modulus or coefficient of elasticity ; but as this title is now employed in a different sense, we shall follow the current usage in adopting the name, coefficient of restitution. From this law it follows that the quantity of motion lost by M during the second stage of the impact bears a constant f2 68 Impact and Collision. ratio to that lost during the first ; and similarly for the quan- tity of motion gained by M'. Accordingly M(u-v)=eM{V-u), M\v'-ii) = eM'tu- V). Hence we get v'-v = e(V-V'\ (2) and M V + M'V'=Mv + MY. (3) These equations enable us to determine the velocities, v, v' 9 after impact, when those before impact are given, as also the masses IT, M', and the coefficient of restitution. It should be observed that equation (3) expresses that the total quantity of motion of the two bodies is the same after impact as before. This result is a particular case of a gene- ral principle which shall be subsequently considered (see Art. 83). The result contained in (2) may be stated thus : In direct collision* of two spheres the relative velocity after collision bears a constant ratio to the relative velocity before collision. This law was established by Newton, as the result of ex- periment (see his Leges Motus, scholium) ; and the coefficients of restitution for several substances, such as glass, ivory, steel, &c, were determined by him. In more recent times a number of careful experiments were undertaken by Hodgkinson on the laws of restitu- tion. The results are to be found in the Report of the British Association, 1834, and also in the Transactions of the Royal Society. His conclusions agree in the main with the law laid down by Newton, given above. Some of the more important results of Hodgkinson's experiments may be briefly stated as follows : — The coefficient of restitution diminishes slowly as the * This result is by some writers taken as the basis of the rational theory of collision. However, the method here given is that more usually adopted ; it has the great advantage of connecting the problem directly with the consideration of force, and of illustrating the principle (Art. 67) that impulsive forces must be regarded as forces of great intensity whose time of action is very short. Height of Rebound. 69 velocity of impact increases ; it is independent of the relative magnitude of the masses. In impact between bodies differ- ing much in hardness, the coefficient of restitution is nearly equal to that between two specimens of the softer body. No perfectly elastic body exists in nature : glass, however, may be regarded as nearly so, its coefficient being -ff > approxi- mately, as determined by Newton. When the mass M f is at rest, and very great in comparison with M, v is very small, and we have approximately v = - eV. Hence, if a body impinge perpendicularly, with a velocity V, upon a fixed plane, it will return back in its former direc- tion of motion with a velocity represented by e V, where e is the coefficient of restitution. 79. Height of Rebound. — If a body fall from a height h on a fixed horizontal plane, then V, the velocity with which it strikes the plane, is equal to y/2gh. The velocity of rebound is e V, or e ^2gh : hence, if li be the height to which it re- bounds, we have y2gti = e ( 13 ) where v x denotes the common velocity at the instant in ques- tion. Again, the impulse of the tension of the string is measured by the quantity of motion communicated to M'\ and accord- ingly is represented by If the table be rough, since the friction of the table is pro- portional to the weight of IT, it may be neglected in com- parison with the impulsive force, and we obtain the same value for i\ as in the case of a smooth table (see Art. 67). J 78 Impact and Collision. Examples. 1. A sphere, of 30 lbs., moving with a velocity of 45 feet a second, over- takes another, of 27 lbs., moving" 32 feet a second ; if the relative coefficient of restitution be f, find their velocities after collision. Ans. 34if, 43ff . 2. Two spheres meet directly with equal velocities ; find the ratio of their masses that one of them, M, should be reduced to rest by the collision— (1) when perfectly elastic ; (2) for coefficient of restitution e. Am. (1) M = 3.1/', (2) M = 31' (1 + 2e). 3. If two equal and perfectly elastic spheres be dropped at the same instant from different heights, h and h', above a horizontal plane ; determine whether their common centre of inertia will ever rise to its original height. Ans. No, unless A— is a commensurable number. 4. A 101b. shot is fired from a gun of 12cwt, that is quite free to move. The velocity with which the shot leaves the mouth of the gun is 1600 feet per second ; find the velocity of the gun's recoil. Ans. 11*9 feet per second. 5. Three homogeneous spherical bodies, m, m', m", are placed with their centres in a row. If m be projected with a given velocity V towards m; to find the magnitude of m' in order that the velocity communicated to m" by its inter- vention shall be the greatest possible. Let e, e denote the relative coefficients of restitution between m, m, and hetween m', m", respectively. Then, if v' be the velocity of m' after the first collision, we get from Art. 78, , m{l + e)V v = '—. m + m In like manner, if v" be the velocity of m" after the second collision, we have „ _ tn'(l + e')v' _ mm (I + e) (1 + e) V m' + m" "" (m + m) (w' + m") m' , . (*« + ♦»') {m' + m") Accordingly, ; 7V1 — jjz must be a maximum ; or —, ° {m + m') {m' + m ) m mm" ... , mm" • • • is a minimum : i.e. m + m + m" + — — is a minimum, or m + — - is a mini- tn m mum: hence, m = Vw»»", by elementary algebra ; consequently the masses must be in geometrical progression. This reasoning is readily extended to the case of any number of spheres placed in a row ; and, when the first and last are given, the masses must be in geometrical progression, in order that the velocity communicated to the last should be the greatest possible. Examples. 79" 6. Two particles are connected by a string, and laid on a uniformly rough horizontal table, at a distance from each other less than the length of the string. One of the particles receives a given impulse along the line joining them : deter- mine the motion which ensues after the tightening of the string. 7. Find an expression for the vis viva lost in the direct collision of two imperfectly elastic spheres. From equations (2), (3), Art. 78, we have (mV + m'Y') 2 = (mv + m'v') 2 , and mm'e~(V — F') 2 = mm'(v - v') 2 . This latter may be written mm\V - r') 2 =mm'(v-v') 2 + (1 - f)mm'(V- V'f. Hence, by addition, (m + m')(mV 2 + m'V 2 ) = (m + m')(mv 2 + m'v' 2 ) + (1 - e 2 )mm'(V- V'f . Therefore m V 2 + *»' V' 2 = mv 2 + m'v' 2 + (1 - e 2 ) ; ( V- F') 2 . m + m Accordingly, the vis viva lost by the collision is represented by (1 _, ) _=£_ (r _ F7 . v J m + m' v ; 8. Find the loss of vis viva caused by the direct impact of two balls, one weighing 10 lbs. and falling from a height of 20 feet, the other at rest and weighing 301b. ; assuming the coefficient of restitution = -|. Ans. ^-th of the original vis viva. 9. A body, after sliding down a smooth inclined plane of given height, re- bounds from a hard horizontal plane ; find the range on the latter plane. 10. A mass 31, after falling freely through h feet, begins to pull up a heavier mass Mi by means of a string passing over a pulley, as in At wood's machine ; find the height through which it will lift it. Let v\ be the velocity communicated to M\ by the impulsive action ; then by Art. 86 we have v\ = — — ^l2gh. M -f M\ During the subsequent motion Mi is subject to a uniform retardation — — —g, as in Art. 73; accordingly, if ITdenote the height to which iLTi ascends before it is brought to rest, we have 2/ Mr-M- 11. An inelastic particle falls from rest to a fixed inclined plane, and slides down the plane to a fixed point in it ; show that the locus of the starting point is a straight line when the time to the fixed point is constant. (Camb. Trip., 1871). SO Impact and Collision. 12. Two equal balls of radius a are in contact and are struck simultaneously by a ball of radius c moving in the direction of their common tangent ; if all the balls be of the same material, the coefficient of elasticity being e, find the velo- cities of the balls after impact, and prove that the impinging ball will be reduced to rest if 2e= ;" + C \ . (Camb. Trip., 1871.) a d (2a+c) ' 13. Show how to determine the motion of two elastic spheres after direct \S impact, and prove that the relative velocity of each of them with regard to the •centre of mass of the two is, after the impact, reversed in direction and reduced in the ratio e : 1, e being the coefficient of restitution. A series of n elastic spheres whose masses are 1, e, e 2 , &c, are at rest, sepa- rated by intervals, with their centres on a straight line. The first is made to impinge directly on the second with velocity u. Prove that the final vis viva of the system is (1 - e + e n )u 2 . {Ibid., 1875.) 14. An elastic ball makes a series of rebounds from a perfectly smooth inclined plane : to investigate its motion. Let i be the inclination of the plane to the horizon, and suppose the ball projected from the point in the plane, in a direction which makes the angle a with the plane. Let £, jBi, &c, fin be the angles at which the ball strikes the plane at the first, second, . . . n th impacts ; and oi, ct2, • • • a.,,, the angles it makes after rebounding. Then, by equation 9, Art. 56, we have cot £ = cot o - 2 tan i ; cot £ = e cotoi, .-. ecotai = cot a — 2tani. Similarly e cot 02 = cot 01 — 2 tan i ; .-. tf 2 coto 2 = cot a - 2(1 + «)tani; and it is easily seen that we have, in general, e n cot «„ = cot o - 2 (1 + e + . . . + e' 1 ' 1 ) tan i 2(1 -*»). • = cot o \ tan t, 1 - e from which the angle after the n th rebound can be found. Again, the ball will proceed to bound up the plane so long as the angles ai, «*■>,.. . are each less than 90° — i. but, by (10) Art. 80, Examples. 81 If a» be the first of a series of angles which exceeds 90° - i, we will have cot o»< tan t. If cot a is greater than it can be readily shown that for all values of n a n is less than 90° - i ; and in this case accordingly the ball would proceed to ascend the plane by au indefinite series of parabolic paths. But if cot a be less than — — , after a certain number of impacts, the body would proceed to rebound down the inclined plane. T . . . . 2 tan i In the particular case where cot a = , or 2 tan i = cot a ( 1 — e), we have 1 — e v " e cot oi = e cot o ; .'.01 = 0; hence a = a\ = az = . . . = a n ; or, all the angles of rebound are equal to one another ; consequently the series of parabolic paths in this case are similar, and the particle would proceed up the plane with an indefinite number of rebounds. In general, let t\ t fa, . . . t„ be the times of flight for the series of parabolic paths, and v\, 02, . • • v n , the velocities of the successive rebounds ; then by equa- tion (5), Art. 50, we have 2v sin a 2v\ sin a\ B t\ = r, h = — , &c. g cos 1 g cos 1 But if v' be the velocity with which the ball strikes the plane at the first impact, we have v' sin /3 = v sin a ; but by Art. 80, vi sin a\ - ev' sin $ = ev sin a ; consequently h = et\ ; also h = eh = e 2 ti, &c. Hence the times of flight are in geometrical progression, having e for their common ratio. If the intervals of time occupied by the successive impacts be neglected, we get for the time T of describing the first n parabolas, 2v sin o 1 - e n g cos i 1 - e ' Again, let i?i, i? 2 , . . . B,i denote the consecutive ranges" on the inclined plane ; then, by Art. 50, we have -Si = ?gh 2 — : = \gtf- cos i (cot a - tan i). sin a ' Similarly, Hz = \gh* cos i (cot a\ - tan i) = \gt£ cos i {e 2 cot ai - e 2 tan i) = yeh* cos i { cot a - (2 + e) tan i } . G 82 Impact and Collision. And, in general, R n = lgt n 2 cos i (cot ctn-i — tan i) ^ = ye»- 1 ti 2 cosi{cota-(2 + 2e+ . . . + 2e«- 2 + e"- 1 ) tani} (2 — £ n-1 — e" \ 1 ) 2v 2 sin 2 a ( , 2e"~\ . 1 + * . _. .) = \e n ~ l cot a tant + - e 2 »- 2 tantj. y cos i ( 1 — e 1 - e ) Hence the sum of w ranges is found to be n v 2 sin 2 a 1 -«« ( 1 -e» .) = 2 r -r j cot a - tan x \ g cos t 1 - e ( 1 - e ) ( 1 - e n A = t>T sin a ( cot a — ■ tan i ) . \ 1 -e J 2 tan i If cot a be greater than we get the entire range on the inclined plane by making n — oo ; hence the entire range is, in this case, rr • ( * tani v 1 sin a \ cot o I l-« The preceding question was discussed at great length by Bordoni, Mem. della Societa Ital., 1816. See also Walton's Problems on Theoretical Mechanics, pp. 262, 263, 3rd edition. 15. In the preceding example show that the greatest distances of the body from the inclined plane in the successive parabolic paths are in geometrical pro- gression, having e 1 as their common ratio. 16. If two bodies, of the same elasticity be projected with the same velocity from a point on an inclined plane, and if the directions of projection make equal angles at opposite sides of the perpendicular to the plane, prove that the series of parabolic paths described, one up, the other down the plane, will be described in times which are respectively equal in pairs. 17. An imperfectly elastic ball falls from a height h upon an inclined plane ; find the range between the first and second rebounds. Am. 4eh sin i (1 +e). 18. Prove that, in order to produce the greatest deviation in the direction of a smooth billiard ball of diameter a, by impact on another equal ball at rest, the a ll-e former must be projected in a direction making an angle sin -1 - / with the c \ 3 — e line (of length c) joining the two centres ; e being the coefficient of restitution. Camb. Trip., 1873. 19. A bucket and a counterpoise, connected by a string passing over a pulley, just balance one another, and an elastic ball is dropped into the centre of the bucket from a distance h above it ; find the time that elapses before the ball Examples. 83 ceases to rebound ; and prove that the whole descent of the bucket during this imh e interval is — where m, M are the masses of the ball and the 2ji + m (1 — ey bucket, and e is the coefficient of restitution. Camb. Trip., 1875. Let v be the velocity of the ball just before the first impact. The relative velocity after the first impact is ev, and the relative acceleration is g, since the acceleration of the bucket is zero. Therefore the time during which the ball rebounds is 2» # , K 2v e — (e + e 2 + e z + . . . ) = 9 U 1 - e 1 -e\ g e hh Let Pi, r 2 , r 3 , ... be the velocities of the bucket between the first, second, third, . . . impacts. „, „ m(l+e) m(l+e) Then 7 i = v y > Fa = Pi + -± '- ev, &c. , 2M + m 2JI + m and the space described by the bucket is 2v . _ _ Tr , TT , 2mev 2 4mh e — (eVi + e 2 V*+ 6 3 r 3 + . . •) = g v ' ^(2Jf+«i)(l-«)2 21T+m (1 - e) (This proof is taken from GreenhilTs solutions of Cambridge Problems and Eiders for 1875.) 20. A particle is projected with a velocity V, in a direction making an angle a with the horizon, and strikes a vertical wall, at a distance a from the point of starting. Find when and where it will strike the horizontal plane drawn through its initial position. 2 V sin o Ans. T= . The distance from the wall at which it will strike ff the ground = e ( o J , where e is the coefficient of restitution for the particle and the wall. 21. A large number of equal particles are fastened at unequal intervals to a fine string, and then collected into a heap at the edge of a smooth horizontal table, with the extreme one just hanging over the edge. The intervals are such that the times between successive particles being carried over the edge are equal : prove that if c H be the interval between the n ih and tbe (n + l) th particle, and v n the velocity just after the (n + l) th particle is carried over, then — = — = n. Ci Vi Professor Wolstenholme, Educ. Times. If v be the velocity acquired by the first particle during its fall through the interval ci, we get immediately, from the conditions of the problem, the two series of relations vi = §0, r 2 = § («j + v) = f v, v s = | (i'2 + v) = %v, &c. 2gc x = v\ 2gc 2 = (n + vf - vj = 2v 2 , 2gc z = (r 2 + vf - v? = 3v 2 , &c, Hence vi : vt : Vs : &c. : v n = a : e%\ c* : &c. : Cn = 1 : 2 : 3 : &c. : n. G 2 84 Circular Motion. CHAPTER V. CIRCULAR MOTION. Section I. — Harmonic Motion. 87. Uniform Circular Motion. — If a point P describe a circle with a uniform motion, the radius of the circle is called the amplitude of the motion, and the time of making one revolution is called its period. If the arcs are measured from a fixed point A, and the time counted from the instant the moving point passed through a fixed point E, then the angle A OE is called the angle of epoch, or briefly, the epoch. Also the ratio which the arc PE, at any instant, bears to the cir- cumference of the circle is called the phase of the moving point at that instant. The arrowheads on the figure denote the direction in which the motion is supposed to take place, and such a rotation as there repre- sented, i.e. in the opposite direction to that of the hands of a clock, is con- sidered a positive rotation : that in the opposite direction, or clockwise, being considered negative. Let to be the angular velocity of P, or the circular measure of the arc described in one second, e the circular measure of the epoch AOE, and that of AOP, we have = wt + e. (1) Again, if T denote the period, we get u = —, and hence, if desirable, we should write but we shall generally employ the form 6 = wt + c, being more compendious. Harmonic Motion. 85 88. Harmonic .notion. — If PM be drawn perpendi- cular to the diameter AA\ then as P moves uniformly round the circle, the point M moves backwards and forwards along the line AA\ and is said to have a simple harmonic motion. The amplitude, period, epoch, and phase of the harmonic motion are the same as those of the corresponding circular motion. If 021 = x, then the position of M at any instant is given by the equation x = a cos (iut + a), (2) where a represents the amplitude, and e the epoch of the motion. The angle wt + e is called the argument of the motion, and the distance x is said to be a simple harmonic function of the time. Again, if PN be perpendicular to OB, and y = ON, we have y = a sin (cot + e) = a cos (tot + e - §7r). Hence the point iV has also a harmonic motion, and we infer that a uniform circular motion is equivalent to two simultaneous rectangular harmonic motions, of the same amplitude and period, but differing one-fourth in phase : and conversely. Again, if the point M be projected on any line, the pro- jected point plainly has a harmonic motion of the same period and phase, but having for amplitude the projection of the amplitude of M. If we differentiate equation (2) we get clx . v = — = - aw sin [wt + e). do Consequently the velocity of a point which has a simple har- monic motion is a simple harmonic function of the time ; and its maximum value is equal to the velocity in the circle. Again, the acceleration /is given by the equation j. dv „ . / = — = - u)~ a cos (iut + t ) = - w'x. dt Consequently the acceleration at any instant is propor- tional to the distance from the middle point of the motion, and is always directed towards that point. The acceleration at either extremity of the motion is - u> 2 a. 86 Circular Motion. Any number of harmonic motions of equal periods in the same line are equivalent to a single harmonic motion. For let x = a cos (iot + t) + a cos (iot + e') + &c. Then x = A cos w£ - 5 sin tot, where A = 2a cos e, and B = ^a sin e. Hence x - C cos (o>£ + 7), where C = v^ 2 + By an( i tan 7 = — . This result admits also of a simple geometrical demon- stration. 89. Elliptic Harmonic Motion. — If a circle be pro- jected orthogonally on any plane its projection is an ellipse, and the projection of any point which moves uniformly on the circle is said to have an elliptic harmonic motion. An elliptic harmonic motion may be resolved into two simple harmonic motions, of the same period but differing in amplitude, along any two conjugate diameters of the ellipse, these motions differing one-fourth in phase. This follows immediately from the property that rectangular diameters in the circle are projected into conjugate diameters in the ellipse. Conversely, any two simple harmonic motions, in different lines, of the same period and differing one-fourth in phase, compound an elliptic harmonic motion, having the lines for conjugate diameters. Examples. 1. A point P describes a circle with uniform velocity. If M be its projective on any fixed diameter, prove that the velocity of M varies as PM, and that its acceleration varies as OM ; being the centre of the circle. 2. If two harmonic motions in the same line have equal amplitude (a) and equal periods, but different epochs, e, e', find the amplitude of tbeir resultant motion. Ans. 2a cos £ (e - e'). 3. If the difference of phase in the last passes continuously from to 2tt, find the mean value of the square of the amplitude of the resulting vibration. Ans. 2a 1 . • /• Examples. 87 The mean value is represented by the definite integral {Int. Calc, Art. 238), 77 8« 2 f T n , — eos-(pa owing to the centri- petal force. This force is directed towards the centre of the circle, and may be regarded as constant in magnitude and direction, during the indefinitely short time At. Hence, if /denote its acceleration, we have, by Art. 36, QN=i/(At)\ But, in the limit, PN* = 2QN.PC, where C is the centre of the circle ; .-. V\Atf=2QN.PC. Centrifugal and Centripetal Force. 89 V 2 Hence / . (1) r Or, the centrifugal acceleration f is a third proportional to the radius of the circle and the velocity of the particle. mV 2 The centrifugal force is accordingly represented by ■ — — . If it be required to calculate the pressure in pounds due W to the centrifugal force, we substitute — for m, and the pre- W V 2 ceding expression becomes — — . Since the centripetal force is always directed to the centre of the circle, and is consequently at right angles to the direction of motion, it has no effect in altering the velocity of the mov- ing particle. Hence, if no other force act on the particle, its velocity will be constant during the motion. Conversely, if a particle m describe a circle of radius r, with a uniform velocity F, we infer that the resultant of all the forces which act on it passes through the centre of the m V 2 circle, and is represented by . Again, as the velocity in the circle is uniform, if T denote the number of seconds in which the circumference is described, we have V=- 7 z r . Hence, in this case, we have /=4*'£. (2) Consequently, in uniform circular motion, the centrifugal force varies directly as the radius of the circle, and inversely as the square of the time of revolution. Again, if w be the angular velocity of the radius CP, we have (jj = — : accordingly, in terms of the angular velocity and the distance, we have /--V. (3) 90 Circular Motion. Examples. 1. Calculate the centripetal acceleration of a particle which moves in a circle of 5 feet radius with a velocity of 10 feet per second. Ans. 20. 2. A particle performs 20 revolutions per minute in a circle of 1 foot cir- cumference : find its centrifugal acceleration. Ans. -6981. 3. A hody of 1 lb. mass revolves, in a horizontal plane, at the extremity of a string 10 feet long, so as to make a complete revolution in 2 seconds ; find the tension of the string in pounds. 107r 2 9 4. A railway carriage of 1 ton weight is moving at the rate of 60 miles an hour round a curve of 1210 feet radius : find the centrifugal pressure on the rails. Ans. 448 lbs. 5. In the last example, find how much the outer rail should be raised in order that the total pressure should be equally distributed between both the rails, the distance between the rails being 4 feet. Ans. 9 \ inches, approximately. 91. Circular Orbits. — In the case of uniform circular motion, since the centrifugal force acting on the particle at each instant is directed from the centre of the circle, we may suppose the particle to be kept in its circular orbit by the action of an attractive force always directed to the centre r of the circle, and whose acceleration is 4?r 2 — . Hence, if the magnitude of the acceleration directed to a fixed centre of force be known, we can determine the conditions that a particle should describe a circle, having the fixed point as its centre. For, if/ be the acceleration caused by the central force at the distance r of the particle, we have f = ^., and therefore v = */fr. This determines the velocity r at each point in the circle. Conversely, if the particle be projected at the distance r from the centre of force, at right angles to the radius vector, with a velocity v = yV^> it will proceed to describe a circle freely round the centre of force. 2-rrr Also, the time T of describing the circle will be — — , or T= 2- 4 Centrifugal Force at Earth's Equator. 91 For example, if the attractive force be directly propor- tional to the distance, we have / = fir, where ju is some con- stant ; and consequently, in this case, T-2± (4) Hence, for this law of attraction the time of revolution in a circular orbit is independent of the distance ; and we infer that the times of revolution for all circular orbits round the same centre of force are equal. Again, let the attraction vary inversely as the square of the distance from the centre, that is to say let/= — . In this case the velocity in the circular orbit is represented by /-, and the time of revolution by 2?r /— . Hence we see that in different circular orbits round the same centre of force (which varies as the inverse square of the distance), the squares of the periodic times vary as the cubes of the distances from the centre of force. This establishes Kepler's Third Law for circular orbits. The preceding are particular cases of general results con- nected with the problem of Central Forces, which will be treated of in a subsequent Chapter. 92. Centrifugal Force at Earth's Equator.— We now proceed to consider the centrifugal force arising from the rotation of the Earth on its axis. Let r be the number of feet in the Earth's radius ; T the number of seconds in the time of a complete rotation on its axis ; / the acceleration due to centrifugal force at the Equa- tor : then we have /=4 .2 '• The most convenient method of determining /is by compar- ing its value with that of g at the Equator : thus f —% (5) 9 , the velocity of a point whose distance from the axis is p is represented by pw. Rotation of a Rigid Body. 95 96. If a plane lamina rotate about a fixed axis at right angles to its plane, the centrifugal forces of the different elements of the lamina are equivalent to a single force, passing through its centre of mass, and which is the same as if the entire mass were con- centrated at that po int. Let the plane of the paper represent that of the lamina ; and take the point 0, in which the fixed axis meets the plane, as the origin of a pair of fixed rectangu- lar axes OX and Y. Suppose w to be the angular velocity of the lamina at any instant ; then, since each point in the lamina describes a circle round 0, the centrifugal forces for all its elements pass through that point ; these forces accordingly are equiva- lent to a single force. To find the value of this single resultant; let OP = r, and let dm denote the mass of an element at the point P; then the centrifugal force of the element is wPrdm, acting along the line OP produced. This force can be decomposed into two, ufxclm and o> 2 y dm, parallel to the axes of x and y respectively. Suppose the centrifugal forces of the other elements re- solved in like manner, then the entire system is equivalent to the forces u> 2 ^x dm and o 2 2y dm, parallel to OX and Y. But 2# dm = Mx, 2y dm = My, where x, y are the coordinates of the centre of mass of the lamina. Hence the resultant of the entire system of centrifugal forces is the same as that of the two forces w~ Mx and id 2 My ; or to the single force w 2 Md, where d denotes the distance of the centre of mass of the lamina from the fixed axis. 97. A similar theorem holds for any uniform rigid body turning round a fixed axis, provided the body has a plane of symmetry passing through the axis. For the body may be conceived divided into a number of 96 Circular Motion. indefinitely thin parallel plates, by planes perpendicular to the fixed axis, and the preceding theorem holds for each of these plates or laminae. Again, if m i9 m 2 , m 3 , . . . &c, denote the masses of the plates; and^i, p 2 , Pz> . . . &c., the distances of their respective centres of mass from the fixed axis; then as the body is supposed uniform and symmetrical, the centres of mass of each of the plates all lie in the plane of symmetry ; therefore the forces w 8 mipi, h> 2 m 2 p2, . . . &c, form a system of parallel forces. They consequently have a single resultant, equal to their sum, or to to 2 Md, where IT denotes the mass of the body, and d the distance of its centre of mass from the axis of rotation. Hence, the stress on the fixed axis produced by the cen- trifugal force is in this case represented by w 2 Md. If the fixed axis pass through the centre of mass, the stress on the axis becomes a momental stress or a couple : as will be shown also in the following Article. 98. Centrifugal Forces arising from Rotation for a Rigid Body. — Next let us consider the case of any rigid Suppose a plane drawn y body rotating round a fixed axis, through the centre of mass, per- pendicular to the fixed axis, and meeting it in the point 0. Take as the origin, the fixed axis as that of z, and two rectangular axes as those of x and y } respectively. Let dm denote an element of mass at the point P, whose co- ordinates are x, y, z ; then, by Art. 96, the centrifugal force for the element dm is equivalent to the forces, io 2 xdm and w 2 ydm, acting at P, parallel to OX and z OY, respectively. Transferring these forces to the point N, they are equiva- lent to the forces w 2 xdm, ufgdm, parallel to OX and OF; along with the couples w 2 xzdm, ufyzdm, parallel to the planes XZ and YZ, respectively. The resultant of the forces ufxdm, w 2 ydm, acting at N, is obviously directed to ; and Rotation of a Rigid Body. 97 consequently if it be transferred to 0, it can be resolved into ufxdm and urydm, acting along OiTand OY, respectively. If each centrifugal force be resolved in like manner, the whole system is equivalent to the forces hflLxdm and uf'Eydm, or to u>~Mx and w 2 My, acting along OX and OY, respectively; together with the couples w^xzdm, ufSyzdm, acting in the planes of XZ and YZ, respectively. If the fixed axis be a principal axis relative to the point (Int. Cede, Art. 214), we have *2xzdm = 0, and *2yzdm = 0. Hence, in this case the strain on the axis produced by the rotation is the same as if the entire mass was concentrated at the centre of mass of the rigid body. If, further, the fixed axis be one of the principal axes passing through the centre of mass of the body, the cen- trifugal forces arising from the rotation produce no strain on the fixed axis. And accordingly, if, from any cause, a rigid body commence to rotate about such an axis, it will continue to rotate permanently round the axis, provided the only external force be that of gravity. For example, if we suppose a homogeneous sphere, whose centre is fixed, to receive any impulse, it will commence to rotate around some one of its diameters ; and, as every dia- meter is in this case a principal axis, it follows, from the preceding, that it will continue to revolve permanently round that axis, if we suppose no external force but gravity to act on it. On account of the property established above, it is of im- portance, in order that any machine should work smoothly, that the centre of mass of any wheel, or portion which ro- tates rapidly, should lie on the axis of rotation, which should be a principal axis ; for otherwise the centrifugal forces would cause strong disturbing vibrations. The theorems of this section are particular cases of im- portant general results, which will be discussed in a subse- quent chapter. H 98 Circular Motion. V Examples. 1. A string of 5 feet length is just capable of supporting a weight of 10 lbs. ; find the greatest number of revolutions per minute that a weight of 4 lbs. attached to the extremity of the string is capable of making in a horizontal plane without breaking the string. Ans. 38. 2. A mass of 81bs. is suspended from the extremity of a string 10 feet long : find the least velocity that should be given to it in order to break the string, if its breaking tension be 12 lbs. Ans. 12 "64 feet per second. 3. Two balls weighing 6 lbs. each are fixed at the extremities of a rod of 10 feet length, which revolves 100 times in a minute around a central vertical axis; find the tension of the connecting rod. Ans. 102 lbs. 4. If two equal bodies moving on a rough horizontal plane be connected by a string of invariable length «, but without weight ; find the longest time- that one can continue to move after the other has been stopped by friction. Ans. >U' where p is the coefficient of friction. 5. A bodym sliding on a perfectly smooth horizontal table is connected by a string passing through a smooth hole in the table, with another body m' which hangs freely ; find the condition that m' should remain at rest, and also the time of revolution of m in its circular path, supposed of radius a. Ans. Velocity of m should be \ m 9 a ; time of revolution Im'ga \~nT 2tt fa m \ m'g 6. If a body, attached at its centre of mass to one end of a string of length r y the other end of which is attached to a fixed point on a smooth horizontal plane, makes n revolutions per second ; prove that the tension of the string is to the pressure on the plane as Air 2 n-r to g. Prove that at the Equator a shot fired westward, with velocity 8333, or east- ward, with velocity 7407 metres per second, will, if unresisted, move horizon- tally round the earth in one hour and twenty minutes, and one hour and a-half respectively. Camb. Trip., 1878. 7. A rig^Aody of any form revolves freely round an axis fixed in space: required thewtiditions under which the centrifugal forces of its several elements will have — (a) no resultant ; (b) a resultant pair ; (c) a resultant single force ; (d) a resultant pair and single force. Lloyd Exhib., 1872. Section III. — Motion in a Vertical Circle. 99. Velocity in a Smooth Vertical Curve Before the dis- cussion of motion in a circle we shall consider some properties of the motion of a particle, under the action of gravity, on any vertical curve. Take OX a horizontal, and 07a vertical line in the plane as axes of coordinates ; and suppose x, y the coordinates of P, the position of the particle at the y r i p = 2g(li + y Q -y) = 2gPL. (3) Consequently the velocity at any point P is the same as that acquired in falling from the horizontal line DR. This is an extension of the result given in Art. 49, and is itself a case of the general principle of work which shall be treated of in the next chapter (sec Art. 132). 100. Motion in a Vertical Circle. — If a particle be constrained to move in a vertical circle under the action of gravity, its velocity at any point, by (2) , is the velocity due to falling through a certain height from a certain horizontal line, or level. The motion will be one of complete revolution if this right line lies altogether outside the circle. If the line cut the circle the motion will be oscillatory. We proceed to consider the latter case in the first instance. In this case we may either consider the particle as moving in a smooth circular tube, or as attached by an inextensible string to a fixed point in the centre of the circle, the weight of the string being neglected. H2 100 Circular Motion. When the arc in which the oscillation has place is but a small portion of the circumference we get what is called a simple pendulum. From this statement the student will see that a simple pendulum can only be approximately repre- sented. However, a small leaden ball suspended from a, fixed point by a very fine wire may be regarded approxi- mately as a simple pendulum. 101. Simple Pendulum. — Let C be the centre of the circle ; its lowest point ; A the point from which the particle may be sup- posed to start ; P its position at the end of any time t ; v the corresponding velocity, Q=LPCO, a=LACO, estimated in circular measure, s = AP, Z=OC. Then, since the velocity at P is that due to falling from a horizontal line drawn through A, we have v 2 = 2gl (cos - cos a) ; It. but # = - \dt therefore = ^(cos0 COS a) i sin' ■n-g. Consequently ^ = ± 2 ^/f V sin ^ ~ - sin* 2' Again, since in the motion from A to 0, 6 diminishes as t increases, — is negative. Accordingly we have ut §-4MM- Time of a Small Oscillation. 101 l~ CO " 7fi' ' ' ' "' j ' - "' • * ' Hence we get tf / f = - — -===-. (4) 102. Time of a Small Oscillation. — The preceding definite integral, which represents the time of describing a circular arc, cannot be expressed in finite terms by means of the ordinary algebraic or trigonometrical functions ; however, when the amplitude of the oscillation is small we can easily get an approximate value for t, as follows : — When a is so small, that we may neglect powers of a and 9 beyond the second, we have 4(Wg-sin 2 ^ = a 2 -0\ Hence (4) becomes ?< do jo cos" 1 /a 2 - 2 \« No constant is added as = a when t = 0. Consequently we have acosjf = a COS /| *. (5) Accordingly is a simple harmonic function of the time (Art. 88). Again, when 6 = 0, we have Jj t = - ; hence the time of descent to the lowest point is represented by o J~- The particle, after arriving at the lowest point, plainly moves up the other side of the arc, and if the whole time of a small oscillation, expressed in seconds, be denoted by T, we have r-w£ (6) Since this expression is independent of a, it follows that 102 -Circular Motion. the time ' cf'ia smrttt oscillation is the same for all arcs of vibra- tion in the same circle.- From this property the vibrations of a pendulum are said to be isochronous. Also the time of a small oscillation at any place varies as the square root of the length of the pendulum. 103. The Seconds Pendulum. — A pendulum which oscillates once in every second is called a seconds pendulum. If L be its length, since the corresponding value of T is unity, we have g = 7T 2 X. (7) Hence the value of g can be determined for any place whenever the corresponding value of L is obtained. This gives the most accurate method of finding the value of g at any place, since that of L can be determined with great accuracy by observation. Any rigid body made to vibrate about a fixed horizontal axis is called a compound pendulum . It will be shown sub- sequently {see Art. 135) that in every such case there is an equivalent simple pendulum which would vibrate in the same time as the actual pendulum under consideration. This cir- cumstance renders the consideration of the ideal pendulum above discussed of the utmost practical importance. The length of a seconds pendulum at London is found to be 39*1416 inches, approximately; hence the corresponding value of g is 32*1908 feet. Pendulum observations furnish the most accurate proof of the fact that the force of gravity acts with equal intensity on all substances, as it will be seen that the length of the simple pendulum equivalent to any compound one .depends merely on the shape of the latter, but not on its material, provided it be homogeneous. Again, if T, T' be the times of small oscillation for two pendulums of different lengths, / and V; and if n and ri be the number of their respective vibrations in the same time (a day suppose), we shall have •-■^- P. (8) Hence, if the length I of any simple pendulum be known, Change of Length. 103 and also the number n of its vibrations in a day, the length L of the seconds pendulum at the place can be calculated. For, since the number of seconds in a day is 86400, we have, from formula (8), H«) 2/ - (9) The time T of vibration of a pendulum varies either— (1) by altering the length I of the pendulum, or (2) by changing the place of vibration. We shall consider these causes independently. 104. Change of Length. — Adopting the same notation as before, we get *L - r w' 2 " I ; n 2 - n 2 V - I , n 2 V - I hence T , — = — r~ 5 •'• n ~ n = ? — T~- n- I n+ n I When the change in length is a very small fraction of the whole length, n and n are nearly equal, and we have, approximately, n + n 2 Accordingly, in this case, n-n=- T ; (10) where £l denotes the change of length of the pendulum. If the pendulum be lengthened, i.e. if Al be positive, n - n' is positive, and hence the number of vibrations in a given time is diminished when the length of the pendulum is increased, as is otherwise evident. In the case of a seconds pendulum we substitute L for J in the preceding ; and since n = 86400, we get for the dimi- nution in the number of vibrations in a day, 43200 ^. JU 104 Circular Motion. Hence we can determine the number of seconds gained or lost by a seconds pendulum in a day when its length is slightly altered. As bodies in general expand slightly with an increase of temperature, an ordinary clock should go slower in hot weather, and faster in cold. The different methods of compensation for correcting the error arising from this cause will be found in practical treatises on the subject. The amount of expansion for an increase of temperature for different substances has been accurately determined, and registered in Tables. If AL denote the change in the length of a seconds pen- dulum arising from this cause, the corresponding loss or gain can be determined by (10). We add a few examples for illustration. Examples. 1 . Calculate the length of a pendulum beating seconds in London, assuming ^ = 32-19. 2. If the bob of a seconds pendulum be screwed up one turn, the serew being 32 threads to the inch ; find the number of seconds it should gain in the day in consequence, assuming L = 39'14 inches. Ans. 34'7 seconds. 3. A heavy ball, suspended by a fine wire, makes 885 oscillations in an hour. Find the length of the wire approximately, assuming the length of the seconds pendulum to be 39' 14 inches. Am. 54 feet. 4. Find the error in one day produced by an increase of 15° F. of tempera- Al 1 ture in a steel seconds pendulum ; assuming that — for 10° F. = . Ans. 4*15 seconds. 5. A seconds pendulum is lengthened -r -th of an inch ; find the number of seconds it will lose in one day. Ans. 110*4. 105. Change of Place. — The acceleration g varies* from place to place, and consequently the number of vibra- * For places at the sea level, this arises from two causes— one, the variation of centrifugal force already considered (Art. 94) ; the other, that the Earth is not an exact sphere, but is more nearly an oblate spheroid of revolution round its axis of rotation. From each of these it arises that the value of g diminishes in proceeding from the pole to the equator. It was from the observation by Richer, in 1672, that a clock lost two minutes daily when taken to Cayenne, lat. 5° N., and that when the corrected pendulum was brought back to Paris it gained an equal amount, that the variation of the force of gravity on the Earth's surface was first established. The explanation is due to Huygens. Change of Place. 105 tions in a given time will vary with the place for the same pendulum. Suppose n and ri to represent the number of vibrations made in one day by the same pendulum at two places, at which g and g are the corresponding accelerations, we have ?L-L m t or n l-t n T Sg' n* g' Hence, as before, for one and the same pendulum, From this, if L and II be the lengths of the seconds pendulum at the two places, we get nL'-L n-n^-JT- (12) It is shown by theory, and verified by observation, that the variation in the length of L, and consequently in g, at the sea level, is proportional to the square of the sine of the latitude (compare Art. 94). Thus, if L denote the length of the seconds pendulum at the equator, II that at latitude A', we have Z/ = Z + ™sin 2 A'. (13) Hence, if L x be the length of the seconds pendulum at 45° latitude, we have L x = L + — . Eliminating L, we get L' = L l -^cos2X. (14) Again, if L" be the length corresponding to the latitude A", and g" the corresponding value of g, we have m - f (cos2A"-cos2r) ST 2L' — sin (A' + X') sin (A' - A"), approximately. 106 Circular Motion. By accurate observation of the number of vibrations lost by a pendulum which beats seconds at the latitude A, when 111 taken to a latitude A', the value of y- can be determined. Such observations give -=- = zr^=, approximately, and Zi= 39-118 inches. Hence, we get If =39-118-^ cos 2\'. Again, suppose a pendulum, beating seconds at any place, taken to the height h above the Earth's surface at that place ; and let g be the value of g for the new position ; then, since the force of gravity varies as the inverse square of the distance from the Earth's centre, we have g ' = 9 (r + hf " * ( * ~ 7/ a PP roximatel y> where r denotes the length of the Earth's radius ; therefore g - g = 2A 9 r* Hence, when - is a very small fraction, the number of seconds lost in a day by the seconds pendulum is 86400 -. Suppose, for example, h = 1 mile, and r = 3956 miles, then the number of seconds lost in a day will be 22, approxi- mately. In this investigation the attraction on the pendulum of the part of the Earth above the sea level has been neglected. Examples. 1. If a pendulum, beating seconds at the foot of a mountain, lose 10 seconds in a day when taken to its summit; find approximately the height of the ^1 mountain, assuming the radius of the Earth 4000 miles, and neglecting the ^ attraction of the mountain. Am. 2444 feet. 2. How much would a clock gain at the equator in 24 hours if the length of the day were doubled. Am. 112| seconds, approximately. Airyh Investigation of Mean Density of Earth. 107 106. Airy's Investigation of the Mean Density of the Earth.— A series of important pendulum experiments were undertaken by Sir Gr. B. Airy, in the Harton coal mine, for the purpose of determining the mean density of the Earth. He found that a pendulum beating seconds at the surface gained 2| seconds a-day when taken to the bottom of the mine, 1260 feet deep. The calculations employed in arriving at this result, and in determining from it the Earth's mean density, are very intricate ; they will be found in the Eoyal Society's Transactions for the year 1856. The following is a method of arriving, approximately, at the result : — Let g, g denote the accelerations due to gravity at the surface and at the bottom of the mine ; then, by equation (11), we have g 43200 19200 Again, let r and r denote the distances of the upper and lower stations from the centre of the Earth, supposed spherical. Suppose a concentric sphere described through the lower station, then the attraction of the Earth at the upper station may be regarded as consisting of two parts — one due to the interior sphere, the other to the concJw or shell, bounded by the two spheres. Again, if we suppose this shell to be of uniform density, it exercises no attraction on the pendulum at the bottom of the mine. This can be easily ^ seen from elementary geometrical considerations (Minchin, Statics, Art. 319). Hence the part of g due to the attraction of the inner sphere is represented by /o o 7 r If /denote the acceleration at the upper station due to the attraction of the shell, we have <7=/+^=/+ We now proceed to find a general expression for the time T of vibrations for any amplitude. From (4), since \T represents the time to the lowest point on the circle, we get '■M J sin "r sm 2 Now, assuming* sin - = sin- sin <£, we get dd 2fty 2 Also, when = we have = 0; and when = a we have 7T = 2* Consequently 2 1 = 2 J- P ^ (it; 2 sm- ^ Again, substitute A; 2 for sin 2 - ; then, since (1 - A< 2 sin 2 tj>)~* = 1 + \ k 2 sin 2 tf> + ^| ¥ sin 4 tf> 1.3.5 2.4.6 and (Jw*. Cfcfc., Art. 93), 7t 6 sin 6 + &c, 1.3.5...(2m-l)ir r 2 l . J r r 2.4.6... 2m 2 we get r-4.m*t then the constant is, plainly, \xa 2 ; fdx\ 2 therefore ( -j ) = fi(a 2 - x 2 ). Hence ' 7 „ = vM^j y/ a- - x~ or sin -1 - = t^fi + a, ci where a denotes an arbitrary constant. Consequently x = a sin if^ii + a), (23) where a, a are arbitrary constants, to be determined in each case by the conditions of the problem. It may be observed that x is a simple harmonic function of the time (Art. 88). The preceding solution admits of being also written in the form x = C cos t ^/fx + C sin t y/ji, (24) where C and C are two arbitrary constants. Either of the latter equations may be regarded as the complete integral of the differential equation d 2 x It 2 Integration of —^ ± fix = 0. 113 2nd - aV =fxX ' Proceeding as before, we get in which a is an arbitrary constant ; dx or . = at . /.. . therefore, Vm + a = J /^=g = log ^ + -/^A in which a is arbitrary. Hence # + */» a - # 3 = &■** = Ae u * where A is arbitrary. Again, since (x + yV - a 2 ) (x - */x 2 - a 2 ) = a 2 , , a 2 .- we get x -, (25) when C and C are two arbitrary constants, to be determined, as before, by the conditions of the problem in each particular case. i 114 Circular Motion. 110. The equation (Px _ + ^ + v = is immediately reducible to the preceding, for it may be written d 2 x ( v\ A If we substitute z for x + -, this becomes ■=-= + jtes = ; jjl ax consequently we have ® = ~ ~ + C cos t^Z/j. + C sin £ v p. In like manner the solution of d 2 x n is x = Y. + Qe u » + C'e- U *. 111. Time of Oscillation in Cycloid. — Returning to equation (22), Art. 108, and substituting s for x, and— for /* in equation (24), we find for its integral . ... oo. *j£ + •*.*,/£. (26) In order to determine the constants c and c', suppose the particle to start from rest, at the distance s' from the ver- tex (measured along the curve) ; then we have s = and — = 0, when t = 0. Making these substitutions in (26), dt as well as in the equation derived from it by differentiation, we get. c = s% and c = ; therefore * = »' cos t^. (27) Conical Pendulum. 115 Again, when s = 0, we get or t 3 this gives the time of descent to the lowest point. If T de- note the time of an oscillation, we have -V 2a (28) Since this result is independent of the length of the arc of vibration, it follows that the time of vibration is the same for all arcs of the cycloid; accordingly the property of tautochro- nism, which in the circle holds only for very small arcs, holds in all cases for the cycloid (compare Art. 88). The foregoing value of T is the same as that for a small oscillation in a vertical circle of radius 2a. Moreover, as 2a is the radius of curvature at the vertex of the cycloid' {Biff. Calc, Art. 276), the duration of an oscillation in a vertical cycloid is the same as that of a small oscillation in the circle which osculates it at its lowest point ; as is manifest also from other considerations. It is readily seen that the time of an indefinitely small oscillation about the lowest point in any plane vertical curve is the same as that in the osculating circle at the lowest point ; and its duration is accordingly represented by irJ- 9 where p denotes the radius of curvature at the point. 112. Conical Pendulum.— Suppose the pendulum, in- stead of moving in a vertical plane, to describe a right cone around a vertical axis; and let P be the position of the revolving particle at any instant; the point of suspension ; PiV'the perpendicular let fall on the vertical axis. Also let OP = 1, L PON = 0. Then the motion of P may be considered as n taking place in a horizontal circle, whose centre is N, and radius PiV or I sin 9. i2 116 Circular Motion. Now, in order that this motion should take place, it is necessary that the resultant of the tension of the string and the weight of the particle should act along PN, and be equal and opposite to the centrifugal force ; i.e. that the resultant of the weight W, and the centrifugal force, — —. — r., should act & ' 9 IsmO in the line OP. This gives W v 2 W:—-r^- a =ON:PN, g Ism t) or _ . ,PN ; sin 2 ON v cos 6 ' therefore v = sin Bj—# (29) A COS U This gives the velocity in terms of 6 and /. Again, if T be the time of revolution, we have T ^PN ^ (30) This determines the time of revolution when the angle 0, which the pendulum makes during the motion with the ver- tical, is known. It is evidently the same as that of a double oscillation in a simple pendulum of length /cos0 or ON. The tension of the string is represented by IFsec 6. The preceding is a particular case of the motion of a particle on a smooth sphere, a problem which will be considered in Chapter VIII.. 113. Watt's Governor. — The principal of the conical pendulum was employed by Watt, in the instrument called a governor, for the purpose of regulating the supply of steam so as to maintain, approximately, a steady motion in a steam- engine. Its construction, under a form which is commonly employed, is as follows : — Revolution in a Vertical Circle. 117 Let AB represent a vertical spindle rotating with an angular velocity, whose speed is so regulated as to be always propor- tional to that of the machine. CP and CI* are rigid rods, jointed at C and C upon the revolving spindle, and having massive equal balls, P and P', fixed at their extremities. FJD and F'D' are two rods also jointed at D and I) f to the rigid rods, and jointed at F and F' to a collar, movable freely on the spindle. The collar at F, sliding freely up and down the spindle, is united to a lever which opens or closes the valve that regu- lates the supply of steam to the cylinder of the engine. When the shaft AB turns too fast, the balls P and P' fly from it, raising the collar F, and thus diminishing the supply of steam, and consequently re- ducing the speed. For a more complete discussion the student is referred to works on practical mechanics. 114. Revolution in a Vertical Circle. — We now re- turn to the question of the revolution of a particle in a vertical circle under the action of gravity. Suppose DR to be the horizontal line to the distance below which the velocity at any point is due, and let AD=h; then, by Art. 99, the velocity at any point P is given by the equation v~=2g(h-AN) = 2g(h-2asm 2 ±0), where PCA = 0. i Hence, denoting — by k 2 , and substituting a 2 ( — ) for v\ we get dt 118 Circular Motion. therefore ^=-7 /Vl-Fsin^fl, dt k\a 2 ' in which k is less than unity. If = Z. P-B^4 = J0, the time of describing any arc of the circle is represented by the definite integral (31) .y/l - /rsin 2 ^ where a and /3 are the values of

+ ;r- * =/> (it" Ad and we get, by Art. 110, * = — + ccos ^/^ * + c'sin ^~^t. Examples. 123 , ds If, when t = 0, s = s and — = 0, we get 2a/ /, 2«A |7, ,«& /. 2«A (7 • (7, *=— — + I* - cosJ — £, and -r =- ( s J — - sin — £. y \ £ / \2« dt \ g )\2a \2« This vanishes when J ^ = 7r; accordingly the time of an oscillation is n J — ; the same as when unresisted. 1 1 . A heavy particle is connected by an inextensible string, 3 feet long, to a fixed point, and describes a circle in a vertical plane about that point, making 600 revolutions per minute ; find, approximately, the ratios of the ten- sions of the string when the particle is at the highest and lowest points, and when the string is horizontal. 12. A body hangs freely from a fixed point by an inextensible string 2 feet in length. It is projected "in a horizontal direction with a velocity of 20 feet per second. Compare the tensions at the highest and lowest points of the circle which is described, assuming g = 32. -4ws. 29 : 5. 13. Show that the time of a small oscillation of a pendulum which vibrates in the air is unaffected by its resistance. The resistance is usually assumed to vary as the square of the velocity. It (dd\z can accordingly be expressed by a term of the form fx ( — J , where fi is a very small fraction ; hence in this case the equation of motion may be written d°-9 g ldd\ 2 dt 2 I (dQ\ Since /t is small, as also ^, we get as a first approximation = o cos J- t, dQ_ dt* 'de\ 2 as before. If this value be substituted in p l — \ , in accordance with the method of successive approximations, the differential equation becomes de dt t = 0, is The integral of this, subject to the condition that = o, and ^ = = 1/xd 2 + (a - f ,ua 2 ) cos t. ^+iaVccs2^ 124 Circular Motion. Also ^ = ~yjj sin wf [ a " Ia "* 2 + * *"* cos wf ) ' Hence, since — = at the end of one vibration, if T be the corresponding value of t, we have smTJj = 0, or T= irJ-- Accordingly, the duration of the oscillation is not affected by the resistance. Also, since we have in this case, cos t A- = - 1, the corresponding value of is - (a- f ^o 2 ) ; accordingly the resistance of the air reduces the amplitude of the oscillation by f jxa 2 . The successive angles of oscillation diminish according to the same law, but the time of oscillation remains the same for each. ( 125 ) CHAPTER VI. WORK AND ENERGY. 118. Work. — In all cases where force is employed in over- coming resistance so as to produce motion, work is said to be performed. Hence the conception of work involves both motion and resistance ; and therefore a corresponding effort or force to overcome the resistance. In general, work may- be defined as the act of producing a change in the configu- ration of a system in opposition to forces which resist that change. We proceed to consider how the amount of work performed in any case is to be estimated. 119. Measure of Work. — The simplest idea of work is derived from raising a weight through a vertical height ; in which case the attracting force of the Earth is the resistance overcome. The amount of work in such cases evidently in- creases in proportion to the weight of the body raised and to the height to which it is raised. For example, the work done in raising one ton through a height of 10 feet is ten times that of raising it one foot, or twenty times that of raising one cwt. through 10 feet; and so on in all cases. Hence it is readily seen that the work performed in such cases is measured by the product of the weight into the height, i.e. by Wh, where W represents the number of units in the weight, and h that in the height. In general, if we confine our attention to a single point which is moved in direct opposition to a constant resisting force, the work done is estimated by the product of the force and the distance through which the point is moved, i.e. by Pp f where P represents the force, which overcomes the equal and opposite resisting force, and p the distance passed over. 120. Gravitation Unit of Work. — From the ordi- nary units adopted in this country we derive the unit of work called a footpound, i.e. the work performed in raising 126 Work and Energy. one pound through one foot in height. This is the unit usually adopted in practical local application of work, and is called the Gravitation Unit of Work (Art. 65). The corre- sponding unit in the metric system is called the kilogram- metre, or kgm. That is the work of raising a kilogramme through the height of a metre. A kilogrammetre is 7*233 foot-pounds. The unit of work in this system varies slightly from place to place with the value of g, and this should be remembered if numerical or scientific accuracy were required (Art. 39). 121. Absolute Unit of Work. — In the absolute sys- tem the unit of resistance is that already adopted (Art. 64) as the unit of force. Thus, if we take a poundal as the unit of force, the corresponding unit of work is that done by a poundal acting through a foot. This is sometimes called the foot-poundal. It is obvious that a foot-pound is g times a foot-poundal : accordingly, any result in the former system is reducible to the latter at any place by multiplying by the corresponding value of g. Again, adopting the definition of a dyne given in Art. 64, the work done by a dyne in working through a centimetre g is called an erg ; and a foot-poundal is 421,394 ergs. In such measurements as are required in electrical and magnetic investigations, the absolute unit of work is always adopted, and the erg is the unit usually employed. 122. Horse-power. — Although in our definition of work we have taken no account of the time occupied in its performance, yet time becomes a necessary element when we come to compare the efficiency of different agents. For in- stance, if one agent working uniformly performs an amount of work in one hour which it requires another 5 hours to accomplish, the former is said to be five times as efficient. In comparing the work done by a steam-engine or other agent we usually adopt as our unit the horse-power defined by Watt. Thus an engine is said to be of one-horse-power when it is capable of performing 33,000 foot-pounds of work in one minute of time, or 550 foot-pounds in one second, and so on in proportion. Horse-power. 127 Continental writers employ horse-power as 75 kgm., that is, 542*475 foot-pounds, per second. 123. Again, the work performed in raising a body of weight W to any height h is the same whether the body be raised vertically up or brought up by any other course. The whole work is still represented by Wh, where h is the space through which the weight has been moved, estimated in the vertical direction, i. e. in that in which the resistance of gravity acts. And, generally, the work done by any uniform effort or force, acting in a constant direction against an equal and opposite force P, is measured by the product of the force into the space through which its point of application is moved, estimated in the direction in which the force acts. Thus, if a force P be supposed to act at A, and to move its point of application to B ; then if BM be drawn perpendicular to AP, the work done is estimated by Pp, or by PAs . cos 0, where p = AM, As = AB, and = L BAM. A M The work done is, therefore, regarded as positive or negative according as the angle 0, which the direction of the force makes with that of the motion, is acute or obtuse. If 6 = \tt, the direction of the motion is perpendicular to that of the force, and the work done is zero. If two or more forces act on a system, the whole work done is the sum of the works done by each force separately. If any number of forces be in equilibrium, it can be readily seen that the total work done by them for any small dis- placement is zero : from this the statical principle of virtual velocities can be immediately deduced. Examples. 1. Prove that the -whole work done in raising a system of heavy bodies, each through a different height, is the same as that of raising their entire weight through a height equal to that through which their centre of inertia is raised. 2. Find the work performed in moving a ton along 100 yards on a uniformly rough horizontal road, the coefficient of friction being -rV. Ans. 67,200 foot-pounds. 3. Show that the same work is expended in drawing a body up an inclined plane, subject to friction, as would be expended upon drawing it first along the base of the plane (supposing the coefficient of friction the same), and then raising it up the height of the plane. 128 Work and Energy. 4. "What time will 10 men take to pump the hold of a ship which contains 30,000 cubic feet of water; the centre of inertia of the water being 14 feet below the point of discharge, and each man being supposed to perform 1500 foot- pounds per minute ; assuming the weight of a cubic foot of water to be 62^ lbs. ? Ans. 29 hrs. 10 mins. 124. Work done by a Variable Force. — If the force be not constant, we may suppose the path described by its point of application divided into portions so small that for each the force may be considered constant. Hence, for the displacement ds of its point of application, Pds is the corre- sponding element ofivork s and the total work in moving through any space s is represented by the definite integral Pds. J If the direction of P makes an angle with ds, the cor- responding element of work is P cos 6ds, and the total work is represented by Pcos Qds. Again, let x, y, z, be at any instant the coordinates of the point of application of the force P, referred to a system of rectangular axes ; and let X, Y, Z y be the components of P parallel to the coordinate axes respectively ; then we have Pcos ds = Xdx + Ydy + Zdz. Hence the total work done by P in moving its point of application from one point to another is represented by {Xdx + Ydy + Zdz) taken between the two points. If the expression Xdx + Ydy + Zdz be an exact diffe- rential, i. e. if Xdx + Ydy + Zdz = du 9 where u is a function of x, y, z, then the integral \(Xdx + Ydy+ Zdz), taken between any two points, is a function of the coordinates of those points ; and the work done is accordingly a function Forces directed to Fixed Centres. 129 of the extreme coordinates solely. When this is so, the mutual forces between the parts of a system always perform or always consume the same amount of work during any motion whatever by which it can pass from the one particular configuration to the other ; hence such a system is called a conservative system of forces. In general, for any system of forces acting at different points, the total work W done for any finite displacements is represented by TT=S Pdp = 2 {Xdx + Ydy + Zdz), (1) where the summation extends to all the forces of the system. 125. Forces directed to Fixed Centres. Potential. — If the force F be directed to a fixed centre, and if r be the distance of its point of application from the centre, then the corresponding element of work is represented by Fdr; and the total work, when the point is moved from a distance / to a distance r", is represented by Fdr. Jr If F be a function of r represented by n'(r), then the value of this integral will be m!*(0- *('•'))• In the law of attraction which holds in nature we have F=- — ; and the expression /u( — , J represents the corre- sponding work in moving a unit of mass from the distance / to the distance /'. Hence the work done in the motion of a unit mass from an infinite distance to the distance r is represented by -. The function 2 — in the case of the ordinary law of gravi- tation is called the potential of the system of attracting masses. This potential function is usually represented by V ; and if dm be the element of attracting mass, and r its distance from a point P, then V, the potential at P, is denoted by dm V =*- (2) extended through all points in the attracting system. K 130 Work and Energy. Again, if a number of forces F, F\ F'\ &c, be directed to fixed centres, and if r, /, r", &c, be the corresponding distances, then the total work is represented by \Fdr + \F'dr f + \F"dr" + &c, taken between the limiting positions. If the forces be each a known function of the distance from the corresponding centre of force, the result can, in ge- neral, be immediately integrated, and the work is a function of the initial and final positions of the points of application solely. Consequently such a system of forces is always a conservative system. Example. If m. m be the masses of two particles attracting each other with a force u*^- where r is their distance apart, show that the work done when they have r 2 ' mm moved from an infinite distance apart to the distance ris /x ■ -. 126. Potential of an Attracting Spherical Mass.— If each element of the surface of a sphere be divided by its distance from an external point, and the sum taken over the entire surface, this sum is readily shown by elementary integration to be equal to S ~$ where S is the whole surface of the sphere, and R the distance from its centre to the external point. Hence, if a mass m be uniformly spread over the surface of the sphere 8, we have a *. » r Jx From this it follows at once that in a solid sphere of mass M, for which the density is constant through each concentric couche, we have r-x£-J (4) r Jx That is, the potential is the same as if the whole mass were concentrated at the centre of the sphere. Consequently the work done by an attracting sphere M, Work done by a Stress. 131 in moving a unit of mass from the distance R' to the distance jR, measured from the centre, is It may be remarked that it can be readily seen from (4) that a homogeneous sphere attracts an external mass as if the whole mass of the sphere were concentrated at its centre. 127. Work done by a Stress. — If two equal and oppo- site forces, each represented by F, act respectively at the A r B points A and B, along the f\~~ A line connecting these points, p M A B N F to find the element of work for a small displacement. Suppose A! and B to be the new positions for an indefinitely small displacement, and let fall the perpendiculars AM and B'N on the line AB ; then the elements of work are represented by F. AM and F. BN. Hence their sum is F{AM + BN) = F(AB' - AB), or FAs, where As denotes the indefinitely small change in the distance between the points of application of the forces. Hence, if the points A and B be rigidly connected, as the distance AB is invariable, the total work done by the forces for any displacement is zero. Also the point of application of a force maybe transferred from any one point to any other on its line of action without altering the work done, provided the distance between the two points is invariable. The pair of equal and opposite forces that two bodies exert on one another in accordance with the general prin- ciple of action and reaction is called in modern treatises a stress. When the forces act away from each other, as in the figure, the stress is called a tension ; when they act towards each other it is called a pressure. Hence the work done by a stress is positive or negative according as the change of distance between the points of application is in the direction of the mutual action of the forces or in the opposite direction. Also in the case of a rigid body it follows that the total work done by the internal forces of stress is always zero, k2 132 Work and Energy. 128. Body with a Fixed Axis.— To find the work done by a force acting on a rigid body which is capable of turning round a fixed axis. Suppose the force R resolved into two components — one parallel, the other perpendicular to the fixed axis. The former does no work, since it is perpendicular to the direction of motion of every point in the body. Let the latter component be repre- sented by P, and suppose it to act in the plane of the paper ; the fixed axis being perpendicular to that plane, and meeting it in the point 0. Let N be the foot of the perpendicular drawn from to the line of action of P ; then by the last Article we may take i^as the point of application of P. Suppose now the body to receive a small angular displace- ment Ad round the fixed axis in the direction of the arrow ; then, if ON ' = p, the displacement of N will be p&6, and the corresponding element of work is P/;A0, or A0 multiplied by the moment of the force R with respect to the fixed axis. Again, if we suppose a pair of equal, parallel, and opposite forces to act on the rigid body ; then, provided the plane of the pair is perpendicular to the fixed axis, the work clone by the pair is evidently, from what precedes, represented by the moment of the pair multiplied by the small angle of rotation. And if the pair continue to act on the body, the work done by it during any rotation is represented by the product of the moment of the pair by the angle, in circular measure, through which the body has rotated. Example. A pivot or screw turns round a central axis and presses against a rough, plane ; find an expression for the work expended on the friction which acts on the circular end of the pivot in one revolution round its axis. Let Q denote the entire normal pressure between the pivot and the plane, /j. the coefficient of friction, supposed constant, a the radius of the end of the pivot. This end may he regarded as consisting of an indefinitely great number of concentric circular rings. If r he the radius of one of the rings, dr ^ its breadth, then the area of the ring is 27rrdr, and the corresponding friction, taken over the entire ring, is represented by — ~rdr. Hence the corresponding Measure of Kinetic Energy. 133 -work for one revolution is — —r' 2 dr. Integrating, we get %ir/j.Qa for the en- re- f tire work expended. In this investigation the nonnal pressure Q has been sup- posed to be uniformly distributed over the end of the pivot. 129. Energy. — Energy is the capacity of doing work. For instance, a spring when bent by pressure contains a cer- tain amount of energy stored up in it ; thus the mainspring of a watch, by the energy which it possesses, maintains the motions of the works until that energy has been expended. Again, a quantity of air, when compressed into a smaller volume, possesses energy, and can perform work when occasion requires; for example, in projecting a bullet from an air-gun. Also a raised weight is capable of doing work, and is therefore said to possess energy. For instance, the motion of a clock is maintained by the energy of its descend- ing weights. The energy of a weight IF raised to a heights above the ground is measured by Wh, that is, by the work it is capable of performing by its descent to the ground. In general, when the configuration of a system is altered, it has a tendency to return to its former state, and in effecting this return is capable of doing a certain amount of work. This capacity of doing work, arising from change of configuration or of relative position in a system, is called potential energy ; the work employed in producing this change being in a sense accumulated. For example, if two bodies which attract one another are separated, they have a tendency to rush together, and in so doing are capable of overcoming a certain amount of resistance. Again, a body in motion possesses a certain amount of energy which is measured by the work it is capable of per- forming before being brought to rest. This latter is called the Kinetic energy of the body. We proceed to consider how its amount is measured. 130. Measure of Kinetic Energy. — The measure of the kinetic energy of the mass m moving, without rotation, with the velocity v, is easily found. For, suppose the mass acted on by a uniform resistance R in the direction of its motion, and let R = mf; then, if v be the initial velocity and s the space described before coming to rest, we have, by Art. 37, v 2 = 2/8 ; hence \mv 2 = Rs. J 134 Work and Energy. Accordingly, the work which a mass m moving with the velocity v is capable of performing before being brought to rest is J mv 2 . Hence its kinetic energy is equal to half its vis viva, and is represented by \ mv 2 . Examples. 1. A train of 60 tons, moving at the rate of 15 miles an hour on a horizontal railway, runs, when the steam is shut off and the breaks applied, through a quarter of a mile before stopping. Find in lbs. the mean resistance, and its time of action. Ans. 770 lbs.; 2 minutes. 2. The breadth of a river at a certain place is 100 yards, its mean depth is 8 feet, and its mean velocity 3 miles an hour. Calculate its horse-power, as- suming a cubic foot of water to weigh 62| lbs. Here the quantity of water which passes per minute is 633,600 cubic feet ; and the required answer is easily seen to be 363 horse-power. 3. A shot of 1000 lbs., moving at 1600 feet per second, strikes a fixed target. How far will the shot penetrate, the target exerting on it an average pressure equal to the weight of 12,000 tons? Ans. \\ ft., approximately. 4. Determine in ergs the kinetic energy of a mass of one hundred pounds moving with a velocity of one foot per minute. Ans. 5853. 5. A heavy particle resting on a rough inclined plane, and attached by a string to a fixed point on the plane, is projected from the lowest point of the circle in which it moves in the direction of the tangent, (a) Find the velocity necessary to carry the string to a horizontal position ; (b) If the particle descending from this position reach the lowest point and remain there, deter- mine the coefficient of friction. 6. A ball moving with a velocity of 1000 feet per second has its velocity f- reduced by 100 feet in passing through a plank. Through how many such y ■ planks would it pass before being stopped ; assuming the same amount of work to be performed in overcoming the resistance of each plank ? Ans. 5^. 131. Energy due to a Variable Foree. — If a va- riable force F act at the centre of inertia of a mass m, in the direction of its motion, we have, by Art. 68, _. dv dv F = m — = mv—, dt ds or Fds = mvdv; accordingly, if V and V\ be the initial and final velocities of m, we have i» ( Pi* -F.V ('*'*• (6) Jo Energy due to a Variable Force. 135 From this we infer that if a variable force F act on a mass 0t, in the direction of its motion, the work done by it is measured by half the corresponding change in the vis viva of the moving body, or by the change in its kinetic energy. In general, let X, F, Z, as before, denote the components, parallel to the axes of x, y, z, of the force acting on the mass m ; then, by Art. 68, we have _ d 2 x _ d 2 y „ d 2 z X=m— 9 Y=m-f f Z=m—. dt % dtf dt- Multiply the first by dx, the second by dy, and the third by dz, and add ; then Xdx + Ydy + Zdz = mi-jdx + -~ dy + -^ dz J sHIHS)') = \ md (v 2 ) . Hence, if V Q and Vi be the initial and final velocities, im{V? - V 2 ) -\{Xdx + Ydy + Zdz), (7) the integral being taken from the initial to the final position of the centre of inertia of m. Hence we infer that in this case also the work done by the forces during any motion is measured by half the change in the kinetic energy of the moving mass. If after the lapse of any time the velocity of m become equal to its original value, the work done in that interval by the forces which accelerate the motion is equal to that done by the forces which retard it. In the case of a central force, represented, as in Art. 125, by fj.(r)- = -, and M, the mass of the fly-wheel = 7 '25 x f x 62^ . tt . lbs. 5 Also (Int. Calc, Art. 201), i"= M (Q) ; hence the required answer is 805 foot- pounds, approximately. 138 Work and Energy. 2. A rod of uniform density can turn freely round one end ; it is let fall from a horizontal position ; find its angular velocity when it is passing through the vertical position. Am. /_£, where a is the length of the rod. 3. Two masses if and 31' are connected as in Atwood's machine (Art. 78) ; find the acceleration when the mass fi of the revolving pulley is taken into account. If v be the common velocity of 31 and 31' at any instant, and fik 2 the moment of inertia of the pulley; then the entire vis viva of the system is repre- sented by (M + 31') v 2 + fxk- u> 2 . Hence, if z be the distance fallen through from rest, we have (M + 31 ' ) v 2 + (i&a 2 = 2g {31 - M') z. Also v = aw ; .'. v 2 {{M + 3T) a 2 + fik 2 } = 2ga 2 (M - 31') z. Again, the acceleration therefore If the pulley be supposed a homogeneous cylinder, k 2 = ^—, and /becomes /= /= dv "di~ ga'< dv az {M- 31') {31 + 31') a 2 + fd? M+M'+fr 4. Find in the same case the tensions of the strings. v 2M'a i + l xk 2 ^ 23fa 2 + nk 2 AnS ' M9 {M+M')a^^ Mg {M + W)a 2 +^ For a homogeneous pulley these become __ W+fl i3f+ ix Mg — , and M. g 2{3f+3f') + fJ .' "2{M + M') + fi 5. A homogeneous cylinder, of weight IF, is rotating round its axis, sup- posed horizontal, with an angular velocity w ; find to what height it is capable of raising a given weight F, before coming to rest. r 2 ; a- + k z at . ' 2ga i s sin hence ( — ) = — „ , , 3 (ds\ 3 2gd l s si It) = a 2 + k therefore, by differentiation, d-i a 2 g sin i a- 1 a- + k 2 140 Work and Energy. This shows that the acceleration down the plane is constant. Hence the velocity acquired, and the space described in any time, can at once be determined. If the cylinder be homogeneous, we have k 2 = \a 2 {int. Calc, Art. 201), and the acceleration/ in this case is f g sin i. This shows that the velocity of the centre of gravity of the cylinder is f that acquired by a particle, in the same time, in sliding down a smooth inclined plane of the same inclination. If the cylinder be hollow, k = a, and accordingly f=\g sin i. 2. A mass M draws up another, M', on the wheel and axle; find the motion. Let a be the radius of the wheel, a' that of the axle ; then, as in Ex. 3, Art. 133, it is easily seen that we get l^pj "(Ma°~ + M'a" 2 - + fuW) = 2g{Ma - M'a') 6 + const. Hence, by differentiation, d 2 9 _ g(M a-M'a') d6 Accordingly, if = 0, and — = 0, when t = 0, we get for the angle turned at through in the time t, x Ma — M'a' e= * 9t Mtf+M'a't + frifi' 3. Find the tensions of the strings in the same case. M'a { a + a) + ^ 2 w Ma ( a + a ') + M 2 Am ' Mg Ma^M'a'* + nk* 9 Ma? + M' 'a'* + M 2 ' 4. Find the velocity acquired by the centre of a hoop in rolling down an_ in- clined plane of height h. -Ans. s j l jj lm 135. Work done by an Impulse. — If a mass H moving with a velocity V receives an impulse in the direction of its motion, and if V f be its velocity after the impulse, then the change in its kinetic energy is \M{J"- F 2 ) =M(V'- V).i{V'+ V). But M{ V - V) measures the impulse. Hence the work done by the impulse is measured by the product of the momentum, which measures the impulse, by half the sum of the velocities before and after the impulse. For example, a bullet m in passing through a plank expe- riences a definite amount of resistance, measured by the thickness and by the resisting force ; but this equals half the loss of vis viva of the bullet, or \m (v~ - v n ) = m [v - v) . \ [v + v), Compound Pendulum. 141 where v and v are the velocities with which it meets and leaves the plank. Hence the momentum m (v - v) commu- nicated to the plank varies inversely as v + v : consequently the greater the velocity of impact the less the momentum imparted. This explains how a bullet with a high velocity can pass through a door without moving it on its hinges. 136. Compound Pendulum. — A solid body oscillating under the action of gravity, around a fixed horizontal axis, is called a compound pendulum. The motion of such a body is readily reduced to that of the corresponding simple pendulum, as follows : Let the plane of the paper re- present that in which the motion of G, the centre of inertia of the body, takes place, and let be the point in which the fixed axis intersects that plane. Draw OY vertically downwards, and let GO = a, M = mass of the body, let L GOT = 9. Suppose the pendulum to start from rest, when 9 = a ; then, in the time t, the point G will have descended through the vertical height a (cos 9 - cos a) . Also the vis viva of the body at the same instant (Art. 133) is represented by Also ;f)W, Hence, by the principle of work, Art. 132, we have /(— -) = 2 Mga (cos 9 - cos a). If the moment of inertia I be represented by MR 2 , the latter equation becomes 2ga (cos if - cos o K- dt where IT is the radius of gyration of the body [Int. Cede., Art. 197), relative to the axis of suspension. 142 Work and Energy. Hence, by differentiation, $,£-•-«■ (13) Comparing this with the corresponding equation for the motion of a simple pendulum (Art. 101), we see that the motion is the same as that of a simple pendulum of length a Again, if Mk 2 be the moment of inertia relative to an axis through the centre of inertia parallel to the axis of suspension, we have {Int. Calc, Art. 196), K 2 = a 2 + k 2 ; K 2 k 2 hence I = — = a + — . (14) a a v J The point is called the centre of suspension. If OG be produced until OC = I, since the body moves as if its entire mass were concentrated at the point C, that point is called the centre of oscillation. Again, if through C a right line be drawn parallel to the axis of suspension, all the points of this line move like the point C, i.e. as if they were freely sus- pended from the axis of rotation. This line is called the axis of oscillation. Again, since OG . GO = k 2 , the axes of suspension and oscillation are interchangeable, i. e. the time T of an oscilla- tion is the same for both, viz., T = tr / . ; V <*g By varying the axis of suspension, the time of a small oscillation will also, in general, vary. For parallel axes, T obviously is a minimum when a = k, and the corresponding time of a small oscillation = tc \ In order that this should be the smallest possible, the axis of suspension must be parallel to that axis round which the moment of inertia is least {Int. Calc, Art. 217). Determination of the Force of Gravity. 143 If the axis of suspension of a compound pendulum be inclined at an angle a to the vertical, it is readily seen that the preceding investigation holds good, provided g sin a be substituted for g throughout. Again, as in Art. 101, the time of any motion of a com- pound pendulum is represented by an elliptic integral. Also, if a solid body make a complete revolution round a horizontal axis, the time of revolving through any angle can be reduced to that for the corresponding oscillatory motion of a particle. Examples. 1. A uniform circular plate, of radius a, makes small oscillations about a hori- zontal tangent ; find the length of the equivalent simple pendulum. Ans. £ a. 2. Find the position of the axis with respect to which a uniform circular plate will oscillate in the shortest time. Ans. The axis is at a distance of half the radius from the centre. Length of the equivalent pendulum = a. 3. Find the centre of oscillation of a homogeneous sphere, of radius a, oscil- lating round a horizontal tangent to its surface. Ans. At a point f a below the centre. 4. Find the ratio of the times of oscillation of a homogeneous solid sphere, and of a spherical shell of equal diameter, each being taken with reference to a horizontal tangent. j± ns% -^21 : 5. 5. A sphere of radius a is suspended by a fine wire from a fixed point, at a distance I from its centre ; prove that the time of a small oscillation is repre- I5P + 2a 2 sented by it j — — (1 + |sin 2 ^a), where a represents the amplitude of the vibration. 6. If the semiaxes of a uniform elliptic disc be 2 feet and 1 foot, and it be suspended from an axis perpendicular to its plane through one of its foci, find the time of a complete oscillation under gravity. Ans. V3 \ 9 137. Determination of the Force of Gravity. — We have already seen (Art. 103) that the value of g at any place can be determined from the length of the seconds pendulum at the place. To apply this it is necessary to know the nu- merical value of Two methods have been devised for this purpose — one employed by Borda, Arago, Biot, and others ; the other first 144 Work and Energy. used by Bohnenberger, and afterwards brougbt to great per- fection by Captain Kater. In the first method the compound pendulum, supposed made of a material of uniform density, has such a shape that its radius of gyration can be calculated mathematically, as also the distance of its centre of inertia from the fixed axis. The second method depends on the reciprocity of the centres of suspension and oscillation. Kater' s compound pendulum consisted of a heavy bar having two apertures at opposite sides of the centre of inertia, through which knife edges passed, on either of which the body could be supported. On the bar was placed a ring capable of being moved up or down by means of a screw. Kater moved the ring until the times of oscillation round the two axes were equal ; in which case, by the preceding, the distance between the axes is equal to the length of the equi- valent simple pendulum. The distance, /, between the axes having been accurately measured, the value of g was calcu- lated from the formula g = — , where T denotes the time of an oscillation. Kater published an account of his observations in the Philosophical Transactions, 1818, 1819. For a more detailed account of this method the reader is referred to Bouth's Rigid Dynamics, Arts. 100-108. 138. Motion of a Rigid Body round a Fixed Axis. — In general, let a force P, in a direction which is at right angles to the fixed axis, act on a body; then for a small angular motion cl9 the work done by P is, by Art. 128, re- presented by PpdO. Again, as this work is equal to the corresponding change in the kinetic energy of the body, we have PpdQ = im z d(^j= Ml/~ di Hence we get d~9 _ Pp Moment of impressed force .- -. d? = Mtf = Moment of inertia ^ ' W ft W+jF \ V Motion of a Rigid Body round a Fixed Axis. 145 Examples. 1. A uniform circular plate of 1 foot radius and 1 cwt. revolves round its axis 5 times per second ; calculate its kinetic energy in foot pounds. \f Ans. 863, approximately. 2. A bent lever ACB rests in equilibrium when AC is inclined at the angle o to the horizontal line ; show* that when this arm is raised to the horizontal posi- tion it will fall through the angle 2a, C being supposed fixed. 3. A homogeneous cylinder, of mass M, and radius a, turns round a hori- zontal axis ; a fine thread is wrapped round it, and has a mass M' attached to its extremity. Find the angular velocity of the cylinder when M' has descended through the height h. A 2 _ ± M '0 h KS ' W a?{M+2M')' 4. A right cone oscillates round a horizontal axis, passing through its vertex and perpendicular to the axis of the cone ; find the length of the equivalent simple pendulum. Ans. — — — , where h is the height of the cone, and b the radius of its base. oh 5. If in the last example the cone be let fall from the position in which its axis is horizontal, find its angular velocity when in the lowest position. 4A 2 + b 2 6. In the same case find the pressure on the fixed axis, at the lowest position of the body, arising from centrifugal force (Art. 98). Ans. — 7T-75 — r^, where W represents the weight of the cone. 2 4/r + 0- 7. A thin beam, whose mass is M and length 2a, moves freely about one ex- tremity attached to a fixed point whose distance from a smooth plane is b, (b < 2a) : the other extremity rests on the plane, the inclination of which is o. If the beam be slightly displaced from its position of equilibrium determine the time of its small oscillations. Indian Civil Service Exam., 1860. In this case the beam may be regarded as turning round the perpendicular on the plane. 8. A bullet weighing 50 grammes is fired into the centre of a target with a velocity of 500 metres a second. The target is supposed to weigh a kilogramme, and to be free to move. Find, in kilogrammetres, the loss of energy in the impact. Lond. Univ., 1880. Ans. 635'6. 9. "When the w-eight P of the pulley is taken into account, show that equa- tion (9), Art. 76, becomes in which the pulley is supposed to be of uniform density and thickness. L 146 Work and Energy. 10. If the motion of a solid body acted on by attracting forces be a pure ro- tation, the velocity « of rotation at any instant will be given by the equation JKF («*-»>) = 2 (F- To), where V represents the potential of the attracting forces. 11. A hollow cylinder rolls down a perfectly rough inclined plane in 10 mi- nutes ; find the time a uniform solid cylinder would take to roll down the same plane. Ans. 5 \/Z minutes. 12.*The particles composing a homogeneous sphere of mass M and radius It were originally at an infinite distance from each other : find the work done by their mutual attraction. Suppose the sphere in question to have been formed by the condensation of an indefinitely diffused nebula ; and imagine the sphere divided into a number of concentric spheres. Let M' be the mass contained in the sphere whose radius is r ; then we have M' = M— • IP Also, if dM'he the mass bounded by the spheres r and r + dr, then Accordingly the work done in condensing dM' , in consequence of the attraction of the interior mass M', is, by (5) Art. 126, fx — dM' =3fi-^- r*dr. Hence the whole work done in the condensation of If is Jf 2 M* f* 7 3 M °~ ( 147 ) CHAPTER VII. CENTRAL FORCES. Section I. — Rectilinear Motion. 139. Centre of Force. — We next proceed to consider motion under the action of a force whose direction always passes through a fixed point, and whose intensity is a func- tion of the distance from that point. The fixed point is called the Centre of Force; and the force is said to be attrac- tive or repulsive according as it is directed towards or from the centre. If we assume that two particles of equal mass, placed at the same distance from a centre of attractive force, are equally attracted towards the centre, when they are conceived placed together, the whole force acting on them — considered as one mass — will be double that which acts on one of the particles. Similarly, if any number (n) of equal particles be placed together, the whole force will be n times that which acts on a single particle. Hence it follows that in such cases the whole attracting force is proportional to the number of par- ticles, i. e. to the mass of the attracted body — provided the attracted mass be of such small dimensions that the lines drawn from its several points to the centre of force may be regarded as equal and parallel. Accordingly the force, in this case, is proportional to the attracted mass ; consequently the acceleration produced by it is independent of the mass attracted, and is a function of the distance from the centre of force only. 140. Attraction. — The acceleration due to an attractive force, at any distance, is called the attraction of the force, and is, as we have seen, independent of the mass of the attracted particle. Consequently the measure of an attractive force at any distance is the velocity per second which the L 2 148 Rectilinear Motion. central force could generate in one second, in its own direc- tion, if it were conceived to act uniformly during that time. For instance, g, i. e. the velocity acquired in one second by a falling body (Art. 38), measures the attractive force of the Earth, at any place, and is, as already stated, the same for all bodies at that place. 141. Rectilinear Motion. — If the particle acted on be originally at rest, or be projected in the line joining its posi- tion to the centre of force, its motion will take place in that right line. Taking this line for the axis of x, and the fixed centre as origin, we have for the equation of motion (Art. 21), £--*• (1) where F represents the attraction at the distance #, which is taken with the negative sign because it tends to diminish the velocity. We shall illustrate equation (1) by applying it to a few elementary cases. 142. Force Varying as the Distance. — If the force be proportional to the distance from the fixed centre, we may assume F = fix ; then, for attractive forces, the equation of motion becomes d*x d 2 x . ^ = -"*' or ^ + "* = - (2) This equation has been already considered in Art. 109, and accordingly we have x= C cos t yjl + C" sin t y/fi. (3) The constants C and C are determined from the initial circumstances of the motion. For example, if the particle start from rest, at the distance a from the centre of force ; then, when t = 0, we have x = a and — = : this gies C = a, and C"=0; Inverse Square of Distance, 149 and consequently x = a cos t ^/ju. This determines the posi- tion of the particle at any instant, and shows that the motion consists of a simple harmonic vibration. Again, if (f - t) ^//ul = 2tt, it is evident that the values of x rfv and of — are the same at the end of the time f as at the dt time t : this shows that the motion is oscillatory, and that the time of a complete vibration is ——. (Compare Art. 111.) y/fl For a repulsive force the equation of motion is w-~ (4) Accordingly (Art. 109), we have x = Ce u ~* + C f e~ u K To determine the constants : suppose, as in the former case, the particle starts from rest, at the distance a ; then a - C + C, and C - C = 0. Hence x = \a [e u * + e* 1 *) . (5) 143. Inverse Square of Distance. — In the case of the law of nature, in which the attractive force varies as the inverse square of the distance, we have F= — ; and the dif- x 1 erential equation of motion is df x 2 Multiplying by 2dx, and integrating, we get 'dxY 2 M — 7 = const. dt J x 150 Rectilinear Motion. Hence, if the particle be supposed to start from rest, at the distance a, This equation determines the velocity at any distance from the centre of force. Again, extracting the square root, and transforming, we get V^tf-.-^Lr- (7) \x a The negative sign is taken since, in the motion towards the centre of force, x diminishes as t increases. To integrate this equation, assume x = a cos 2 9 ; then = — — , and dx = - 2a sin 9 cos 9 cffl ; consequently */%dt = 2a % cos 2 9 d9 ; ^- (9 + \ sin 29) + constant. Again, the constant vanishes, since t and 9 vanish when x = a ; .:t = J^(0 + isin20). (8) Hence the time of motion from the distance a to the distance x is •_ * = ii ( a oos ' 1 J^" + v'* («-*))• (9) Also the time of motion to the centre of force is 2 >J2/ Application to the Earth. 151 Again, if the body be supposed to start from an indefi- nitely great distance we have, making a = go in (6), *> 2 = ^. (10) iff 144. Application to the Earth. — We have seen, in Art. 126, that the attraction of a homogeneous sphere is the same as if its mass were concentrated at its centre. Hence, the results of the last Article can be readily applied to the approximate determination of the motion of a body falling from any height above the Earth's surface, all resistance of the atmosphere being neglected. In this case g measures the Earth's attraction at its sur- face ; hence, if R denote the Earth's radius, we have fx = gR 2 > and if this value be substituted for /u, we can readily deter- mine the velocity and time of motion in any particular case. For instance, the velocity Fwith which a body falling from the height h would reach the surface of the Earth is given by the equation Also, by (9), the time of motion in seconds is IB + hLR + h . , I h \h ) where R and h are expressed in feet. If R = nh, this becomes h (1 + n) { 1 + n . , 1 , sm" 1 , + 1 When n is a large number this becomes, approximately, if (• * k\ 152 Rectilinear Motion. If the body be supposed to start from an infinite distance, the velocity with which it would reach the Earth is given by the equation v* = 2gR. (13) 145. Comparison of Attraction of Different Sphe- rical Bodies. — Let If, M' denote the masses of two spheres ; 3, £' their mean densities ; r, r their radii ; f,f their attrac- tions at their surfaces, respectively : then we have MM' For example, if D be the mean density of the Earth, and R its radius, then/, the attraction at the surface of a planet of radius r and mean density §, is given by the equation f=9~- (14) If the mean densities be the same for both, we have J = g R- If we assume the mean density of the Sun to be one- fourth that of the Earth, and its radius 104 times that of the Earth, then the velocity acquired in one second by a falling body at the Sun's surface is approximately represented by 26g. In the case of the mutual attraction of two spheres, it is often convenient to assume the origin at their common centre of gravity, which remains a fixed point during the motion. For instance, if two equal spheres, each of radius r, be placed at a given distance apart, and left to their mutual attraction, this method may be employed to find the time they would take to come together. Let 2a be the initial distance between their centres, and assume the origin at the middle point of the line joining the centres. If x be the distance of the centre of either sphere from at any time ; then j-j represents the corre- Examples. 153 sponding attraction, and the time required is, by (9), repre- sented by the expression where fx can be determined by the equation r 2 J 9 B 11 ' Examples. 1. If k be the height due to the Telocity F"at the Earth's surface, supposing / its attraction constant, and S the corresponding height when the variation of gravity is taken into account, prove that 1 J__J_ h~ H~ r' 2. If a man weigh 10 stone on the Earth's surface, calculate, approximately, \f his weight if he were transferred to the surface of the Sun. Arts. 1 ton, 13 cwt. 3. Calculate, approximately, the velocity with which a body falling from an ^ indefinitely great distance would reach the surface of the Earth, neglecting all forces besides the Earth's attraction, and assuming R = 4000 miles. Ans. 7 miles per second. 4. Calculate, in like manner, the velocity with which a body falling from an indefinitely great distance would reach the surface of the Sun. Ans. 364 miles per second. 5. In a work erroneously attributed to Sir Isaac Newton, it is stated, that if two spheres, each one foot in diameter, and of a like nature to the Earth, were distant by but the fourth part of an inch, they would not, even in spaces void of resistance, come together by the force of their mutual attraction in less than a month's time. Investigate the truth of this statement. Sch. Ex., 1883. Equation (15) gives in this case for the time, in seconds, sin- 1 (-) 49 . , /1\ 1 — sur 1 [ - ) -f — - 90 \7/ 8V3 This gives about 5 minutes and 38 seconds. If the question be solved on the assumption that the attraction is constant during the motion, and equal to that when the spheres are touching, the time required is readily found to be, approximately, = 100 VTl = 5 m. 32 sees. It may be observed that the former result follows from this immediately by application of formula (12). 154 Rectilinear Motion. 6. Show that if a sphere, of the same density as the Earth, attract a particle placed at the n th part of its radius from its surface, the time of motion to the surface is the same as that of a particle moving to the Earth from a distance equal to the n th part of its radius. 7. What is meant hy the Astronomical Unit of Mass ? The astronomical unit of mass is that mass which attracts a_ particle placed at the unit of distance so as to produce in it the unit of acceleration in the unit of time. 8. If a foot and a second be taken as the units of length and time, calculate, approximately, the number of pounds in the astronomical unit of mass. Let M denote the mass of the Earth, and m that of the astronomical unit ;. then we have M , M — : m = g : 1, or m = — , fi » gr * where r is the radius of the Earth in feet. Now assuming D, the mean density of tbe Earth, to be h\ tbat of water, the mass of a mean cubic foot of the Earth is 344 lbs. approximately. If we assume the radius of the Earth to be 40 00 miles, we get ?L = _ — x 344 = 951,000,000 lbs., approximately. gr> 3 g 9. Taking the value of gravity as 981 in centimetres and seconds, and the Earth's radius as 6-37 x 10 8 centimetres : find the Earth's mass in astrono- mical units. ^-ns. 398 x 10 18 . 146. Force any Function of Distance. — If the force be attractive, and vary inversely as the n th power of the dis- tance, the equation of motion becomes Multiply, as before, by 2dx, and integrate ; then fdxY 2 in 1 — ] = — !-— — - + const., \dtj n-lx 71 - 1 or 2 M 1 n-1 x n ~ + const. If the attracted particle start from rest at the distance ct f we have n-\ W'- 1 a" >-*Tfi- = )- (16) Elastic Strings. 1,j5 This determines the velocity at any distance from the centre. In general, if F= fi(jS(x) 9 we have and, proceeding as before, we get 'dx\ 2 , + 2ju J 2b. In this case x vanishes, and conse- quently T also, when b + (c-b) cos Jjt = 0. The corresponding velocity is easily found to be J go {c - 26) b As the tension of the string vanishes at this instant, the body may be regarded as projected upwards with the fore- going velocity. The height, h, to which it would ascend is given by the equation *-£(«-»). (25) The body will afterwards fall to the origin, and the subse- quent motion will be as before. 149. Weight Dropped from a Height. — Next sup- pose the weight attached to the string, and dropped from a height h, vertically above the lower extremity of the string when hanging freely and unstretched. The solution is con- tained in the preceding investigation : for the maximum ex- tension c of the string is given by (25), and is represented by c = b + yb{b + 2h). (26) In practice it is found that Hooke's law does not hold beyond certain limits which are attained long before the string is broken. It is interesting to consider whether in any particular case the string will be broken or not by the fall, assuming Hooke's law still to hold. A given string is capable of supporting only a certain weight, called its breaking weight. Denote this weight by B ; Weight Dropped from a Height. 159 then e, the corresponding extension of the string, is found, by Hooke's law, from W £=— e, (27) and the string will break or not according as the maximum extension, given by the preceding an alysis, is greater or less than e ; that is, according as i + ^/b [b + 2h) is greater or less than e. Again, if b and e be both given, the least height of fall, h, in order that the string should break, is got by substituting e for c in (25), and is h = e ±zm. (28) Suppose the weight W to be the n th part of B, i. e. let e = lib, and we have h = e [%n - 1). Thus, for instance, a weight \ of the breaking weight, dropped from the height e, should suffice to break the string. The preceding analysis applies also to the vertical oscilla- tions of rods supporting heavy weights ; and many interesting practical questions are explained thereby — for instance, the danger to the stability of a suspension bridge arising from the steady march of troops over it. — See Poncelet, Mecanique Industries, 'Arts. 332-345. Examples. 1. A heavy particle attached to a fixed point by an elastic string is allowed to fall freely from this point. Show that the elastic force at the lowest point is given by the equation m total fall- F= 2W— : t-7-^j extension of string where W is the weight of the particle. 2. A heavy particle attached to a fixed point by an elastic string hangs freely, stretching the string by a quantity e. It is drawn down by an addi- tional distance/; determine the height to which it will rise if p - e- = iae, a being the unstretched length of the string. Am. 2a. 3. A heavy body is attached to a fixed point by an elastic string, which passes through a fixed ring, the natural length of the string being equal to the distance between the ring and the fixed point. (a) If the body receive an impulse, it will describe an ellipse round the place it would occupy if suspended freely. (h) When does this ellipse become a right line ? 160 Central Orbits. 4. A particle is attached by a straight elastic string to a centre of repulsive force, the intensity of which varies as the distance ; the string is at first at its natural length. Find the greatest distance from the centre of force to which the particle will proceed, and the time the string takes to return to its natural length. 5. Two bodies, TFand TV, hang at rest, being attached to the lower end of a fine elastic string, whose upper end is fixed : supposing one of them, W , to drop off, find the subsequent motion of the other. Let a be the natural length of the string ; b its extension of length for the weight W; c that for the weight W ; then, at the end of any time t, from the commencement of the motion x, the depth of W below the fixed point is given by the equation x = a + b + c cos t J 6. Two particles, connected by a fine elastic string, are moving in the direc- tion of the line joining them with equal velocities, their distance being the natural length of the string ; if the hinder particle be suddenly stopped, find how far the other will move before it begins to return. r ' ■ : — Section II. — Central Orbits. 'W 150. Plane of Orbit. — If we suppose a particle acted on by a force directed to a fixed centre to be projected in any direction, it is easily seen that its subsequent path will lie in the plane passing through the centre of force and the direc- tion of its projection. For, since the force acts towards the fixed centre, it has no tendency to withdraw the particle from that plane at the first instant, nor at any subsequent instant during the motion ; because the motion of the particle at each instant is got by compounding its previous motion with that due to the central force. We shall accordingly take this plane, called the plane of the orbit, as the plane of rectangular coordinate axes ; the fixed centre of force being the origin 0. 151. Differential Equations of Motion. — Suppose the force attractive, and P 'the y position of the attracted particle at the end of any time t. Let ON=x, PN=y, OP=r, /_XOP=0. Suppose F to represent the acceleration due to the attractive force have then, by Art. 68, we Equation of Orbit, and Periodic Time. 161 (1) — = -i^cos = F- ar r The complete determination of the motion for any law of force depends on the solution of these simultaneous equations. In the case of a repulsive force it is necessary to change the sign of F. The path described is evidently always concave to the centre of force for attractive forces, and convex for repulsive. 152. Law of Direct Distance. — There is one case in which the differential equations can be immediately inte- grated, viz., when the force varies directly as the distanoe from the fixed centre. Let F= fir ; then, for attractive forces, we have d 2 x _ N (2) dt :jf + /«y = The integrals of these equations, by Art. 109, may be written x = A cos t y a + B sin t a/ju ) ,- /- • ( 3) y =A'costyfi + B'smt^/n ) The arbitrary constants in this, as in all other cases, can be found from knowing the position, velocity, and direction of motion at the first instant. 153. Equation of Orbit, and Periodic Time. — If we solve the preceding equations for cos t ^/p and sin t \Zfi> and add the squares of the results, we get (Ay - A'z)* + {By - B / x) 2 = (AB f - BA') 7 (4) This equation represents an ellipse, whose oentre is at the centre of force. 162 Central Orbits. Again, if 2-nr + t*/ji be substituted for t^fx in equa- tions (3), the values of x and y remain unaltered ; hence, if (f - t)^/n = 2tt, the body will occupy the same position at the end of the time t' which it occupied at the time t. Accord- ingly, if The the time of a complete revolution in the orbit, we have o T = -— T is called the periodic time, and is the same for all orbits round the same centre of force, since it depends only on ju, the intensity of the central force, i.e. the acceleration at the unit of distance, and not on the initial conditions of the motion. 154. Determination of the Arbitrary Constants. — Let a, b be the coordinates of the particle at the instant from which the time is reckoned, V the initial velocity, and a the angle which the initial direction of motion makes with the axis of x; then, making t = in equations (3), we get A = a. A = b. Again, by differentiation, we have — = B y }x cos t ^/ju - A v/ju sin t y/fx > ctt ■j- = B^\/ijl cos t y/fi - A'^/fx sin t »/ p. Hence Fcos a = B ^//x, Fsin a = B'^/li ; Fcos a ._ ~\ (5) consequently, x - a cos t v^u + ~7=~ sin t '(r) dr + const. = - 2/*0(r) + const. Again, let V be the velocity at the distance B, and we get V 2 = - 2{i(j>lR) + const. ; therefore r 2 - V 2 = 2 M { (R) - (r) ) . (14) 168 Central Orbits. For instance, for the law of nature, we have *-r , -*G-i} (15) Hence we see that the velocity at any distance from the centre of force is independent of the path described, and is the same as if the body had been projected, with the initial velocity, directly towards the centre of force (compare Art. 131). Again, if /= — , we have v *- p.^SL/JL _ JL\ (16) If y = 0, when R = , i.e. if the velocity at any point in the path is that which the body would acquire in moving from rest from an infinitely great distance towards the centre of force, we have '--^4 ( 17 ) n- 1 r n * For instance, if the force vary as the inverse square of the distance, we have in this case v> = &. (18) T Again, if the force be repulsive, and vary directly as the n th power of the distance, we have F = - fir n , and (14) be- comes ^ _ V z = -^ (r n+1 -iZ n+l ). (19) n + 1 If F = when i£ = 0, i.e. if #?e velocity at any point be the same as that acquired in moving from the centre of force, 2m i*» (20) + 1 To prove the Relation F=-'^-. 169 p* dr 161. I*aw of Inverse Square. — If F= ^, equations (1) v become s~e; i-^}. (2D 1 f) Also from (8), we have, — = - ; / /i hence, equations (21) become h r h Integrating, we get, x = - r sin 9 + a y = | cos + P (22) in which a and /3 are constants, whose values can be found by the aid of the initial circumstances of the motion. Again, substituting these values of x and y in the equa- tion xy - yx = h, we get t r +(3x-ay-h = 0. (23) From this it follows that the orbit is a conic section having the centre of force at its focus. Further discussion of this law of force is postponed to Art. 166, in which will be given another demonstration that the orbit is a focal conic. h 2 dp 162. To prove the Relation .F=— -f . p? dr Equation (13) gives, by differentiation, ^.^=-14(1)^1 (24) dr dr\p 2 J p* dr 170 Central Orbits. This result admits of a useful transformation ; for, if 7 denote the semichord of curvature drawn through the centre of force, we have y=p d £' [Biff. Calc, Art. 235.) Hence the previous equation becomes v 2 F=-. (25) 7 This result can also be readily deduced from the conside- ration that the centrifugal acceleration, — , at any point in the orbit, must be equal and opposite to the component of the cen- tral acceleration taken in the normal direction (Arts. 25, 90). Examples. 1. Prove that the velocity at any point in a central orbit is the same as that acquired in moving from rest along one-fourth the chord of curvature at the point, under the action of a constant force, equal in intensity to that of the central force at the point. 2. A particle describes a circle freely under the action of a force whose direction is constant; determine the law of force. Taking the centre of the circle as origin of rectangular axes, the axis of y being parallel to the constant direction of the force, we have dx dx dy dt ' dt y dt hence, dy = _ x dt a y ^ d-y a I du dx\ a 2 a 2 hence, Y= -£ = — (x-£ - y — \ = r * dfi y z \ dt dt J y z 3. Apply equation (24) to find the law of force directed to a focus in an ellipse. In this case we have To prove the Equation -^r + u = —— . 171 dO" nrur 1 dp a 1 , _ ah 3 •"• ir = 7TTi hence, -F = -— . p s dr b- r- b 2 r 2 4. Find the law of force in the curve r m = a m cos Ww> Here we have {Diff. Calc, Art. 190) r w » Tl = a m p. XT t, («J+l)A 3 a 2 »» Hence, F= v ^ 5. Prove that the force under whose action a bodj T P revolves in any orbit, about a centre of force S, is to the force under whose action the same body P can revolve in the same orbit, in the same time, round another centre of force B, as SP. RP~ : SG 3 , where SG is the straight line drawn from S parallel to PP, meeting in G the tangent at P to the orbit. Principia, Sect, n., Prop, vii., Cor. 3. ,7 2,. Tp 163. To prove the Equation -^+w = T5— ?■ In the equation if we regard r as a function of 0, we have - 2F- ^ - _ ill ^M f/r eft* f/0 Moreover, from (12), we have ~W~ M dd[ u + d¥J Substituting in the preceding, we get d 2 u F -77S + W = ; — ; • (26) dd 2 h\u* v ; This important result can also be proved as follows : — Substituting - jFfor P, in equation (11), Art. 28, we get Central Orbits. *■-- d 2 r df + (d9\\ '{dtp fdOV h % u\ by (8) dr dt = hu 2 dr , ST"* d0 172 but also d 2 r d (du\ _ dO d 2 u _ a <£u ■'■ 5»""" A 5"A5eJ'" dtaW" hu dd* ; consequently F- h 2 u 2 (-^- 2 + uj. The discussion of central orbits comprises two distinct classes of questions. In the one it is required to find the equation of the orbit when the law of force is known ; in the other the orbit described is given, and the law of force, directed to a fixed point, is required. In the latter case, if the origin be taken at the fixed centre of force, the equation of the orbit can, in general, be d 2 u expressed in terms of u and 0, from which the value of -^ can be determined. If this be substituted in the equation the resulting value of F determines the required law of force. 164. Application to Ellipse. — For example, to find the law of force which will cause a particle to describe an ellipse round a centre of force situated in one of its foci. Here the equation of the orbit is 1 + e cos 9 " = — L ' where L is the semi latus-rectum. Laic of Inverse Square. 173 Hence d 2 n e cos d6 2 ~ L ' therefore dhi 1 U + dW " £' and consequently 7 n _h 2 u i _ h* L a (1 - e 2 ) )• (27) Accordingly the force varies inversely as the square of the distance from the centre of force. Examples. Find the law of force, directed to the origin, in the following curves :— 1. r = ae ad . 2. u = ae a9 + be- a9 . 3. r = ae a9 + be' a9 . Ans. Land 2.-. 3.*«£(~ ^-) 165. Case where the I*aw of Force is given. — When the law of force is given, the determination of the orbit depends on the solution of a differential equation ; for, if F=fi -£. This result may be exhibited in another form by aid of equation (18), as follows : — The velocity at any point in an ellipse is less, in a para- N 178 Central Orbits. bola equal to, and in a hyperbola greater than, the velocity which the body would acquire in moving to the point from an infinitely great distance, under the action of the central force. 171. Construction of Orbit. — The preceding equation shows how to construct the orbit when we are given the absolute force, the initial velocity, position, and direction of motion. For, suppose P the initial position, PT the direction of motion, and S the centre of force ; let V= velocity of projection, SP = P ; then — (1) if V 2 < -jj the orbit is an ellipse whose semi-axis a is given by the equation a R fi Again, draw PIT, making the angle TPH = L SPT, then the second focus H lies on this line, and its position H is found by taking PH = 2a - P. Consequently, as the two foci and the axis major are known, the ellipse is completely determined. (2) When ^ V 2 the orbit is a parabola, which can be easily determined by drawing &ZV perpendicular to the direc- tion of motion at P, inflecting ST= SP, and dropping NA per- pendicular to ST. The parabola described with S for focus, and A for vertex, will be the required orbit. (3) When V 2 >\ the orbit is AX a hyperbola, whose semi-axis a is given by the equation 2_ B Effect of a Sudden Change in Absolute Force. 179 The second focus, II, can be easily constructed, as in the first case, but lies on the opposite side of the direction of motion from the centre of force S. Again, as the value of the semi-axis a is independent of the direction of projection, we infer that if a number of bodies be projected from a point with the same velocity, in different directions, and be attracted by a common centre of force, the mean distances, and consequently the periodic times, will be the same for all the orbits. It may be remarked that the orbit will be a circle, pro- vided the angle SPTis right, and V 2 = ~ (compare Art. 91). The formulae in this and the preceding Article are of importance in the discussion of focal orbits. We add a few elementary applications. Examples. 1. Calculate, approximately, the periodic time of a planet if its mean dis- v^ tance from the Sun is double that of the Earth. Ans. 1033 days. 2. If a body be projected with a given Telocity about a centre of force which varies as the inverse square of distance, find the locus of the centre of the orbit described. Here, since the locus of the empty focus is a circle, the locus of the centre is also a circle. 3. In the same case, show that the length of the axis-minor varies directly as the perpendicular drawn from the centre of force to the direction of pro- jection. Since r and r' are each constant, p is to p in a constant ratio ; consquently b varies as p. 4. Show that there are two directions in which a body may be projected from a given point A, with a given velocity V, so as to pass through another given point B. Since the axis-major %a is given, the position of the second focus is deter- mined by the intersection of two circles, with A and B for centres. Hence there are two solutions — one for each point of intersection of the circles. 5. Prove that the time of describing an arc of a parabolic orbit, bounded by a focal chord of length c, varies as c*. 172. Effect of a Sudden Change in Absolute Force. — A body is revolving in a focal orbit ; if when it arrives at any position the absolute force /j. be suddenly altered, to determine the subsequent path. N 2 180 Central Orbits. Let R and V represent the distance and velocity at the instant in question, and let ft be the new value of the absolute force, and d the semi-axis major of the new orbit ; then, as the velocity receives no sudden or instantaneous change, we have, by (33), 2ft n 2ft ft R~a = -R~a" (36) The value of a\ and consequently the position of the new orbit, can be immediately determined from this equation. For example, suppose the original orbit a parabola, and the central force suddenly doubled in intensity. Here // = 2/x, and our equation becomes 2ft = 4ft _2p R " R a'' hence a' = R; and, consequently, the new orbit is an ellipse having the extremity of its axis major at the point. If the change in ft be very small, and represented by Aft, and the corresponding change in a by £a> it is plain that we have *—?*■ II- 3- (37) Hence, if the central force (or the attracting mass) be in- creased slightly, the axis major will be diminished; also, if the force be diminished the axis major is increased. The corresponding change in the periodic time is readily found; for, by (31), we have 2 log T + log ft = 2 log 2tt + 3 log a ; 2&T 3Aa Aft hence „ = ; 1 a ft therefore — = - -^ f— - 1 J. (38) Again, if the centre of force be supposed suddenly trans- ferred to a new position, the subsequent path can be readily constructed, as in Art. 171. Application of Method of Hodograph, 181 Examples. 1- A number of bodies are projected from a point with the same velocity, but in different directions ; prove that the centres of their orbits are situated on the surface of a sphere. 2. A body is describing a circle under a central force in its centre ; if the V force be suddenly reduced to one -half, find the subsequent path of the body. Am. a parabola. 3. In the same case, if the central force be suddenly increased in the ratio of m : 1, find the eccentricity of the subsequent path. m - 1 Am. . m 4. Two equal perfectly elastic particles describe the same ellipse in the same period, in opposite directions, one about each focus ; prove that the major axis of the orbit is a harmonic mean between those of the orbits they will describe after impact. This result follows immediately, since the vis viva is the same after collision as before (see Art. 81). 5. Prove that there are two initial directions for the projection of a particle with a given velocity, so that the axis major of its orbit may coincide in direc- tion with a given line. 6. If, when the Earth is at an end of the minor axis of its elliptic orbit, a meteor were to fall into the Sun, whose mass is the m th part of that of the Sun ; find the resulting change in the Earth's mean distance, and also in the length of the year. a 2T Am. Aa = , At = — — - . m m 173. Application of Method of Hodograph.— The method of the hodrograph ("Art. 26) furnishes a simple mode of determining the law of force in a focal ellipse. For, since the velocity at any point P varies inversely as the perpen- dicular SL, it varies directly a' as the perpendicular .fiTVdrawn from the second focus ; since SL x HN = b°\ Consequently the hodograph is similar to the locus of JV, when turned through a right angle. But the semicircle de- scrihed on the axis major as diameter passes through N, con- sequently the hodograph is a circle. Again, to find the law of force, let Pi denote the position of the movable at the end of an indefinitely small time At, / 182 Central Orbits. and Ni the corresponding position of N ; then (Art. 26) —— is proportional to the central attractive force. Join the centre C to N and to Ni ; then, by an elementary- property of the ellipse, CN is parallel to SP, and CN^ to fifPi. Let SP = r, lCSP=9, SL=p, fflST^p; then lNCN x =lPSP,= A0. .' iVi^ A0 drA Also (by 8), _ = „_ = _. Hence the force varies inversely as the square of the dis- tance. Again, since v = - = — p', we have " V ~KT = & a * r 3 ' Consequently, if ,u represent the absolute force, i. e. the force at unit of distance, we get _ tfa as in (30). Again, since the velocity at P is proportional and per- pendicular to jBTZV; and CN, CH are constants, it follows that the velocity at P can be resolved into two constant velo- cities — one perpendicular to the radius vector, the other to the axis major. _ h Also, since the velocity at P is represented by — HN, the component velocity perpendicular to SP is represented by — , and that perpendicular to the axis major by — : i.e. by j and j e, or by /— and J~ e, respectively. That the hodograph is a circle in this case appears also at once from (22). For if x\ tj be the coordinates of the Lambert's Theorem. 183 point in the hodograph which corresponds to the point xy in the orbit, we have x' ='x, y = y ; hence, substituting in (22). and eliminating 6, we get for the equation of the hodograph (/-«)' + (//-/3y- = p which is the equation of a circle. We may here observe that in any case of the motion of a particle, if we can find an equation connecting the velocities .r, //, z of the motion, with constants, that equation may be regarded as that of the hodograph, in which x, y, z are the current coordinates. (See Art. 26.) Example. A particle moving in an ellipse under the action of a force directed to a focus has a small velocity n y impressed on it in the direction of the focus ; find the corresponding changes in the eccentricity, and in the position of the apse. 174. Lambert's Theorem.— In Art. 140, Int. Calc, it has been shown that the area of the elliptic sector PSQ is represented by \ab J0-0'-(sin0-sin^J), ^ where

are given by the equations . 1 rfri+r 2 +c\$ . 1 , ./n + ra-cx" sm|f/) = | , sin | (p = j ' in which SP = r lf SQ = r 2 , and PQ = c. Accordingly, if t represent the time of describing the arc PQ, we have . 2areaP/SQ fa z \ h , , ,. . ,a, , oon t = = ( - ) {^ -

As observation shows that Kepler's third law is very nearly exact for all the planets, we conclude that the mass of the Sun is very great in comparison with that of any of the planets. In fact the mass of Jupiter, which is the largest of them, is less than a thousandth part of that of the Sun. This conclusion will appear more clearly from the follow- ing method of comparing the mass of the Sun with that of a planet where the planet has a satellite : — 186 Central Orbits. 177. Comparison of Hasses of Sun and Planet. — Let S denote the mass of the satellite, $ its distance from the planet, t its periodic time ; then, since the satellite revolves round the planet we have, as in last Article, P+S fSVfT S + P \aj \t \a (42) When the calculations are made, it is found that in all cases [ - ) f — J is a very small fraction : and hence also — . If S be supposed very small in comparison with P, as P is in comparison with S, we can, by (42), obtain the ratio of the planet's mass to that of the Sun, approximately. Again, for two planets, P and P', if the masses of the satellites be neglected, we have p_ _ /sy (C v ~ \$) U 178. Mass of Sun.— When applied to the Earth and its satellite the .Moon, the preceding formula gives a means of comparing the mass of the Sun with that of the Earth. Let E and 31 represent the masses of the Earth and the Moon, r their distance, then equation (42) becomes S + E~\a) \tj r 1 Now, as a rough approximation, we assume - = j^q » i. e. that the Sun's distance from us is 400 times that of the T Moon. Also we take — = 13*4, or that the year is, approxi- mately, 13*4 times the periodic time of the Moon. This gives |±|- 6 Y 7 9 °56 00 = 356 > 420 approximately. E Moreover, as determined by tidal calculations, M = ^ > hence we get S _ „ fi - .„~ E Mean Demit ij of Sun. 187 This result represents very closely the ratio of the Sim's and Earth's mass as determined by more exact investiga- tions. The foregoing calculation shows the enormous mass of the Sun in comparison with that of the Earth. In like manner the relative masses of Jupiter, Saturn, and other planets which have satellites can be found, approximately. Examples. 1. Prove that the mass of Jupiter is nearly 270 times the mass of the Earth from the following observations : — Jupiter's fourth satellite is at a mean distance \^ of 25 radii of Jupiter, and its periodic time is 16 days 18 hours; Jupiter's mean radius is 11 times the mean radius of the Earth ; the mean distance of the Moon is 60 radii of the Earth, and a mean lunation is 28 days. 2. Prove that the mean density of Jupiter is a little greater than that of water, and that the mean value of g on the surface of Jupiter is about 71, taking the mean density of the Earth as 5-67. 179. Mean Bensity of Sun.— The ratio of the mean density of the Sun to that of the Earth can be determined, as follows : — From (42) we have, approximately, ■m 8 E Again, let p, p x denote the radii of the Sun and Earth, and o- the ratio of their mean densities; then, assuming them spherical bodies, we have K-gi 8 E = '($ ■<) m or -m- where a denotes the Sun's mean apparent semi-diameter, and P the Moon's mean horizontal parallax. 188 Central Orbits. T If we substitute 16' for «, and 57' for P, and take — as t before, we get «r = 0*23, i.e. the Sun's mean density is about one-fourth that of the Earth. It should be observed that this result does not require a knowledge of the Sun's distance ; and, as the constants in (43) can be obtained with great accuracy, the ratio of the mean densities of the Sun and Earth can be determined with great precision. 180. Planetary Perturbations. — The previous deduc- tions respecting the planetary motions are only approximate for another and a more important reason, namely, that in them we have neglected the mutual actions of the planets on each other. However, since the Sun's mass is very great in comparison with that of all of the planets, their attractions on any member of the solar system may be regarded as small disturbing forces, and the planetary orbits as approximately ellipses. The usual method of treatment, accordingly, is to regard each jolanet as moving in an ellipse, in which the elements* are subject to very slow changes, arising from the perturba- tions or disturbing effects of the other planets. In this manner the problem has been discussed by Lagrange, Laplace, and other great writers on Physical Astronomy. We shall not enter into this discussion, as it is beyond the limits contemplated in this treatise. There is, however, one mode of considering the effects of a disturb- ing force, which may be here introduced. This consists in sup- posing the disturbing force resolved into two componentsf, * The elements by which a planet's path is determined are — (1) its mean dis- tance from the Sun ; (2) its eccentricity; (3) the longitude of its perihelion; (4) the inclination of its plane to a fixed plane; (5) the angle which the intersection of these planes makes with a fixed line ; (6) its epoch, or the instant of the planet's being in perihelion. f There is in general a third component, perpendicular to the plane of the orbit. It is not proposed to consider the effects of this component here. This method of treating the disturbing forces is discussed in a masterly and lucid manner by Sir John Herschel, in his Outlines of Astronomy, ch. 12 and 13. Normal Disturbing Force. 189 one along the tangent, the other along the normal to the orbit, and in treating their effects separately. 181. Tangential Disturbing Force. — Suppose P the position of a planet, moving in the ellipse BPA, in which S and H are the foci ; then, since a tan- gential disturbing force alters the velocity, but produces no effect on the direction of motion, it is easy to find the corresponding changes in the elements of the path. For the new position, H', of the second focus will still lie on the line PH. Again, if v denote the velocity at P, we have, as before, 2u When the change in v 2 , caused by the tangential disturb- ing force, is known, the corresponding change in a can be found ; and hence the position of H\ and consequently that of the new axis major. Thus if Ev be the small change in v, due to the disturbing force, we have 2vhv = — r ; ... §a = — v$v; .-. HH'=2Sa = —vSv. (44) fx /x If the tangential force act in the direction of the motion, and consequently increase the velocity, a will also be in- creased, and the perihelion A' will consequently move towards P. Again, the eccentricity e will be increased when 811' is greater than SH, i.e. when P is between the perihelion A and the extremity of the latus-rectum drawn through H. 182. Normal Disturbing Force. — Next, if a normal 190 Central Orbits. disturbing force act at P, inwardly, it does not alter the velo- city, but it changes the direc- tion of motion, through a small angle Sep. As the velocity is unchanged, the length of the A semi- axis major a is unaltered, a\ while the angle SPTis altered by the quantity §<£. Therefore the angle RPR', between PR and the corresponding line PR' in the new orbit, is 280 ; also PR = PR'. In this manner the position of R' is found when the angle Btp is known. Again, join SR', and produce it at both ends, then the line A'R' will represent the direction of the axis major of the new orbit. Through R draw DD' perpendicular to SR. The points D and D' are called the quadratures of the orbit. When P lies between D and the perihelion A, the line AB, called the line of apsides (see next Article), moves in the same direction as the planet, and is said to advance. The eccentricity in- creases at the same time. If the planet be between aphelion B and D, the eccentricity continues to increase, and the line of apsides recedes. Again, in moving from A to D' , the disturbing force still acting inwards, it is easily seen that the line of apsides advances, and the eccentricity diminishes. Hence, in the motion from quadrature to quadrature, through perihelion, the apse continually advances, in the case of a normal dis- turbing force acting inwards ; the eccentricity increases during the first half of the motion, and diminishes during the second. The contrary effects have place for a normal disturbing force acting outwards. In like manner in the motion from quadrature to quad- rature through aphelion, the apse recedes ; the eccentricity increases during the first half and diminishes during the second. 183. Apsides. — A position for which the moving body is at a maximum or a minimum distance from the centre of force is called an apse. The corresponding distance from the centre of force is called an apsidal distance, and the line join- ing the centre of force to an apse is called an apsidal line. Equation for Determination of A}) sides. 191 Since r, and consequently u, attains a maximum or a minimum value at an apse, we have at such a point ^ = It is easily seen that the orbit is symmetrical at both sides of an apse, provided the force is a function of the distance only. For, if a particle be supposed projected from a point A in a direction perpendicular to the line OA drawn to the centre of force, it is obvious that for the same velocity of pro- jection we must have exactly similar paths, whether it be projected in any given direction or in that exactly opposite. Moreover, if the velocity were reversed at any point, the body would proceed to describe the same orbit, but in an opposite direction. From these considerations it follows that the central orbit must be symmetrical at both sides of an apse, since at that point the motion is perpendicular to the central radius vector. 184. An Orbit can have but Two Apsidal Dis- tances. — For, suppose A and B to be two apsides, and the body to move from A to B ; then after passing B it will, by the preceding Article, describe a curve similar to BA ; and so on. Hence the apsides are constantly repeated, and the angle between two consecutive apsidal distances is the same for all positions of the orbit. This angle is called the apsidal angle of the orbit. It is plain that a central orbit cannot be a closed curve unless the apsidal angle is commensurable with a right angle. 185. Equation for Determination of Apsides. — Let F= fi(ji(u), then we have, by (13), r = 2fM W& du + o, where the value of C is determined by the initial conditions ; f(«) therefore h* ( u* + I — J ) = 2fj. du + C. (45) 192 Central Orbits. Hence, as -^ = at an apse, the equation for determining the apsidal distances is h'u^zJt^du+C. (46) If we suppose F= ftu", equation (45) becomes \\ddj ) (n-l) and the equation for the apsides tfu^-^rU^ + C. (48) n - 1 The form of the latter equation shows that it cannot have more than two positive roots, which therefore correspond to the two apsidal distances. For example, let the force consist of two parts, one vary- ing as the inverse square of the distance, the other as the inverse cube, or F=nu* + f i'u>, (49) then h- ir = 2fiu + p'u* + C. Accordingly the apsidal distances are in this case deter- mined by a quadratic equation. If ju = 0, there is but one apsidal distance. 186. Case of Velocity due to an Infinite Distance. — The integration of equation (47) in a finite form is in general impossible ; there is, however, one case in which the equation of the orbit can be readily determined, viz., when the velocity at any point is that acquired in moving from an infinite distance under the action of the central force. For we have, m this case, by (17), r = - u B-I therefore * + (gj.-3fa«». (50) Case of Velocity due to an Infinite Distance. 193 Hence ~jz = u y/au n ~* - 1, writing a instead of dO ' (n-l)h 2 ' therefore = J u yau n ~ 3 - 1 To integrate this, let au n ~ 3 = — , then — - - du and we get f- J u u n - 3 dz v/W 1 " 3 -! n-Z) q .\ + (5 = — — -o cos"^, or s = cos — — (0 + j3), where /3 is an arbitrary constant : hence r ~ = ij ^ cos ^-? (0 + /3). (51) If « denote the apsidal distance, and be measured from the apsidal line, the preceding may be written «-3 n-3 n _ 3 r 2 = « s cos _^ 0. (52) This is the polar equation of the orbit. For example, when n = 2, we get the parabola rz cos i 9 = ah. Again, when ^ = 5, it becomes r = a cos ; a circle having its centre on the circumference. o 194 Central Orbits. For n = 7 we get the lemniscate r = « 2 cos 20, and so on. Equation (52) fails when n = 3 ; in this case, however, (50) becomes which gives £0 = log w + const., where A- = /— - 1, or u = Be k o. This is the equation of a logarithmic spiral. 187. Approximately Circular Orbits.— If the orbit described round a centre of force be nearly a circle, its equa- tion can be found approximately, as follows : — Assume F= fiu 2 f(u), then equation (26) becomes + « = &/(«). d6* K If the orbit were an exact circle we should have A dh > ft therefore a must satisfy the equation a = £/(«). (53) When the orbit is approximately circular we may assume w = #, + z, where s is always very small. Hence ^ + a + s = ^/(« + s), or C di 2+a + Z== tf ^ + ^^ ' ' Approximately Circular Orbits. 195 By (53) this becomes, neglecting s 2 and higher powers of s, 8-(l-£/'M) = 0; or, substituting — - for £, If Jx = 1 - — — , this becomes ./ («) g + fa = 0. (54) When k is positive, the integral of this, by Art. 109, is Z = C COS [By/li + a), or u = a + c cos (0 y^ + a), (55) when c and a are arbitrary constants. The greatest value of u is a + c ; consequently, in order that the orbit should be approximately circular, it is necessary that c should be very small in comparison with a. Again, supposing c positive, the greatest value of u has place when 9 */k + a = 0, and the least when 6 x/k + a = it ; consequently the apsidal angle is 7T IT or If k be negative, i.e. if — — > 1, the integral of (54) is of the form z = Ae e '~ k + Be- 6 ^, and therefore z would either increase or diminish indefinitely 02 196 Central Orbits. with ; and accordingly the orbit cannot be approximately circular in that case. The value of k depends on the law of force : for example, if the force vary inversely as the n th power of the distance, then /(«) = „«•-*, and ^W-»-2. Accordingly, in this case, k = 3 - n. Hence a nearly circular orbit, having the centre of force in the centre, is impossible for laws of force which vary inversely as a higher power than the cube of the distance. When n is less than 3, the angle between the apsides is y3-n For instance, if n = 2, the angle is tt ; this agrees with what has been already proved, as the orbit is a focal conic in this case. Again, if n = - 1, the angle is \ir, as it ought to be, since the orbit is a central ellipse. 188. Movable Orbits. — If a central orbit be made to move in its own plane with an angular velocity propor- tional at each instant to that of the radius vector in the orbit, we can easily show — (1) that the new orbit is also a central orbit ; (2) that the difference between the forces in the two orbits varies inversely as the cube of the distance from the centre of force. (Newton, Principia, lib. i., sect. 9.) In a central orbit we have, in general, > V. In this case 1 — — is positive — equal A" 2 , suppose — and the equation may be written d 2 u A" The integral of this is of the form u = A cos (k9 + a). A is plainly the maximum value of u ; and therefore corresponds to an apsidal distance. Let a be this distance, and, if 6 be measured from the apsidal line, the equation of the orbit is r cos kd = a. (1) M (2) Let V sin a> = V\ then I - — = 0, and we have — -0- d0~ ~ ' this gives u = A (0 + a) ; and the equation of the orbit is reducible to r0 — constant, (2) which represents the hyperbolic spiral. (3) Let V sin 1, the d9 h* orbit has or has not an apse according as V is less or greater than V . Hence, if the initial velocity be less than that in an equidistant circle, the orbit is apsidal. Examples. 201 Suppose a to be the corresponding apsidal distance, then F'2 - F 2 =MM and, making ^ — 1 = k-, equation (3) becomes \dd) a 2 v ' therefore — - = - V a~n 2 - 1 ; or = Arf0. d0 Va 2 w 2 - 1 The integral of this is he + a = log («w + V«-w 2 - 1). But if a be measured from the apse, -we have = when au - 1. Conse- quently a = 0, and we have au + V« 2 m'- - 1 = <^- Hence «« = |(^ + «"**). Here u increases with 6 ; and consequently the body, after leaving the apse, approaches nearer and nearer to the centre of force. Secondly, if the initial velocity be equal to that in the equidistant circle, (3) becomes du- du — — = & 2 « 2 , or — - = ku ; dd 2 dd this gives u = ae ke , the equiangular spiral (Ex. 7). Thirdly, if V be greater than F', let and equation (3) becomes = * 2 J8 2 , This, when integrated as above, gives w + Vw 2 + £ 2 = -4***, and the curve is represented by the equation 2 « = ^ _ ^' r* A The value of -4 can be readily determined from the initial conditions. 17. In elliptic motion about a centre of force in a focus, prove that Jvcfo, taken through any arc, is proportional to the area subtended by the arc at the empty focus. 202 Central Orbits. 18. Prove that the expression for the central attraction for any law of force may be written in the form F= — -r. r 3 If we change the sign in the expression for the acceleration along the radius vector in Art. 28, we get F=rO°~-r. h This assumes the proposed form on substituting for 6 its value -^ . 19. "What would be the motion of a projectile if the force of gravity varied inversely as the cube of the height above a horizontal plane ? Here the path evidently lies in a vertical plane. If the line of intersection of this plane with the horizontal plane be taken as the axis of x, and a vertical line as the axis of y } the equations of motion may be written dt 2 ' dfi v/ 3 ' therefore — =«, ^-_ + /, where e and c' are constants which depend on the initial circumstances of the motion. Consequently 'dy\ 2 1 jx + c y 2 (dyy = I \dx) c* ydy dx VV + c' y' 1 c Hence we get V/j. + c'y' 2 = c' - + const. c Consequently the path is an ellipse or a hyperbola according as c' is negative or positive. The path is a parabola if c' — 0. 20. Prove by Newtonian methods that, if two bodies attract one another according to any law, they describe similar figures about their centre of inertia and about one another. Neglecting the obliquity of the ecliptic, and the inclination and the eccen- tricity of the lunar orbit, show that, if we take the Sun's distance as 390 times that of the Moon, the Earth's mass as 79 times that of the Moon, and the lunar synodic period as 30 mean solar days ; then the solar day is, to a near approxi- mation, shorter at full Moon that at new Moon by one 468,000th part of a mean solar day. ' Comb. Trip., 1882. 21. A material particle, moving freely in a plane, being supposed to describe a conic under the action of a central force emanating from any point in the plane ; show that the force varies directly as the distance from the point, and inversely as the cube of its distance from the polar of the point with respect to the curve. Examples. 203 22. In free motion in a plane under the action of a central force varying according to any law, state and prove the effect on the trajectory (and on the motion in it) of an additional force emanating from the same centre, and varying inversely as the cuhe of the distance. 23. An ellipse of eccentricity e and a parabola have a common focus and latus rectum ; and equal particles describe them under the action of forces, to the common focus, of the same absolute intensity. If the particles moving in the same direction meet at one extremity of the common latus rectum and coa- lesce, prove that their subsequent path will be an ellipse of eccentricity §-(1 ±e), according as both foci of the ellipse do or do not He within the parabola ; and find its major axis. What will the path be, if the particles be moving in oppo- site directions when they meet ? Gamb. Trip., 1879. 24. A body is revolving in an ellipse, whose eccentricity is > \, under the action of a force tending to the focus S; and when it is at a distance SP from S equal to the latus i*ectum, a blow is given to it perpendicular to SP, such that its new direction is perpendicular to the major axis. Show that the dimensions of the orbit are unaltered, but that the major axis is turned through an angle SPS, where R is the empty focus. Id., 1882. 25. Find the laws of attraction for which the trajectories described round a centre of force are closed orbits. (Bertrand, Comptes rendus, 1873.) **m F(u) + const., equation (23) gives o lduY~ 1 „, , where c is an arbitrary constant ; therefore du - F(u) +e-tfi Again, let a, /3 represent the values of u which correspond to the apsidal distances, then a and £ are roots of the equation ^ F{u)-l c-n*=0. Accordingly we must have i*(a) + c-a*=0, i-F(j8)+*-/3* = 0; and if O be the apsidal angle, we get, abstraction being made of the sign, ,.£ du 00 }- r F(u) + c-u i >a\J A- Assuming >n9o = 7r, then, for a closed orbit, m must be a commensurable number (Art. 184). 204 Central Orbits. If — and c be eliminated by aid of the two preceding equations, we obtain du TV w , *»-*'(«) 9 F(u)-F(») P F(fS)-F(a) ° F(/3)-F(a) an equation which should hold for all values of o and £. To determine the form of the function F, we suppose o and # very nearly equal, in which case the orbit is approximately circular. Hence, from Art. 187, we get J% I 1 aF " {a) or aF " {a) 1 Let F'(a) = F, then ^T = ^^ = (i - nfi) — ; hence p .r (a) a where C is an arbitrary constant. (d) From this we get F(a) = ■. o 2 -"' 2 + const. ° 2 - w 2 We may assume the latter constant to be zero, since it disappears when we substitute in equation (c). Again, since 2/t ^ = F'(u), we have £(«) = fl m 3 -"' 3 , where Ci is arbitrary. "We next proceed to determine m from the condition that (c) must be satis- fied for all values of o and £. (1) Let ?n 2 < 2, and make o = 0, and /3 = 1 ; then F(.) = 0, flfl-j^ 5- Substituting in (c), we obtain dw IT i: Jo VV-™ - W 2 Again, if u™* = z, Jo w v^^ri " ™ 2 Jo v^r^) The condition gives — = — - : therefore m — 1. Accordingly, the forc< mm 2 1 varies as «*, or as — - • r 2 Examples. 205 (2) Let or > 2 ; then, if o = 0, we have F(a) = F(0) = - oo ; and if j3 = 1 (y we have F(&) = F(l) = . Substitute in (c), and it becomes 1 — w l' 1 dti IT IT IT 1 - = — ; or — = — ; ,\ m = 2, Jo V 1 — ur m 2 m in which case the force varies directly as the distance. Hence, as M. Bertrand observes, "parmi les lois d'attraction qui supposent Taction nulle a. une distance infinie, cella de la nature est la seule pour laquelle un mobile lance arbitrairemoit, avec une vitesse inferieure a une certaine limite, et attire vers un centre fixe, decrive neoessairement auteur de ce centre une courbe fermee. Toutes les lois d'attraction permettent des orbites fermees, mais la loi de la nature est la seule qui les impose.'''' 26. Investigate the condition of stability of a circular orbit described about a centre of attraction in the centre of the circle. Prove that if the attraction varies inversely as the fourth power of the dis- tance, a particle describing a circle of radius a freely will be found ultimately describing either the curve cosh e + 1 cosh 0-1 >• = a — ■ , or r = a — : • cosh e - 2 cosh + 2 206 CHAPTEE VIII. CONSTRAINED MOTION — MOTION IN A RESISTING MEDIUM. Section I. — Constrained Motion. 189. Motion on a Fixed Curve. — When a particle is constrained to move, without friction, on a given fixed curve, the problem reduces to the determination of the velocity at any instant, as well as of the normal reaction of the curve. The motion may in this case be regarded as free by the introduction of the force of reaction of the curve, in addition to the external forces. Hence, if JV represents the normal reaction, the general equations of motion may be written, when referred to a rect- angular system of axes, m-r^ = X + Ncosa, m -^ = Y+N cos/3, m — ■=■ = Z + N cos 7, dt' dt' dt (i) where a, /3, 7 are the angles the normal reaction makes with the axes of coordinates ; and X, Y, Z are the components of the external force, parallel to the axes of coordinates, respec- tively. If the first equation be multiplied by dx, the second by dy, and the third by dz, we get, on addition, "' (J? * + S rfy + 8 * ) ~ x (%, y, z) ; then if v be the velocity at the point ,/, y\ z y we have \m {f - v 2 ) =

(of, y\ z). (4) Hence the velocity at any point is independent of the path described ; and, accordingly, if different curves be drawn joining any two points, a particle starting from one of these points with a given velocity would arrive at the other point with the same velocity whatever path it described ; friction being neglected. Two of the preceding equations (1) are sufficient for a plane curve ; for in this case N acts in the plane of the curve, and, by taking the axes of x and y in that plane, the third equation will disappear. In the case of a central force, represented by ju<£'( r )> we have, as in Art. 131, * . i«(*-O = M(*(r)-#(r0). Again, as in Art. 116, it is readily seen that the pressure on the curve in any case is the resultant of the centrifugal force and the normal component of the external forces. The particle will leave the curve at the point for which the normal reaction becomes zero. Examples. 1. A particle is constrained to move in a circle under the influence of a re- pulsive force, acting from a point on the circumference, and varying as the ^ distance : find the pressure on the curve, the initial position being at the centre of force, and the particle starting from a state of rest. j£ nSt J^L } where r is the distance from the centre of force, and a the radius % a of the circle. I 2. A particle is constrained to move in a logarithmic spiral, and is attracted to the pole of the spiral by a force varying inversely as the square of the dis- tance. If the particle start from rest at the distance a from the pole, find the time of describing any portion of the curve. 208 Constrained Motion, Let ix denote the absolute force ; then, by (5), we have ds rr x 1 Again, if r = ce* be the equation of the spiral, we have ds dr , -, therefore — == / — — A dt \ 1 + k~ \ r a Integrating, as in Art. 140, we get for the time of motion from the distance a to the distance r, -J*£fi(.-j2+v^) Also the whole time of motion to the centre is - A / — • 2 \ 2yU It is readily seen that the problem of constrained motion in a logarithmic spiral, under the action of any central force directed to its pole, is reducible to free rectilinear motion under the action of a corresponding central force in the line of motion. 3. A particle under the action of gravity moves down the inner side of a smooth ellipse whose axis major is vertical. Being given its initial velocity, find where it will leave the ellipse. Taking the centre as origin, and the axis major as axis of x, the value of x at the required point is given by the equation 2d = Zx - & -4, where d is the height above the centre of the level line to which the velocity at each point is due. 4. In the same question find the least velocity at the lowest point of the ellipse in order that the particle should make a complete revolution in the curve. Am. \/(/a (5 — e 2 ). 190. Theorem of M. Ossian Bonnet. — If masses m, m', m", &c, respectively subject to the action of forces, F, F\ F'\ &c, and starting all in the same direction from a point A, with velocities t? , v ', v", &c, describe the same curve Theorem of M. Ossian Bonnet. 209 ACB ; then the same path will also be described by the mass M, when projected from the same point in the so;me direction, and subject to the action of all the forces, F> F', F", &c, provided the initial vis viva MV 2 is equal to c mv 2 + m'i\? + m"v" % + &c, the sum of the vires vivce of the different masses. (Bonnet, IAouville's Journal, 1844.) For, suppose the particle M constrained to mov^e in the curve ACB, and let iVbe the normal reaction at any point ; then, if the components of F, parallel to a rectangular system of axes, be respectively represented by X, F, Z, those of F\ by X\ Y\ Z\ &c. ; from (1), we have M-£ = X + X' + X" + &c. + i^cos a. dt~ M^ 2 = Y + Y' + Y" + &e. + iV^cos j3, M^ = Z + Z' + Z" + &o. + i^cos 7 , at" and, as in (2), we have d(MV 2 ) = 2dx2X + 2dy^Y+2dz^Z. But if v, v\ v'\ &c, be the velocities in the partial movements of m, m', m\ &c, at the same point, d(mv 2 ) = 2 (Xdx + Ydy + Zdz), &c, &c, &c. Hence d (MV 2 ) = d (mv 2 + mv n + m'v" z + &c.) ; therefore MV" = 2(m# 2 ) + constant, or M V 2 = 2w# 2 , from our hypothesis. It is now easy to prove that the normal pressure iV is zero at each point, and consequently that M would describe the curve ACB freely, under the combined action of all the forces. p 210 Constrained Motion. For the force N is equal and opposite to the resultant of MV 2 the centrifugal force, , and the several normal compo- nents of the forces, F, F\ F" , &c. Again = — + + + &c. ; (o) P P P P but — , - — , &c.j are respectively equal and opposite to the P - P normal components of F, F\ F" 9 &c, because m, m\ &c, describe the path A CB freely. Hence there is equilibrium between the centrifugal force MV 2 and the total normal component of F, F\ F r \ &c. ; and P consequently N = 0. In general, if the initial velocity of M do not satisfy the equation MV 2 = ^nv 2 , the normal pressure on the path ACB mil vary directly as the curvature. For, from the preceding analysis, jsr= = . (6) p p Also, if one of the forces (F f suppose) be changed into its opposite, it is readily seen that the preceding theorem still holds, provided we change the sign of the corresponding term (mV 2 ) in the expression S(m© 2 ). Examples. 1 . A particle constrained to move in an ellipse is acted on by an attractive force directed to one focus, and a repulsive force from the other, whose intensi- ties vary as the inverse square of the distance : if the absolute intensities of the forces be equal, find the pressure on the ellipse at any point during the motion. 2. Hence show that a particle placed at equal distances from two such centres of force will describe a semi-ellipse, under their joint action. 3. A particle moves under the attraction of two forces directed to the fixed points A and B, each varying according to the law of nature, and a third force, varying directly as the distance, directed to C, the middle point of AB ; show that the particle can be projected from any point so as to describe an ellipse having A and B as its foci. Lagrange, Mec. Anal., t. 2, § 83. Ans. The initial velocity v is given by the equation t'o 2 = —z + -j, + /* J J > Motion on a Fixed Surface. 211 where /x, /x', ll" denote the ahsolute forces for the centres A,B, C, respectively ; /,/' the initial distances from A and B ; and a the semiaxis major of the ellipse. The initial direction of motion must bisect the external angle formed by the lines joining A and B to the point of projection. 4. In the same case, if the particle he constrained to move in the ellipse, find the reaction R at any point during the motion. Am. Rp = m (~- -f ^- + n"ff - tv j , where p is the radius of curvature at the point. 5. If a material particle, moving freely under the action of gravity, he dis- turbed by the action of a central force varying inversely as the square of the distance ; determine the circumstances of its projection from a given point, in order that it may describe a parabola in a vertical plane having its focus at the centre of force. 191. Motion on a Fixed Surface. — If a particle be constrained to move on a smooth surface, the general equa- tions of motion are plainly, as in (1), d 2 x _ „ (Pu _ , T _. d 2 z _ __ w— = X+Jy cos a, m — = F + iV cosp, m — = Z + N cos y, C(l~ CIL 1(0 where a, /3, y ar © the direction angles of the normal to the surface. It is obvious that in this case also the velocity at any point is determined by the equation j>mv 2 = J [Xdx + Ydy + Zdz) + const. (7) If gravity be the sole acting force, and the axis of z be taken in the vertical direction, our equations may be written d ~x d^u d~z — r ^i^cosa, -j~ = Ncos(3, — = Ncosy - g. (8) c(t" (it" (It When the surface is one of revolution round a vertical axis, the normal at each point intersects that axis ; and if n denote its length, we have x „ II cos a = - , cos p = - • n n Hence the two former equations give dhj d'x . P 2 212 Constrain ed Mo tion . or, on integration, dy dx where c is a constant. This equation shows that the point of projection on a horizontal plane describes equal areas in equal times round the point in which the axis of revolution meets the plane. 192. Motion on a Spherical Surface.— We shall apply what precedes to the motion of a particle under the action of gravity on a smooth sphere. This contains the general question of the motion of a simple pendulum, and is called the problem of the spherical pendulum. Taking the centre as origin, and the positive direction of the axis of z downwards, the equation of the sphere is x 2 + y 2 + z 2 = a 2 , where a is the radius. Also the general equations of motion may be written x=N-, y = N-, z = N- + g, a a a adopting Newton's notation (Art. 23). From the first two equations we get, as before, xij -yx = c. (9) Also, as in (7), x 2 + y 2 + z 2 = V,?+2g(z-a), where V represents the velocity corresponding to z = a. Again, differentiating the equation of the sphere, xx + yij + zz = 0, or xx + yy = - zz. If this be squared and added to (9), when also squared, we get {x 2 + y 2 ) (x 2 + y 2 ) =c 2 + z 2 z 2 . Motion on a Spherical Surface. 213 Hence (a 2 - z 2 ) { V 2 + 2g {z - a) - z 2 j = c 2 + z 2 z 2 , or a 2 z 2 = (a 2 - z 2 ) { V 2 + 2g(z-a))- c\ (10) The subsequent investigation is simplified by supposing V to correspond to the lowest point in the path of the particle ; for, since the motion at that point is horizontal, we have z = when z = a, and consequently c 2 = {a 2 -a 2 )Vo 2 = 2gh{a 2 -a 2 ), if h be the height to which the velocity V is due. Substituting this value for c 2 in (10), we get // a 2 z 2 = 2g{a-z)\z 2 + h(z + a)-a 2 }. Again, the expression z 2 + h (z + a) - a 2 may be written (z-f5){z + y), where a 2 -fi 2 a 2 + afi h m __ and y . __. (11) Accordingly therefore a'z = a — = - s /2g{a - z) (z - [5) {z + y). (12) a v The negative sign must be taken since % diminishes with t, which is reckoned from the instant the particle is in its lowest position. Also, when s=(3we have z = 0, and the motion is again horizontal. It is readily seen that during the motion z must lie between the limits a and /3 ; and consequently the path of the particle is a tortuous curve lying between two horizon- tal lesser circles on the sphere ; we accordingly may assume z = a cos 2 <£ + (3 sm 2 (j), (13) and, substituting in (12), get 2a -j~ = */2g (a cos"^ + j3 sin'^ + y), ao 214 Constrained Motion. Hence, since t = when

a|3+a 2 )J o V^(« 2 + 2a/3+a 2 )J o ^/i _/^ s in 2 a 2 - a 2 It may be observed that when a = ]3, we have A = — ~ — , and the question reduces to that of the conical pendulum, already considered in Art. 112. Next let \p be the angle that the vertical plane, passing through the centre and the position of the particle at any instant, makes with the plane of z%, then y = x tan \p ; and consequently dl J „ dx ^ d (V\_ ^_ 2 ., dx P c = x-tt - y— = a? 2 — - =# 2 sec 2 i/> , dt J dt dt\x r dt « + /)f-(^--D«. (15) Also c =yw^?)= h^-aw-m . (16) \ a + p a W~* \ a /3 '^> . _ ^ •(... ) (,. i3 )(, + 7 , «, and the angle /?. (19) Next, if z = a cos 0, a = a cos O , j3 = « cos 0i, the equa- tion z = a cos 2 <£ + /3 sin 2 gives 2 = o 2 cos 2 <£ + 1 2 sin 2 <£, neglecting powers of 0, O , t beyond the second. Also (16) gives in this case \}a .-. by (15), we have dt = '' 2 \a 0o 2 oos 8 + 0i 8 sinV\a Consequently, by (19), # _ 0Q0i <:/<£ O 2 cos 2 <£ + 0i 2 smV Hence, by integration, . 0i. tan \p = 7j- tan 0, "o 216 Constrained Motion. sin \L 0, sin or -f = r ; COS \p O COS (f) hence we have . , 0i sin Sm * = — ' , O COS and cos ^ = — ^ — . u Moreover, to the same degree of approximation, we have x = a 9 cos \p, y = a 9 sin \f, ; accordingly, # = « O cos $, y = a9i sin ; ••• g? + Jr«- (20) This shows that the horizontal projection of the path is, approximately, an ellipse, whose semiaxes are a9 and adi. The next approximation is given in the following examples. The general problem of the spherical pendulum appears to have been first fully discussed by Lagrange : see Nee. Anal., t. 2, sect. 8. Examples. 1. If a particle perform small oscillations about the lowest point on a sphere, investigate its motion to an approximation of the second order. It is here more convenient to transfer the origin to the lowest point on the sphere, and to take the positive direction of z upwards. Accordingly, we sub- stitute z = a-z', a = a-a, = a-/3', when equation (12) becomes Examples, Hence, removing the accents, we get dz 217 \ff)a Vfr-w-^-s+ssfefl)' where £ and o represent the distances of the highest and lowest points in the path from the plane of xy. Again, if a and /3 be both so small that their higher powers may be neglected, we obtain dz V'J.^-«)o-,)(i-^) 2 \9„a+ —I (a cos 2 + /3 sin 2 ^) cfy> \y Ay/agJo Consequently, if T be the whole time of motion from one lowest position to a consecutive one, we get '•'£( l + '-& Again, to find the apsidal angle to the same degree of approximation. Transforming the origin in equation (16) to the lowest position, we readily obtain * = "Vaj8 dz z(2 --■^-oM'-i&^S) 218 Constrained Motion. Hence, since as before we may take we get 2« -a-j8 1 (2a- a) (2a - dz 0) 2a' if? = «S V2aj3 J a (2a-«)SaV(z- a) (j8 - z) J«iV(«-o)0-«) J««V(*-a)(fl-«) 8 a J„ ,V(«-«)0B-») neglecting the subsequent terms as before. Substituting o cos 2 + j3 sin 2 <£ for z, we obtain 3Voj8 , / 1 \ 3 Vaj3 tan-^-taa^+j— *. Hence, taking

(v) , the equation of motion becomes where Fis the external force acting along the right line. It is usual to assume, with Newton, that (p (r) = ^v 2 , where ju is a constant depending on the density of the me- dium and on the area (8) of the greatest section of the body taken perpendicular to the direction of motion. Hence we get do m Tt = F-fiv\ If we suppose F constant, and make F = nV\ we get dv m— = dt M (F 3 -^; (1) (2) If the initial velocity be less than V, it is obvious that the velocity increases so long as it is less than V : this gives V 220 Constrained Motion. as the limit to which the velocity approaches. For this rea- son V is called the terminal velocity of the body. Also, since 1 1(1 1 ) V 2 -v z 2V\ V+v V-vY the preceding equation gives , mV, fV+v\ ._. t= W l0 %(v^v)' ^ No constant is added since we suppose t reckoned from the position of rest. Equation (3) shows that, while v increases with t, yet when v = V we should have t = oo . Accordingly the body requires an infinite time before arriving at its terminal ve- locity. 195. Vertical Motion. — One of the most important cases is that of a body falling vertically in a resisting medium. In this case F = mg, and equation (3) becomes (4) ~TT . .. lot This gives Hence Again, since we get when x is measured from the position of rest. I' V+v r-v" 2yt V V at gt e v - e v ft _ st ~ e v + e' v dx v = Tt> Ranh gt V X = 7 l0 < 2 gt z > Vertical Motion. 221 This may be written in the form x - — log cosh — (6) Again we may write /ul = AS, where A is a constant de- pending on the density of the medium. Hence from (1) we get £-J -£-, (7) AS' K } where W denotes the weight of the body. This shows that, W remaining the same, the value of V can be increased by diminishing the area of the transverse section. In the case of a homogeneous sphere of radius r, we have W= iirr^p, where;; is the weight of a unit of volume; also S = wr* ; therefore 4pr SA I Hence we see that for spheres of the same density that of the greater radius has the greater terminal velocity, and we can readily compare the vertical motions of different spheres in the same resisting medium. Next, for a body projected vertically upwards in a resist- ing medium the equation of motion is V 2 civ whence at = Accordingly, if V be the initial velocity, we find t = — tan -1 -77 - tan l — g\ V V 222 Constrained Motion. From this equation the velocity at any instant can be de- termined. Also, since v = at the highest point, the time of ascent to that point is represented by — tan -1 — . V 2 vdv Again dx = ~Y^TV 2 ' Hence, if x be measured upwards from the point of projec- tion, we have _ F% Fo 2 + V 2 X ~2g g v* + V 2 ' If h be the height of ascent, we get ^Yg^K—V 2 -)- (8) If the time t be reckoned from the instant at which the ' body is at its highest point, we have *=Ftan^. (9) The downward motion is given by the former investigation. Examples. 223 Examples. 1. Find a vertical curve such that the time of describing any arc, measured from a fixed point, shall he equal to that of describing the chord of the arc. Taking the origin at the fixed point, the time down a chord r, whose incli- nation to the vertical is 0, as in Art. 46, is J: •lr g cos0 Also the time of descending: the arc is V2g where O is the value of when r = 0. Hence, since the times are the same for all chords, we get, by differentiation, dr r sm + cos — (19 cos 9 J~£) 1 dr This gives - — = cot 26 ; ° r tf0 hence we get r 2 = a 2 sin 20, where a is a constant. Accordingly the curve is a Lemniscate. 2. Investigate the corresponding problem when the acting force is propor- tional to the distance from a fixed point. Let A be the position of the fixed point, the point of departure of the par- ticle, P its position at any instant, 6 — L POA, OA = a ; then we find, without difficulty, that the time h, of describing OP, when the absolute force is taken as unity, is given by . , r — a cos 9 ir h = sm- 1 — + - • a cos 9 2 Also the time of describing the arc OP is 2 *2 = J Hence, since t\ = h, we have W-® V2ar cos tf0. i = d I . ,r-acos9\ 9 (dry 224 Rectilinear Motion in a Resisting Medium. therefore (- r tan 9 -J**©' from which we get r 2 = a 2 sin 20. This represents a lemniscate also, as in the previous question. 3. If the motion of a conical pendulum be slightly disturbed, prove that the period of a vibration is —=■ /-, and the corresponding apsidal angle a it , where b is the distance from the centre to the plane of the conical Vtf 3 + Zb 2 r pendulum. 4. A particle is projected from a given point in a horizontal direction along the surface of a smooth sphere ; find the velocity of projection in order that the particle should rise to a given height on the surface before commencing to descend. 5. A particle is constrained to move in a smooth circle, under the action of a central force which varies directly as the distance. If the time of describing any arc be constant, prove that its chord envelops a circle. Townsend, Eduo. Times, 1875. 6. If a particle describe a curve freely under the combined action of the forces F, F', &c, where F, F\ &c, act along r, r', &c, prove that the equation must be satisfied at every point of the curve, where This theorem plainly contains as a particular case that given in Art. 190. Examples. 225 7. Apply the preceding to the case of a conic described under the action of forces, F, F', directed to its foci. TT A* » A 4 ' Here $=-?>

7 = 7 ; r- r - therefore — — ( Fr 2 ) dr + -~ — IF' / 2 ) dr = , r 2 f/r r* dr or, since rfr + dr' = 0, -4 £(Ft*) = 4/r (^"2). r 2 ^/- r 2 dr ' This is satisfied by the equations ^^ (i? ' ; ' 2)=/l(r)+/2( ^-" ) ' ^/^-.(F'r' 2 )=Mr')+ma-r'), where /1 and /a are both arbitrary functions. If we assign the same form (/) to /1 and / 2 , we obtain as a particular solution F =^-Sr 2 {f{r) +f(2a - r)} dr, &e. If any particular form be assigned to/, a corresponding form of F y as also of F', will result. 8. As an example of the preceding, show that a particle can be made to de- scribe an ellipse freely under the action of forces, A 4 , A 4 ' Ar + ~, Ar + — , r- r - directed to its foci. The student is referred to Professor Curtis' Paper for additional applications. 9. A spherical particle moves within a smooth rectilinear tube, which re- volves about one extremity with a uniform angular velocity in a horizontal plane ; find the motion of the particle. Let w be the angular velocity of the tube, and r the distance of the particle, at any time t, from the fixed extremity of the tube ; then, since the force acting on the particle is always perpendicular to r, we have (Art. 28), d-r I lc-- r \ d0\ 2 n d-r — 1 = 0, or -rw- <»' r = °- dt ) ' df- 226 Rectilinear Motion in a Resisting Medium. dr Hence r = ce^ { + c'e-^K If r = a, and — = b, when £ = 0, we get 2wr = (aa> + b) c"* + (ctca - b) e-"*. 10. Consider the same prohlem if the tuhe he supposed to revolve uniformly in a vertical plane. Here, if the time he reckoned from the instant that the tuhe was horizontal, the equation of motion is — - - dP-r = - g sin at. air The integral of this is r = Ce^t + CV W < + -/-z sin at, and the constants can he determined from the initial conditions. 11. Two spheres of the same diameter, hut of different weights, fall freely in air ; find the ratio of the maximum velocities they will attain, stating clearly what assumptions you make. Lond. Univ., 1881. 12. Explain what is meant hy the terminal velocity of a hody in a resisting medium. If the resistance vary as the square of the velocity and the hody move in a vertical line, prove that at the time t, reckoned from the instant at which the hody is at its highest position, its depth x helow this position is given hy rhen ascending, and by co*. gt x = — log sec — , x — — log cosh — , when descending ; u> denoting the terminal velocity in the medium. Lond. Univ., 1883. 13. If a hody be projected vertically upwards in a resisting medium with its terminal velocity for the medium, determine the height of its ascent, and the time of reaching the highest point. Prove that, if an engine can pull a train of W tons at a velocity V on the level, against resistances varying as the square of the velocity, the engine exert- ing a constant pull of P tons : then up an incline o to the horizon the maximum velocity will fall to FV(1 - W sin a I P), and that down the incline without steam the terminal velocity is FV( JFsina / P). Prove that, if on a long railway journey, performed with average velocity V, the actual velocity v varies from its mean value by a periodic function of the time, say v = V+ Usinnt, the average horse-power and consumption of fuel is to that required to take the train with uniform velocity V as 1 + | Z7 2 / V°- : 1. Lond. Univ., 1887. ( 227 ) CHAPTER IX. THE GENERAL DYNAMICAL PRINCIPLES. 196. D'Alembert's Principle. — If a system of mate- rial points connected together in any way, and subject to any constraints, be in motion under the influence of any forces, each point of the system has at any instant a certain accele- ration. If now to each point an acceleration were applied equal and opposite to its actual acceleration, the velocities of all the points of the system would become constant — in other words, each point would move as if free and unacted on by any force whatever ; that is, the applied accelerations, the external forces, and the constraints and mutual or internal forces of the system, would equilibrate each other. Stated in algebraical language, the principle which is given above may be enunciated as follows : — If the coordi- nates of any particle m of a material system be a?, //, z, and the external forces there applied X, Y, Z; the system of forces, d 2 x x d~y, d 2 z, Xl " Wl ^' Y '~ nh -dF> Zl - mi ~df> d 2 x 2 d 2 y 2 d 2 z % x *- nh HF> Y *- ,)h liF> ^->^> &c -> acting at the points %it/iZ u x 2 y 2 z : , &c, will be in equili- brium, in virtue of the constraints and mutual reactions of the system. d 2 x d 2 t/ (Pz The force whose components are - m — , - m , 1 dt? 9 dt 2> m d is called the force of inertia of the mass m, and D'Alembert's Principle (as stated in Article 71) simply expresses that — The applied forces and the forces of inertia in any system are in equilibrium. q2 228 The General Dynamical Principles. In applying D'Alembert's Principle, we may, as in Statics, consider the constraints of the system either as geometrical conditions, or else substitute for them unknown forces. In the algebraical statement just given, the former plan has been adopted ; but if we choose to adopt the latter, we have merely to make X, Y, Z, &c, include not only the applied forces, but also the stresses arising from the constraints. If the Statical Principle of Yirtual Yelocities be employed, we have for D'Alembert's Principle the concise mode of expression given by Lagrange in his Mecanique Analytique, viz. : — s j(x-.g)fc + (p:-«S)*r + ^-S)fcJ-o. (1) This equation may also be written Sw [xlx + j/By + zBz) = S {XSx + YSy + Z$z), (2) a form which is often more convenient than (1). If the forces X, Y, Z, &c, constitute a conservative system, Art. 124, we may write 2(X&-+ YSy + ZSz) =SY, and (2) becomes in this case Sw2 (£& + ydy + zh) = SY. (3) 197. D'Alembert's Principle for Impulses. — As has been stated already in Article 66, an Impulsive or In- stantaneous Force is a force which produces a finite change of velocity in a time so short that in it no sensible change of velocity is produced by the action of the forces which are not impulsive. If the constraints and connections of a system be regarded as giving rise to forces, these forces may be im- pulsive or not, according to the nature of the constraint. For example, a blow given to a body which is resting on an im- movable surface produces an impulsive reaction, provided the blow is not tangential to the surface ; but a sudden jerk to a body attached to the end of an extensible elastic string pro- duces no impulsive reaction. It is important to observe that D'AIembert's Principle for Impulses. 229 each point of the system may be regarded as occupying the same position in space at the end as at the beginning of the time during which the impulsive forces have acted. In other words, the velocities of the various points may change by a finite amount, but the positions can only change by an infi- nitely small amount during the time under consideration. If u, v', w be the components of the velocity of any point, whose coordinates are a?, y, s, before the action of the impulsive forces; and u, v, w the corresponding velocities after their action ; and X, F, Z be the components of the im- pulse which has acted at this point, D'AIembert's Principle as applied to impulsive forces may be expressed in the form — Sm ( [u - u') Sx + [v - v') hj + (w - w) Ss) = 2 [XBx + Ydy + ZSz). (4) The truth of the Principle in the present case can be established by reasoning similar to that employed in the preceding Article. It may also be derived from the Principle applied to continuous forces, by considering the impulsive forces as continuous forces of great magnitude acting for a very short time. In fact, if we multiply >the equation -S)a. + .(r-^^ + (*-«S)*|-° by dt 9 and integrate between the limits t and f; if the interval t-t'he sufficiently short, the system has not sensibly altered its position, and therefore Sa>, &c, are the same at the end of the time as at the beginning, and we have V (dx _ fdx\\] _jt Xdt - m \Tt-{jt njfc+M-o- Now, if X be the component of a continuous force, ^ Xdt is insensible ; and if Xbe the component of an impulsive force, it Xdt is the component of the impulse along the axis of x, 230 The General Dynamical Principles. which may be denoted by X ; hence, as dx fdxY , we immediately obtain equation (4). 198. Initial Motion. — If a system start from rest under the action of given impulses, equation (4), Art. 197, becomes 2w {uSx + v$y + wdz) = 2 (XSx + Tdy + Zdz), (5) where u, v, w are the components of the initial velocity of the point xyz. Now as Bx, Sy, $z are any arbitrary displacements of this point, consistent with the conditions of the system, we may, if the equations of condition do not involve the time explicitly, substitute for 8x, Sy, §z the actual displacements of the point (see Art. 200). Hence, as actual displacements when divided by the element of time become velocities, we may substitute for Bx, By, Bz the components u' , v', w', of the velocity of xyz in any actual motion of the system. Thus we obtain 2m {uu + vv + ww) = S (Xu' + Yv' + Zw'). (6) Examples. 1. If the same system be set in motion successively by two different im- pulses applied at the same point, each impulse is proportional to the velocity in the direction of the other which it imparts to its point of application. Let these velocities be q and jt/, and let X, Y, Z; X', Y', Z' be the compo- nents of the impulses P arid Q, and u, v, w ; u', v', w' the components of the initial velocities of the point of application, then, Xu' + Yv' + Zw' = ~Zm{uu + vv' + ww) = X'u + Y'v + Z'w ; but Fp = Xu' + Yv' + Zw', and Qq = X'u + Y'v + Z'w, whence P : Q::g : p'. 2. In any system at rest, if we suppose an impulse P applied at a point A, and an impulse P' applied at a point B ; prove that P: P' = v :v\ where v is the component, in the direction of P', of the velocity of the point B due to the impulse P ; and v' is the similar component of velocity of the point A. Energy of Initial Motion. 231 199. Energy of Initial Motion.— If T be the initial kinetic energy of a system set in motion by given impulses, by substituting u, v, w for &r, hj, $z (in 5) we obtain 2T = Sw O 2 + v 2 + w 2 ) = 2 (Xu + Yd + Zw). (7) BertramVs Theorem* — If a system start from rest under the action of given impulses, every additional constraint diminishes the initial kinetic energy. Let ii ', v\ w be the initial velocities of the point xyz under the action of the given impulses when the additional con- straints are imposed ; and u, v, w the initial velocities when the system is free from these constraints, then, u'dt, v'dt, w'dt are possible displacements in the unconstrained as well as in the constrained system. Hence, substituting u', v, w for &r, $y, cz in equation (5) we obtain 2w (uu + vv + ww') = 2 (Xu' + Yd' + Zw'). But, by (7), 2w (u'°~ + v' 2 + w' 2 ) = 2 [Xu* + Yv' + Zw) ; thus we have Sm { (u - it)" + (v - v'y ■¥ (to - w'Y\ = "2m (u 2 + v 2 + iv 2 ) - 22w (uu' + vv' + ww') + "2m (it 2 + v' 2 + w 2 ) = 2T- 4T' + 2T' = 2T- 2T'. (8) Hence, we see that the energy of the unconstrained exceeds that of the constrained motion by the energy of the motion which must be combined with either to produce the other. Thomson's Theorem. f — If impulses are applied only at points where the velocities are prescribed, additional con- straints increase the initial kinetic energy. Here, when additional constraints are imposed, the im- pulses are supposed to be altered in such a manner as still to produce the prescribed velocities in the assigned points ; then, u', v\ w being, as before, the velocities belonging to the constrained motion, we have, since in the present case the * Liouville, tome septieme (1842), p. 165. t Proceedings of Royal Society of Edinburgh, April, 1863. 232 The General Dynamical Principles. velocity of every point at which an impulse acts is sup- posed to remain unaltered, S {Xu' + IV + Ziv) = S [Xu +Yv+ Zw) = 2T. Hence by (6) we obtain Sm { [ii - uf + {v f - vf + (to' - ivf } =2T'-2T, (9) and therefore T exceeds T by the energy of the additional motion. Examples. 1. A system is set in motion by an impulse which is measured by the momentum of a mass of 60 lbs. moving with a velocity of 24 feet per second. The impulse imparts to its point of application a velocity of 8 feet per second in a direction inclined to that of the impulse at an angle of 60°. Find in foot pounds the initial kinetic energy of the system. Ans. 90. 2. If the initial velocities of certain points of a system be given, prove that its initial kinetic energy is least when tbe system is set in motion by impulses passing through these points. 200. Equation of Vis Viva.— A first integral of the equations of motion can very frequently be obtained directly from D'Alembert's Principle, as follows : — where &r, hj, Bz, &c. are arbitrary displacements consistent with the conditions of the system. If the equations of con- dition do not contain the time explicitly, dx (the actual movement of the point along the axis of x during an infinitely short time) is always a value which may be legitimately assigned to Sa? ; for the fact that it is an actual displacement shows that it is consistent with the equations of condition, and therefore possible, provided these equations do not alter with the time ; that is, do not contain the time explicitly. If they contain the time explicitly, dx is not in general a possible value of &e. In fact, IT = being an equation of condition, if U is a function of the coordinates simply, &r and dx must satisfy the same equation, viz., dx dy Equation of Vis Viva. 233 If, however, TJ contains t explicitly, $x has to satisfy the equation 4?&e + &o. = 0, ax where t is treated as constant; but dx has to satisfy the equation -7- dx + &c. + — - at = 0. dx at This is so, because dx is the interval between two successive positions of a point, at consecutive instants of time ; whereas he is the interval between two simultaneous infinitely near possible positions of the point. In the great majority of problems dx is a possible value of Sx ; and the same holds for the other displacements. Assuming then that the transformation is legitimate, let us assign to Sx, 8y, &c. the values dx, dp, &c. ; D'Alembert's equation becomes then 2ra (~ dx + ^ — , and clt c(t negative if ~ > — ~. In the first case the mutual action 5 dt clt tends to increase the relative velocity in its own direction, and in the second case to diminish this velocity. Also, if the mutual action tends to diminish — - 1 similar reasoning applies. 202. Effect of Impulses on lis Viva, — The change of vis viva resulting from impulses may be investigated by means of equation (4), Art. 197. In general for any displacement c\ whose components are Ex, $y, dz, we have X Sx + Yhj + Z%z = R or, where R is the impulse whose components are X, Y, Z, and oY is the projection of cs on the direction of R. Hence, if the direc- tions of R and &s are at right angles to each other, XSx+ Y§y+Z$z= 0. 236 The General Dynamical Principles. Again, if two equal and opposite impulses occur at points whose coordinates are Xi, y Xi z x ; x 2 , y 2 , z 2 , the corresponding terms in 2(X&r+F8y + Z&) are X {Sx, - $x 2 ) + F(S^ - fy 2 ) + Z (dz, - &,), or R (h\ - Sr 2 ), from which we conclude, that if the relative displacement of two points be perpendicular to the direction of the mutual impulsive reaction at those points, the corre- sponding terms in 2 (XSx + TBy + ZSz) vanish. We can now prove the following theorems : — If a system be acted on by external impulses, the vis viva is diminished by the vis viva of the additional motion when the impulse at each point is perpendicular to the subsequent velocity of that point, but increased by the same amount when the impulse is perpendicular to the antecedent velocity. Similar results hold good for internal impulsive reactions when each mutual impulse is perpendicular to the relative velocity of the points between which it acts. For the same notation being adopted as in Art. 197 — 1° when each impulse is perpendicular to the subsequent ve- locity of the point at which it acts, we have 2{Xu+ Yv + Zic) = 0; and 2° when it is perpendicular to the antecedent velocity, S(Ij/+ Yv'+ Zw= 0. In the first case, substituting u, v 9 w for Ex, By, Bz in equa- tion (4), we get 2 m [ (u - u') u + (v - v) v + (w - vf) w } = ; from which we obtain 2 m ( [u - uj + [v - vj + (iv - to') 2 } = 2 m (u 2 + v' 2 + w 2 ) - 2 m (u 2 + v 2 + to 2 ). (13) In the second case, substituting u', v\ w for $x, By, Sz in (4), we obtain in like manner 2 7ft { (u - uf + (v - v) 2 + (w - tof) = 2 m (n 2 + v 2 + w 2 ) - 2 m [u 2 + v 2 + w 2 ). (14) Effect of Impulses on Vis Viva. 237 Bertrand's Theorem (Art. 199) is obviously included in the first case of the above theorems. The impulses resulting from the impact of inelastic bodies against fixed obstacles, or against one another, as well as those produced by sudden pulls on inextensible strings, come under the first case considered above. To the second case, on the other hand, belong impulses due to explosions, or to the process of restitution which takes place in the second period of the impact of elastic bodies. As has been already stated (Art. 78), in the impact of such bodies there are two periods. In the first, the mutual action reduces the relative normal velocity of the colliding points to zero. In the second, a force of restitution is developed, which acts at each point in the same direction as the original force, and produces an impulse which bears a constant ratio to that belonging to the first period. This constant ratio is called the coefficient of restitution. As the special equations which determine the changes of velocity in terms of the corresponding impulses are obtained by equating to zero the coefficients of the independent varia- tions in equation (4), Art. 197, we see that these equations are always linear. Moreover, in the impact of elastic bodies the geometrical conditions are the same in the periods of compression and of restitution ; but each impulse in the latter period is equal to the corresponding impulse in the former multiplied by the coefficient of restitution. Hence we con- clude that this holds good likewise for the corresponding changes of velocity in the two periods. Examples. 1. If a system be acted on by any set of impulses, prove that the increase of its vis viva maybe expressed in the form 2{ X(u + u') + Y{v +- v) + Z(w + w')} t where X, Y, Z are the components of the impulse acting at any point of the system; and «, v, w, u', v', w' the components of the velocity of this point after and before the impulsion, respectively. 2. If a system be acted on by a set of impulses which reduce to zero the velocities, in the direction of the impulses, of the points at which they act, find the change of vis viva in terms of the impulses and the antecedent velocities of the points at which they act. Ans. ^{Xu' + IV + Zw). 238 The General Dynamical Principles. 3. The ends of a string passing over a smooth pulley are attached to two masses, of which one rests on a horizontal plane, and the other is dropped through a height h , the masses of the string and pulley heing neglected, determine the loss of kinetic energy caused hy the impulsive tension of the string. If in and in' he the masses, v\ and v> the velocities of the dropped mass m hefore and after the chuck, and 7 the loss of kinetic energy, mm 2,7 = m iv\ — ro) 2 + m vz 2 . Hence 7 = - ah. v ' m + m J 4. If any system of smooth imperfectly elastic hodies having a common coefficient of restitution collide, show that the loss of vis viva is 1 — e 2m {(u-u') 2 + (v -v') 2 + (to - tv') 2 }, 1 + G where e is the coefficient of restitution, m the mass of any particle, and u\ v', to', u, v, w the components of its velocity hefore and after the shock. Let U, V, Whe the components of the velocity of in at the end of the first period of impact; then by equations (13) and (14), Art. 202, if 7 be the total loss of kinetic energy, 27=2m{(U-u')* + {r-v') i +{W-w')*}-2m{(u-U)*+(v-jrf+(w-W)*}; but u — U=e(U— n'), &c, and therefore, u — u' = (1 +e)(U— u'), &c. Hence, 27 = (1 - e 2 ) Sm { ( U- u'f + ( V- v') 2 + { W- w') 2 } 1 — ' e 2 = — y 2 Smj{ (m -ti') 2 + {v -v') 2 + [w-w') 2 } . The theorem contained in this Example is due to Carnot. 5. Two weights are connected by a fine inextensible string passing over a smooth pulley. The lesser hangs vertically, and the other descends a smooth inclined plane, starting without initial velocity from a point vertically under the pulley ; determine how far it will descend ; and state the limit of the ratio of the weights within which finite descent is possible. If z be the height which the lesser weight W ascends, and s the distance along the inclined plane traversed by the greater weight W, we have, by the equation of vis viva, when the system comes to rest, TVs sin i - W'z = 0, and W therefore z = \s sin i, if — = A. Also, if h be the vertical distance of the pulley from the inclined plane, we have geometrically, since the string is inex- tensible, to o ^t • • TT 2(A— l)sine, (h -t z)~ = //- -f s- -f 2hs sm x. Hence s = -^ . h. 1 — X z sin-i In order that finite descent should be possible, W >W r sin i. 6. Two equal spheres, A and B, starting simultaneously from rest, descend down two equally inclined planes ; the one plane quite smooth, the other perfectly rough ; find the ratio of the velocities of the centres of the spheres at the end of any time. Let v\ be the velocity of the centre of A, v% that of the centre of B, and w the angular velocity of B at the end of any time t, then v% — aw (see Ex. 1, Examples. 239 7 Art. 134), also, mvi 2 =2ffmzi, and- ma 2 » 2 = 2gmz2, where z\ and z 2 are the 5 distances through which the centres of A and B have descended. Now, if S] and s 2 he the distances which the centres have moved parallel to the inclined plane at any time, ds\ da* . . . . Vi = — , t'2= — , and :i = *i sin ?, r 2 = S2smi. dt dt Hence, substituting and differentiating, we have d n 'Si . . 7 d' 2 s 2 • • tt 5 - — = l + ira J 9 A uniform bar, of length 2a, is suspended from a fixed, parallel, and equal horizontal bar, by strings of equal length joining the adjacent extremities of the bars An angular velocity a> is imparted to the snspended bar round a vertical axis through its centre of inertia. Determine the vertical height through which its centre of inertia will rise. As each extremity of the bar moves on the surface of a sphere to wnicn tne attached string is radius, the tensions of the strings do not appear in the equation a 2 co 2 of vis viva ; hence h = — - . 240 The General Dynamical Principles. 203. The General Equations of Motion of a Rigid Body apply to every System. — If the forces acting on any system be in equilibrium, the equilibrium is not disturbed by rendering the mutual distances of the points of the system invariable — in other words, by making it rigid. Hence the equations of motion of a rigid body are, in their most general form, applicable to any system whatever. The special reductions which may be applied to the forces of inertia in the case of a rigid body cannot, however, be em- ployed in other cases. 204. Equations of Motion of a Rigid Body. — By means of D'Alembert's Principle we can at once write down the equations of motion of a rigid body. We have, in fact, merely to write down the six equations of equilibrium, taking into account, not only the applied forces, but also the forces of inertia as denned in Art. 196. Hence the six equations of motion are — (15) »£-2X, *.fjf-ar. a-9-s* ~(*£-S)-*-^ ^(.S-S)-^--*)-* >, ^iU d ^-y d ^\=^{xY-yX)=N (16) where Z, M, N are the moments of the applied forces round the axes. For impulses, the corresponding equations are — ^m [y{w - vf) -z(v-v')} = ^(yZ-zY)=L \ ^m{z{u-v!)-x{w-w')} = 2{zX-xZ) = Jf > • (18) 2m [x (v -v)-y (u - u) } = 2 [x Y- yX) = N J These equations hold good for any system which is Constraints and Partial Freedom. 241 altogether free, i, e. unacted on by any constraints, and not subject to geometrical conditions external to itself. In the case of a system subject to external constraints, the constraints must, in general, be replaced by the stresses to which they give rise. Equations (15) and (17) may be put into a simpler shape as follows : — 205. Motion of the Centre of Inertia of a Free System. — Let x, y, z be the coordinates of the centre of inertia, and 9ft the mass of the entire system ; then, for continuous forces, Wise = 2w -j-r = 2X I dt Wly = 2»» df d-z S7 3tts = * m aW Also, for impulses, m {u - u) = 2w(m m{v -v) = 2m (v Wl {re - of) = 2»i(?c = 2Z (19) ii ) = 2X \ tO=2F,l. (20) w) = 2Z ) These equations give the motion of the centre of inertia of a free system acted on by any forces. From them it appears that — The centre of inertia of a free system moves as if all the forces were applied to the entire mass concentrated there. 206. Constraints and Partial Freedom. — If a system be subject to external constraints, we may apply equations (19) and (20), provided we suppose the constraints replaced by the forces to which they give rise. If a system, though not entirely free, be such that equal and parallel displacements of arbitrary magnitude can be given to each of its points in a definite direction, and if the axis of x be taken in that direction, we have still the equation m = 2X R 242 The General Dynamical Principles. 207. Internal Forces. — Any force by which two parts of a system act on each other is said to be internal. Since action and reaction are eqnal and opposite, the components of internal forces destroy one another in the sums EX, E F, and S^. Hence in any system Interna I forces, whether continuous or impulsive, have no effect on the motion of the centre of inertia. 208. Case of no External Forces. — It follows from Art. 207, that if a system be acted on by no external forces, its centre of inertia is either at rest or moves in a straight line with a constant velocity. This theorem is sometimes termed The Principle of the Conservation of the Motion of the Centre of Inertia. Kesults similar to those of the preceding Articles hold good for impulses. 209. Motion of a Free System relative to its Centre of Inertia. — If £, rj, Z be the coordinates of any point of a system referred to axes through its centre of inertia parallel to fixed directions, x = x + £, y = y + rj, s = z + Z- Substituting in D'Alembert's equation, we have *(ax- (»,) j£)+*(ar- <*0 §} »(**-(*-) g) o. .,((x-5f)«*(r-5)**(,-5)^ Now 2w£ = Emij = Sm£ = ; hence, by equations (19), Art. 205, we obtain It follows from this equation that — The motion of a free system relative tojts centre of inertia is the same as if this point were fixed in space, the applied forces Moments of Momentum. 243 being unaltered as regards magnitude, direction, and point of application. The theorem just stated holds good as well for impulsive as for continuous forces. This readily appears by applying the transformation employed above to equation (4), Art. 197, and making use of equations (20), Art. 205. 210. Moments of Momentum. — It is readily seen that at any instant the expression 2m (xv - yu) or 2m (xy - yx) represents the entire moment of the momenta round the axis of z of all the elements of the system at the instant : and similarly 2m (yz - zy) and 2m (zx - xz) represent the corre- sponding quantities relative to the axes of x and y, respec- tively. These moments of momenta are of fundamental impor- tance in the discussion of the motion of any system, and we shall accordingly represent them by distinct symbols. Thus let S x = 2m [yz - zy), S 2 = 2m (zx - xz), S 3 = 2m {xy - yx), (22) then equations (18) may be written in the form J S 1 -S l '=L, H 2 -H:=M, H z -Hs' = N, (23) in which IZi', Si, Si represent the moments of momenta of the system before, and S lf S 2 , S z those after, the impact. If the body be at rest when acted on by the impulses, these equations become S X = L, S 2 = M, S, = N. (24) Hence, in this case, the moments of momenta generated by the impulses are respectively equal to the impulsive moments applied. Next, since xy -yx = — (xy - yx), dS x we have — = 2m (xy - yx), r 2 244 The General Dynamical Principles. and it follows that equations (16) may be expressed in the following form : — ^- = i ' ^r =Jf ' ^ =if - (25) The quantities H lf H 2 , H 3 admit of an important trans- formation, as follows : — If \ It 3 dt represent the elementary area described round the origin by the projection of the point xyz on the plane of xy, then xy - yx = h z . Hence H* = S;w/i 3 , and, likewise, representing the pro- jections on the planes of yz and xz by a similar notation, Hi = SffiAi, H 2 = "2,mh 2 . Accordingly equations (25) may be written Sm-rr=i:, Sm-^ = M, ^m-^ = N. (26 dtf atf ^ The corresponding equations for impulses are 2mA, = X, 2w/i 2 = if, 2mA 3 = JV". (27) If the system is in motion when the impulses act, the three latter equations should be written ^mhi = L + ^m/h\ ^mh 2 = M+ %mk 2 ', *2mh z = N+ 2mA 3 ', (28) where hf, h 2 , h{ are the values of hi, h 2 , h 3 the instant before the impulses act. The quantities h iy h 2 , h 3 , &c, are the areal velocities, relative to the origin, of the different points of the system ; and —, &c, are the areal accelerations (see Art. 29). av In any system in motion the three moments H h H 2 , H z , if they were regarded as moments of forces or couples acting on the same rigid body, would be equivalent to a Moments of Momentum. 245 single moment H round a line whose direction cosines TT TT TT are ~, — - 2 , -=^ ; H being given by the equation a a n E 2 = H? + Hi + -ff 3 3 . This line is called the momentum axis of the system relative to the origin. As it is the axis of the couple which is the resultant of the couples corresponding to the moments of the momenta of the different elements of the system, it is plain that its direction is independent of the directions of the co- ordinate axes. If Sdt be twice the sum of the projections of the elementary areas described by all the points of the system round the origin, each multiplied by the corresponding element of mass, on a plane whose normal makes an angle 6 with the momentum axis, then S = Hcos 6. (29) This may be proved in the following manner : Let hdt be double the elementary area described by the element whose mass is m round the origin ; and let a, /3, y be the cosines of the angles its plane makes with the coordinate planes; then, A, /u, v being the direction cosines of the normal to the plane of 8, S = , 2mh (a\ + (5fi + yv) = X^mha + ju2w/*|3 + v2mhy = A^ 1 + j u^ 2 + v5- 3 = ^JAj + / xJ 2 + vJ 3 j = ^cos0. Hence, the multiple sum of the projections of the elementary areas on the plane at right angles to the momentum axis is a maximum. This plane is called the Principal Plane relative to the origin. From what has been just proved, we see again that its position is independent of the directions of the axes. If g, n, Z be a second set of rectangular axes through the origin parallel to directions fixed in space, and if the direc- 246 The General Dynamical Principles. tion cosines of £, referred to x, y, z, be a h a 2 , a 3 ; of r\, b ly b 2 , b 3 ; of £, Ci, c 2 , c 3 ; we have, as particular cases of what has been proved above, 2m(ij£ - £rj) = E x a x + E 2 a 2 + E :i a 3 \ 2m(Z% - &) = E x b L + E 2 b 2 + E z h . (30) 2m (ty - r)%) = E x c x + E 2 c 2 + E z Ci ) The preceding theorems of this Article are true for any system of moving points, and whether the origin be fixed or movable. Again, to find the moments of momentum of a system round axes intersecting at a point whose coordinates are a, b, c. Let Ei, E 2 , E 3 be the moments of momentum of the system round axes parallel to the coordinate axes, and inter- secting at the point abc ; then, we have m=-2m{[y-b)i-(z-c)y) = 2m (yz - zy) - (b^mz - c'Emy) ; but x, y, z being the coordinates of the centre of inertia of the system, 2my = my, Sms=3ffs; hence we obtain E;=E x -m(bz-~cT/) \ e: = E 2 -m{cx- a i) [ • ( 31 ) E 3 '=E 3 -Wl(ay-bic) ) Again, the moment of momentum of a system round an axis, through any point 0, is equal to the moment of the momentum relative to the centre of inertia round a parallel axis through that point, together with the moment of momentum round the axis through of the entire mass of the system supposed to be con- centrated at the centre of inertia, and moving with it. Internal Forces. 247 Take the axis through as the axis of %, and make use of the transformation employed in Art. 209, then 2m (yz - zy) = Sm { {[/ + n) (k + £) - (z + Z) (y + y) ) = m(fi-zf / ) + 2m( v t-Zv); (32) since 2mr} = HmZ, = 0, Smrj = Sm§ = 0. The student will observe that £, 17, &o., denote relative, not absolute, velocities. If the origin be the centre of inertia of the system, equations (23), (24), and (25) hold good whether be fixed or moving (Art. 209), the axes being parallel to lines fixed in space. In the deduction of equations (23), (24), and (25), we have supposed that the system is free, that is, unacted on by constraints external to the system itself. 211. Constraints and Partial Freedom. — A system which is not free may be regarded as free, if the external constraints be replaced by the forces to which they give rise. In general, we can ascertain whether a given constraint affects the validity of equations (23), (24), and (25), by con- sidering its influence on the conditions of equilibrium of a rigid body. If one point of a rigid body be fixed, we know that for its equilibrium the moments of the applied forces round three rectangular axes meeting at the point must each be equal to zero. Hence we conclude — If there be one point of a system fixed, equations (23), (24), and (25) hold good for this point as origin. Again, if there be a fixed line in a rigid body, the con- dition of equilibrium is that the moment of the applied forces round this line should be zero. From this we infer — If there be a fixed line in a system, the rate of change relative to the time in the moment of momentum of the system round this line is equal to the moment of the applied forces. 212. Internal Forces. — Since internal forces occur in pairs, each pair consisting of two equal and oj)posite forces 248 The General Dynamical Principles. having a common line of direction, the moment round any line of the whole set of internal forces must be zero. Hence the moments of momentum of any system are unaffected by forces internal to the system. 213. Conservation of Moment of Momentum. — If a free system be unacted on by any forces external to itself, its resultant moment of momentum, relative to any point fixed in space, is constant, and has for its axis a line whose direction is invariable. A similar result holds good for the centre of inertia even though this point be not fixed in space. If a system, otherwise free, contain a point or a line fixed in space, and be unacted on by external forces, the resultant moment of momentum of the sj^stem relative to the fixed point, or the moment of momentum round the fixed line, is constant. The theorems enunciated in this Article together consti- tute what has been often termed The Principle of the Conser- vation of the Moment of Momentum, or The Principle of the Conservation of Areas. As the moment of a force round an axis intersecting the line of direction of the force is zero, we see that — If the lines of direction of all the external forces which act on a free system be met by the same space axis, the moment of mo- mentum of the system round this axis is constant. If the space-axis be fixed in the system, which is other- wise free, the theorem above still holds good. In a similar manner we may conclude that — If a system receive an impulse, the moment of momentum- of the system round an axis fixed in space, and passing through any point on the line of direction of the impulse, remains the same as before. Examples. 1. In any system in motion, show that the moments of momenta round three rectangular axes are equal to the moments of the impulses which would impart to the system if at rest its actual motion. 2. If |, ?j, £ he the coordinates, relative to the centre of inertia, of any point of a free system, show directly, that if the system start from rest, »»(^'-^) = 2(^-fr), &c, Examples. 249 and that during the motion, Ey equation (32), Art. 210, we have but 2fty = 2F, and 27*2 = 2/?; therefore, &c. Again, differentiating each side of the equation (32) of Art. 210, we have m{y't- zy) + 2m{r,C- Cv) = Zm{y~-zy) = Z = 2{ (y + rj) Z- (» + {)Y} ; and as Wty = 2F, m'z = ^Z, we obtain the required result. 3. A satellite of mass m is moving in a circle whose radius is r, round a planet whose mass is M, and which rotates round an axis perpendicular to the plane of the orbit with an angular velocity n. If Cbe the moment of inertia of the planet, and /x the attraction between unit masses at the unit of distance, show that the moment of momentum of the system round its centre of inertia is cln + p^{M+m)-±f*\. 4. A heavy particle moves on a smooth surface of revolution whose axis is vertical ; prove that the moment of momentum of the particle round the axis is constant. 5. A number of mutually attracting particles are acted on by forces passing through the same fixed point ; prove that their resultant moment of momentum relative to this point is constant, and that the direction of its axis is invariable. 6. A system is acted on by no external force except gravity ; prove that its moments of momenta round axes parallel to fixed directions in space, and inter- secting at its centre of inertia, are constant. 7. Show that the centre of inertia of the universe is either fixed in space or else moves in a straight line with a constant velocity. 8. A man walks from one end to the other of a uniform plank which is placed on a smooth horizontal table ; determine the displacement of the plank. Let a be the length of the plank, P its mass, M that of the man ; the dis- placement is — a. 9. A uniform plank is placed on a smooth inclined plane, so as to be perpen- dicular to the intersection of the inclined plane with the horizon ; determine the 250 The General Dynamical Principles. time in which a man should go from the upper to the lower end of the plank in order that it should remain unmoved. Let t he the time required. The displacement of the centre of inertia of the M system in the time t in space is \gt 2 sin i, and relative to the plank is — — -a. If the plank remain unmoved these must he equal. Hence f 2M a M+ F g sini 10. The base of a smooth homogeneous circular semi-cylinder rests on a hori- zontal plane. A particle m is placed at a point on the surface of the semi- cylinder, situated in a vertical plane containing its centre of inertia and perpen- dicular to its axis. Show that the particle will describe an ellipse. Let the axis of x be the intersection of the vertical plane, in which the particle moves, with the horizontal plane on which the semi-cylinder rests ; the axis of y being vertical. Let x, y be the coordinates of the particle, x' the co- ordinate of the centre of inertia of the semi -cylinder, m its mass, and a its radius. Considering the whole system as one body, we have (Art. 206), d 2 x , d 2 x' m — — + m — - = 0. dt 1 dt 2 Hence, since the system starts from rest, mx + m'x is constant, or the pro- jection on the horizontal plane of the centre of inertia of the whole system remains fixed in space. Taking this point for origin, we have mx + m'x' = 0. Again, since the semi-cylinder is homogeneous, we have, from the geometri- cal conditions, (x — x') 2 + y 2 = a 2 . Substituting for x', we obtain (m + m') 2 x 2 + m' 2 y 2 = m' 2 a 2 . 11. Two particles, connected by a rigid rod whose weight is negligible, are projected along a smooth horizontal plane ; determine their motion. The position of the centre of inertia at any time is given by the equations x = mt + a, y=nt+ b, and the inclination of the rod to the axis of x by the equation 9 = tot + e, where m, n, a, b, w, and e are constants. 12. Two equal particles are connected together by a fine inextensible string ; one of them is placed on a smooth table, the other just over the edge, the string being at fnll stretch at right angles to the edge ; find the interval of time from the instant at which the particle originally on the table leaves it to the instant at which the string first becomes horizontal. The acceleration of the particle moving on the table is \g. Hence, if c be the length of the string, the particle leaves the table with a horizontal velo- city v, where v 2 = gc. At this instant the middle point of the string has a horizontal velocity \v, and the lower particle has no horizontal velocity. Hence Examples. 251 the moment of momentum of the system round a horizontal axis through the centre of inertia is \mcv. This remains constant (Ex. 6), and therefore twice the area described round the centre of inertia in any time t is \mcvt. If t be the interval of time during which the string passes from a vertical to a horizontal position, we have, therefore, J?rc 2 = \cvt, and substituting for v its value, we obtain -*£ 13. A sphere is projected with a velocity v along a uniform smooth tube within which it fits exactly. The tube rests on a smooth horizontal plane, and its axis forms a circle ; determine the motion. Let m be the mass of the sphere, ml that of the tube, and a the radius of the circle formed by its axis. The common centre of inertia of the tube and sphere moves parallel to the direction of projection of the sphere with a velocity , and the centres of the tube and sphere describe circles round m + m v with an angular velocity -. 14. A spherical shell rests upon a smooth horizontal plane; a particle is placed at the lowest point of the internal surface of the shell, which is then projected with a horizontal velocity V. The internal surface of the shell being smooth, determine to what height the particle will ascend. Let x and y be the coordinates of the particle, m its mass, and v its velocity ; x' and y the coordinates, and v the velocity of the centre of the shell, m being its mass. Take as axis of x the intersection of the smooth horizontal plane with the vertical plane of motion ; then, Art. 200, mv z + m'v' 2 = m'V 2 - 2mgy, and, by Art. 206, mx + mx' = m' V. Also, as the particle remains on the sphere whose radius is a, we have (x-x') 2 + {y - y 'f = a i ; whence, differentiating, and remembering that y' = 0, we have x - x' = when y = 0. Hence, substituting, we obtain m r 2 2 (m + m') g This result may not hold good if the value of y given above exceed a. 15. A smooth tube, movable in a horizontal plane about a vertical axis, is charged with a number of balls at given intervals ; an angular velocity Q. is communicated to the tube ; determine the velocities of the tube and of the balls at any assigned distances of the latter from the axis. Let mi, mo, &c. be the masses of the balls, a\, a 2 &c. their initial distances 252 The General Dynamical Principles. from the axis, n, r 2 , &c. their distances at any instant, w the angular velocity, and Mk 2 the moment of inertia of the tube about the axis ; then (Arts. 213, 200), (mi n 2 + m% r 2 2 + &c. + Mk % ) w = (mi «r + m% a%- + &c. + Mk 2 ) n, mi h 2 4- ^2 h 2 + &c. + (mi rr + m% r£ + &c. + Mk 2 ) a? = (m\ a{* + mia£ 4- &c. + Mk 2 ) n 2 . (f 2 Ti d 2 ?*o Again (Art. 28), — - n « 2 = = — — r 2 a> 2 , c? 2 n d 2 r 2 whence r 2 —7-: — n -=-r- = 0, and integrating, Hence we have dt 2 dt 2 dri dr% r 2 — n — - = constant = 0. dt dt r\ «i — = constant = — , and therefore also — = — , with similar equations for the other distances and r 2 a 2 velocities. Substituting in the equations of momentum and vis viva, and putting mi «i 2 4- mz «2 2 + &c. = 7, Mk 2 = I', we obtain (7?-i 2 + 7'«i 2 ) a? = (7+ 7') «i 3 a, (ln 2 + r ar) h 2 = (7+ 7') «i*(n a - «i 2 ) n 2 , &c. 16. An indefinitely great number of thin cylindrical shells are revolving in the same direction about their common axis, the angular velocity of each shell being proportional to a positive power of its radius. If the system of shells be suddenly united into a solid cylinder, find the angular velocity of the cylinder about its axis. Let w be the angular velocity of any shell, r its radius, fl and R being those of the outermost shell, then w = Ar», and before the shells are united, the moment 2ttA r» +3 dr. Jo of momentum of the system is If to' be the angular velocity of the united svstem, its moment of momentum is fi — — w . Equating these two, we obtain a) = -. n + 4 17. A uniform horizontal stick falling to the ground strikes at one end against a stone ; compare the blow it receives with what it would have received had both ends struck simultaneously against two stones, the blows being sup- posed to be perpendicular to the stick. Examples. 253 Let v and v be the velocities of the middle point C of the stick, before and after it receives the single blow at the extremity A ; let za be the length of the stick, m its mass, and P the impulse of the blow. The moment of momentum of the stick round a horizontal space axis through A remains unaltered by the blow. Before the blow the whole moment of momentum is due ( (32), Art. 210) to the motion of the centre of inertia, the stick having no motion relative to it. After the blow the stick is rotating round A (since this point is reduced to rest) with an angular velocity ». Hence f ma 2 , x = x - (£ sin \)y + »j cos \p) w (2) if = y + (? cos \fj - 7] sin xp) id Or, x = x - (y - y')w, y = y + {x - x) w. (3) These equations show that the velocity of the point xy is made up of two parts — one a velocity of translation, the other a velocity of rotation, as in (1), round an axis through xy. For any other definite point, x"y" of the figure we have, in like manner, x = x" - {y - y") J\ y = y"+(z- x") <»". Equating these values of i and y to the former, and compar- ing the results with the equations x =x - (y -y )to , y =y + [x - x ) w , Pure Rolling. 261 we see that w" = w, or the velocity of rotation to be attributed to the body, is the same whatever be the point through which the axis of rotation is supposed to pass. 223. Instantaneous Centre, Body Centrode, Space Centrode. — If we put x = 0, y = in equations (2), we get the coordinates of the instantaneous centre of rotation, referred to axes fixed in the body. In like manner equations (3) give the coordinates of the same point referred to axes fixed in space. If we call the coordinates of the instantaneous centre £oj *?o ; ffoj i/o, respectively, we have &> = - {% sin \p -y cos \p), 7j = - (x cos \p + y' sin \p), (4) W (t) , 1 ./ , 1 ./ x = x - - y, y = y +-x. (5) If x, y, (i), and \p are known functions of the time t, we can find from equations (4), by eliminating t, the path described in the body by the instantaneous centre. From equations (5) we can find in the same manner the path described by the instantaneous centre in space. The former of these curves is called the body centrode, the latter the space centrode. The student must carefully distinguish between the in- stantaneous centre and the point of the body which coincides with it at any instant. The latter has no velocity at the instant either in space or in the body ; the former (the instantaneous centre) has in general a velocity both in space and in the body. 224. Pure Rolling. — In pure rolling the points of one curve or surface come into contact successively with those of another, the relative tangential velocity of the points of con- tact being zero. If one curve or surface be fixed in space, the motion of the other consists of a series of rotations round axes through the successive points of contact (Differential Calculus, Art. 295). In the case of one plane curve rolling on another, this appears as follows : — 262 Kinematies of Rigid Body Moving Parallel to Fixed Plane. Let QQ' be the curve fixed in space, and PP f the one which rolls on it, P, P r being two consecutive points on the latter. By hypothesis, P has no velocity along the tangent at P Q, and at the end of an infi- ^~o^ nitely short time P' coincides with Q\ and the distance between P and Q is then an infi- nitely small quantity of the second order. Hence, while other points of the body have received infinitely small dis- placements of the first order, P has received one of the second order, and has, therefore, no velocity in any direc- tion. Hence, during the instant under consideration, the body must be rotating round an axis through P (Art. 217). It is obvious that the acceleration of P in the direction of the tangent at Q is zero ; and it can be easily seen that its acce- leration in the direction of the normal is in general finite, and equal in magnitude to Uto, where w is the angular velo- city of the body, and U is the velocity of the instantaneous centre of rotation, this point having moved during the instant from Q to Q' in space, and from P to P' in the body. 225. Geometrical Representation of the Motion of a Body moving Parallel to a Fixed Plane. — When a body is moving parallel to a fixed plane, if we can deter- mine the space centrode and the body centrode, the motion of the body is completely determined, as it consists of the rolling, without slipping, of the body centrode on the space centrode. The geometrical applications of the principles laid down in the present and preceding Articles are numerous and im- portant ; but as they do not fall within the scope of the present treatise, the reader is referred for them to Chap. xix. of the Differential Calculus, and to Minchin's Uniplanar Kine- matics, Chap. in. 226. Velocity of any Given Point of a Body. — In Kinetics the motion of a body is usually regarded as made up of a motion of translation v, and a motion of rotation w, round an axis through the centre of inertia G. It is sometimes important to determine the velocity of a given point A of the Examples. 263 body. In the case of motion parallel to a fixed plane this is readily done analytically by equations (3). Otherwise, geometrically : — let p be the distance from A to the axis of rotation through G, then, owing to the rotation, A has a velocity pco perpendicular to the plane passing through A and the axis of rotation, and this, combined with the velocity of translation v, gives the velocity of A. Examples. 1. Show directly that if a body have two equal and opposite velocities of rotation round two parallel axes, the velocity of any point is at right angles to the plane containing the parallel axes, and is equal to the distance between the axes multiplied by the angular velocity. Draw a plane through the point at right angles to the two parallel axes. Describe round the axes circles passing through the point. The component velocities of the point are perpendicular and proportional to the radii of these circles, and the resultant velocity is, therefore, in the direction of the common chord, and proportional to the line joining the centres. 2. Prove that a velocity of rotation round any axis is equivalent to an equal velocity of rotation » round a parallel axis, together with a velocity of transla- tion wa along a line at right angles to the plane containing the axes, the distance between which is a. 3. A body receives, in a given order, finite rotations round two parallel axes fixed in space. Determine the magnitude of the equivalent rotation, and the position of its axis. 4. If the parallel axes round which the body receives successive rotations be fixed not in space but in the body, determine the single rotation which would bring the body into the same position. If A, B are the intersections of the nxesjixed in space, with a plane at right angles, £ that of the resultant axis, and o, £, x, the magnitudes of the rotations 264 Kinematics of Rigid Body Moving Parallel to Fixed Plane. round them, then BAR = — f a, ABR = § #, and the resultant rotation x = a + £, or, (a - /3), according as a and jSare in the same or opposite directions. In the latter case its direction is the same as the greater of the two. If A and B' are the positions of the axes fixed in the hody, B'AR = ^ a, AB'R — — ^ $. 5. Two equal and opposite finite rotations round parallel axes hring a body into the same position as a single motion of translation. Determine the direc- tion and magnitude of this translation. The direction of translation is at right angles to a line which makes with AB or AB' an angle equal to - J a, or J o, and the magnitude of the translation = 1AB sin | a, or, 2AB' sin \ a. 6. If the direction of the motion of each point of a hody be always parallel to a fixed plane, the motion is equivalent to a succession of rotations round the generating lines of a cylinder fixed in space, which is at right angles to the fixed plane. 7. A plane area receives a motion of translation in its own plane whose com- ponents, parallel to the axes, are a and b ; and a rotation 6 round the point in the body which, at the beginning of the motion, coincides [with the fixed origin. Determine the coordinates of the point, a rotation round which would bring the body into the same position. a sin|0- b cos|0 b sin i0 + a cosi0 Ans. x = — : — : =-, y = = s_. 2sin£0 ' * 2sini0 8. Show from these expressions that the amplitude of the rotation is the same as before. \ a 1 _j. £2 If (p be the amplitude, sin \

2 = OC. o) ; hence — = -— . — — (Thomson a.xdTait). «2 AB CO - 10. A bar AB moves in one plane with given angular velocity rounds, while at B it is freely jointed to another bar BC, whose extremity Cis con- strained to move along a fixed straight groove passing through A ; find the velocity of C in any position. Examples. 265 Draw a perpendicular to AC at C, and let it meet AB in ; then is the instantaneous centre of rotation of BC. If v he the velocity of C, and w the t9C angular velocity of AB, v = AB . — . a> = AP. w, where ^P is drawn at right OB angles to AC to meet PC in P. For the further discussion of this question the reader is referred to Minchin, Uniplanar Kinematics, p. 47, or Goodeve, Elements of Mechanism, Chap. i. The arrangement of machinery mentioned in this example is called the crank and connecting rod. 11. A har moves in a horizontal plane with uniform angular velocity round one extremity. To the other extremity a horizontal circle is attached. If the circle he regarded as rotating round its centre, what additional motion must it be considered to have ? A velocity of translation at right angles to the har, and equal to aw, where a is the distance of the centre of the circle from the fixed end of the har, and co the angular velocity. 12. If two definite points of a plane figure are constrained to move along two straight lines in its plane, which are fixed in space, the space centrode and the body centrode are circles, the former being double the latter {Differential Calculus, Art. 295). 13. In Peaucellier's arrangement find the relation between the velocity of the point describing the straight line and that of one of the adjacent corners of the parallelogram. M. Peaucellier, in 1864, first succeeded in transforming circular into recti- linear motion by the following arrangement : — A and B are fixed points ; AP and AQ are two equal bars which can turn freely round A ; BR is another bar turning freely round P, and equal in length 'to AB ; QRPX is a jointed parallelogram composed of four equal bars turning freely round their points of intersection. If a motion be imparted to the system, the points P, Q, P describe circles. That the point X describes a straight line may be shown as follows : — In any position of the system, since L PRX = L QRX, and L PEA = L QRA, XR and RA are in one straight line ; then XPR being an isosceles triangle, and PA a line drawn from the vertex to the base, AR . AX = AP*-RP 2 = const. ; wherefore X describes a curve which is the inverse, with respect to A as origin, 266 Kinematics of Rigid Body Moving Parallel to Fixed Plane. of that described by\R. Now the point R describes a circle which passes through A ; hence X describes a straight line, perpendicular to AB at the point 8, where AS.AD = AP* - RPK We proceed to find the relation between the velocities of P and X. Draw XO at right angles to SX ; then is the instantaneous centre of rotation of the bar PX. Let AP = a, PX = b, BR (in former figure) = c ; then « being the angular velocity of AP, a' that of PX, and v the velocity of X ; we have, since is the instantaneous centre, v = OX . «', and OP . io' = AP . w ; therefore = °*.AP. OP AT. w. ur Again, if PAT=B, PTA = cp, we have AT = a sin 6 (cot 6 + cot ) o>, where (p is given by the equation a cos 6 + b cos 2c 14. A plane area is moving in its own plane ; determine the accelerations of any point in it parallel to the tangent and the normal to the space centrode at the instantaneous centre of rotation. Let xo, yo be the coordinates of a point fixed in the lamina, |, 77 those of any point in it referred to xo, yo as origin, and to axes parallel to those of x, y ; then d£ drj It w being the angular velocity of the body ; whence dx dxo ■ Tt = *°> dt dt -7JW, dy dy dT = -dT + ^ Examples. 267 d-x d 2 %o o du> d 2 y d 2 y dot — = A t — CD" 7). Call the centrode fixed in space C, that fixed in the body, r. The velocity of the point of the body which coincides at any instant with the instantaneous centre of rotation is zero. At the next instant the instantaneous centre of rota- tion has moved to the consecutive position on each of the curves C and r. At the end of this instant Ohas a velocity in the normal to C equal to FIco, where I, T are consecutive positions of the instantaneous centre on the tangent to C. Hence the acceleration of along the tangent to C is zero, and along the normal to C is w 2 * if we pu t /' /= dff, and w = — . Xow if p and p be the radii of cur- dO dt 1 l l -. • -i a 4. dd 1 vature of C and I\ and, if we put = -, it is easily seen that — = -. p p jk aa it Hence, if xqi/q coincide with 0, and we take as axes the tangent and normal to C, we have d 2 x dt 2 = — co'- 1 doi ~d7 v d 2 y dt 2 oP~R + dca ~di*~ 0T7J. 15. Determine the points of the body which have at any instant (I) no acceleration parallel to the tangent to C at the instantaneous centre of rotation ; (2) no acceleration parallel to the normal. These points consist of two straight lines in the body at right angles to each other, the first of which passes through the instantaneous centre of rotation. 16. Determine at any instant the position of the point in the body having no acceleration. It is the intersection of the two lines mentioned in the last example. If a be the angle which the line of non-tangential acceleration (Ex. 15) makes with the axis of x, the coordinates of this point may be expressed in the form | = R sin a cos a, 77 = R sin 2 a. These expressions readily follow from the equations of Ex. 14. This point is called the acceleration-centre. 17. The acceleration of any point of the body is the same as if the body were turning round the acceleration-centre as an absolutely fixed point. 18. All points of the body which have a common acceleration lie on a circle having the acceleration-centre as centre. 19. Find the points of the body for which the acceleration normal to the path described by the point is zero. Take the centre of rotation as origin of £77 ; any point is describing a circle round it ; hence the line joining the origin to £tj is the normal to the path of 268 Kinematics of Rigid Body Moving Parallel to Fixed Plane, the latter; and if N be the normal acceleration, and r the distance from the in- stantaneous centre of rotation, '-H-^-ff') + r(-" + ?«-")- £ 2 + 7T 7? •r f- - a/ it. r r Hence, at any instant, the points for which N = lie on the circle e + t = R-n- This circle passes through the instantaneous centre of rotation, touches the curve C, and has a radius = Ji?. For the reason stated in Ex. 21 it is called the circle of inflexions. — Differential Calculus, Art. 290. 20. Determine the points of the system for which the acceleration along the path is zero. They lie on a circle whose equation, referred to the centre of rotation as origin, is and which passes through the instantaneous centre of rotation and cuts the curve C orthogonally. The theorems of the last two examples are due to Bresse {Journal de Vecole poly technique, t. xx.). 21. Determine at any instant the points of the body which are passing over points of inflexion on their respective paths. v 2 . They are the points having no normal acceleration (Ex. 19) ; for, as is then zero, and v not zero, p must he infinite. 22. Determine the coordinates of the acceleration centre referred — (1) to axes fixed in space ; (2) to axes fixed in the body (see Article 223). Let x\, y\, |i, t?i be the coordinates in question, then, %' , y' being the space- coordinates of the point of intersection of the body-axes, we have {co 2 + co 4 } [x\ — x') = - co y' + oj 2 x', {or + co 4 } (yi —y') = obx' + co 2 y, {or + co 4 } |i = co (if sin i// - y* cos ij/) + a 2 (x" cos if/ -f y' snuj/), {c6 2 + co 4 } ?ji = ci (if cos -if + y sin \p) — co 2 (x sin \p - y cos \p). Section II. — Kinetics. — Constrained Motion. 227. Special Cases of Motion. Degrees of Freedom. — In order to transform the general equations of motion in such a way as to be of use in particular problems, it is necessary to know something of the special conditions of the problem which it is required to solve. We have seen in Article 214 that six conditions are re- quired to fix the position of a rigid body, and we have found Kinetics, Constrained Motion. 269 accordingly six equations of motion for a body perfectly free. Such a body is said to have six degrees of freedom (Art. 215). We have obtained the equations for this case in their most general form (Art. 204), but we shall now adopt the reverse method of procedure, and consider the special equations to be employed for a body having one degree of freedom. 228. One Degree of Freedom. — A body is said to have one degree of freedom when its position is limited in such a way as to depend on a single indeterminate quantity. It will be shown subsequently that the variations of the co- ordinates of any point of a body entirely free are linear func- tions of six undetermined quantities. If these six quantities are connected together in such a way that one being given all the rest are determined, the body has one degree of freedom. The simplest cases of one degree of freedom occur when some of the six undetermined displacements are zero. We shall consider here only two cases. (1). If the motion of the body be limited to a series of pure translations, and the path of one of its points be as- signed. (2) If the motion of the body be limited to a rotation round an axis fixed in space. In the first case the problem is readily reducible to that of the constrained motion of a particle. This reduction is most easily effected by employing D'Alembert's Principle as expressed by Lagrange. In fact we have *-4>"(r-'"S'K(*-'''SH- - Now, by the conditions of the question, $x, c^ Sz must be the same for every point of the body, and ds being the arc of the curve described by the centre of inertia, ds ds ds 270 Constrained Motion of Rigid Body Parallel to Fixed Plane. Making these substitutions, we obtain the single equation of motion, d % x\dx ( d 2 i/\dy / d 2 z\dz = ( S X)| + ( S F)| + (^)| ; or, as ds z = da? + dy~ + dz*, we have finally, if we put tyfl for the whole mass of the body, 3»S - a, (i) where # is the sum of the components of all the applied forces along the tangent to the path of the centre of inertia ; but this is obviously the equation required for determining the constrained motion of a particle. 229. Hotion of a Body round an Axis fixed in Space. — The condition of equilibrium of a rigid body having a fixed axis is, that the moment of the forces round this axis should be zero. Take the fixed axis as axis of x, then the single equation of motion is the first of equations (18) or (16), Art. 204, according as the forces acting on the body are impulsive or continuous. Adopting the notation of Art. 210, the equation of motion is thus : Let p be the perpendicular on the axis from any point P of the body, a> its angular velocity at any instant, and / its moment of inertia round the axis; then, since pu) is the velocity of the particle P, its moment of momentum is mp 2 w, and H y = wSmp 2 = Iw. Substituting this value for Hi, and remembering that / is constant, we obtain as the equation of motion in the case of impulses J(w - w) = L, (2) Examples. 271 and in the case of continuous forces Equation (3) was obtained before in Art. 138 by a different method. 230. Equation of Vis Viva for a Body moving round a Fixed Axis.. — The expression for the vis viva of a body moving round a fixed axis has been given already, Art. 133. If we take the fixed axis for the axis of a?, we have, as the equation of vis viva, Lo 2 = 22j(Fd> + Zdz) + c. (4) Examples. 1 . To the ends of a thin light piece of wood are fastened spheres of lead •whose masses are P and P'. The piece of wood turns on a horizontal axis through its middle point. Its length being 21, and its mass negligible, deter- mine the time of a small oscillation, the spheres being so small that the squares of their radii are negligible as compared with /. A \l JP+ P' Ans. '^At= P' By changing P, and comparing the times of oscillation, an apparatus of the kind mentioned can be used to verify the Laws of Motion. 2. A heavy pendulum, capable^ of revolving round a horizontal axis, is struck when at rest by a bullet moving in a horizontal direction at right angles to the fixed axis. The bullet remains in the pendulum. If b be the distance of the extremity of the pendulum from the axis, c the distance traversed by that extremity under the influence of the shot, a the distance from the axis at which the bullet penetrates, v the velocity of the bullet at impact, m its mass, M that of the pendulum, k its radius of gyration round the fixed axis, and p the distance of the latter from the centre of inertia ; prove that v = — ^J{g{MTc 1 + ma?)(Mp + ma)}. A pendulum such as that described above is called a Ballistic Pendulum. It has been employed by numerous Physicists to determine the velocity of bullets. 3. A plane area is made to rotate with an angular velocity w' round a fixed axis in its own plane by the expenditure of a given amount of work. When rotating it strikes a sphere of mass m, at a distance a from the fixed axis, whose 272 Constrained Motion of Rigid Body Parallel to Fixed Plan e. velocity at the instant of impact is zero. Determine the moment of inertia of the plane area round the fixed axis in order that the velocity imparted to the sphere should be a maximum. If R be the impulse on the sphere in the first period of impact, v its velocity, and oj the angular velocity of the lamina at the end of this period, mv = R, I(u) -«')=— aR, aw - v, lalca' whence R = /+ ma 2 Tbe whole impulse given to the sphere is (1 + e)R. Hence R is to be a maxi- y/l mum ; but Iio' 2 = given constant ; therefore = maximum ; and therefore 1+ ma 1 I = ma 2 . 4. In Atwood's machine, if the pulley be not perfectly rougb, and slipping takes place, determine the motion : the weight of the rope and the friction of the pulley on the axle being neglected. If an acceleration equal and opposite to that by which it is actually animated were applied to each element of the string it would be in equilibrium ; but the mass of the string being negligible, the force corresponding to this acceleration is zero g.p. Hence the other forces acting on the element of the string are in equilibrium, and fj. being the coefficient of friction, and T, T' the tensions of the two ends of the rope (Minchin, Statics), T = Te'^ =\T. If z be the height from the ground of the ascending weight W, M the mass of the pulley, A' its radius of gyration, a its radius, the angle through which it has turned, we have also T - W" W 9 _d 2 z "dt 2 W-T - W 9 a(T- T). a 2 If the pulley be homogeneous, k 2 = — , and we have finally, a 2 WW d 2 z KW - W -9, \W+ W 1 dt 2 \W+ W" d-0 .., . WW' dt 2 v 'M{\W+ W'Y 5. Taking into account the friction on the axle, and supposing the outside of the pulley to be perfectly rough, and the inside to slip on the axle, determine the motion. The mass of the string being neglected, we may, as in the last example, regard it as acted on by a system of forces in equilibrium. Hence (as this equilibrium would not be disturbed if the string were rigid) the tensions ^and T at its extremities must equilibrate the pressure and friction exerted by the pulley against the string ; and, conversely, T and T must be equivalent to the Moments of Momentum. 273 pressure aud friction exerted by the string against the pulley. Hence we may consider the pulley as acted on by the forces T, T 1 , and its own weight ; and if jPbe the horizontal, and Q the vertical, pressure on the axle, and fx the coefficient of friction, since the centre of inertia of the pulley is at rest, we have (Art. 206), P = fiQ, Q = T 4 T' + Mg — /xP. The moment of the couple resulting from the friction is /x(P + Q)a, where o is the radius of the axle, and may therefore be written in the form &(T+ T + Mg), where (1 + fj 2 ) = fi(l + fi) a. d 2 9 Substituting for the equation 31k 2 — = a(T-T') of Ex. 4, the equation Mk 2 — = a(T- T) - &(T+ T + Mg); nd remembering that as the pulley is perfectly rough, a — - = — , we obtain, if j8 a 2 we put v = - and assume that k 2 = — , a 2 (l+2u)Mg + 4(l + u)W Mg + 2(1 - v )W+2{l + v)W s (l-2p)Mg + ±(l- v )W Mg+2(\-v)W+2{\-rv)W' ' &z _ (1 - v) W- (1 + v) W - vMg dP ~ (1 - v) JF+ (1 + v) W + \Mg* 9 ' 6. If the pulley be not perfectly rough, and slipping of the string on the pulley takes place, determine the motion, taking into account the friction on the axle, and supposing the inside of the pulley to slip as before. In this case, as in Ex. 4, the acceleration of the weights is quite independent of the mass and size of the pulley, and we have T= ^WW ffiz _ \W- w \W+ W' ' dt* \W+ W' 9 ' d 2 9 U{\-v-A(\ + v)WW> d€~ ( Mg(\W+W) 231. ^loiuents of Moineutuiu of Body having iixed Axis. — The expression for the moment of momentum of a rigid body round an axis fixed in space was found in Art. 229. Adopting the notation of that article, we shall now, by a more general method, obtain expressions for the moments of momentum round each of the three coordinate axes. T 274 Constrained Motion of Rigid Body Parallel to Fixed Plane. We have (Art. 222), since the body is supposed to be rotating round the axis of x, x = o, y = -sw, s = ycv ; whence by (22), Art. 210, H x = w2m [y 2 + z 2 ), H 2 = - to^mxy, H z = - w2«s. (5) Also, by differentiation, and substitution of their values for x, y, and s, we obtain tfJ^ du -tt = - it Swa^ + co 2 Zmxz, I . (6) dt dt J If the axis of rotation be a principal axis for the origin, equations (5) and (6) become g where ^4 is the moment of inertia of the body round the fixed axis. 232. Acceleration of any Point of a Body having a Fixed Axis. — If we differentiate the expression for x, y, and z given in Art. 231, and then substitute in the results thus obtained the values of x, y, z already employed, we get x = 0, y = - • N- A T o = - w%mxz / (10) When io has been found from the first of equations (10) the remaining five equations determine the stresses. If the fixed axis be a principal axis at the origin the last two equations become M-M o = 0, iV-iv"o = 0. Hence, if a body having a fixed axis, which is a principal axis for one of its points, be set in motion by an impulsive couple whose plane is perpendicular to the axis, there is no impulsive stress couple. t2 276 Constrained Motion of Rigid Body Parallel to Fixed Plane. From this we infer, that if a body, having a fixed point 0, be acted on by an impulsive couple in one of the principal planes at 0, it will commence to turn round the axis perpen- dicular to the plane of the impulsive couple. Again, if the body be acted on by an impulse whose line of direction is situated in one of the principal planes at 0, it will commence to turn round the normal to this plane. For a free body, likewise, having for its centre of inertia, these results are true ; but, in the case of the second, the body has also an initial motion of translation. If the body, before the action of the impulses X, &c, be already rotating round the fixed axis with an angular velocity a/, equations (9) and (10) still hold good in their final form, provided w - id' be substituted for w. If we suppose the origin to be the centre of suspension, or point in which the fixed axis is met by the perpendicular p from the centre of inertia G, and take the axis of y to coincide with this line, and if we denote the sum of the components of the applied impulses parallel and perpen- dicular to OGhy P and Q, and the corresponding impulsive stresses by _P and Q , equations (9) become IX - X = 0' y P - P = 0, Q - Q = Wpu. (11 j 235. Centre of Percussion. — If a body receive a blow which makes it begin to rotate round a fixed axis, without causing any impulsive pressure on the axis, the point in which the direction of the blow intersects the plane containing the fixed axis and the centre of inertia is called the centre of percussion. In order that such a point should exist, both the axis and the line of direction of the impulse must fulfil certain conditions, which we proceed to investigate. In this case, the fixed axis being, as before, the axis of x, we have, by hypothesis, X, = 0, Y = 0, Z =0, M = 0, N = 0. If we denote the components of the impulse due to the blow by X, P, $ ; and the components of the impulsive couple which it produces by X, M, N; equations (11) and (10) become X = 0, P = 0, Q = Sfeto, | ^ ^ L = Iw, M = - u&mxy, N = -u^mxz Stress on Fixed Axis during Rotation. 277 Since X = 0, aud P = 0, the centre of inertia must lie in a plane through the fixed axis, at right angles to the direction of the impulse. Again, since X = 0, the direction of the blow may be supposed to lie in the plane of yz, and therefore the resulting couple has no components in the plane of zx or of xy ; accordingly, M = and N = 0. Hence, we have %nxy = 0, and %mx% = ; consequently, the axis of rotation must be a principal axis for the point in ichich it is met by its shortest distance from the line of direction of the impulse. If, now, qbe the distance from the fixed axis of the line of action of the blow, L = Qq, and therefore tylpq = I. If Wlk 2 be the moment of inertia of the body round an axis through its centre of inertia parallel to the fixed axis, / = Wl (k~ + p 2 ). (Integral Calculus, Chap. X.) Hence q = — . Accordingly, the distance of the centre of percussion from the fixed axis is the same as that of the centre of oscillation. (Art. '136.) Moreover, if £, rj, £ be the coordinates of any point rela- tively to the centre of inertia, jxzdm = Sfflxz + \%Zdm ; hence, if the axis of suspension be parallel to a principal axis through the centre of inertia, x = 0, and the shortest distance between the direction of the blow and the fixed axis passes through the centre of inertia, and the centre of percussion coincides with the centre of oscillation. 236. Stress on Fixed Axis during Rotation. — In accordance with Art. 233, and following the analogy of Art. 234, we shall suppose the stress at any instant to consist of a force passing through the origin, whose components are X , F , and Z , together with a couple whose components round the axes of y and z are M and N . 278 Constrained Motion of Rigid Body Parallel to Fixed Plane. In this case, by Arts. 231 and 232, equations (15) and (16), Art. 204, become 2X - X = 0, -| 27-7o = -»-^ w 2 , y (13) %Z-Z»= mydy-Wlztv 2 } - 6D%mxy + to 2 %mxz = M - M > X (14) i - 6)%mxz - to'^mxy = N- N Q J If the axis of rotation be a principal axis for the origin, the last two equations reduce to M - M Q = 0, JV - iVo = 0. If also the couple resulting from the applied forces be perpendicular to the axis of rotation, we shall have M = 0, and JST = 0. Accordingly, in this case, the stress couple vanishes when the axis of rotation is a principal axis for the origin. If the axis of rotation pass through the centre of inertia of the body, we have 2X-X = 0, 2F-F = 0, 2Z-Z = 0. Accordingly, if a body be rotating round a principal axis through its centre of inertia, no external forces being supposed to act, there is no stress on the axis, and the body will continue to rotate round that axis with a uniform angular velocity. This result was obtained before in Article 98. If we make the same hypotheses as those at the end of Art. 234, and adopt a similar notation, equations (13) become %X-X o =0, P-P = -mpto\ Q-Q = Wpd>. (15) These equations of motion of the centre of inertia can of course be obtained directly from the consideration that this Examples. 279 point is describing a circle round the origin with an angular velocity w. In general, w and to can be determined from the first of equations (14), and the stresses can then be found from the remaining equations of this Article. Examples. 1. A rigid body is turning round a fixed axis under the influence of a couple, whose axis is parallel to the axis of rotation : what condition must be fulfilled in order that the axis should suffer a pressure at only one point ? (Schell, Theorie der Bewegung und der Krufte.) The axis of rotation must be a principal axis at this point. The pressure is then at right angles to the axis. 2. If the pressure at the fixed point vanishes, what further condition must be fulfilled ? The point must be the centre of inertia. 3. If a homogeneous cubical mass at rest receive an impulse, determine the resulting motion. 4. A body starting from rest turns under the action of gravity round a fixed horizontal axis, which is a principal axis at the centre of suspension. Find the stress on the axis. Take the centre of suspension (Art. 136) for origin, and the fixed axis for that of x. Let 6 be the angle which the line joining the origin to the centre of inertia makes at any instant with a horizontal line perpendicular to the fixed axis, then o) = — , and the axis of x being a principal axis at the origin, the stress couple dt is zero. Again, m being the mass of the body, L = mpg cos 0, and therefore, dco _ d-d _ gp cos m dt ~ df~ ~ k*+p z ' whence, by integration, ■©■ F+p- (sin 0- sin a), where a is the initial value of 0. Finally, _P= nig sin 0, and Q = mg cos ; whence, substituting their values f or P, Q, w-, and a in equations (15), we obtain - sin 9 - 7 , , sin o [ , Qo= mg p— — - cos 6. + P * v &+p* y ™ * &+p 5. In Ex. 3 find the position of the body in which the stress on the axis is a minimum. 280 Constrained Motion of Rigid Body Parallel to Fixed Plane. From the expressions for Po and Qo> w e obtain P 2 + Q 2 - ** g {k* + 2k 2 p~ sin 6 (3 sin 6 - 2 sin a) + p l (3 sin 9 - 2 sin a)'- } , and, since 9 is never less than a, this expression is a minimum when 6 = a. 6. A bar, revolving with an angular velocity n round a fixed axis perpendi- cular to its length, strikes perpendicularly against a fixed obstacle ; find the impulses against the obstacle and the axis, and the angular velocity of the bar, after collision. Let be the point in which the fixed axis meets the bar, G its centre of inertia, A the point at which it strikes the obstacle, m its mass, and h its radius of gyration round an axis through G parallel to the fixed axis ; let R' and Q' be the magnitudes of the impulses produced by the obstacle and the axis in the first period of impact, R" and Q" those produced in the second period, and ca the angular velocity after collision ; then, if OG = a, GA= b, since the velocity of the bar is reduced to zero in the first period, we have R' + Q' = man, R'(a + b) = m (k* + a 2 )n ; whence, R' = -^ r-i— ; Q' = m — fl. a -t- o a+ o Again, since in the second period the bar starts from rest, we have R" + Q" = maw, R"{a + b) = m(&+&)*, and also (Art. 78), R" = eR', whence Q" = eQ' y w = en, since Q" and o> are the same functions of R", which Q' and n are of R' . It is to be observed that in the equations above the algebraical signs of the angular velocities have not been taken into account, and that the direction of e=(i + ,)» TTT a. When ab = k 2 , the point A is the centre of percussion and Q = 0. This agrees with the result arrived at in Art. 235. 7. A bar, revolving as in Ex. 6, strikes against a sphere whose centre is moving with a velocity U in a direction perpendicular to the bar ; find the magnitudes of the impulses, and the velocities of the bar and sphere, after collision. Let M be the mass of the sphere, u' and u the velocities of its centre, and «' and ca the angular velocities of the bar, at the end of the first_ and of the second period of impact ; then, since the impulses tend to diminish both the velocity of the centre of inertia and the angular velocity of the bar, we have R' + Qf = ma (n - a/), R'h = m {k 2 + a 2 )(n - »')> R' = M(n' - V), where h = a + b. At the end of the first period of impact the relative velocity of the colliding points is zero, and therefore, ha = u'. Examples. 281 Let fl - w = zs', then we have h h and also m(& + a 2 ) w = J/%' - 17) - if/* (W - Z7) = ifA{A(n - w> - U] , hence {w(F + efi) + Mir} W = J/7, (An - tf). Again, E u + Q'' = ma{n-ET) W= ^iT U+{1 + e) "«(*»+ «*) + JfA 2 ' Here, as in the former Example, Q = when the impact takes place at the centre of percussion. 8. Show that the results in Ex. 6 can he deduced immediately from those in Ex. 7. Make U= and M = oo in Ex. 7. 9. Find at what point of its length the har should strike the sphere in order that the impulse of the hlow should he a maximum. If we put m (/c 2 + a 2 ) = I, we have to determine h, so that An- u I+Mh? shall be a maximum. Hence, to determine h we have the quadratic equation Malr - IMUh - la = 0. Bv assuming U = rCl, and I = Mp 2 , this equation becomes A 2 - 2rh - p~ = 0. We have then the following construction for the two values of h. At erect OP perpendicular to the bar, and make it equal to p, take OG in the direction of G equal to r, with C as centre, and CP as radius describe a circle; it will meet 282 Constrained Motion of Rigid Body Parallel to Fixed Plane. the bar in the points required. The value of h, which is greater than r, makes the expression for R a maximum ; the other value of h makes this expression a minimum, but at the same time makes R negative. Thus both values of h make R irrespective of sign a maximum ; bnt one impulse is opposite in direction to the other. • B O If the sphere, when struck, has no velocity in a direction perpendicular to the bar, we have h = p when R is a maximum. 10. Find the point of impact in order that the impulse on the fixed axis should be a maximum. If we put (k 2 -f- a 2 ) = K 2 , we have to determine h so that {ah-K 2 )(hn-U) mK 2 + Mh 2 shall be a maximum. We have then for h the quadratic E 2 {MU- maQ) mK 2 ~~ 2 M(aU + IT-ti) ~~M~~ ' r'M{aU+ K 2 D.) = K 2 {MU - man), and (as in last Example), By assuming the equation for h becomes Mp 2 = mK 2 , h 2 - 2r'h - p 2 = 0, and the two values of h are determined by a construction similar to that of the last Example. If the fixed axis pass through the centre of inertia we have a = 0, and the points for which Q is a maximum coincide with those for which R is a maximum. If aU+ K 2 Q. = 0, one value of h is zero, and the percussion on the axis is a maximum when the sphere strikes at the axis. Free Motion Parallel to Fixed Plane. 283 Section EH. — Kinetics — Free Motion Parallel to a Fixed Plane. 237. Equations of Motion. — The motion of a body relative to its centre of inertia consists at any instant of a rotation round some axis through that point. _ Moreover, in the case here considered, this axis must be at right angles to the fixed plane, and is fixed in space if the centre of inertia be regarded as invariable. Now, by Art. 209,^ the motion relative to the centre of inertia is the same as if that point were fixed in space, the forces remaining unaltered. Hence, taking the plane of yz for the fixed plane, we have, to deter- mine the motion of the body, the equations 3»^|=sr, 3»^=Si?, 3M 2 2=£, (i) dt~ dt" at where y and z are the coordinates of the centre of inertia, k the radius of gyration round an axis through it at right angles to the fixed plane, and L the moment of the applied forces. If the axis of rotation through the centre of inertia be always parallel to a line fixed in space, it is plain that the last of these equations holds good no matter whether the whole motion of the body be parallel to a fixed plane or not. In the latter case the only difference will be that an addi- tional equation, viz., will be required to determine the motion of the centre of inertia. In any case, therefore, the motion of the body is determined, when we know the motion of its centre of inertia, and the angular motion relative to that point. 238. Connexion of the Angular Telocity with the Velocity of the Centre of Inertia. — As the motion is parallel to a fixed plane, the parallel section of the body passing through the centre of inertia must at each instant be rotating round a point in its own plane (Art. 219). If p be the 284 Free Motion of Rigid Body Parallel to Fixed Plane. distance from this point (the instantaneous centre of rotation) to the centre of inertia, s the path of the latter, and w the angular velocity, then pw = — , as is obvious. Also w = — . Examples. 1. A body is moving parallel to a fixed plane under the action of forces which are in equilibrium : show that the locus of the instantaneous centre of ro- tation in the body is a circle, having the centre of inertia for centre, and a radius v — , where v is the velocity of the centre of inertia, and a> the angular velocity. 2. The locus of the instantaneous centre of rotation in space, under the circumstances of Ex. 1, is a straight line parallel to the path of the centre of inertia, and at a distance from it equal to -. CO 3. If a body move parallel to a fixed plane, and be acted on by a constant couple, lying in the plane ; show that the locus of the instantaneous centre of rotation in space is an equilateral hyperbola. 4. An inextensible string, whose mass is negligible, passes over the line of intersection of two smooth inclined planes. Each end of the string passes under and. round a smooth circular homogeneous cylinder, to which it is attached, and which rests on one of the inclined planes. The line of intersection of the inclined planes is parallel to the axes of the cylinders, and perpendicular to a vertical plane containing their centres of inertia and the string. Determine the tension of the string. As in Ex. 4, Art. 230, the portion of the string wrapped round one of the cylinders may be regarded as in equilibrium under the action of the tensions at its extremities and of the pressure produced by the cylinder. Hence all the forces exerted by the string on the cylinder are equivalent to the tension T act- ing at the point of contact of the cylinder with the inclined plane. If s and s' be the distances at any time of the points of contact of the cylin- ders and inclined planes, from the point of intersection of the latter with the vertical plane perpendicular to them ; and 0' the angles through which the cylinders have turned from their initial positions ; a and a' their radii ; m and m their masses ; and i and % the inclinations of the inclined planes to the horizon, the equations of motion are Ta, Ta'. If a be the distance the string has slipped at any time along the inclined m ■— = mg sm % - -T, « 2 d°~d m — — 2 dt- ,d*s' , . ., tn -j-j- = mgsmi - T, , a" 2 - dH' m rrr 2 dt- Vis Viva, 285 planes, and b and b' the initial values of s and *', we have, since the string is inextensible, s = b + aQ + \ Again, if y and z be the coordinates of any point referred to that space point as origin which coincides with the instan- taneous centre of rotation, Art. 238, then dy f , dz' hence the equation of vis viva assumes the form therefore * £(/„,») = j, (3) Zu) at where J is the moment of the applied forces, round the in- stantaneous axis of rotation. 286 Free Motion of Rigid Body Parallel to Fixed Plane. 240. Moment of the Forces of Inertia. — If b and c be the coordinates of any point, fixed or movable, the moment of the applied forces, round an axis through it parallel to the axis of x, must be equal and opposite to the moment of the forces of inertia round the same ; hence, calling the former moment J, we have aw | &r _j ) __ ( ,. )^|.j: If, as in Art. 209, we put y = y + tj, z = z + Z, we get, by omitting the terms which vanish, ^ ( , _ . d"z . _ , d~y 10 d(u) , fAS **{<*-*)&-{— )£+»0\->r- (*) If we suppose the point b, c to coincide with the origin fixed in space, and to lie in the plane of the motion of the centre of inertia, this equation becomes, if we call r and x ^ ne polar coordinates of the centre of inertia, 241. Moments of Momentum relative to any Point. — Since the body is supposed to be moving parallel to a fixed plane, its motion at any instant is a pure rotation. If we take a line coinciding with the instantaneous axis of rotation as axis of x, then x, y, z being the coordinates of the centre of inertia, we have, by Art. 222, x = 0, y = - iw, i = yta. Substituting these values in (31), Art. 210, and introducing the values of Hi, R 2 , E 3i given by (5), Art. 231, we obtain H\={I-m{by + cz)}u>, \ H' 2 = [Way- ^mxy}w, [ • (6) R'* = \Wloz - 2mxz}u> J Equations of Motion for Impulses. 287 Examples. 1. The motion of a body consists of a pure rotation ; find the conditions that it should be brought to rest by a single impulse. Take the axis of rotation as the axis of x, and a perpendicular p on it from the centre of inertia G as that of y, then the whole velocity of G is parallel to the axis of z, and is equal to^cu, where co is the angular velocity of the body. Hence the impulse which reduces the body to rest must be parallel to the axis of z, and is given by the equation Z= - 2Jty«. Let b, c be the coordinates of the point in which the impulse Z meets the plane of xy ; the moments of momentum relative to be are each zero after the body is reduced to rest ; but, since the impulse passes through be, these moments are the same as they were before the action of the impulse. Hence, originally H{ = H Z ' = H,' = 0. Substituting for H\, &c, their values from (6), we have, if jBTbe the radius of gyration round the axis of rotation, K" - bp = 0, Xmxy = 0, 2mxz = 0. Hence we conclude that the axis of rotation must be a principal axis at the point in which it is met by the perpendicular from G, that the impulse must be perpendicular to the plan containing G and the axis of rotation, and that its shortest distance b from the axis is given by the equation bp = K 2 . (Compare Art. 235.) 2. A uniform circular plate whose centre is fixed lies on a smooth horizontal plane. An insect starts from the centre of the plate, and returns to the same point after describing a circle whose diameter is the radius of the plate ; find the angle through which the plate has turned. Let cp be the angle through which the plate has turned at any time, a its radius, m its mass, m' that of the insect, r and its polar coordinates in space, r and \p its polar coordinates relative to the plate ; then a 2 dd> „ dd 1,1 ~ Tt + m '"dt^ 0, ^ = e ~ *' V = a C0S ^* __ f 2m' cos 2 \L Hence - on + 2m' cos 2 ^/ and the angle required is _/_-_\*J. \m + 2m) ) 242. Equations of Motion for Impulses. — In the case of impulses the changes of velocity which they produce are determined by the equations (Arts. 204, 209, 229), m [v - t>') = SF, m {w - uf) = 2Z, Wr {w - a/) = Z, (7) where a/ is the angular velocity of the body, v and w the 288 Free Motion of Rigid Body Parallel to Fixed Plane. components of the velocity of its centre of inertia, before the action of the impulses ; and w, v, and w the corresponding quantities after their action. 243. Impact. — When impact occurs between two smooth bodies, a mutual impulsive force is developed in the direction of the common normal. In the first period of collision this force reduces the relative normal velocity of the colliding points to zero. In the case of motion parallel to a fixed plane, there are for two bodies seven unknown quantities, viz. the changes in the two components of the velocity of the centre of inertia, and in the velocity of rotation for each body, and the magnitude of the mutual impulse. There are like- wise seven equations to determine these quantities, viz. the six equations of motion, and the equation which expresses that the relative normal velocity of the colliding points is zero at the instant of greatest compression. In the second period, a new mutual impulsive force is developed, whose impulse bears a constant ratio to that of the former, and can therefore be found. The changes of velocity which it produces can then be determined. If the bodies which collide be perfectly elastic, the im- pulse developed during the period of restitution, or second period, is equal to that developed during the period of com- pression. What is here stated is merely a generalization of the theory given in Articles 78 and 202. Examples. 1 . A bar, which is rotating round an axis perpendicular to its length, and whose centre of inertia is moving in a plane at right angles to the axis of rota- tion, strikes perpendicularly against a fixed obstacle ; determine the impulse of the blow, and the subsequent motion. Let m be the mass of the bar, k its radius of gyration, V the velocity of its centre of inertia G in a direction perpendicular to its length, and n its angular velocity before impact ; also let v and «', v and a> be the corresponding velocities at the end of the first and of the second period of impact, respectively ; and let h be the distance from G of the point A at which the bar strikes the obstacle ; then, if R' be the impulse of the blow during the first period of impact, and if we suppose the velocity of A due to the motion of translation to be in the same direction as that due to the rotation round an axis through C7, we have, since the blow diminishes both the velocity of translation and the angular velocity of the bar, JR f = m{V- v'), R'h = m& (a - «') ; Examples. 289 but also, since at the instant of greatest compression A is reduced to rest, v' + ha = 0. Hence we obtain (A 2 + #)«' = #0- h V\ . . , A(r+/*n) , _, m#(F+*n) therefore n - co = _ — , whence R = — — — — -. k l + h- k 2 + h 2 Now, as in Ex. 5, Art, 236, R={l + e) R', and fl - « = (1 + e) (fl - «'), ^_ v = (1 + e>) ( V - v). Hence we have (k 2 + h 2 ) R = (1 4«)«A 2 (r+ An) ; (h 2 - ek 2 ) V-{1 +e)k 2 ha {& - eh 2 ) a-(l +e) hF consequently v = — ^ + ^ -, « = - — ^-^ . 2. Find the point at which the bar in Ex. 1 should strike the obstacle in order that the impulse of the blow should be a maximum. We have here to determine h so that — — — shall be a maximum, and the k--\-h- y required values of h are given by the quadratic equation h 2 -f 2 — h — k 2 = 0, or h 2 + 2rh - k 2 = 0, if we put r& = V. If C be the instantaneous centre of rotation of the bar, corresponding to V and fl, we have GC = — r, and the points of the bar at which the impact produces the maximum impulse are determined by erecting a perpendicular GP equal to k, and with C as centre, and CP as "radius describing a circle. The points A and B in which this circle meets the bar are the points required. (See Fig. p. 282.) Let R\ and R2 be the values of the impulse R, corresponding to the points A and JB, respectively, we readily find that .Ei = -J^ ma . BG, and R 2 = - l -^- ma . GA. l 2 The negative sign of R% shows that in this case the impulse must act at the opposite side of the bar ; hence, if we consider magnitude only, without regard to sign, each impulse may be regarded as a maximum. 3. A bar moving as in Ex. 1 strikes against a sphere of mass M, whose centre has a velocity Uin a direction perpendicular to the bar ; find the impulse of the blow, and the subsequent motion. Let u' and u be the velocities of the centre of the sphere at the end of the first and of the second period of impact, then, the bar beirig supposed to over- take the sphere, we have R = m (V- •), R'h = mk 2 {a - «'), R' = M (uf - U) ; (a) and also, v' -f hu' = u' . (b) If we put a - co' = zs'j from equations (a) we have _ , A 2 , . _ m k 2 7-v .-*,.-*--*, 290 Free Motion of Rigid body Parallel to Fixed Plane. whence, substituting in (b) for v , «', and «', we obtain zj '= — r^> and therefore R - (1 + e) — . Consequently the motion after collision is given by the equations & , TT /, ^ m k °~ (1 + ^w', v=F-(l + e)-?o', w = *7+(l+e) — 4. Find in Ex. 3 at what point of its length the bar should strike the sphere in order that the impulse of the blow should be a maximum. The values of h which make R a maximum are given by the quadratic equa- tion A 2 + 2rh - (K±J?\ k- = 0, where rfl = V- U. The points of the bar at which the impulse of the blow is a maximum may be determined by a construction similar to that of Ex. 2. In the present case, C is the point whose velocity perpendicular to the bar i s equal to that of the , I /M+ m\ sphere. The perpendicular to be erected at G is now kj I — — — 1 . 5. In Ex. 1 find the loss of kinetic energy due to the impact. If 7' be the kinetic energy lost during the first period of impact, we have, by Ex 2 Art. 202, 27' = R'(V-\- h£i), but if 7 be the total loss of kinetic energy, „ , ox mk°-(V+ha) 2 7 = {l- e*)7' (see Ex. 4, Art. 202). Hence 27 = (1 - e 2 ) — r^i • 6. Find at what point the bar should strike the obstacle in order that the loss of kinetic energy should be a maximum. ^ ^ _ &b 7. In Ex. 3 find the total loss of kinetic energy of the system ; and determine at what point the bar should strike the sphere in order that this loss should be a maximum. Here, if 7' be the kinetic energy lost by the system during the first period of impact, by Ex. 2, Art. 202, 27 , = R'(V+hn)-R'U, Mmk\V+h£i-Uf hence 27 = (1 - O mk2 + M{h2 + k2) • m + M &Q This expression is a maximum when h = — — — . 8. Find at what point the bar should strike the sphere in order that the gain of kinetic energy by tbe sphere should be a maximum.^ The required points are those at which R is- a maximum. Examples. 291 9. Find the loss of kinetic energy by the bar. - If 7' be the loss of kinetic energy during the first period of impact, we have, Ex. 1, Art. 202, 2?' = X'(V+hn + v' + ha'); TV but v'+h'w' = «*' = U+jpt and therefore, since 27 = (1 - e 2 )7', we have 27={i-(?)^{it , +M(v+ha+ir)} /-, ««- „( 70/ v+hn-u \ 2 (F+An) 2 -t7M = (1 - e 2 ) Mmk 2 I mk z ( — ] + — N f . 10. In Ex. 1 find the conditions that the whole motion of the bar should be destroyed by the collision. Am. k 2 & = hV, and e = 0. This is also easily seen from first principles. See Ex. 1, Art. 241. 11. A body is moving parallel to a fixed plane, when a line AB in the body perpendicular to the plane becomes suddenly fixed ; determine the subsequent motion. Let m be the mass of the body, Jits moment of inertia round AB, k its radius of gyration round a parallel axis through its centre of inertia 67, D. the angular velocity of the body, and V the velocity of G just before the line AB becomes fixed, p the shortest distance between the line of motion of G at this time and AB, and w the angular velocity of the body round AB just after this line is fixed, then we have Iv = m(Vp + k 2 a). 12. A plane lamina is moving in its own plane when one of its points becomes suddenly fixed ; determine the subsequent motion. Let us suppose that the lamina is constrained to rotate round a perpendi- cular axis through 0, then, adopting the same notation as in Ex. 11, we have, by (10), Art. 234, since the axis of rotation is a principal axis at 0, /« = m[Vp + k 2 n), M Q = 0, JV = 0. Hence the actual motion of the lamina when O is fixed is a rotation round a perpendicular axis, and the angular velocity w is given by the first of the equations above. 13. A bar moving in a vertical plane impinges upon a smooth horizontal plane ; find the motion immediately after impact. If the horizontal and vertical components of the velocity of the centre of inertia G of the bar be represented by U and V immediately before the impact, and by u and v immediately after, if n and w be the corresponding angular velocities, a the distance from Cr'to the point of impact of the bar, and a the TJ2 292 Free Motion of Rigid Body Parallel to Fixed Plane. angle which it makes with the horizontal plane at the instant of impact, the values of v and a are obtained by substituting in the equations of Ex. 1, a cos a for h. Accordingly we have {a 2 cos 2 a — eh" 1 ) V— (1 + e) k 2 aCi cos a u = U, v = ' , — - , k- + a~ cos 2 a _ (k 2 - ea 2 cos 2 a) n - (1 + e) a V cos a Ic 2 + a 2 cos 2 a If the bar be homogeneous, 3& 2 = a 2 , and we get (3 cos 2 a - e) V — ( 1 + e) aCl cos a 1 + 3 cos 2 a (1 — 3> m jt = »*ff - T sin ■ d ~9 d2 z , d2 d°~9 294 Free Motion of Rigid Body Parallel to Fixed Plane. d 2 Hence, eliminating -— , we get d 2 z d?y . . nS d 2 9 A sin j8 — - + cos j8 — + «sin(a - 0) ■— = 0. dt 2 df- dt~ Substituting from (a) and (b), and putting a and £ for and (/> in those equa- tions, we get for To, the initial value of the tension, To = mg k* sin j8 A- 2 + « 2 sin 2 (a-£) 2. A body, whose centre of inertia is G, is suspended by strings attached to two of its points A and B, and fastened to two fixed points and 0' . The plane AGB is a principal plane at G, the string O'B is cut; determine the initial tension of the other. We may here suppose the body compelled to rotate round an axis through G, whose direction is fixed in space, and is perpendicular to the initial position of the plane AGB. Since this axis is a principal axis at G, we find, then, Art. 236, equation (14), that the components of the stress couple on this axis are zero, and therefore that the body rotates round it freely. Hence the whole motion of the body is parallel to the vertical plane which is the initial position of AGB, and the question becomes the same as in the last example. 3. A circular disk is hung, with its plane horizontal, from a fixed point vertically over its centre, by means of three equal strings attached to three fixed points in the circumference of the disk at equal distances from each other. One of the strings is cut ; determine the initial tensions of the other two. The two tensions along the threads OA and OB may be replaced by the single force F along OS, where F = 2 T cos AOS, S being the middle point of the chord joining the fixed points A aniLB. In this case F takes the place of T, and the point S of A in Ex. 2. Then, & being the initial value of the angle which OS makes with the horizontal line which is the initial direction of SG, the length of the'latter being a, we have, since SG is originally horizontal, k 2 sin £ F=m » e + oW Examples. 295 If l be the length of OS, the expression for F may be put into the form k~ sin £ F= mg A-- + £-sin 2 /3 cos-)3 4. Determine in Ex. 9, Art. 202, the initial tensions of the strings, and their tensions when the bar is at its greatest height, the length of each string being 2a. If 9 be the angle one of the strings makes with a vertical line at any time, z the vertical coordinate of the middle point of the bar, i// the angle the bar makes with a horizontal line parallel to the fixed bar, and T the tension of one string ; then or d 2 ib „ _ . . , m --?- = - 2a T sm 6 cos ^, 3 at- m — = 2Tcos dt~ mg ; also, from the geometrical conditions, 2 i|/, as 2asin0 = 2«sin^, z = 2a(l-cos0). Substituting £ \\> for in the last equation, differen- tiating twice, and observing that initially \p = 0, and — = a>, we get. for the d$ initial tension of one string T = J mg + f maw To get the tension when the bar is at its highest position, make -j- = . , 2a - h cos h ^ = 2a T=mg -, where h has the value in Ex. 9, Art. 202 ; then 4« 3 288<7 3 = mg ( 1 2g - aw 2 ) { 48^ 3 +ft» 2 (24ag - a? w 2 ) } {2a-h){ia 2 +Sh(4a-h)} 5. In Ex. 1, find the values of a and £, in order that T shall be the greatest possible. Ans. £ = -. The corresponding value of T is mg, i.e. the weight of the lamina. 6. If £ be given, find a, so that T shall be a minimum. Here sin (a- 0) =max., and therefore a - £ = |, or -40 is perpendicular to CL4. 7. If the initial position of AG be horizontal, find £, so that To shall be a smjS 1 Here Ave have to find /3, so that — — — 1- -=-: — - ¥■ c 2 sm)8 may be a minimum. There- fore, k = a sin £ = p , where p is the initial value of the perpendicular from G on OA. The result here obtained holds good for Ex. 3, if OS be substituted for OA, and F for T. 296 Free Motion of Rigid Body Parallel to Fixed Plane. 245. Friction. — Friction [see Art. 60) is a tangential force passing through the point of contact of two rough surfaces, which tends to prevent the one from slipping on the other. If there be slipping, the friction is in an opposite direction, and takes its greatest possible value, which is in a constant ratio to the normal pressure between the surfaces. If the motion be pure rolling, just enough friction is exerted to maintain pure rolling. The force of friction is then usually less than its maximum value, and is determined, as if it were an unknown reaction, by means of the equations of motion and the geometrical condition which expresses that the motion is pure rolling. If the value thus found for the force of friction does not exceed its maximum value, and pure rolling be consistent with the initial condi- tions, it will be the actual motion. When there is slipping, the friction, which is then a maximum, and therefore de- termined, tends to make the motion pure rolling. If pure rolling be attained, the friction at the instant pure rolling commences changes in general its value, and must be de- termined in the manner stated above. It is to be observed, as already stated in Art. 60, that the maximum value of friction, when slipping actually takes place, is, in general, less than its maximum value when there is no slipping, and friction is acting against a force which tends to produce slipping. When a surface is said to be perfectly rough it is under- stood that ncr slipping can take place between it and any other surface with which it is in contact. The amount of force which it is capable of exerting by means of friction is, in this case, unlimited. Examples. 1. A homogeneous cylinder, having its axis horizontal, rolls without slipping down a rough inclined plane ; determine the amount of friction brought into play (see Ex. 1, p. 139). The equations of motion are M^ = Mgsmi-F, Mk^ = aF; dt 2 * dt 2 the axis of y being a line in the inclined plane at right angles to its intersection Examples. 297 with the horizon. Also, adO= dy ; whence k 2 sin i F^Mg p, we have 0o > 0i, and therefore in this case the mass will slip before the lamina begins to slip. On the other hand, if h < p, we have 0i > Oa and slipping begins at the edge AB. Examples. 299 5. In Ex. 3, if any number of masses mi, m 2 , &c, be placed on tbe lamina at points D u D 2 , &c, on the line OG, investigate the motion. Let ODi = h h 01*2=}% &c, then, d(o Mp + mi 7*i + m 2 h 2 + Sec. = ; — — - g cos 0, dt M(k 2 +p 2 ) + mi hr -t- m 2 hi~ + &c. Mp + tnihi + m 2 h 2 + &c. . W= Jf (& 2 +i? 2 ) + mi fa 2 + m 2 h 2 2 + &c. 9 Sm If we put Mp + mihi + m 2 ?i 2 + Szc. _ Ai _ \ 2 _ M(k 2 +p 2 ) + mi 7*r + mzfa 2 + &c. ~ h ~ h 2 ~ '' we have and also Mp + mihi + rnhz + Scc. . ,, c = vihi = v 2 h 2 = &c. , M + mi + w?2 + &c. Pi = (1 + 2Ai) mi^ sin 0, Qi = (1 — Ai) miy cos 9 ; JP 2 = (1 + 2\ 2 )m 2 g sin 0, Q% = (1 - A 2 ) m 2 <7 cos ; &c, &c. ; p 1 + 2Aij/i = 1 + 2A 2 n= &c. (M + mi + m 2 + &c.) g sin , - — = 1 — \\v\ = 1 — A 2 J/ 2 = &C (M + mi + m 2 + &c.) g cos The rest of the investigation is the same as in the last example. _ 1 - Ai 1 - A 2 If hi > h 2 , then Ai > A 2 , and <- — -, 1 -f- Z\i 1 + ZA 2 and therefore 9\ < 6 2 , or mi slips before m 2 : that is, the mass farthest from the edge begins to slip first. 6. If a hoop rolls down a rough inclined plane without sliding, show that tan i < 2/jl ; the initial position of the hoop being in a vertical plane at right angles to the intersection of tbe inclined plane with the horizon. Take the initial position of the centre of the hoop for origin, and the inter- section of the inclined plane with a vertical plane at right angles thereto as axis of y, its positive direction being downwards. Let the positive direction of rota- tion be from the upper side of the inclined plane towards y positive. Then, y being the coordinate of the centre of the hoop, m its mass, a its radius, and Pthe friction brought into play, the equations of motion are n d& dry . . -, ma 2 — = Fa, m — r = mg sm i - F; at at- 300 Free Motion of Rigid Body Parallel to Fixed Plane. but the motion being pure rolling, aw = — ; hence, eliminating we obtain, F = sini but F< fiing cos i ; therefore tan i > 2fi. 7. A homogeneous circular disk, whose radius is a, rolls inside a rough ver- tical circle whose radius is b ; the motion is pure rolling under the action of gravity ; show that the rolling forward and backward of the disk is isochronous with the oscilla- tions of a simple pendulum whose length is f {b - a). ¥e have, I being the moment of inertia of the disk round an axis through P, w the angular velocity, and 9 the angle between GA and GP, - - {lor) = 2mga sin 9 (0 being co at reckoned from the vertical line GA, where G is the centre of the vertical circle, and w being the angular velocity of the disk rolling down). As the instantaneous centre of rotation lies, in this case, on the circumference of the disk, I remains constant throughout the motion ; there- jlw fore I dt mga sin but I = (Integral Calculus, Chap. X.), and aw = — (b — a) — , since either represents the velocity of 0. Hence dt d~9 § (b — a) — = - g sin 9 ; .'. &c. The student will observe that the friction at P does not enter this equation. 8. A uniform sphere, resting on a rough horizontal plane, is set in motion by an impulse applied in a vertical plane passing through its centre. Show that, when sliding ceases, the rolling motion will be direct, stationary, or retrograde, according as the direction of the impulse intersects the vertical diameter above, at, or below the point of contact with the plane. Let v be the velocity, at any time, of the centre of the sphere parallel to the intersection of the horizontal plane with the vertical plane containing the impulse ; the direction of the latter making an acute angle with the positive direction of v. Let w be the angular velocity of the sphere, counted from the vertical towards the direction of v positive : then V and n, the initial values of v and w, are determined by the equations mV = Y, ink- n = Yb, where Y is the horizontal component of the impulse, and b the distance from the centre, at which its line of direction intersects the vertical diameter of the sphere. Eliminating Y, we obtain a = b_V £ 2 ' Examples. 301 For the subsequent motion, if Fbe the force of friction, we have the equations dv -„ *o do _ m — = F, mk~ — = - Fa ; dt dt dv 7 . dw whence a — + r — =0. Integrating, we obtain av + k 2 co= constant = a V + £ 2 n = (a + b) V. "When sliding ceases v = aw. Substituting for v in the preceding equation, we have (a+ b)V <0 = 5 rr-. Hence, since Fis necessarily positive, w, when sliding ceases, is positive, zero, or negative, according as a + b> 0, « 4 J = 0, or a + b < 0. The first condition holds good, if b is either positive, or negative and less than a in absolute magnitude ; the second, if b = — a ; the third, if b is negative and greater than a in absolute magnitude. The results of this example may be extended to other solids of revolution. 9. A circular plate rolls down the inner circumference of a rough circle under the action of gravity. The plane of the plate coincides with that of the rough circle, which is vertical. Determine the amount of friction brought into play if the plate start from rest, the motion being pure rolling. (See Ex. 7.) If co be the angular velocity of the plate, the equations of motion are da d 2 Q . ,., \ma 2 — = Fa, m i h " a )j^ = - m 9 sm d + F together with the equation of condition dQ hence F= \mg sin 9. 10. Show that the plate in the last example will ascend to the same height as that from which it started, and that the motion will go on for ever. The vis viva = 2mg (z - z ) : this will vanish when z = z ; therefore, &c. 11. Determine the velocity of rotation of the plate at any time. Ans. u) 2 = i *"„ ■ (cos 6 - cos O ). v- 302 Free Motion of Rigid Body Parallel to Fixed Plane. 246. Tendency of a JRod to Break. — When a body is under the influence of any forces, it experiences pressures or tensions, which tend to alter the relative positions of the molecules. This tendency is resisted by the mutual action of the molecules. Under such circumstances the body is said to be in a state of stress. If we consider a small rectangular parallelepiped in the body, the stresses acting on one of its faces may be resolved into three forces at right angles to each other — one normal, and two parallel to the face under consideration. To ascertain the tendency of a body to undergo a rupture in any part, we must consider the stresses to which it is sub- jected in that part. If the mutual cohesion of the molecules is unable to resist these stresses the body must give way. The question is, in general, one of great complication, and for its full discussion the reader is referred to treatises on Elasticity and Strength of Materials. If the body under consideration be a rod, that is, if two of its dimensions are at each point very small, the question be- comes much simplified. The axis of the rod may be a straight line, or may form a curve of any kind. We shall suppose that this curve is not closed, that it lies in one plane P, and that the rod is in equilibrium under the action of forces in this plane. If we consider a section at right angles to the axis of the rod, at any point A of its length, the action of the molecules at one side of this section on those at the other must equilibrate all the forces acting on the rod at the latter side. These may be reduced to a force F, passing through A, and a couple G, round an axis a at right angles to the plane P. This force and couple, therefore, are equivalent to the stresses acting on the rod through the section containing A. That the tendency of the rod to break results chiefly from the couple may be shown as follows : — The stresses in the plane of the section cannot give any couple round the axis a, since a either meets them or is parallel to them. Hence the couple G must produce stresses, parallel to the axis of the rod at the point A, whose moment round A is equal to G. If iV be the value per unit of area of the Tendency of a Rod to Break. 303 greatest of these stresses, and a be the distance from A of the most remote point of the section, whose area may be denoted by 8; the moment round A of the stresses parallel to the axis must be less than NSa. Hence, if we assume G = Fp, we have F p NSa > Fp, and therefore N > -=■-. o a If we now seek for the stress per unit of area caused by XT' n the force F, we have N' = -; .: N' < - N. S p Hence, if a is very small compared with p, J¥' is unim- portant compared with iV. Accordingly, in general, the tendency of the rod to break at any point A depends simply on N, i.e. on G, the moment round A of the forces acting on the rod at one side of A. We have hitherto supposed the rod to be at rest. If it be in motion, we can, by D'Alembert's Principle, consider it as in equilibrium under the action of the applied forces and the forces of inertia, and the question of stress, or the tendency to break at any point, becomes the same as before, except that we must now add the forces of inertia to the other forces acting on the rod. If the rod be acted on by impulses, the impulsive tendency to break at any point is obtained in a similar manner, and the preceding investigation holds good provided the impulses be substituted for the applied forces, and the resulting changes of momentum for the forces of inertia. To find the couple which measures the tendency of a rod to break at any point P. Let G be the required couple, L' the moment round P of the forces applied to the portion of the rod on one side of this point, tyl' the mass of this portion of the rod, k f its radius of gyration round its own centre of inertia C, and A' the moment of the acceleration of C round P, theu by (4) , Art. 240, e -J? -*(*'♦*•$} (8) 304 Free Motion of Rigid Body Parallel to Fixed Plane. In the case of impulses, if G be the impulsive couple cor- responding to G, we have G = L'-m'{A'+k'> (w-o/)), (9) where A' is the moment round P of the change of velocity of C due to the impulses, and w and a/ are the angular veloci- ties of the rod after and before the action of the impulses. Another expression for G which is often useful may be found as follows : — Let //, z be the coordinates, referred to a fixed origin, of the centre of inertia C of the whole rod ; b, c those of P ; y, z those of C\ and r,, Z those of any point of the rod referred to axes through C parallel to the fixed axes, then, G= L'-Z'm{(y-b)z- {z-c)ij}. But & = $+'*, z = z + Z; substituting, and remembering that ^!my = 9W//, Sm = Wl' z\ we obtain G = L'-W{(y-b)'i-(z'-c)$}-2'm{(y-b)Z-(z-cyri}. (10) In the case of impulses applied to a rod at rest, G = 2/-3RV- 6)5- (s' - e) f}-Xm{{y-b) £- («- *)$}. (11) If the rod be in motion when the impulses are applied, we must substitute in (11) for f, £, 17, and £ the changes in their values due to the action of the impulses. Examples. 1. A uniform straight rod AB rotating round a perpendicular axis passing through one extremity A is struck perpendicularly at a point Q ; find the ten- dency to break at any point P. Let H be the impulse of the blow, a the length of the rod, m its mass, «' and &> its angular velocities before and after the blow ; also let C be the middle point of PB, and let AP= r, AQ = h ; then, L' = P.PQ, A' = PC'.AC {co-o}'), M'* = PC' 2 , m'a = 2mPC , ) Examples. 305 hence we have n. - 7? vn da 2«i G = E.PQ-'— PC'HZAC + PC)(co - «') But m— {(o -u') = H. AQ = Rh, u a — r . „, a + r PQ = h-r, PC = -^- i AC=^-. Substituting these values in the equation for G we obtain G = £r {2a*(h - r) - k(a - r)» (2a + r)} 2 a 6 2 a 6 2. In Ex. 1 find the position of the point at which the tendency to break is a maximum. If 3A > 2a, the tendency to break is a maximum when 3. A uniform rod is turning in a vertical plane round a horizontal pivot A, at one of its extremities. Find the tendency to break at any point P. Adopting the same notation as in Ex. 1, and denoting by the angle which the rod makes with the horizontal line, we have a ~ r , L = — - — m g cos 9. z Moreover, since C is moving in a circle round A as centre, its acceleration has two components — one at right angles to PC, which is a+ r d°-d 2 W and the other along PC. The latter gives no moment round P ; hence a + r a - r d 2 9 A = ~~2 r~ w and G^-j-mgcme-m | __—+*»—[; but | w« 2 — = £»#« cos 0, and A- - = — ^— ; (a - rY- whence G = -mg ■ 4q2 ♦* cos 0. X 306 Free Motion of Rigid Body parallel to Fixed Plane. 4. A cracked hoop rolls on a perfectly rough horizontal plane. Determine the inclination to the horizon of the line joining the crack to the opposite point when the tendency to hreak at this point is the greatest possible. In this case the centre of inertia of the hoop moves in a straight line with a constant velocity. Hence its acceleration is zero ; also if a be the radius of the hoop, cos 9 + — sin 9 ) , since CO = — , J= T g \ a O being the centre of inertia of the semi-hoop comprised between the crack Q and the opposite point. Now, since the angular velocity round a horizontal axis through G is con- stant, the system of forces m'TJ, m(, &c. in (10), are equivalent to a single force STar . GO in the direction of CO. The moment of this force round P is M^— a-, which is independent of 9. The tendency to break at'P is given by IT the equation ^ «- ( /cos sin0\ a 1 a 2 ) ff .jr| 4rtar. This condition appears by considering when G attains its greatest magnitude, irrespective of sign, if it should become negative. 5. In Ex. 3 find at what point of the rod the tendency to break is a maxi- mum. Ans. r = ia. 6. A semicircular wire, of radius a, lying on a smooth horizontal table, turns round one extremity A, with a constant angular velocity o>. Find the tendency to break at any point P. Let be the centre of inertia of the arc PB, and let PGA = (p. Join AO, AP, and PO ; then, since the angular velocity is constant, the acceleration of is co 3 . AO. Consequently — is double the area of the triangle APO ; but since AP and GO are parallel, the triangle APO is equal to the triangle AGP; hence A' = a 2 or sin = -n - the angular velocity of the rod after the action of the impulse ; let CP = r, CQ = h, then L=R.PQ, {y'-b)-z-{z'-c)y- = PC'.v, 2' m {(y-b) C- {z-c)-h} = M' {PC. CC'a + £' 2 a>) ; a~ and also, Mv = R, M u> = Rh ; 6 hence from (11) we obtain by substitution G = — } |> 3 (A - r) -(a- rf (« 2 + 2ha + hr)] = ~- 3 {r + a) 2 {2ah - a 2 - hr). 8. In Ex. 7 find at what point of the rod the tendency to break is a maximum. dG I 2a \ The value of r which makes — - = is a ( 1 - — ) . If we substitute this value dr \ oh' &G d 2 G of r in G and in — —, , we find that G is positive and -~ negative when 3A > a ; dr* > ar~ d 2 G also G is negative and — j positive when 3A < «. Hence in any case the ten- / 2a\ dency to break is a maximum when r — a 11— — 1 . x2 308 Free Motion of Rigid Body parallel to Fixed Plane. 247. Impulsive Friction. — When two rough surfaces collide, the investigation of what takes place is, in general, somewhat complicated. We must regard R and F, the im- pulses of the normal reaction and friction, as variable quan- tities, connected, at each instant of the impact, by linear equations with the coexisting values of the velocities of rota- tion of the bodies and of the velocities of translation of their centres of inertia. The laws which regulate the impulse of friction may then be stated as follows : — (1) The direction of the elementary impulse dF due to friction is opposite to that of the slipping of the point of contact, if there be slipping ; and if there be no slipping, is such as to prevent slipping. (2) The magnitude of dF is, if possible, just sufficient to prevent slipping, and when slipping takes place dF = fxdR, ju being the coefficient of dynamical friction. The equations of motion for impulses (Art. 242) show that the relative normal and tangential velocities of the points of the bodies in contact are, at each instant, of the form AR + BF + C, where A, B, and C are constant during the impact. The value of R is at first zero ; when it becomes R x (at the end of the first period of the impact), the relative normal velocity is zero ; and the maximum value of R, which it assumes at the end of the whole impact, is (1 + e)R x . These principles afford a sufficient number of equations to determine the motion ; and, in the case of motion parallel to a fixed plane, the equations are always soluble. If the bodies which collide are perfectly rough, the relative tangential velocity of the colliding points, or the velocity of slipping, is always zero ; and when R = R x , the relative nor- mal velocity is likewise zero. Hence we have two equations to determine R x and the corresponding value of F. At the end of the impact R = (1 + e)R x ; and the relative tangential velocity being still zero, the corresponding value of F can be de- termined. If the bodies slip on each other in the same direction during the whole of the impact, dF is always equal to /mdR ; hence F= fxR throughout. R x is then determined from the equation expressing that the relative normal velocity is zero ; and the Examples. 309 final values of R and F, which determine the motion after the impact, are (1 +e)E l and /u(l + e)R x . For a discussion of the problem in more complicated cases the reader is referred to Routh, Rigid Dynamics. If a sphere impinges against a fixed surface, or if two spheres collide with each other, the relative tangential velo- city v depends upon the velocities of rotation of the spheres, and the velocities of their centres parallel to the common tangent. It is therefore independent of the normal reaction, and the relative normal velocity in like manner is independent of the friction. In this case, if v become zero it must re- main zero, as friction cannot initiate a relative tangential velocity in its own line of direction. Hence v must be either zero at the end of the impact, or in the same direction as at the beginning. Moreover, the value of R x is independent of F. The problem is, therefore, reducible to one of the two cases treated above. If we assume at first that there is no slipping, and obtain the final value of F on this hypothesis, the solution is correct, provided the value of .Fso obtained does not exceed fx(l + e)Ri. If this value of i^does exceed ju(1 + e)R l9 then slipping takes place in the same direction throughout the impact, and the final value of F which determines the subsequent motion is Examples. 1. A box, placed on a rough horizontal table, carries two vertical rods which support a horizontal rod from which a mass m is suspended. A fine string, fastened to the box, and passing over a pulley at the edge of the table, is attached to a mass M ' which, when set in motion, causes the box and suspended mass m to move with a uniform velocity. The string which supports m is now cut, and m falls into the box. If its velocity after m has struck it be equal to its original velocity, and if the friction on the axle of the pulley be neglected, show that the coefficients of impulsive and continuous friction are equal. Let M be the mass of the box and frame-work, v' its original velocity, /j. the coefficient of dynamical friction, R the impulse of the normal reaction, and F the impulse of the friction, developed between the table and box when the latter is struck by m. Since the box originally moves with a constant velocity, we have M'g = /x (M + m)g. After the string supporting m is cut, the box is acted on by an acceleration/, during the time t, in which m is falling. If I ho the moment of inertia of the pulley, and a its radius, / is given by the equation (M' + M+j\f=(M'-fiM)gi=iimg. (a) 310 Free Motion of Rigid Body parallel to Fixed Plane. The velocity of the box when struck by m is v' + ft. Hence its velocity v after the impact is given by the equation (m + m + M' + - 2 ) v = (m+ M' + -J (v' + ft) + mv' - F. (*) If v = */, from (b) we get I M + M' +—)ft = F; this, by (a), is reduced to F '■= fxmgt', but R = mgt, and therefore F= fiR. This example is a description of the experiment by which Morin showed that impulsive and continuous friction obey the same law, and have the same coefficient. If the friction on the axle of the pulley were taken into account, the terms arising from thence in the above equations would each contain as a factor the quantity -, where o is the radius of the axle. But as o is very small compared with a, these terms may be neglected. 2. A sphere, rotating with an angular velocity n round a horizontal axis at right angles to the plane of the trajectory of its centre, impinges on a perfectly rough horizontal plane : find the motion immediately after impact. Suppose the sphere is moving from left to right before impact with a velocity V, whose direction makes an angle i with the plane of the horizon. Let co be the angular velocity in the direction of the motion of the hands of a watch, and v the horizontal velocity of the centre at the instant after impact. F being the impulse arising from friction, the equations of motion are Mv = MV cos i + F, i-Ma i a, = %Ma 2 Cl-aF. The geometrical condition for no slipping is v — aco = ; F whence — = -${Vcosi-ati), M v = aw = fFcosi'+f an. If V cos i = an, no impulsive friction is called into play. If V cos i > an, the horizontal velocity of the centre of the sphere is diminished, and the sphere re- bounds at a greater angle than if there were no friction. If V cos i < an the horizontal velocity of the sphere is increased, and the sphere rebounds at a smaller angle than if there were no friction. In this case friction accelerates the horizontal velocity of the centre of the sphere. If n is opposite in direction to the motion of the hands of a watch, v = fVcosi — fan. The velocity of the centre of the sphere along the horizontal line is dimi- nished, and the sphere will rebound at a greater angle than if there were no friction. If 5 V cos i=2aQ, the sphere will rebound vertically. If 2an>5Vcosi the sphere will hop back. This explains the effect of slow under-cut in tennis. The numerical factors for a tennis ball may of course be different from those given above. Rolling and Twisting Friction. 311 The magnitude of the total normal reaction between the sphere and the plane , . , 2(Fcosi- ad) . is M(l + e) Tsin i. Ilence, m any case m which /x> — rj=~. — :, the pre- / ( 1 -J- cj v sin i ceding investigation holds good, even though the plane he not perfectly rough. If n be counter-clockwise its sign must be changed in the above expression for the limiting value of /j.. 3. If the plane in the last example be imperfectly rough, so that the impul- sive friction is not sufficient to destroy the whole tangential velocity of the point of contact of the sphere with the plane, determine the motion. The equations are, if V cos i>«n, Mv = MVcos i - /jl(1 + c)MVsmi, fMarw = i-Mara + p(l + e) MVa sin/. The sign of /j. must be changed in these equations if V cosi < an, and the sign of n if its direction be counter-clockwise. 248. Rolling and Twisting Friction. — In questions relating to friction, if great accuracy be required in the determination of the motion, it is necessary to take into account not only the tangential force of friction, but also what is called the couple of rolling friction, which is a couple having for its axis the tangent to the rough surface round which the body is rotating. Its maximum value is the normal pressure multiplied by a linear constant, and is generally small in amount, so that in solving questions con- nected with friction this couple is usually neglected. The direction in which the couple of rolling friction tends to turn the body is opposite to that in which it is actually rotating. If the body be not actually rotating, but be acted on by forces tending to make it rotate, the couple of rolling friction tends to prevent rotation round a common tangent to the two rough surfaces, j If the surfaces have a relative angular velocity about the common normal, then, besides the tangential force of friction, and the couple of rolling friction, there is also a couple, having the normal as its axis, called the couple of twisting friction. This couple likewise is usually small in amount. Examples. 1. Taking into account the couple of rolling friction, and supposing the motion to be still pure rolling, determine in Ex. 9, Art. 245, the amount of friction brought into play, and the angular velocity in any position. 312 Free Motion of Rigid Body parallel to Fixed Plane. Let R be the normal reaction between the plate and circle at any time, and fR the couple of rolling friction ; then the equations of rotation of the plate are du> d 2 \mc& — = Fa -fR, m (b - a) — - = F- mg sin 0. Alsoi2 = m{gcos6+ (b-a)0 2 }, and (b-a)6 = -aa>. Hence, putting - = v, F=lm J<7sin0 + 2vgoosB + 2v{b-a)e~ J ; consequently we obtain §{b-a) — = AgcosO + (b-a)eA - g sin 0. If we change the independent variable by means of the symbolic equation d . d q — = — , and put r 1 — = n, we get dt d0 * b - a — J 02 j = a j n ^ v cos _ s i n fl) + y02 The solution of this differential equation is of the form 02 = ce*" 9 + Dcos0 + F sin 0, where C is an arbitrary constant. Determining the constants D and F, we obtain 02 = c £v0 + _i!!_ ( (3 _ 4 „2) cos e+7u s i n e ) . 9+16j/ 2 ( ) If 0o be the initial value of 0, we have, since 0o = 0, 0=- 9 3 (3 - 4i/ 2 ) cos 0o + 7? sin O When is determined, = - (b - a)0. 2. A circular plate is projected along a rough horizontal plane, with an initial velocity V of translation, and an angular velocity n, round an axis through its centre, at right angles to its plane. Determine the motion, neglecting the couple of rolling friction. Let o> denote the angular velocity — , and v the velocity of the centre, at any dt time, and let x, the horizontal coordinate of the centre, be measured in the direc- tion of V, as in the figure ; then the whole velocity of P is v - aw, where a is the radius of the plate. X P X Different phenomena present themselves according to the values of Fand A (1) ft positive, and V> aCi. Examples. 313 Since V- a& is positive, P begins to slip along PX ; therefore F= fxMg, and the equations of motion are M _? = - fiMff, \Ma~ — = iiMgn. dt~ at V- an Pure rolling commences when v — aw = 0, i. e. at a time to equal to — ; then aw = v = | F+ £ aft. The equations for the subsequent motion are v = aw, Mp-=-F, \Ma*% = Fa, dt 'at where F is the amount of friction brought into play. Hence F=0, and the disk will roll on with a constant velocity of rotation round the instantaneous axis. (2) n positive as before, Y 2 V, both v and w will be negative, that is, the motion of translation of the centre will be in the direction opposite to that originally imparted, and the rotation will be in the same direction as the initial rotation. 3. Discuss the same problem, taking into account the couple of rolling friction. Here we have M^. = - fiMff, \Ma^ = nMga -fMg. at at f Putting - = v, we find then that pure rolling commences when V-aO, {Zix-2 v )g At this instant v = — — — — — = t>o- 3/j. — 2j/ After this the equations of motion become *£?--■*• '"i-*"* along with v = aw ; whence F= *vMg. This expression shows that the friction brought into play varies inversely as the radius of the plate, provided its mass be constant. The plate will come to rest at a time ,, _ 3f " 2r 9 * where f is counted from the instant when pure rolling begins. 314 Free Motion of Rigid Body parallel to Fixed Plane. In order that the motion should hecome pure rolling it is necessary that M > %v. The student will have no difficulty in investigating cases (2) and (3) of Ex. 2, when the couple of rolling friction is taken into account. 4. A sphere is projected down a rough inclined plane, along a line of greatest slope of the plane. The sphere has an initial velocity of rotation round a horizontal axis parallel to the inclined plane ; determine the motion— (1) neg- lecting the couple of rolling friction ; (2) taking that couple into account. Let the line of projection be the axis of x, and let x positive he measured to the right, and a>, the angular velocity, be in the direction of the motion of the hands of a watch. Let F"be the initial velocity of translation of the centre of the sphere, and D the initial angular velocity. (1) The equations of motion are, ~ r d' 2 x ,, . . _ „ _, n d(a _ M-— = Mg sin * - F, f Ma 2 — = Fa, dt* at and the condition for pure rolling is v — au = 0. If ^ be the time at which pure roiling begins, then 2{V-aD) 2 (an- V) (7ucosi- 2sini)g' (7/xcos i+ 2 sin i)g* according as V> aD, or aD > V, where u is the coefficient of dynamical friction. If V— aD > 0, we must have 7,u cos i > 2 sin i in order that pure rolling should be attainable. If V — aD. = 0, pure rolling will continue, provided 7fi' cos i > 2 sini (where /x is the coefficient of statical friction). If V - aD. < 0, pure rolling will be reached necessarily, and will then continue, provided 7/*' cos i > 2 sinr. If vo and w be the values of v and « when pure rolling is attained, bu. V cos i — 2 (sin i — u cos i) aD. aw = v = \ " . — , 7/i cos j-2sini bu V cos i + 2 (sin i + u cos i) aD Ifi cos % + 2 sin % according as V — aD is positive or negative. It may be observed that the equations for the latter case can be obtained from those for the former by changing the sign of u. After pure rolling begins, if it continues, F= f Mg sin v = aw = y (t — to) g sin i + aa> . (2) The equations of motion are M — = Mg sin i - F, f Ma 2 -^ = Fa -fMg cos i. ut at Examples. 315 Hence, putting - = v, we have, when V> aQ., n being positive, a 2(V-aa) tn = { 7fi — bv) cos i — 2 sin i) g ' and in order that pure rolling may be possible, l\i - bv > 2 tan i. b( n-v) V cos i - 2 (sin t - ju cos i) aH Again, r = «a> = (7/1 _ 5„) cos i - 2 sin * ' and at any time after pure rolling is established, aw = « is positive, and 2v x n { l i-p)aa+(fi + v)V t -h+ |( 7/i _5„) cos j_2sini}#~ " {fi + v){{7p- 5v)cosi- 2sini}/ ^ _ „ 5 (/z + v) F cos i 4- 2 (sin i - fx cos t) aft and v = #w = . tz r~: : ~ : : • ° M + v (7/i - 5v) cos x - 2 sin t 5. A number of spheres are projected in different directions with different initial velocities along a rough horizontal plane ; find the path of their common centre of inertia. Am. A series of parabolas, and finally a straight line (see (1), Ex. 2). 316 Free Motion of Rigid Body parallel to Fixed Plane. 6. A hollow cylinder filled with water is projected without initial rotation in a direction perpendicular to its axis, along a rough horizontal plane ; deter- mine the time at which pure rolling hegins, the amount of friction subsequently brought into play, and the time at which the cylinder comes to rest. Let M be the mass of the cylinder and contained water, I the moment of inertia of the cylinder round its central axis, a the radius of its external surface, fx and /the coefficients of sliding and rolling friction, Fthe initial velocity of the common centre of inertia G of the cylinder and contained water, t\ the time at which pure rolling begins, F the friction subsequently brought into play, vi the velocity of the point G at the time h, t 2) the time at which the cylinder comes / to rest. Then, putting - = v, we find ci fi + A (n - v) g 1 + A /i + \(p-v) ' vg where a/= Ma 2 . As A increases, F increases, and so in general does vi, whilst ti diminishes, and to in every case remains constant, being the same as in the case of a solid cylinder (see Ex. 3). ( 317 ) CHAPTER XL MOTION OF A RIGID BODY IN GENERAL. Section I. — Kinematics. 249. Motion of a Body having one Point fixed. — If a rigid body have a fixed point, a spherical surface S fixed in the body, with this point as centre, must move about on the surface of an equal concentric sphere fixed in space. The position in space of S, or of any definite great circle on it, determines that of the body. Hence the motion of a body having a fixed point is reducible to the motion of a spherical figure on a sphere fixed in space. The position of such a figure is determined by the positions of any two definite points A and B in it. If the points A and B move into new positions A' and B\ arcs of great circles bisecting AA' and BB' at right angles will meet in a point 0, and the angle AOA! = BOB'; but the great circle OA can be moved into the position OA' by turning it through the angle AOA' round the axis CO (C being the centre of the sphere) ; and since AOA' = BOB ', the same rotation brings OB into the position OB'. Hence a rotation round OC brings the spherical figure, of which A and B are definite points, from the first position into the second. The point is called the pole of rotation (Differential Calculus, Art. 300). Consequently, a rigid body having a point fixed can be moved from any one position into any other by a rotation round an axis through the point. 250. Composition of Rotations round Axes meet- ing in a Point. — If a body receive rotational displacements round two axes fixed in space, passing through the same point, the resultant displacement may be effected by a rota- tion round a single axis. If the displacements be infinitely small, it appears, as in Article 220, that the order in which they are effected is in- 318 Kinematics of a Rigid Body. different, and also that it is indifferent whether the axes be fixed in space or be axes fixed in the body, whose positions at the commencement of the infinitely small motion coincide with those of the axes fixed in space. If the two displace- ments be regarded as simultaneous, the resultant rotation is the actual motion of the body. Hence we see that — A velocity of rotation round a single axis is equivalent to velocities of rotation round two axes meeting the axis of the resultant rotation in the same point. Beiny given the velocities of rotation of a rigid body round two axes meeting in a point, to determine the velocity of the re- sultant rotation and the position of its axis. Let OA and OB be the axes of the component rotations, and R a point on the axis of the resultant rotation. As R is at rest during the motion, its , displacement from the rotation / round OA must be equal and / opposite to that from the rota- / tion round OB. Hence the / circles passing through R, and / having their planes at right jf angles to OA and OB, and their \7~~ A ,/r centres on those lines, touch ati£. ~~~ " Hence OA, OB, and OR lie in the same plane. This appears readily from the fact, that if two small circles of a sphere touch, the arc of a great circle joining their poles passes through the point of contact. Again, AR multiplied by the angular velocity round OA is equal and opposite to BR mul- tiplied by the angular velocity round OB. If these angular velocities be denoted by a and |3, we have a _ P sin B OR sin A OR' To find a, the angular velocity of the resultant rotation, consider the motion of A. It is unaffected by the rotation round OA, and may be regarded indifferently, as rotating round OB with angular velocity j3, or as rotating round OR with angular velocity w. Composition of Rotations round Axes meeting in a Point. 319 If perpendiculars AP and AQ be let fall on OB and OR, we have then AP. [3 = AQ.w. Hence (x) (5 a sin A OB sin A OR sin BOR Hence, finally — The axis of the resultant rotation lies in the same plane as the axes of the component rotations, and makes with each an angle whose sine is proportional to the velocity of rotation round the other ; and the velocity of the resultant rota- tion is proportional to the sine of the angle beticeen the axes of the component rotations. Accordingly, velocities of rotation are compounded in precisely the same manner as velocities of translation, or as forces meeting in a point. By reversing the reasoning above, it can be shown that a point R, taken as above, remains at rest under the influence of two velocities of rotation round OA and OB ; whence we have an independent proof, that infinitely small rotations round two intersecting axes are equivalent to a single one round an axis lying in the plane of the two former, and passing through their point of intersection. We have already seen, Article 221, that velocities of rotation round parallel axes are compounded in the same way as parallel forces. Hence, in general — Velocities of rotation are compounded like forces, whose directions coincide with the axes of rotation, and whose magnitudes are proportional to the velocities of rotation. The attention of the reader has been directed in Article 221 to the algebraical signs of velocities of rotation. In addition to what was there stated, it may be observed, that the axis of a rotation may be made to represent the rotation both in magnitude and direction. In this case the axis is drawn so that the rotation round it is always positive. For example, instead of speaking of a negative rotation round the axis of X, we may designate it simply as a ro- tation round the axis of X negative. When the axis of a rotation determines the direction of the rotation, the latter is always understood to be in the positive direction round this axis, that is, according to the convention, counter-clockwise. 320 Kinematics of a Rigid Body. When rotations are compounded by means of their axes, like forces, the direction of the axis determines in this way the direction of the rotation. For example, rotations a>i, w 2 , w 3 round three rectangular axes produce a resultant rotation w which is always positive ; but the direction of its axis is determined by — , — , — 3 , the (1) It) (s) cosines of the angles made with the coordinate axes ; and these again depend on the signs of wi, o> 2 , and w z , as well as on their magnitudes. 251. Geometrical representation of the Motion of a Body having a Fixed Point. — When a body has a fixed point, its motion may be represented in a manner analogous to that mentioned in Article 225. In the present case the curves which correspond to the space centrode and the body centrode are spherical curves lying on the surface of the same sphere. The motion of the body is represented by the rolling of a cone fixed in the body on a cone fixed in space (see Differen- tial Calculus, Article 301). 252. Motion of a Body which is entirely Free. — A rigid body can be moved from any one position into any other by a motion of translation, combined with a motion of rotation round an axis through any arbitrary point A of the body. Let A x , A 2 be the two positions in space occupied by A in the different positions of the body. Give to every point of the body a motion equal and parallel to AiA 2 : this brings A into the required position, and a rotation round an axis through A will then (Article 249) complete the body's change of place. If two positions of a body in motion are infinitely near each other, any infinitely small displacements, by which it can be moved from the first of these positions to the second, may be regarded as the actual motion of the body. The actual motion of a rigid body during an infinitely short time is, therefore, a motion of translation together with a motion of rotation round an axis through any arbitrary point of the body. The initial and final positions of a body being given, the mag- nitude of the rotation, which is required to make it pass from one to Motion of a Body having a Fixed Point. 321 the other, and the direction of its axis are determined ; but the motion of translation varies according to the point through which the axis of rotation is supposed to pass. First, let the axis of rotation be supposed to pass through a point A, whose initial and final positions are A x , A 2 . The motion of translation A X A 2 is composed of two parts — one^^I' in the direction of the axis of rotation through A, and the other AA 2 at right angles to it. By means of the first a defi- nite plane section of the body, passing through A and at right angles to the axis of rotation, is moved into the plane in space in which it lies in its final position, and the subsequent motion of the body is therefore parallel to this plane. If, now, the axis of rotation be regarded as passing through another point B of the body, whose initial and final positions are B x , B 2 , we can suppose the translation B X B 2 made up of two parts — one, B X B\ equal and parallel to A X A ; the other, B'B 2 , which depends on the position of the point. B x B f brings the body into the same position as A X A. Hence, a translation B f B> and a rotation round an axis through B are equivalent to an equal rotation round a parallel axis through A and a translation A 'A, (Art. 219). The translation B X B 2 is the resultant of B X B' and B , B 2 ; A X A 2 is the resultant of A X A ' and AA 2 ; B x B f is equal and parallel to A X A; but B'B 2 is not in general either equal or parallel to AA 2 . 253. Analytical Treatment of the Motion of a Body having a Fixed Point. — Suppose three rectangu- lar axes fixed in the body passing through a point ; and three others fixed in space, which at the beginning of the motion coincide with the former. Let the coordinates of any point of the body referred to the former be ?, *?, £, and referred to the latter, ,r, y, z. Let a u a 2y a 3 ; b x , b\, b :i ; c ls c iy c z be the cosines of the angles which £, »j, £ make with x, y, z, respectively ; and let the angles themselves be a l9 a 2 , <*3 ; /3i, /3 2 , ^s ; yi, y 2 , y& If the point be fixed, we have at any instant x = a l Ii+b l ti + c 1 Z i g = a 2 ^+b 2 i]+c 2 Zy z = (h% + b 9 ri + e 3 Z' If at this instant any other point of the body besides occupy the same position in space as at the beginning of the 322 Kinematics of a Rigid Body. motion, for this point, a? = ?, y = t», % = ?, and therefore we should have a x - 1 £>t c t tfo ^2-1 i = 0, b 2 = 1, 6 3 = 0, we have db x + da 2 = 0. In like manner dc x + da 3 = 0, db z + dc 2 =0. Motion of a Body having a Fixed Point. 323 Let now da 2 = dip, db^ = dO, dc y = d$ ; then dx=-i)d\p+ Z>d$ = - ydxp + zd

/ = -Zd9, dz = r } d9; d(p round y would give dz = - %d(p, dx = Z>d

p. Hence the most general infinitely small displacement the body can take, remaining fixed, is equivalent to rotations round any three rectangular axes through 0. Moreover, from the values of dx, dy, dz, given above, it appears that for a point whose coordinates fulfil the condi- tions -tq = — = — the displacements are zero. Hence the three rotations dO, dcp, d\p, round the axis x,y,z, are equivalent to a single rotation round an axis whose posi- tion is defined by these equations. If we put dO = dx cos A, d(p = d% cos fx, dip = dx cos v, where dx = V^dO 2 + d§~ + dip 2 , the equations of the fixed axis are cos A cos fi cos v Also, for any point of the body, dx 2 + dy 2 + dz 2 = [(»? cos v - Z cos /m) 2 + (£ cos A - £ cos v) 2 + (£ cos f.i - ?j cos A) 2 ] dx 2 =P 2 dx 2 y if p be the perpendicular from the point on the fixed axis. Hence d\ is the magnitude of the resultant rotation. Y2 324 Kinematics of a Rigid Body. Whence infinitely small rotations, and therefore velocities of rotation, are compounded like forces meeting at a point. 254. Motion of a Body entirely Free. — If the point of intersection of the axes fixed in the body be itself in motion, and if its coordinates, referred to axes fixed in space, be x\ y, z' ; then, for any point xyz of the body, x = x + «!$ + bit) + dZ, y = y + a.£ + b 2 ri + c 2 Z, s = z + a 3 £ + b z r\ + CzZ; whence dx = dx + 5 da x + r}db i . + Z dc h dy = dy' + % da 2 + rj db 2 + Z dc 2y dz = dz + %da z + ridbs + Zdc 3 . If we suppose the axes of £, n, Z parallel to those of x, y, z at the beginning of the motion, we get, as in the last Article, dx = dx' - y\d\p + Zd(j) = dx' - (y - y) dij/ + (z - z') d(p dy = dy--ZdO + &ty = dy'-{z-z)dd+{x-x')dil, L (2) ! dz = dz' -Z,d/") d^" + (z' - z") df. Subtracting, we get dx = dx - (y - y')d$' + (z - z)d$"\ Velocity of any Point of a Body. 325 but again, dx = dx - (y - y') dif/ + (s - z) d' = dtf ' . In like manner dd' = dQ"; hence the rotation remains un- altered in magnitude and direction. 255. Velocity of any Point of a Body. — Infinitely small displacements divided by the element of time during which they are effected become velocities. If the axes of x, y, z be three rectangular axes fixed in space, and if the velocities of rotation round parallel axes meeting at the point x'y'z\ be w x , w y , w 2 , we have, from equations (2), dx dx / /\ 1 dt dt dy dt dy ~dt {y-y)<*>* + (*- (S - JO €* + (*- X )(i)r dz _dz__ ~dt~~di~^ X ) »>y + (y - y') X )o>. (3) If the point x f yz be fixed in space, and be taken for the origin, we have dx dy w z y w x z y- dz 7t = W x y - (t)yX (4) If we suppose the axes fixed in space to coincide at the instant under consideration with axes fixed in the body, and if the angular velocities round the latter be wi, wz, a*, we have o) x = m, <*> v = w 2 , w z = w,. Consequently, if S, v, 2 be the coordinates of any point, referred to axes fixed in the body, 326 Kinematics of a Rigid Body. and if u, v, w be the components of its velocity parallel to these axes, we have u v = u)£ W = Wit] (I) K (5) Equations (3), (4), and (5) hold good for every instant, whereas the equations x = £, &e., w x = w„ &e., — = u, &c, (it hold good only for one particular instant. If A, jjl, v be the direction cosines of a definite line in the body referred to axes parallel to fixed directions in space, we have, as an immediate consequence of (4), tfA dt (DyV - W 2 jU dn dt = w z X - w r v i dv dt = WxjU - U)y\ >■ (6) The motion of a body relative to the space in which it is moving is unaltered if we attribute to the latter the motion of the body reversed, and suppose the body itself to be at rest. Hence, if /, m, n be the direction cosines of a line fixed in space referred to body axes, we may regard the latter as fixed in space, and the line Imn as moving round them with angular velocities - an, - a> 2 , - a> 3 . Accordingly, from (6), we have dl -\ = — him. 4- num. dt = — u) z n + oj-jn dm dt dn Jt = - h) Z l + d)\n y • -— = - wim + w 2 / (7) Acceleration of Rotation. 327 \ 256. Acceleration of Rotation. — If w l5 w 2 , w 3 , be the angular velocities round three rectangular axes, OA, OB, 00 fixed in the body, and w x , w y , w z the velocities round axes OX, OY, OZ fixed in space ; and if at any instant we suppose OX, OY, OZ to coincide with the positions occupied at the instant by OA, OB, OC, then not only is on equal to w x , w* to wu, and o> 3 to w z , but also du>\ dw x dw z d(M)y dwz dw% "dt = Ht 9 dt ^Hf' lit = ~di This may be proved as follows : — Let w be the velocity of rotation round a line fixed in the body, which passes through 0, and makes angles with the axes OX, OY, OZ, whose direction cosines are A, ju, v ; then (A) = W z \ + WyfJ. + U) Z V ', . . da) dw x A dwy du) z therefore — - = — A + — n + — v dt dt dt dt dX dfx dv + Ux-77 + WyTT + ^3 -JT • dt dt dt Hence, by (6), = A — + ^ — + v — . (8) This equation shows that the acceleration of rotation round a line is the differential coefficient, with respect to the time, of the angular velocity round the same line even though it is in motion, provided it be fixed in the body. Thus, in particular, we have in the case supposed above, du)i d(x) x du) 2 dwy dw 3 ^ diD z ,qv lU = Hi' dt == It' lit "~~~ lit' ^ ' The same may be proved geometrically as follows : — The body at any instant is rotating round a certain axis with an angular velocity w. Draw a line through the fixed origin in the direction of the instantaneous axis, and measure off on it a portion 01, proportional to o) ; then the projec- 328 Kinematics of a Rigid Body. tions of this line on the axes fixed in space represent w Xi w yf w z ; and its projections on the axes fixed in the body represent &>i, w 2 , ws- At the next instant the body is rotating round another line with a velocity &>', represented by OT, and the projections of OT represent w/, w/, «/; oj/, oj 2 ', w 3 '. But the projection of 01' is equal to the sum of the projections of 01 and IT. Hence dw x = w x ' - h) x = projection of IT on axis of x fixed in space, d(t>i = wi - (1)1 = projection of IT on axis of ? fixed in the body. At the first instant the axes of x and S coincide ; and at the next the two projections of IT differ only by a quantity infinitely small compared with IT y which is itself infinitely small of the first order. Hence du x and dw v differ by an infinitely small quantity of the second order ; dwi dw x dwz dd) y dw% ~dt == ~aT' ~dt = ~di i ~dt A line passing through parallel to IT is called the axis of angular acceleration. If we put — — = y 2 + d>~ 2 ), as it is the resultant of the three accelerations w x , w y , and cb z . 257. Accelerations of a Point, parallel to three Axes fixed in the Body. — If u, v, w be the velocities of a point parallel to axes fixed in the body, its velocity-component V y along a line whose direction cosines referred to these axes are /, m, n, is ul + vm + ten. If we suppose this latter line fixed in space, the accelera- dV tion of the point parallel to it is — -, and we have at dV 7 du dv dw dl dm dn -jr = I -77 + m— + n — + u — + v — + w -—. dt dt dt dt dt dt dt Complete Determination of the Motion of a Body. 329 „ dl dm , dn , . Substituting the values of — , — , and — , given by (7), we obtain . at df df df dV du \ fdv \ fdw \ " > 2 + Vl*>\. ). -- / 1 — - Vw 3 + Wb) 2 ) + m \-T, ~ w *»i + li(J) o j + ni — - ?/w. Let us now suppose OX to be the fixed line, then dV du / = 1 m = n = 0, and therefore -r— = — - £w 3 + ww 2 ; but dt dt — is now the acceleration of the point parallel to one of dt the axes fixed in the body ; hence we have, for the accele- rations of a point parallel to three rectangular axes fixed in the body, the expressions du dv dw — - I'lVz + t€Uf 2 , -T. ~ U'toi + «W:i, -TT dt dt at where it, v, iv are the velocities of the point parallel to the axes fixed in the body. 258. Complete Determination of the Motion of a Body. — Every motion which a rigid body can take is re- ducible to a motion of translation and a motion of rotation. In order then to determine the motion of the body, a point in it is selected (usually the centre of inertia), and the motion of the body is reduced to the motion of this point, together with the rotatory motion of the body round it. Geometrically the motion may be represented by the rolling of a cone, fixed in the body, on a cone unattached to the body, except at one point (the common vertex of the cones), the latter cone undergoing a motion of trans- lation. If the two cones and the rate at which the one rolls on the other are known, as well as the position in the body of their common vertex, its velocity at each instant, and the path which it describes, then the motion of the body is completely determined. 330 Kinematics of a Rigid Body. It is usually most convenient to consider the motion of translation and the motion of rotation separately. The in- vestigation of the former motion is, as we have seen (Art. 205), reducible to the problem of the motion of a particle. The latter motion is completely determined if we can assign at each instant the position of the body and its velocities of rotation in reference to axes, through the centre of inertia, whose directions are fixed in space. The equations of Kinetics usually give the velocities of rotation round axes fixed in the body ; but in order fully to determine the motion, it is necessary to ascertain the effect of these velocities when the position of the body is referred to axes whose directions are fixed in space. As the points of intersection of these two sets of axes coincide, the velocities of rotation have no effect on the motion of this point ; and therefore, so far as the angular velocities are concerned, we may regard as fixed, not only in the body, but also in space. Call the space-axes OX, OY, OZ; the body-axes OA, OB, OC, each set being rectangular. Round the point as centre describe a sphere, and let the axes meet it at the points X, Y, Z, A, B, C. Three independent angles are required to determine the position of the body in space. Those which are probably the best adapted for the solu- tion of the problem are the angular coordinates of the point C, or of the line OC, and the angle 0, which the plane CO A makes with the plane ZOC. It is obvious that the position of OC fixes the plane A OB, but does not determine the position of the lines OA and OB in this plane. Hence, when C is fixed, if the angle $ which the plane CO A makes with the plane ZOC be given, the position of the body is completely determined. The angular coordinates of OC are 0, the angle which it makes with OZ, and \p, the angle which the plane COZ makes with the plane XOZ. Suppose now that the body has three velocities of rota- Complete Determination of the Motion of a Body. 331 tion : wx, round OA ; w 2 round 01? ; and w 3 round OC, in the direction of the arrow heads. We have to express — , ^, and ^ in terms of these velocities, remembering that dt dt dt the changes of 0, <£, and \jj are caused solely by w„ w 2 , w 3 . The motion of the point C on the sphere is unaffected by w 3 . If the radius of the sphere be unity, the point C has two velocities, wi and w 2 , along the tangents to the great circles BC and CA. Eesolving these velocities along the great circle ZC, and at right angles to it, we have dO dt o)-> cos

; and since SZ and 8 A are at right angles, and S lies on ZC at a distance 90° -f from Z, the velocity of S along AB is cos Q-tt, therefore n d\L dd> , d(f> 6j 3 = cos -jj + -77, whence -~ dt dt dt (*)* dt The angles made use of by Laplace in his solution of the problem of Precession and Nutation are somewhat different from those considered above. Laplace supposes that a point which is moving from X to Y approaches nearer to C after passing E, and he further places E behind A and X. In this way the various lines and planes assume the positions represented in the accompanying diagram. The angles employed by Laplace are ZOC, which we may denote by 0', EOA, or $', and XOE, or ;//. the last being positive when E is behind X. Taking into account the mode in which Laplace supposes the axes to be situated, we have, Screws and Twists. 333 then, 0' =- 0, $' = <£ - |tt, ^'= Jr - if/, and equations (10), (11), and (12) become, "by substitution, — = w 2 sin - oil COS sin 0' — - = wi sin <£' + ai 3 cos cj>' y • (13) -£- = ai 3 + COS -~ dt dt j 259. Screws and Twists. — It was shown in Article 252 that a body can be moved from any one position into any other, by a translation combined with a rotation, round an axis through any arbitrary point of the body. The translation may be resolved into two — one parallel to the axis of rotation, and the other at right angles thereto. The latter translation, along with the rotation, may be re- placed by a pure rotation round a parallel axis, and so the whole motion will consist of a translation parallel to a certain fixed line and of a rotation round it. Such a motion is simi- lar to that of a nut on a screw, and is called a Twist. Hence a body can be moved from any one position into any other by means of a twist. In order to determine a screw it is necessary to specify — (1) the position and direction of the line round which the rotation is effected, or the cans of the screw ; and (2) the ratio of the translation to the rotation. This last is a linear magnitude, and is called the pitch of the screw. In order to determine a twist, we must, in addition to the screw round which it is effected, specify its amplitude, i. e. the magnitude of the rotation. The twist by which a body can be moved from any one position into any other is in general unique. This readily appears from considering that if two posi- tions of a body are given, the magnitude of the correspond- ing rotation and the direction of its axis are invariable ; and that if two positions of a plane figure in its own plane are 334 Kinematics of a Rigid Body. given, the position of the corresponding centre of rotation is thereby determined. The same thing is proved directly by Sir Eobert Ball (to whom the Theory of Screws is principally due), as fol- lows : — Any point of the body, which lies on the axis of the twist, must continue thereon after the motion. If, therefore, the motion could be effected by two different twists, there would be two different lines along which points of the body would continue throughout the motion. In order that this should be possible, the lines must be parallel, and the motion one of pure translation. If two successive positions of a body in motion are infi- nitely near each other, the twist by which it can be brought from the one position to the other is the actual motion of the body. We see then that the most general motion of a rigid body consists of a succession of twists. The screw round which it is twisting at any instant is called the instantaneous screw. As the position of a straight line in space is determined by four independent quantities, five magnitudes must be assigned to determine a screw. In order to determine a twist, its ampli- tude, and the pitch, as well as the position of the axis, of the corresponding screw, are required. Hence the motion of a rio-id body in general depends on six independent variables, and we see, as in Article 215, that a rigid body entirely un- restrained has six degrees of freedom. 260. Composition of Twists. — If a body receive in succession two twists whose amplitudes are infinitely small, the order in which they are effected is indifferent, and the resulting change of position may be produced by a single twist, which is the resultant of the two former. More symmetrical results are obtained, if instead of seek- ing for the twist which is the resultant of two others, we inquire how three twists having infinitely small amplitudes must be related, in order that the position of a body, after being affected by them, may remain unaltered. The question proposed may be solved directly, but the method of solution devised by Sir Eobert Ball leads to results of a more instructive character. This mode of solution will be found in Example 14. Examples. 335 Examples. 1. Determine the velocity with which the plane of the horizon, at a place whose latitude is given, turns round a vertical axis. Ans. w sin A, where a> is the earth's angular velocity, and A the latitude. 2. If the velocities of rotation of a body round three rectangular axes are given in terms of the time, show how to determine— (1) the velocity of rotation round the instantaneous axis ; (2) the position of the instantaneous axis ; (3) the equation of the cone which is the locus of the instantaneous axis. 3. If the velocities of rotation round three rectangular axes are proportional to the time which has elapsed from a given epoch, the position of the instantaneous axis is fixed. 4. If the accelerations of rotation round three rectangular axes are constant, the instantaneous axis lies in a fixed plane. 5. If 6, 2, »3 have the same significations as in Art. 258, show that de . „ d* Q}\ = smd> sin cos d> ~. Y dt r dt ' de d\b o> 2 = cos

- L , dd> d\L at at 6. A body is rotating round a fixed point 0. If OX, OY, OZ he rectangular axes fixed in space, and OA, OB, OC rectangular axes fixed in the body ; and if the direction cosines of the latter referred to the former be, respectively, o\, <*2> «3 ; bit i>i, bz ; 0i, ci, cz ; show that dai — =b\0)z — Ci<»2, dt db x — -=Cl«l-0lft>3, dt dc\ at dai , — - = Or, 0>3 - C-i, 0>z, at db 2 -j~ — C21, dt where «i, a>2, (oz are the angular velocities of the body round OA, OB, OC. 7. Deduce equations (10), (11), (12), Art. 258, from equations (7), Art. 255. 8. A body receives in a given order rotations of finite magnitude round two axes fixed in space, or in the body, and meeting in a point. Find the posi- tion of the axis, a single rotation round which would bring the body into the same position, and determine the magnitude of the resultant rotation. 336 Kinematics of a Rigid Body. This question is solved in a manner similar to that employed in Examples 3 and 4, Art. 226 ; the construction in the present case being on the surface of a sphere instead of a plane. When the rotations round the given axes are in the same direction, the resultant rotation is double the supplement of the vertical angle of a spherical triangle, whose base and base angles are the angle between the axes and the semi- amplitudes of the rotations round them. 9. A rigid body receives a motion of translation, whose components, parallel to the axes, are a, b, c, and a rotation round an axis fixed in the body, which, at the beginning of the motion, coincides with the axis of z. Determine the position and pitch of the screw, a twist round which would bring the body into the same position ; and find the amplitude of the twist. The screw passes through a point whose coordinates are a sin h9 b sin \-d + a cos hd Pitch of screw = 2 sin %d Amplitude of twist 10. A body receives, in succession, rotations of finite magnitude round two non-intersecting axes a, b, either fixed in space or fixed in the body : if d be the shortest distance between the lines a and b ; Q and Q' the amplitudes of the rotations round them ; e the angle between them ; tp the amplitude of the twist equivalent to the motion ; and p the pitch of its screw ; prove that \p§ sin^ = d sin^0 sin |0' sin e. (This theorem is due to Rodrigues : I^iouville, T. 5, p. 390.) Take the shortest distance between a and b for axis of y ; the point of inter- section of this line with b lor origin ; and a parallel to a for axis of z. After the body has received its rotation round a, suppose it receives in suc- cession two equal and opposite rotations round OZ, the first of these being equal and opposite to that round a. These rotations, being equal and opposite, do not change the position of the body. Examples. 337 First, suppose a and b to be fixed in space, then so also is OZ. The rotation round a and the equal and opposite one round OZ are (Ex. 5, Art. 226) equivalent to a translation, whose magnitude is 2d sin \9, and whose direction lies in the plane XOY, and is at right angles to a line OP which makes with OY an angle — \B. Describe, a sphere round as centre, and let B be the point in which it is met by b, then ZB = e. The axis of the rotation, which is equivalent (Ex. 8) to the rotations round OZ and b, meets the sphere in B, and the direction of translation meets it in T ; where TX = l9, BZX=\9, ZBB = \9'. Then, by Ex. 8, TBB = \ sin |0 = d sin |0 sin ZB sin \

z x -j- w y y + w, z) - (a z x + z — y — dt- at d 2 y — - = w y (ux x + a>y y + ws z) - co 2 y + x — dt z <*t dt dco x z (o>x x + coy y -f- w 3 z) - a> 2 z + y dco x da. dt dt ' remembering that .,2 = oo x 2 + toy* + &>* 2 Let us now suppose the axis of z to coincide with 01, the instantaneous axis, then oo x = 0, w,j = 0, w t = «. Let the plane of xz pass through 01', the consecutive position of the instantaneous axis. Measure off 01 proportional to co on OZ, and take OF proportional to the cor- responding angular velocity u + doo ; draw IP perpendicular to 01 ; then w + do* round 01' is resolvable into OP round OZ, and PP round OX. Let JT'OP = dty ; then PP = OP dty ; therefore dbOx ~dt 'dt = w^, if the angular velocity of the instantaneous axis be denoted by fy. Also £-•* — , since d«> z oz IP = OF - 01. dt Introducing tbese, we obtain d*x dt* duo • X dt d?-y doo , . d 2 z . 16. Find the position of the acceleration-centre in a body rotating round a fixed point. The only acceleration-centre which in general exists is [the fixed point itself. 17. A body is moving round a fixed point 0. If perpendiculars, whose lengths are p and q, be let fall from any point A of the body on 01, the instantaneous axis of rotation, and on OJ, that of angular acceleration ; prove that the total acceleration of A is the resultant of two components, oo' z p along p and tig per- pendicular to the plane AOJ, where a> and o- are the resultant angular velocity and angular acceleration of the body. Examples. 341 If x, y, z be the coordinates of A referred to space axes through 0, and r be the distance OA, the equation for the acceleration x may be written (Ex. 15), x where / tox toy W«\ ) ( 'toy z a> z y\ [x — + y - +z- )-x -for - , \ co co co/ J* \ cr r a r / a 2 = u x 4 toy" + «i>s • If P be the point in which the perpendicular from A meets 01, and if we consider the projection of the triangle OPA on the axis of x, we have projection of AP = projection of OP - projection of OA. From this it is plain that the term by which co- is multiplied in x is the projection of p on the axis of x. Again, if A, fi, v be the direction cosines of the normal to the plane AOJ, and 9 the angle between OJ and OA, we have (b„ z to % y Xsm8=— , and r sin 9 = q ; a r ff r whence it appears that the term by which . xz the tangential acceleration = — r H — = u\p — . p dt 2 p dt 2 p dt p The required locus is therefore the cone dco — (x 2 + y 2 ) - wipxz = 0. dt 19. Show that a point whose normal acceleration at right angles to the instantaneous axis vanishes lies on the cone to {x 2 + y 2 ) + fyz = 0. 20. A body is rotating round a fixed point : determine at any instant the positions of the osculating plane, and of the principal normal, to the path described by one of its points. The normal plane to the path is the plane passing through the point and the instantaneous axis. Hence the perpendicular to the osculating plane is the intersection of this plane with its consecutive position. Again, the direction of the principal normal coincides with that of the resultant normal acceleration ; hence, if v be the angle the principal normal makes with the instantaneous axis, co;; 2 + &jz tan v = : . to 342 Kinematics of a Rigid Body. 21. Find the radius of curvature of the path of any point of the body. If iVbe whole normal acceleration, P = up z N V{(«p3 + ^)8+^yV} 22. A body is moving in any manner. Determine the accelerations of a point parallel and at right angles to the axis of the instantaneous screw. Let xo, l/o, zq be the coordinates of a point fixed in the body, {, 77, £the co- ordinates of any point referred to x , y , z as origin ; then d 2 x dP d 2 x da* do — — - +(Dx(wx£+ wyTj-f uzQ -co 2 ! -f C~r. - -V-Ji dt 2 dt dt **y ^Vo, f t . . fy di? = It 2 " Wy V**l + a,J/7? +Wz &~ u ~ doj s -c do} z ~dT i d~zd 2 z dco x dt Take as xo, 1/0, zo that point of the body which at the instant coincides with the point on the instantaneous screw in space which is nearest the consecu- tive position of the instantaneous screw. If C be the ruled surface in space generated by the positions of the instantaneous screw-axis, will be the point of intersection of the instantaneous screw- axis with the line of striction on C. Let 00' be an element of this line of striction. At the time t + dt the body is twisting round a screw through 0'. Let Tbe the velocity of trans- lation at the time t, and T and w' the velocities of translation and rotation at the time t 4- dt. Now, the velocity of rotation 00' round O'S (screw-axis through 0') is equivalent to o»' round 01' (parallel to O'S), and a velocity of trans- lation oj'.OO' at right angles to 00' and 01'. The velocity of rotation w' round 01' is equivalent to u> round OZ, andwWi// round OX. Hence, at the time t + dt, the point Xoyozo has two velocities of translation : T' along OZ, and (oj'.OO' + T'd$) along OX. Again, as 00' is infinitely small of the first order, the velocity of translation along OZ resulting from 00'. (a' is infinitely small of the second order. At the time t the point x ijqZo had the velocity T along OZ. Hence, if U be the velocity of translation, and ^ the angular velocity of the axis of the instantaneous screw, at the instant, we have = *, d 2 x dt* = .U + T*. ^=0, d 2 z ~~d~F dT dt' Examples. 343 AlSO (I) X =0, (l)y ■ whence dcc x ' dco v = 0, co z =v, — =co*, — = 0, dwz ~dt ' dw ~~ It dH _ _. „,. , and — are supposed to be expressed) ; and, as — = — , by eliminating t an dt dt at equation is obtained between r and s, which is the equation of the curve C; therefore, &c. Kinetics of a Rigid Body. 345 Section II. — Kinetics. 261. Moments of Momentum of a Body having a Fixed Point. — If x, y, z be the coordinates of any point of the body, referred to space axes intersecting at the fixed point ; and H x , H y , H z , the moments of momentum round these axes, we have M x = Sm (yz - zy) . Substituting for y and z their values given by (4), Art. 255, we obtain H x = o)xS(!/ 2 + s 2 ) dm - wylxydm - w z \xzdm. Hence, if a, b, c, i, J, k be the moments and products of inertia of the body at any instant, round the space axes, we have s x = au) T — JCWy -J(Oz By = — k(O x + bWy — iu) z b z = -.M ~ iu) v + Cd) z (1) If the space axes coincide with the instantaneous posi- tion of the principal axes of the body at 0, equations (1) become B, = Am, B 2 = JBuj 2 , B, = Cw 3 , (2) where B ly B 2 , B 3 , wi, w 2 , w 3 are the moments of momentum and the angular velocities round the principal axes at the instant ; and A, B, C are the principal moments of inertia of the body for the point 0. The resultant moment of momentum B is given by the equation B 2 = A 2 ^ 2 + B 2 w 2 2 + C 2 ^. (3) The direction cosines of the momentum axis relative to the principal axes through are proportional to Awi, Bu) 2 , Cw 3 . 346 Kinetics of a Rigid Body. If #,, a 2 , a 3 , b h b 2 , b s , c i9 c 2 , c s be the direction cosines of the principal axes at referred to the space axes, we have H x = Awxcti + Bixjobi + CojzCx \ Hy = Aw x a 2 + Bu 2 b 2 + Cu) 2 c 2 [• (4) H z = Awictz +Bu) 2 b 3 + CwzC z ) If be a definite point of the body not fixed in space, equations (1), (2), (3), (4) still hold good for the motion relative to ; the axes x, y, and z being parallel to fixed directions in space. 262. Motion of a Body having a Fixed Point under the Action of Impulses. — If a body having a fixed point be acted on by any set of impulses, whose moments round the principal axes through are L, M, N; these moments are equal respectively to the changes in the moments of momentum of the body. Hence ( (2), Art. 262), A (o>, - W) = L, B (012 - w 2 ) = M, C (to* - w/) = N, (5) where &)/, w 2 , to*', and u) h to> 2 , to> 3 are the"^ angular velocities round the principal axes immediately before and imme- diately after the action of the impulses. In some cases it may be convenient to use the expres- sions for H x , Hy, H~ given in (1), Art. 261, and the moments G Xi G y , G z of the impulses round the space axes. We have, then, CI (to X - to)/) - k (b)y - to)/) - j (to) z - to)/) = G x \ - JC (to)* - to)/) + b (toly - to)/) - i (to) a - to)/) = Gy > . (6) -j (to)* - to)/) - » (to)*, - to)/) + C (to) z - to)/) = G z I 263. Vis Viva of a Body having a Fixed Point. — As the body has a fixed point, it is at any instant rotat- ing round some axis through it ; whence the vis viva is 7to) 2 , / being the moment of inertia round the instanta- neous axis. Couple of Principal Moments. 347 Again, since — , — , — are the direction cosines of the axis to to to of rotation referred to the principal axes through the fixed point, wr 2 I = A - \ +B - 3 +C - J ; (Int. Calc, Art. 215) ; whence, if 2T or # be the w's mwb of the body, we have 2T=S = Ato? + ZW + Cto Z \ (7) If to z , toy, to- be the velocities of rotation, and «, 6, c, », &c, the moments and products of inertia of the body at any instant round space axes through 0, the general equation of the momental ellipsoid referred to these axes leads to the following expression — 2T= Ito 2 = cito x 2 + btoy + Cto~r - 2ito y to z - 2jto z to x - 2kto x io y . (8) 264. Couple of Principal Moments. — If a body be moving round a fixed point, we may imagine its actual velo- city at any instant to be produced by an impulsive couple acting on it at the instant. By the last Article the compo- nents of this couple round the principal axes of the body are Ato h Btoo, Cto 3 , and the axis of the couple is called the Axis of Principal Moments. This axis coincides at each instant with the momentum axis of the body (Arts. 210, 261). If a tangent plane be drawn at the point of intersection of the instantaneous axis of rotation with the momental ellip- soid corresponding to the fixed point round which the body is rotating, the perpendicular from the centre on this tangent plane is the Axis of Principal Moments. This is obvious, when we remember that the direction cosines of this axis are proportional to Ato h Bto 2 , Civ,, ; and those of the instantaneous axis of rotation to w lf w 2 , w 3 ; and that the equation of the momental ellipsoid is Ax 2 + Bif + Cz 2 = K. If (j) be the angle between the momentum axis and the instantaneous axis of rotation, R the moment of momentum, 348 Kineiics of a Rigid Body. and 8 the vis viva of the body, we have, by the formula for the cosine of the angle between two lines in terms of their direction cosines, Hcj cos ^ = Au)x + Buz + Cwz = 8 ; or whence w cos = -=. (9) Again, if r be the intercept made by the momental ellip- soid on the instantaneous axis of rotation, we have K w! 2 , „ w 2 2 n Wz 8 r* to it) (*) w whence r 2 = -= w 2 . (10) Again, if we draw a tangent plane to the momental ellip- soid at the point where it meets the instantaneous axis of rotation, the intercept p made by this plane on the momen- tum axis is given by the equation p = r cos 0, since the momentum axis is perpendicular to the tangent plane. Hence, if we substitute for r and w cos their values given by (10) and (9), we obtain P - ^p. (ii) Examples. 1. A body is set in motion by an impulsive couple whose magnitude is given ; find the direction of its axis so that the initial vis viva of the body may be a maximum. The axis of the couple must be the axis of least inertia of the body. 2. A body having a fixed point is set in motion by an impulse, passing through a point P, which causes P to move with a velocity having a given mag- nitude and direction ; determine the axis of instantaneous rotation. _ Let the axis of a; be a line through in the direction of the velocity of P, and the axis of z the line OP ; then, if V be the given velocity of P, h the distance OP, and a>„, a> 9 , w z the angular velocities of the body round the axes, we have &>* = 0, h w y — V. Examples. 349 Xow, by Thomson's Theorem, Art. 199, the value of a z must he such as to make To. minimum. But, Art. 263, (8), dT _ — — = — iuy + cco z . Hence cw- = iw y , du~ which determines w z , and consequently the axis of rotation. 3. In Ex. 2, when is the velocity of rotation around OP zero ? Ans. "When OP is an axis of the section of the momental ellipsoid which is perpendicular to the initial motion of P. 4. In Ex. 2, if the magnitude of Vhe given as before, find its direction so that the initial vis viva of the body may be a maximum or a minimum. dT dT dT By Art. 263, 2T = «* — + w y — + w z — ; clwx dca v doc- but Hence dT o>x = 0, and — = (Ex. 2). dco z or h 2 • u - * 2 2 B ' C ' V 11= OCtit/~ — IWutoz = Wu = — c c h' where i?'and C are the moments of inertia of the body round the axes of the section of the ellipsoid of inertia made by the plane yz. The maximum or minimum value of Tis obtained, then, by making c equal to C or to B' ; i. e. the direction of V must be perpendicular to the central section of the ellipsoid having OP as an axis. 5. If a body be moving in any manner, the momentum axis, and the in- stantaneous axis of rotation through a given point of the body, are the radius vector and the perpendicular on the corresponding tangent plane of the ellipsoid of gyration (see Integral Calculus, Art. 216) relative to 0. This is the reciprocal of the theorem given in Art. 264. It can be easily proved directly : radius vector to the point of contact, « = T L . mp~ This is immediately deducible from the consideration that mp^io = !& = mo- ment of momentum round instantaneous axis of rotation = II cos 2 ', w 3 ', wi, w 2 , w 3 the angular velocities of the body round these axes, before and after the action of the impulses, we have A (o), - a)/) = X, B ((o 2 - W2 r ) = JT, C (a> 3 - a)/) = N, (13) General Expression for the Vis Viva of a Body. 351 where A, B, C are the principal moments of inertia. Without having recourse to Art. 209, we may deduce equa- tions (13) directly from (18), Art. 204, and (20), Art. 205, by the method of Ex. 2, Art. 213. From equations (12) and (13) it appears that an impulse whose direction passes through the centre of inertia of a free rigid body produces a motion of translation only, whereas an impulse not passing through the centre of inertia pro- duces both a translation and a rotation. 266. General Expression for the Vis Viva of a Body. — As the motion of a body relative to one of its points must always consist of a rotation round some axis through the point, it follows, from Art. 134, that if a body be free, where %)l is the mass of the body ; Vth.e velocity of its centre of inertia; I the moment of inertia, and w the angular velocity, round the instantaneous axis through the centre of inertia. As was shown in Art. 263, I(o 2 = Aw 2 + Bu 2 2 + Cu 2 . Again, if a, b, c, i,j\ k be the moments and products of inertia for the centre of inertia, round three rectangular axes, which are parallel to fixed directions in space, and w x , o) y , w s the corresponding angular velocities of the body, I(jj 2 = awx + bu) y 2 + cwz - 2i(i)y(v z - %J(Dz(*>x - 2kw x (t)y ; whence we have 2mv 2 = WIV 2 + Am? + JW + CW (14) - %fl V 2 + aw x 2 + bw y 2 + cu) z - 2h y w~ - 2j(u z w x - 2kw x w y . (15) 352 Kinetics of a Rigid Body. Examples. 1 . A free body is set in motion by an impulse. If the initial motion be a pure rotation, show that the directions of the impulse and of the instantaneous axis of rotation are principal axes of a section of the momental ellipsoid relative to the centre of inertia. Since the initial motion is a pure rotation, the initial velocity of the centre of inertia is at right angles to the direction of the instantaneous axis of rotation. The above statement follows, then, from Art. 264. 2. On the same hypothesis as in the last example, show that the instan- taneous axis of rotation is a principal axis of the body, at the point in which it is met by its shortest distance from the line of direction of the impulse (see Ex. 1, Art. 241). 3. If different impulses applied to the same body produce velocities of ro- tation round parallel instantaneous axes, prove that in general these axes lie in one plane containing the centre of inertia, and perpendicular to the lines of direction of the impulses, and that the points in which this plane meets these lines lie on a straight line. 4. If in the preceding example the instantaneous axes are parallel to a principal axis through the centre of inertia, prove that the lines of direction of the impulses lie in the corresponding principal plane at the centre of inertia. The theory of the centre of percussion, given in Art. 235, may be collected from Examples 2, 3, 4. 5. A body is moving freely : under what circumstances can it be brought to rest by an impulse, and what must be the magnitude and position of the impulse ? The direction of the impulse must be opposite to that of the velocity of the centre of inertia, and its magnitude must be; equal to the momentum of trans- lation of the body. Again, the moment of the impulse round the centre of inertia must be equal and opposite to the couple of principal moments. Hence the magnitude and position of the impulse are determined, and the motion of the body must be such that the momentum axis is perpendicular to the direction of motion of the centre of inertia. 6. A free body is set in motion by an impulse of given magnitude, and pass- ing through a given point P of the body ; find the directions of the impulse for which the initial vis viva of the body is a minimum, and for which it is a maxi- mum. Since the impulse is given so is the velocity V of the centre of inertia ; but the total vis viva 1T= m V 2 + S, where S is the vis viva of the motion relative to the centre of inertia ; hence T is a minimum when S is zero, i. e. when the direction of the impulse passes through the centre of inertia. Again, the direction of the impulse for which S is a maximum is found as in Ex. 9, Art. 264, and when S is a maximum so likewise is T. 7. A free body is set in motion by an impulse whose magnitude and perpen- dicular distance from the centre of inertia of the body are given ; find the direc- tion of the impulse so that the initial vis viva of the body may be a maximum. Here S must be a maximum, and therefore, as in Ex. 1, Art. 264, the impulse must lie in a plane passing through the centre of inertia and perpen- dicular to the axis of least inertia of the body. Equations of Motion of a Body having a Fixed Point. ' 353 267. Equations of Motion of a Body [having a Fixed Point. — In the case of continuous forces, if G x , G yy G z be the moments of the applied forces round the space axes, the equations of motion are (25), Art. 210, dR x _ dm dt -**- It '-a. ~-g z . 'y> dt (16) We may substitute for H x , H y . and H z in these equations their values given by (1), or by (4). If we make the former substitution we obtain dt \Ctu> x ~ &*> ■J«>* G x ~) COj J t [~J<»X- t<»y + bu>y — i(o x ) = G v y + c<» z ) = G z ) (17) In the case of homogeneous spheres, as also in that of the initial motion of a body starting from rest, these equations are sometimes useful ; but since in general a, k, J, &c. vary with the time, it is usually necessary to reduce equations (17) to a more manageable form. If we substitute for H x , H yi) H z in (16) their values given by (4) and, after performing the differentiations, suppose the space axes to coincide with the instantaneous positions of the principal axes of the body at 0, we have by (6), Art. 255, and (9), Art. 256, remembering that in this case a 2 , a ly b ly £>,, c l9 c 2 , are each zero, and that a x = b% ■ c 3 = 1, dH x dw x n . dH y ^dujz in A . -df =B Tt- {0 - A ^^> dH z dt L ~di {A. — B) u)i a>2 ; whence, if L, M, N be the moments of the applied forces 2 A 354 Kinetics of a Rigid Body. round the principal axes at 0, the equations of motion be- come A C ^-{B-C)w 2 w 3 = L dco 2 B lli (C- A)w zMl = m y C~-{A-B)u> l w 2 = N etc (18) Equations (18) are due to Euler, and are called by his name. 268. Equations of Motion of a Free Body. — If we denote the mass of the body by 2D?, we have »?-«• ^S= sF ' »£-** (19) where x, y, z are the coordinates of the centre of inertia re- ferred to any three rectangular axes fixed in space ; and "2X, S F, SZ are the sums of the components of the applied forces parallel to these axes. Again, since the motion of the body relative to its centre of inertia is the same as if that point were fixed in space (Art. 209), we have A Cl ~ - [B - C) tt 2 w z = L dd)2 ~di d(i) 2 ~di {C-A)w z w l =M y, -{A-B)muz = iV (20) where o»i, w 2 , w 3 are the angular velocities ; A, B, C the moments of inertia; and L, 3£, iV the moments of the applied forces round the three principal axes of the body at the centre of inertia. Examples. 355 Instead of equations (20) we may use (17), the axes being parallels through the centre of inertia to directions fixed in space. As in the case of impulses, equations (20) may be deduced directly from Art. 204 by the method of Ex. 2, Art. 213. From equations (19) and (20), it appears that a force whose direction passes through the centre of inertia of a free body produces a motion of translation only, whereas a force not passing through the centre of inertia produces both a translation and a rotation. Examples. 1. A body is given a rotation round a principal axis through its centre of inertia, and is acted on by a couple having this line for its axis. Show that the body will continue to revolve round the axis of initial rotation. 2. One end of a uniform rod rests on a horizontal plane and against a vertical wall ; the other rests against a parallel vertical wall. All the surfaces being smooth, if the rod slips'down, determine the motion. Take the intersection of the horizontal and vertical planes passing through the first end of the rod for axis of x, and a vertical plane, at right angles to the -walls and passing through the initial position of the centre of inertia of the rod, for the plane of yz, the axis of z being vertical. Let fi be the angle which the rod at any time makes with the axis of y, 2a its length, 2b the distance between the walls, x\, yi, z\ ; x 2 , y 2 , z 2 ; and x, y, z the coordinates of the two extremities, and of the centre of inertia of the rod. Then y\ = 0, Z\ = 0, y 2 = 2b, y m § {y l + y 2 ) = b : also y = a cos fi, whence cos $ = - ; thus, as fi is constant, the motion of the rod relative to its centre of inertia is a rotation round the axis of y, whose ampli- tude at any time may be denoted by *+P) = 2mg{z -z). 70 tf 2 sin 2 j8 _ . „ Now k 2 = — , z = a sin j8 cos cp, 3 whence, as the initial value of

) ). vV - ft5 Also, .T2 = a sin £ sin cp, which determines the position of the upper end of the rod when (p is known. 2 A 2 356 Kinetics of a Rigid Body. 3. A heavy body is supported in equilibrium by two strings : one is cut ; find the initial tension of the other. The two strings and the centre of inertia G of the body lie at first in the same vertical plane ; let this plane be that of yz, the axis of z being vertical, and its positive direction downwards, and let the origin be the point to which the uncut string is attached. (See figure, p. 293.) Let I be the length of the string OA, and h the distance AG, the direction cosines of OA being a, /3, y, those of AG, A, /j., v ; then, if x, y, z be the co- ordinates of G, we have y = 10 + hp, z = ly + hv ; and, if T be the tension of the string, and m the mass of the body, the equations of motion of G are mx = — To, my = — Tfi, m'z = my - Ty. Differentiating the expressions for y and z twice, substituting, and remembering that the initial values of the differential coefficients, with respect to the time, of o, &, y, A, n, v are each zero, we get d 2 y 7 d 2 v Ty Multiplying the first of these equations by #, the second by y, and addiDg, we have initially I m »{•(?)♦* (3?) since o 2 4 & 2 + y 2 = 1, and initially o = 0, (3MS0- and therefore j8 Now by (6), Art. 255, since A is zero initially, we have d 2 /x dw x d 2 v dw x dP = ~ V ~dt y ~di 2 = ,M ~d7' Hence initially h ($v — y/i) — = yy. dt in If p be the length of the perpendicular from G on the initial position of the string, this equation may be written T po>z = yy- m Motion of a Body under No Forces. 357 Again, if a, b, c, i,j, k be the moments and products of inertia round axes through G parallel to the coordinate axes, we have initially (az) = 0, j- {-ji»x - iwy + CWz) = 0. In differentiating, since the initial values of o} x , z — kuy — jd z = — Tp, — kcbjc + bdly — i(i z = 0, — j(b x — ieoy + cci> s = ; whence, if A be the determinant a - k -j | k b -i j -i c we obtain Au x = -(be - i 2 ) Tp. Substituting for u x , we have finally for To the initial value o T, A70 ABCyo A + mp 2, (be — i 2 ) mg- ABC + mp 2 (be - v mg. 269. Motion of a Body round a Fixed Point, under the Action of no External Force. — In this case equations (18) become in which we shall suppose A > B > C. Multiply the first by o>i, the second by w 2 , the third by w 8 , add, and integrate, and we have Cl»i + Bus + CW = 8. (21) 358 Kinetics of a Rigid Body. Next multiply the first by Aioi, the second by Boj 2 , the third by Cw z , add, and integrate, and we have A 2 w x 2 + B 2 iv, 2 + C 2 w 3 2 = H 2 . (22) In equations (21) and (22) S and H are constants. These equations could have been obtained directly from (3), Art. 261, and (7), Art. 263, by articles 200 and 213. Again, if we multiply the first of the equations obtained from (18) by ^-, the second by -A the third by — s , and add, A -t> O we get don doj 2 dm (B-C - A A - B\ » l W +t * i ~df + " 3 W = \aT + ~b~ + —^-)«^ 2 «» doj (A-B)(B- C)(C-A) or co- = -^ «i<*<*. If we combine the two equations already found with the equation wi 2 + o>2 2 + is) 2 = or, and solve for cm 2 , we get 1, 1, 1 A, B, G A 2 B 2 C' ' ''.'-, BO ( , S(B+Q)-E> ) whenoe "" = (A-B)(A-C) h BO 1 • (23) Of / 7? i /^ _ 77 2 If we denote — ^~ by Ai, and the two correspon- ding quantities by A 2 and A 3 , we have ABO w i 2 =oj 2 BC{C-B)-S{C 2 -B 2 ) + H 2 (C-B); My lx>2 OJz (A-B){A-C)[B-C) v /{(^-A 1 )(A 2 -co 2 )(o> 2 -A 3 )); * w = v ! (Ai " w2) (Az " * 2) (As " " 2) ' ■ (24) Motion of a Body under No Forces. 359 Again, since AS -H 2 = B{A- B)u? + C{A- C)u> z \ it follows that AS is always greater than IP ; in like manner we see that CS is less than H 2 . Hence we see at once that Ai, A 2 , and A 3 are each positive quantities. Also, we have A 3 — Ai A-B [H 2 -CS)> A a -A s = B-C (AS-E 2 ); ABC X ~ ~~" "' "° ABC therefore A 2 is the greatest of the three ; also Ai - A 3 has the same sign as BS - H 2 , and this depends on the initial con- ditions. Again, since on 2 , o> 2 2 , Ai, u> 2 < A 2 , o> 2 > A 3 . Hence we may assume either — (1), w 3 = A 1 sin 2 <£ + A 2 cos 2 <£, or (2) a> 2 = A 3 sin 2 ^ + A 2 cos 2 ^. In the former case, if we select the negative sign of the square root in (24), that equation gives d dt where = \/A 2 -A 3 - (A 3 - Ai) sin 2 = ^/A 2 -A 3 v^l-^sin 2 ^, (25) 72 A 2 -A t to = r- Hence, by (23), we get BO (A-B){A-C) AC A 2 - (A 2 - Ai)cos 2 (A-B)(B-0) AB (A-0)(B-0) In the second case, we get V-(A 2 -Ai)sin s ^ «3 (X„-X s )(l-A 8 sin» f = yx 2 - X, Jl - J sin'*, >• (26) I I J (27) 360 and Kinetics of a Rigid Body. (X) V tjj% = (-1 £p^o) (Aj - Xl)(1_ J sin ^] AG 0)j (A-B)(B-C) AB {A-C)[B-C) (X 2 - A 3 ) sin 2 ^ (A 2 - A 3 ) cos 2 ^ (28) It is obvious that and if/ are connected by the equation sin \\f = k sin (p. We thus see that when either $ or if/ is known the values of wi, o> 2 , a> 3 can be determined. Also, from (25) and (27), we see that

d 2/ v/9J?# W 2 , S yms U>3« (29) Hence, in terms of 0, we have JUB0 (.1 -if) (-4-0) A 3 , i.e. BS > H\ then it is easily seen that the inner circle is the projection by lines parallel to the axis of z. Hence, if M and M' be the positions of the projections of P at any instant, we shall have MOQ = = y\2-\3t+o = ^^- ^ +

2, W3 at any time are given by the equations u\ = u sin 1 cos (A- C \ . . . (A-G _ \ f — a>3 1 + x J 1 «2 = - » sm t sin I — — mi + XJ . (A+ C) S- 3°- o>3 = o) cos », a» 2 = - , where x is an arbitrary constant. AG 270. Conjugate Ellipsoid and Conjugate line- When a body on which no external force is acting is in motion round a fixed point, the squares of the angular 364 Kinetics of a Rigid Body. velocities of the body round its principal axes at the point must fulfil the two independent linear equations An? + B(o 2 2 + CV - S = = | Any other linear equation, 0' = 0, between these variables must be of the form aQ + (3

' = y (^0 - = -g,, r = /-£,- w, p =—-g, — The perpendicular to the tangent plane to i?'at the extremity 366 Kinetics of a Rigid Body. of r corresponds to the momentum axis in the momental ellipsoid, and is called the conjugate line. This Article and the following Examples are taken from a Paper by Dr. Eouth in the Quarterly Journal of Pure and Applied Mathematics for 1888. Examples. 1. If a body on which no external force is acting be moving round a fixed point 0, and a quadric, having as axes the principal axes of the body at 0, be such that the intercept which it makes on the instantaneous axis of rotation at any time is proportional to the angular velocity, and that the perpendicular from on the tangent plane at the extremity of this intercept is constant, the quadric must be either the momental or the conjugate ellipsoid. 2. If P be ? point on the conjugate line at a constant distance R from the fixed point 0, and Q the point of the body which coincides at the instant with P, prove that the velocity of P is double that of Q, and that the directions of these two velocities coincide. Let x, y, z be the coordinates of Preferred to the principal axes at 0; u, v, to its space' velocities parallel to these axes ; and u', v , w' those of Q ; then ll' = (02 Z - 0)3]/, v' — <*z% — WlZ, W' = (till/ — (02%, a = x+ ?/, v = y + v', tv = z + to'. Now, H'x = RA'wi, R'y = P£'w2, H'z = PC'w 2 ; hence, by (41), we have x = ^ r ,A(B + C-A)d, u iR and W = — (B - C) (B + C- A) co 2 o> 3 ; whence, by Euler's equations, Art. 267, we obtain x = u', and therefore u = 2u ; and in like manner v = 2v' } w = 2iv'. 3. Determine the motion of the conjugate line in space. Let 9 be the angle between the conjugate line OP and the invariable line or momentum axis OZ, \|/ the angle which the plane ZOP makes with a fixed plane passing through OZ, (p and sin «£', we have, by Ex. 2, \ (sin 2 6ip + 6°-) = co 3 sin 2 3 2 ). Hence we get H H ' A* «i 2 + 2? 3 a> 2 2 + C 3 m 2 =\E — cos 9. If we combine this equation with (33), and solve for 2 2 and a> 3 2 we get TTTT' ABCar = S{AB + BC+ CA) -±E Z (A + B+ C) — cos 6. Substituting the value for or given by this equation in (b), we obtain sin 2 0i^+ 6*=-^--{S(AB + BC+CA)-±E 2 (A + B+C)} From (a) and this equation and \p can be obtained by quadratures. 271. Stress Exerted by a Body on a Fixed Point. — In order to determine the force exerted by a fixed point on a body we have only to consider the point as replaced by a force, whose components are X , T , Z , passing through it. We may then consider the body as free, and we have, by Article 268, dr with two similar equations. But as the body is rotating round the origin, if we sup- pose the axes fixed in space to coincide at the instant under 368 Kinetics of a Rigid Body. consideration with the principal axes through the origin, we have d 2 x _ dioz _ d(jj z ._ k , 2 2 v . = - y — + z -jj- + wi (yah + S(u 3 ) - (wz + ivs ) x. df dt dt Substituting for ~ and -£r f rom Euler's Equations, we dt dt n _M + Wl (5 + C-A) U~ + S ^)" K + "a 2 ) x get, <# 2 = Now, let ft, ft, ft be the components of the stress on the fixed point at any time, in the directions occupied at the instant by the principal axes of the body, then ft = - X , and therefore ^ = 2r-3Ti[-?j+i^+co 2 (C+^- J B)^ + l9-)-(c 3 2 +^)^ >,(42> ^ = 2^-^[-|f+^+ W3 ^ + ^-^)(^ + ^ 2 )" (&,l2 + &,22) ^l where £, »?, X are the coordinates of the centre of inertia referred to the principal axes through the fixed point, and are absolute constants : 2X is the sum of the components of the applied forces parallel to one of these axes, and L the moment round it of the same forces. 2X, S F, SZ, L, M, N are in general variable with the time. In like manner if ft, ft, ft be the impulses arising from the instantaneous stresses "exerted by a body on a fixed point, in consequence of the action on the body of any system of impulses, we obtain, by Arts. 255 and 265, Centrifugal Couple, 369 272. Centrifugal Couple. — If a body have a fixed point 0, the change produced in its angular velocity round one of its principal axes at in the element of time dt is given, (18), Art. 267, by the equation Adu>i = (B - C) (Diaz dt + Ldt. The first term on the right-hand side of this equation results from the angular velocities already existing round the other two axes. In consequence of these velocities each point of the body, in virtue of its connexions with the other points, exerts a force on the entire body. These forces are in fact the centrifugal forces resulting from the motion of the body, and their moments L\ M\ N' round axes fixed in space may be determined directly as follows : — Let a, j3, y be the angles which the instantaneous axis of rotation makes with the axes of coordinates ; p the perpen- dicular distance from this axis to any point xy% of the body ; q the intercept between the origin and the foot of p ; r the radius vector to the point xyz ; and w the angular velocity of the body round the instantaneous axis. The centrifugal force at the point xyz is mpu> 2 acting along p ; and the component of this force along the axis of x is mco 2 multiplied by the projection of p. If we project the triangle formed by rpq on the axis of x f we have projection of p = projection of r - projection of q = x - q cos a, and q = x cos a + y cos f5 + z cos y ; hence the centrifugal force along axis of x = mco 2 [x - (x cos a + y cos /3 + z cos y) cos a) = mia* {x (cos 2 3 + cos 2 y) - y cos a cos/3 -z cos a cos y J = m{X ((Dy* + (D Z 2 ) - yW X il)y - Z(D X (1) Z ), remembering that w x = <»> COS a, (i)y = (i) COS ]3, w z = u) COS y. 2 B 370 Kinetics of a Rigid Body. In like manner for the force along the axis of y, we have m \y (u) z 2 + tax) ~ z <*> y w z — xiiiytox} j and for that along the axis of s, in \z w x 2 + w y ~) - xw z iv x - yiD Z (jjy) ; whence, taking moments round the axis of x, and integrating through the entire body, we obtain L' = [wy - id z 2 ) jyzdm + m v w z J (z* - y 1 ) dm -w z w z j xydm + d) x w y jxzdm. (44) If we now suppose the axes to coincide with the instan- taneous positions of the principal axes of the body, every term in 11 vanishes except w y w z j (z 2 - y % ) dm, and we get L f = {B-C) 3> &c., is called the centrifugal couple. The axis of the centrifugal couple is at right angles to the axis of principal moments, and to the axis of rotation. For the direction cosines of the axis of the centrifugal couple are proportional to (B - C) Mi w 3 > [C - A) wo w„ [A - B) h)i wo ; whence it is seen at once that the conditions for its being perpendicular to the two other lines are fulfilled. If a central section of the momenta! ellipsoid be taken passing through the instantaneous axis of rotation and the axis of the centrifugal couple^ these two lines coincide with the principal axes of this section. The lines in question are at right angles, and one is parallel to the tangent plane through the point where the other intersects the ellipsoid. Examples. 371 273. Hotion of a Free Body relative to its Centre of Inertia. — As the equations for determining the motion of a body relative to its centre of inertia are the same as if the centre of inertia were a fixed point, the theorems of Arts. 264 and 272, in reference to the instantaneous axis of rotation, the axis of the centrifugal couple, and the axis of principal moments, hold good. Examples. Motion of a Body unacted on by Force. 1. The angular velocity at any instant is proportional to the intercept on the instantaneous axis of rotation through the centre of inertia cut off by the momental ellipsoid. The velocity of the centre of inertia is constant as well as the whole vis viva. Hence the vis viva of the motion relative to the centre of inertia is constant, and therefore, (10), Art. 264, w is proportional to r. 2. The component of the angular velocity round the momentum axis through the centre of inertia is constant. See (9), Art. 264. 3. If a tangent plane be drawn to the momental ellipsoid at its point of intersection with the instantaneous axis of rotation through the centre of inertia, the distance of this plane from the centre is constant. This follows from (LI), Art. 264. If a body have a fixed point, the results of the preceding examples hold good, the fixed point being substituted for the centre of inertia. 4. A body moves round a fixed point : give a geometrical representation of the motion. The momental ellipsoid relative to the point rolls on a plane fixed in space, so that the line joining the centre to the point of contact is always the instan- taneous axis of rotation. 5. A body is moving round a fixed point ; find the locus of the instantaneous axis of rotation in the body. Since — , — , — are its direction cosines referred to the principal axes through CO CO CO the point, its locus is the cone A{W - AS) x* + B (W - BS) y* + C(H* - CS) & = 0. 6. Find the locus of the momentum axis in the body. Its locus is the cone r 2 v 2 z 2 g A B J Hence the curve traced out by this line on the ellipsoid of gyration is a sphero-conic, as already stated in Art. 269. 2B 2 372 Kinetics of a Rigid Body. 7. Determine the curve traced out on the momental ellipsoid by the instan- taneous axis. The equations of the curve are got by combining the equations ot tne ellip- soid with that of the cone given in Ex. 5 ; they are, therefore, Ax> + By 2 + Cz 2 = K, AW + Bhf + C V = — . This curve is called the polhode. The curve traced out on the fixed plane, by the point of contact, is called the herpolhode. 8. The projections of the polhode on the planes perpendicular to the axes of greatest and least moment of inertia are ellipses. Its projection on the plane perpendicular to the remaining principal axis is a hyperbola. This appears at once by eliminating x, y, z successively from the two equa- tions of Ex. 7, remembering that A>B >C. 9. In what case does the hyperbola become a pair of straight lines ? HH~=BS. (See Ex. 1, Art. 269.) 10. If the body be free, give a geometrical representation of the motion. (See The momental ellipsoid relative to the centre of inertia rolls on a plane at a constant distance from the centre of inertia and parallel to a plane fixed in space, the instantaneous axis of rotation being the line joining the centre of inertiato the point of contact, whilst the whole system moves with uniform velocity parallel to a fixed direction. 11. Show that the herpolhode lies between two circles the squares of whose radii are Kf S*\ 3 K( S 2 \ Kl S-\ -^(A 2 --), and _^Ai--j,or j^-^J according as (see Art. 269) Xi is greater or less than A 3 - If p be the distance from the point of contact to the foot of the perpendicular on the fixed plane, we have 7T9 TT K I S 2 \ ? = r^p>\ but^ = ^,andr2 = -a>MArt.264); /.^- -^ a>* - -,). But, Art. 269, co 2 >M, » 2 2 >A 3 . . Hence the greatest and least values of p 2 are comprised between the limits- stated above. 12. If a body be rotating round a fixed point, or a free body round its centre of inertia, the couple resulting from centrifugal forces lies in the plane containing the momentum axis and the instantaneous axis of rotation, and its magnitude is Eu> sin cp, or S tan 2 &>3, (0— A) 0>3 COl, (j4 — B) £01 C02, or w« 2 (b 2 — c-) cos /3 cos 7, wco 2 (c 2 — a 2 ) cos 7 cos a, mca 2 (a 2 — b 2 ) cos a cos /3 ; where a, )8, 7 are the angles made by the instantaneous axis of rotation with the principal axes of the body, and a, b, e are the semi- axes of the ellipsoid of gyration. Jf ;; be the perpendicular from the origin on the tangent plane to the ellipsoid of gyration at the point x'y'z' where it is met by the momentum axis B, double the projection of the triangle formed by the origin, x'y'z, and the foot of jo, is p (x' cos £ - y' cos a) , or (a 2 — b 2 ) cos a cos (3, and double the area of the same triangle is Bp sin c/> ; therefore by Ex. 5, 7, 8, Art. 264, we have the required result. 13. If a tangent plane be drawn to the ellipsoid of gyration at the point where it is met by the axis of the centrifugal couple, the perpendicular on this tangent plane is the axis of the rotation produced by the centrifugal couple. L', M'y N' being the components of the centrifugal couple, and 5coi, 5«2, S0.3, the rotations produced by it considered alone, we have, from Euler's equa- tions, A5m = L'dt, B8w2 = M'dt, C5« 3 = K'dt ; but these equations are of the same form as those connecting the instantaneous axis with the components of the couple of principal moments ; therefore, &c. It follows from this, that the axis of rotation produced by the centrifugal couple is at right angles to the momentum axis ; for {see Fig., Ex. 16) if OB be the momentum axis ; OP the instantaneous axis of rotation ; OB' the axis of the centrifugal couple, and OB' the axis of the centrifugal couple rotation ; OR' being at right angles to OB (Ex. 12), is conjugate to OB : hence OB is parallel to the tangent plane through B' , and therefore at right angles to OB' . Also, OB and OBJ are the principal axes of the section of the ellipsoid made by their plane. 14. The intercept on the momentum axis cut off by the ellipsoid of gyration is of constant length (Ex. 8, Art. 264). 15. The motion of the momentum axis in the body consists of a series of rotations, the axis of each rotation being at right angles both to the momentum axis and the centrifugal couple axis, and the magnitude of the rotation being equal and opposite to the rotation of the body round the same axis. The centrifugal couple tends at each instant to alter the position of the momentum axis, since the new moment of momentum is the resultant of the principal couple at the beginning of the instant and the momentum produced by the centrifugal couple during the instant. The former component is H, the latter Hoc sin dt go sin PI rp dv sin P'l sin RP sin PI sin IR sin IR ' sin PI dv sin

' but whence, finally, r cos (p' } and p = R cos <\> ; r 2 dv = ooR? cos

' S tan

cos

for — , P- for — , and mar for A, mP for B. mc- for C, R mS the equation given above is reduced to — = co cos , p = _ (Art. 2o4), and — = — ^1 +/* ^ J where M - ^^ (AS_-IP)( BS-B*){CS-IP ) (Ex 19) Impact. 379 If we express co in terms of p, on substituting in equation (24), Art. 269, we get i-s-^Ji[-4^^][-^^] A 3 7,(P 2 + P") ]l- 274. Impact, — When two smooth bodies moving in any way collide, the results of the impact are obtained in a manner precisely similar to that employed in Article 243. When the motion is wholly unrestricted there are thirteen unknown quantities and thirteen equations. If A, ju, v be the angles made by the common normal at the point of contact with axes fixed in space ; R the whole impulse of the mutual normal action during the first period of impact ; p and p the perpendiculars on its line of action from the centres of inertia of the two bodies ; a, /3, y, a, /3', y the angles made with the principal axes of the bodies by the axes of the couples produced by R round these points ; twelve of the equations mentioned above are SDct* = R cos A, 9Jte = R cos p., yjliv = R cos V, Wu ' = - R cos A, Wv =-Rcosp, Wl'tv' = - R cos v, Azji = Rp cos a Btu 2 = Rp cos j3 C-& z = Rp cos 7 A'zs{ = - Rp cos a J BV = -ify , cos/3 / O'W = - ify>' COS y' >, (46) where ««, &c, are the changes of the components of the velo- city of the centre of inertia of the first body, parallel to axes fixed in space, produced during the first period cf impact ; _ zs x + &c, the changes of the angular velocities round the principal axes through the centre of inertia produced during the same period ; and u, &c, have similar significations for the second hody. At the end of the first period the actual components of the velocity of the centre of inertia of the first body are u + k , &c., where 4 represents the component of this 380 Kinetics of a Rigid Body. velocity immediately before the impact. In like manner, ts x + Q, x is the actual angular velocity round the first principal axis. "We can then write down, in terms of t# + x 0i - ts l + £2 b &c, the relative normal velocity of the points of the two bodies which are in contact. Equating this relative normal velocity to zero gives a thirteenth equation ; so that te , zs i9 &c, become completely known. If x be the component of the final velocity of the centre of inertia of the first body at the end of the second period of impact, and wi the final angular velocity round the first principal axis, &c, the values of the velocities at the end of the impact can now be determined, by aid of the following equations — x - x = (1 + e) t«, on - Qi = (1 + e) zj u &c, ) . (47) x' - x '= (1 + e) u, u){- Q,i- (1 + e) ot/, &c. ) Since the positions of the two bodies are not sensibly altered during the whole period of impact, it is to be ob- served that throughout this period any lines fixed in either body coincide with lines fixed in space. 275. Impulsive Friction. — When collision takes place between two rough surfaces we can investigate the motion according to the principles laid down in Article 247. The elementary impulse dF of friction, at each instant of the impact, is to be resolved into two components, dP and dQ, along two tangents through the point of contact at right angles to each other. At any instant during the impact, P represents the entire impulse in a given direction due to the action of friction up to that instant. A similar remark applies to Q, and R is the corresponding impulse due to the normal reaction. If at any instant during the impact u, v, w be the com- ponents, along the two tangents and the normal, of the relative tangential and normal velocities of the points of the two surfaces which are in contact, u, v, w can be expressed in terms of the velocities of the two centres of inertia and of the angular velocities of the bodies at that instant ; they are Collision of Rough Spheres. 381 therefore linear functions of P, Q, R. If slipping take place its direction coincides with that of the elementary impulse of friction, and therefore dQ = v ' als ° ^ ^ dF * + d ®) = fxdR ' Initially R is zero, and therefore so likewise are P and Q, except the colliding surfaces be perfectly rough. When R = R l9 at the end of the first period of impact, w = ; and if R 2 be the value of R at the end of the whole impact, P 2 = (1 + e) R,. If the surfaces which collide be perfectly rough., the equa- tions u = 0, v = 0, w = enable us to determine P l9 Q l9 R { . Knowing the value of R z we can find P 2 and Q 2 from the equations u = 0, v = 0, which hold good throughout the impact. If the bodies slip on each other in the same direction during the whole of the impact, the direction of clF is con- stant, and we may take clQ = 0, clP = /mdR. Hence P, =/uR u Qi = : these equations, with w = 0, determine R x ; then P 2 = fi (1 + e) Rl 276. Collision of Rough Spheres. — If a homogeneous sphere impinge against a fixed surface, or two homogeneous spheres collide with each other, by taking as axes of 2 == OJ-i = Q 3 , *Jn+—, , Wo, = ii 2 — -, (1)3= Q/, J > ( 48 > where 0, Wl, and Pare the radius, mass, and moment of inertia 382 Kinetics of a Rigid Body. round its diameter, of the first sphere : x, &c, the compo- nents of the velocity of its centre, wi, &c, the components of its angular velocity : x , &c, and Oi, &c, the values of these components immediately before the impact : and a', 2D?', &c, have similar significations for the second sphere. At the same instant the velocities of the points of the spheres which are in contact are given by the equations TJOJ3 + Z(t) 2 , X* = X - ri'to-/ + ?Vs', &C., where £, ri, Z ', £', v', Z', are the coordinates of the point of contact relative to the centres of the two spheres ; and the relative velocities u, i\ to are then determined by the equa- tions u — x— X, &c. Hence, since £ = 0, r\ = 0, £ = a, £' = 0, r{ = 0, Z' = - a', substituting for x 9 to u &c., their values given by (48), we have , / (1 \ a 2 d 2 \„ u = x - x + aQ 2 + a £2 2 - 1 ^ + —-, + — + j,f, ' ' / „ ;/1 , /l 1 a 2 d 2 \ _ v = V* - Vo - aQ l - a Q l -( ^ + ^p + j + y) & that is m = Mo - /P, # = «?o - JQ, w = Wo - nR, where / and n are constants. Hence die = - IdP, dv = - IdQ, — = -jp. ; dv dQ but, if there be slipping, we have, Art. 275, — = - ; therefore — = -, and accordingly - is constant, dQ v dv v v or the direction of slipping is invariable throughout the impact. Moreover, if either u or v vanish so must the other. All. slipping then ceases, and cannot recommence, as u and v are independent of R, and friction cannot initiate slipping. Since, in the present case, Pi is independent of P and Q, if there be no slipping at the end of the impact the result is the same as if there had been no slipping at all. Ejcamjrfes. 383 Hence, in all cases, either the impulse of friction is a maximum, and the direction of slipping the same throughout the impact, or else the surfaces may be regarded as perfectly rough. If the problem be solved on the latter hypothesis, and the value resulting loi*/(P? + Q 2 2 ) does not exceed fi(l + e) B„ the solution is correct. If z = - aX, x=0, y = 0, 0. Hence =fPij y=0, 2=7^3, «wi=-fr 3 , a o>2=0, flw 3 = fFi At the end of the second, period of the collision five of the six velocities above remain unaltered, but y becomes — eVt. At the second impact the coordinates of the point of contact are S = «, T7 = 0, C = o. Hence, = y + ao>3, z = z — aw2. 384 Kinetics of a Rigid Body. Also, as has teen proved above, Proceeding as before, since y = 0, z = 0, we obtain awi = - f T s , «o> 2 = (y) 2 T 3 , a W3 = f e Fa + f . f Fi, J— *Pi, ?=- *r»-*.*Fi, MfpFs. 2. In the last example, if the walls be not perfectly rough, determine the final velocities of translation and rotation. We first treat the question as before, and obtain, as in the last example, at the first impact, 1 (X 2 +^ 2 )=-A-(Fi 2 + F 3 2 ); if then M (l +^)r 2 >f V(Fi 2 + F 3 2 ) we may assume there is no slipping ; but if this condition is not fulfilled slipping takes place, and the maximum amount of friction is exerted. In this case at the end of the first impact, x= V\— fi{l + e)Vt cos a, y = -eV^ z— F 3 — fi{l + e) F 2 sinff, «o>i = — |/t(l +e)Vz sin a, co 2 = 0, «w 3 = f /x(l +e) F 2 cos a, where tan a = — . 'i The values of x, &c, »i, &c, at the end of the first impact are the values of#o, &c, Xii, &c., at the second impact. If slipping takes place during the whole of the second impact, we have finally z = -e{Vi - fi{l + e)VzCOSa}, y- = _ gf 2 - ^(1 + e){ Vi - fi(l + e)V 2 cos a] cosj3, | = r 3 - /*(1 + e) { r 2 sin a + Vi sin /3 - p. (1 + *) V% cos a sin £}, «Wi = 0, «W2 = f/*(l + «){Fi-/t(l + ^)F 2 cos a} sinjS, 2 = f /*(1 + «){Fi-/i(l + «)F2 cos a} sin 0, «fi>3 = | /*(1 + e){F 2 cosa[l +/*(1 + e)cos/3]- Ficos/3}, 2{ r 3 -/*(l +^)F 2 sina} where tan /3 = 5/a(1 + i, and o> 2 . Hence dt dt X = — yu9% cos a, Y = - ^9% sin a. These are the components of a constant force in a fixed direction. Hence in general the centre of the sphere describes a parabola. If, however, the initial axis of rotation be perpendicular to the direction of the initial motion of the centre, i.e. if V\Cl\ + VzQz—O, the centre of the sphere continues to move in the direction of its initial motion. Substituting the values of X and Y in the equations of motion, we find that 2 ( Vi - ra-) slipping ceases along the axis of x when t = „ - , and along the axis of >/ ** ° 7 /iff cos a ' ° J , 2(V2+rQi) ..Vz+rQi Fi - ril 3 . when t = — : ; but — : = , hence, slipping along each axis 7 /x.ff sin a sin « cos a irr o o ceases at the same time, to, where t g V{(ri-m 2 ) 2 +(r 2 + rfli) a } to = t ' H-ff After pure rolling begins it will continue, since the values which X and Y must take in order to maintain it are zero ; the components of the velocity of the centre are then given by the equations dx _ 5 Vi + 2>-n 2 dy _ 5 Vi- 2rHi ¥ " 7 ' 2 Hence the path, is a straight line. Multiplying the third equation of motion by co i, the fourth by « 2 , and adding, we have dt whence n being the initial angular velocity ra = - | fjL t + ra, Examples. 389 7r 2 a The sphere will come to rest when t = — — . ¥2^wi — widw2 = 0. Hence — = constant =— tan a, where too a is the angle which the path makes with the axis of x. 3. A sphere is projected obliquely down a rough inclined plane, the motion being pure rolling ; determine the friction brought into play, and the path, neg- lecting the couple of rolling friction. Take as axis of x the intersection of the inclined plane with 'a vertical plane at right angles thereto. The equations of motion are dt- dt* dt at ... ..'. dx dy with the conditions — — ra>2 = 0, — 4- ra\ = ; dt at whence Y= 0, X = - f Tig sini. The whole force, therefore, is f Tig sin i parallel to the axis of x, and the centre of the sphere being acted on by a constant fotce parallel to a fixed direc- tion, describes a parabola. Also, since dx dy dt dt the instantaneous axis of rotation is at right angles to the tangent to the path of the point of contact on the inclined plane. This is otherwise immediately obvious, since the motion is pure rolling. 279. Equations of Motion of a Solid of Revolu- tion. — If one point of a rigid body be fixed in space, and two of the principal moments of inertia at the point be equal, the equations of motion of the body can be expressed in a comparatively simple form. Let 00 (Art. 258) be the axis of revolution of the momental ellipsoid of the body, and A and its principal moments of inertia at 0, then, by considering the motion of a point situated on OO, it is plain that the angular velocity of the body round an axis OS perpendicular to OO in the plane ZOO is \p sin 9, and the moment of momentum round OS is therefore A\p sin 9. Hence E z = Axp sin 2 9 + Cw 2 cos 9. (52) 390 Kinetics of a Rigid Body. Again, the angular velocity of the body round OE perpen- dicular to the plane ZOC is 9, and therefore we have 2T = A (^ 2 sin 2 9 + 9 2 ) + CW. (53) If G x , G yy G z be the moments round the space-axes of the applied forces, and Y the force function, we have then, as the three equations of motion of the body, — (Ail sin 2 9 + Cw z cos 0) = G z A («A 2 sin 2 9 + 9 2 ) + CV = 2Y + constant C —jj- = N = sin 9 (G x cos \j/ + G y sin «A) + G z cos 9 >■ (54) We may if we please substitute + ^ cos 9 for w 3 in (52), (53), and (54). Equations similar to (54) hold good for a free body if two of its principal moments of inertia at its centre of inertia be equal. In this case OZ is a parallel through the centre of inertia to a line fixed in space. Examples. 1 . A homogeneous solid of revolution terminating in a cone is placed with the vertex of the cone on a perfectly rough horizontal plane, the initial condi- tions heing given, find the equations of motion. Here the vertex of the cone is the fixed point ; and if a vertical line through he taken as the space-axis OZ, since gravity is the only force, G- and N are each zero. Then hy (54) we have u>% — constant = w, and therefore the first two of equations (54) become Aty sin 2 6 + On cos 6 = X, A (^ 2 sin 2 + 2 ) + On 2 = E - 2mgh cos 0, (a) where h is the distance of the centre of inertia from 0. If we take a point P on 00 the axis of revolution at a distance I from such that mhl = A, this point P is the centre of oscillation of the body for an axis perpendicular to ZOO. Assuming XI = Ona, and {E - On 2 ) I = Imghb, we have, then, to determine the motion of Pthe equations mhP\p sin 2 = On (a - I cos 6) J 2 (ipsin 2 + 2 )= 2g(b DS0) V. (b) I cos 0) ; Examples. 391 2. Give a geometrical construction for the velocity of P in any position. Take on OZ two points D and F such that OB = a, OF=b, and let x he the angle which FD makes with the plane of the horizon, then at any instant the velocity of F is that due to the depth of F helow the horizontal plane through F, and the component of this velocity perpendicular to the plane ZOO is On 3. Show that the motion of the axis of revolution may he represented by that of the conjugate line (Art. 270) in a body not acted on by any force. The equations of motion of 00 in Ex. 1 are of the same form as those of the conjugate line in Ex. 3, Art. 270, and by properly determining the dis- posable constants in the latter they may be made identical with the former. This theorem was first given by Jacobi, but the mode of investigation here adopted is due to Dr. Routh. 4. Determine the limits of the inclination of the axis of revolution to the vertical. Eliminating ^ from equations (b) of Ex. 1 we obtain (Cn\ 2 (a- lcos0\- \mh} \ IsmQ ) r-p = 2?(b-uo S e)-(^-) r-T^P) . (a) Now when attains its limiting value, = 0, and therefore to determine the limiting values of we have the equation 1A 2 g {b - I cos 0){1 - cos 2 0) - CPn 2 (a - 1 cos 0) 2 = F (cos 0) = 0. (b) From equation (a) it is plain that F cannot be negative for any value of attained in the actual motion of the body. Hence, if i be the initial value of 0, F is positive or zero when cos = cos i. Again, it is easy to see that F is negative for cos = — 1, or cos = 1, and positive for cos = oo . We con- clude that the equation F (cos 0) = has three real roots, two between - 1 and + 1, and one between + 1 and oo . This last root is an impossible value for cos 0. In general, then, the angle oscillates between two limiting values a\ and ci2, one less and the other greater than the initial value i. That this oscillation should be possible, it is necessary, however, that F should vanish before any point of the body above the point of support comes into contact with the horizontal plane. If & be the value of for the position in which such contact takes place, in order that an oscillation should be possible, i^cosjS) must he negative, and therefore C 2 n 2 (a - I cos £) 2 > 1A 2 g (b - I cos 0) sin 2 £. (c) In terms of the original constants, iT and F, this condition becomes {A sin 2 £ + C cos 2 0) On 2 - 2KCn cosP>A(F- 2mgh cos ff) sin 2 £ - K 2 . (d) 5. Show that the character of the oscillating motion depends on the relative magnitudes of a and b. 392 Kinetics of a Rigid Body. If in equation (a) Ex. 4, we make I cosO - a, we get l 2 2 = 2g(b- a). If a be less than b this gives the value of when ty = ((b) Ex. 1.) In this case the angular motion of the plane ZOO changes its direction at the point corresponding to a = I cos 0. Again, if a > b, the relation a = I cos G leads to an imaginary value for 0, and consequently ^ cannot vanish during the motion. Hence in this case the axis 00 rotates constantly in one direction round the vertical line OZ. 6. Determine in any particular case of the motion whether a or b is the greater. If a > b, then, by Ex. 1,: — > — . ' : ,J Cn 2mgh 7. If the axis of revolution rotate constantly in one direction round the vertical, and if ^ be the value of \p which corresponds to either the greatest or . , - , • 2mgh least value of 0, prove that i|/ < — — . 8. Find the conditions which must be fulfilled in order that the motion of OC should be steady. In this case, if it can occur, the inclination of 00 to the vertical and the angular velocity of the plane ZOO are constant. If we eliminate between the two equations obtained by differentiation from (a), Ex. 1, we get A0 = Aip 2 sin cos — Cnty sin 9 + mgh sin 0. (a) Hence if = 0, we have sin0 (Afr cos - C>4 + mgh) =0. (b) If sin = 0, we have = 0. In this case the axis is vertical throughout the motion. Again, if we obtain A cos 0ty 2 — Cn^ + mgh = 0, On ± V(C 2 « 2 - 4Amgh cosfl) 2 A cos If then i the initial value of fulfil the condition O 2 n 2 > 4Amgh cos i, and if likewise initially = 0, and On + \J(C 2 n 2 — 4:Amgh cos i) * = ^ 2A7oTi ' {C) all the successive differential coefficients of and ^ must vanish initially, as readily appears from_ the expression for and the first of equations (a), Ex. 1, and therefore and ^ remain constant, and the motion is steady. 9. Prove that if the motion be not steady initially it cannot become so sub- sequently. In order that the motion should become steady it would be necessary that and should vanish simultaneously. is given in terms of by equation («), Ex. 4, which is of the form k0 2 sin 2 = i^cos 6), where h is constant. If we Examples. 393 differentiate this equation, divide both sides by sin G, and then make and each zero, we obtain F' (cos 0) = 0. Hence if and vanish together, the equa- tion i^(cos 0) = must have equal roots. Now (Ex. 4), _F(cos 0) = 2A-gl (cos - cos a\) (cos - cos a 2 ) (cos - A), where A is always greater than 1. Hence if the equation .F(cos 0) = have equal roots, we must have ai = a 2 , and as in the actual motion always lies between ai and a-, the double root must be cos i, where i is the initial value of 0. Consequently, if the motion be not steady originally it can never become so. 10. A peg-top is set spinning on a rough plane, determine the motion. In this case the only initial motion is a rotation round OC, and therefore, if i be the initial value of 0, we have K= Cn cos i, E - Cn 2 = 2mgh cos i. Hence a = b = I cos i, and equations (b) Ex. 1, become mhty sin 2 = Cn (cos • - cos 0), l{& sin 2 + 2 ) = 2g (cos % - cos 9) . {a) The latter equation shows that cos i > cos 0, and therefore that ty has the same sign as ». Hence the rotation of the top round its axis is in the_ same direction as the rotation of the latter round the vertical through the point of support. Again, if we put C 2 n- = IvmghA = ±vm- gh* I, equation {b), Ex. 4, becomes (cosi - cos 0) {1 - 2»/cosi + 2v cos - cos 2 0} = 0. {b) Hence = i or i', where i' is determined by the equation sin 2 i' = 2v (cos i - ops »'), and oscillates between its least value i and its greatest value i', provided V < &, that is 2v > ^-^ . It is plain that the latter equation is what (c) Ex. 4 cos i — cos j8 becomes in the present case. 1 1 . Show that steady motion is impossible in the case of a top, except the initial position of its axis be vertical. 12. Investigate the small oscillations of the axis of a top about its mean inclination to the vertical. We have seen that if v or n fulfil the condition given at the end of Ex. 10, vanishes when = i, and also when = *'. At some position of the axis of revolution intermediate between these two = 0. If zs and a be the values of \p and corresponding to this position, we have A^cos a — CnZS + mgh = ) Azj sin 2 a + Cn cos a = Cn cos i ) The motion would now be steady if were zero. .We have seen that this is impossible ; but as is now of the opposite sign to the axis of the top will oscillate about this position, provided is small. 394 Kinetics of a Rigid Body. To determine these oscillations, let \p = "& + €=0 ' and therefore e =j sin (/xt + 7), where j and 7 are arbitrary constants, and jx is given by the equation A 2 fi 2 = A 2 zf sin 2 o i- (2^4ot cos o - Crc) 2 . From the expression for 8 2 given by (a) Ex. 4, it is easy to see that by properly determining n, we can make small throughout the motion, and thus the condition requisite for a small oscillation can be secured. 13. Find the vertical pressure on the plane on which the top is spinning. If z be the vertical coordinate of the centre of inertia of the top, and P the vertical force exerted on the top by the plane, we have P = tug -\- m'z ; but .. , d (dz\ 2 d I . d9\ 2 z = h -r ( — I —hn sin 6 — ) , 2 dz\dt ) 2 d cos \ dt ) ' and from (a) Ex. 4 and Ex. 10, we have sin 2 2 = ^ (cos i - cos 0) { 1 - 2v cos i + 2v cos - cos 2 0} . Hence P=mg J 1 + - (3 cos^ - 2 (cos i + 2v) cos + 4j/ cos i - 1) j . 14. A solid of revolution, having a great angular velocity round its axis, and terminated by a spherical surface of small radius, is placed, with its axis inclined to the vertical, on a rough horizontal plane. The moment of inertia round the axis of revolution being not less than that round an axis perpendicular thereto, and the distance of the centre of inertia from the lower end being considerable, show that after some time the axis of revolution will become vertical. (Jellett, Theory of Friction, Chapter VIII.) Examples. 395 Let the axis of z through the centre of inertia he vertical, and letOC be the axis of revolution, which must pass through S, the centre of the terminating spherical surface. Accordingly the point of contact T lies in the plane ZOS. The forces acting on the body are gravity, and the resultant of the normal reaction and friction at T. The friction may he re- z solved into two, one along TZ', the other at right angles to the plane ZOC. Calling this latter com- ponent F, and putting TS =a, SO = b, ZOC = 0, the moments of the applied forces round OZ and OC are respectively Fb sine, and -Fa sin 0. Hence, by (54), — (a sin 2 0tH Co* cos J = Fb sin ; also, C = - Fa sin at therefore -I at \ A sin-0^+Ca>3 cos 9 ) ~~a dt Hence, putting n = -, A sin 2 9\p + Ca> 3 cos 9 + nCwz = constant = CD. (cos O + w), where Q and 9 are the initial values of 03 and 9, since ^ = initially. As the force F constantly diminishes the angular velocity, after some time a>3 must become equal to n + cos 0o fi — . n+ 1 When this happens, we have = 0. For, substituting in the previous equation the value just obtained for a>3, we get (1 ( „ . n + cos 0o ) 0) 2^cos 2 i0^ - Cn — - — j— J =0; but as « is greater than 1, ^ small as compared with n, and C not less than A, the second factor of the above expression cannot vanish, and therefore we must have = 0. . , The result obtained here may be regarded as holding good in the case 01 a humming-top. 396 CHAPTER XII. ENERGY AND THE GENERAL EQUATIONS OF DYNAMICS. Section I. — Energy. 280. Energy. — Work and Energy have been defined, Arts. 118 and 129, and the equation of Energy for a rigid body has been obtained by two different methods (Arts. 132 and 200). In the present section we propose to consider the subject in a somewhat more general manner, and to show that on the equation of Energy may be based the whole theory of the action of forces on a connected system. 281. Equation of Energy. — If a system of material points be acted on by any forces, we may suppose the con- straints and connexions of the system replaced by correspon- ding forces, and thus regard each point as entirely free. Assuming then the principles which govern the resolution and composition of forces acting at a point, and the relations between force and acceleration, we have d 2 x x d 2 y x m W = x » m 'UF d 2 x 2 d 2 y 2 " h ^iF = x '-' " h HF where Xi, Y Xi Z x , &c, include the components of the forces which replace the constraints, if any, acting at the points Multiplying the first equation by dx ly the second by dy ly &c, and adding, we have (d 2 x , d 2 i/ , d 2 z _ \ CT/ ^ 7 t, , m\ 2 m( — - dx + -^ dij + — dz J = 2{Xdx + Ydy + Zdz) ; F„ a Z]_ Y,, " h le= z "" Conservation of Energy. 397 or, putting T = J 2m# 2 , — r . dt = 2 (X + Ydy + Zdz + X'dx + Ydy + 71 dz) + c. Since the conservation of energy holds good, T is a function of the relative position of the two points. Again, as they are points, all directions must be supposed indifferent as regards either of them considered alone. Hence their relative position must depend solely on the distance between them, and T is therefore a function of this distance r, or T = $ (r). Equating the two expressions for T, and differentiating, we have X-#' W £ F =*'(,■) g, Z= ¥ (r)%, X-fW* Y'-¥ir)%, W(r)*; 400 Energy. or X- # '(r)fZ^ r_^( r )LZ* Z='(r)~, X'=0'(r)^, T-f\r)t-X Z'=4>'(r) r Hence the point xyz is acted on by a force '(r) in the direction of the line joining xyz to ocyz ; and the latter point is acted on by an equal force in the opposite direction. Conversely it is easy to see that, if two material points acted on each other with a force depending as regards mag- nitude on their mutual distance, but not in the direction of the line join in cj them, they would be capable of producing in each other an ever-increasing velocity, and of thus generating an unlimited amount of energy. In order to bring about this result we have only to suppose the points connected by a rigid rod. The whole system would then be acted on by a constant couple. 284. Forces which appear in the Equation of Energy. — For any system entirely free w r e have obtained the equation dT = S(Xda? + Tdy + Zdz), and have seen that this equation holds good for a system restricted in any way, provided the constraints are replaced by equivalent forces. If the constraints of the system consist of smooth curves or surfaces along which the points are restricted to move ; of rigid bars or inextensible strings connecting the different points with each other or with any external fixed points ; or in general of any connexions such that the distance between each pair of points immediately acting on each other is in- variable, the whole work done by all these constraints and connexions is zero, and may therefore be omitted from the right-hand side of the equation (Arts. 124, 127). If the potential energy (Arts. 129, 282) of any portion of the system be a function of a single variable quantity u 9 the work done by this part of the system in any displacement must be of the form ASw; for V= (u), and therefore dV = '(u)du. If between any points of the system there be a connexion which is capable of being expressed by means of an equation Forces which appear in Equation of Energy. 401 "between their coordinates, such connexion can be effected by means of constraints of an invariable character ; such as smooth fixed surfaces or curves, or rigid bars or inextensible strings. Hence we may conclude that, in any motion of the system, the work done by the forces replacing any connexion between the points of the system which is capable of being expressed by equa- tion* between their coordinates, is zero. A formal proof of this important proposition may be given as follows : — If TJ = be an equation between the coordinates of any points in a moving system, the forces which the corresponding constraint introduces into the system must be functions of the coordinates and of the other forces. Hence, if the latter be conservative, so are the forces caused by the constraint, which for brevity we shall refer to as the constraint TJ. Again, if at any time the condition TJ- be actually ful- filled, the imposition or removal of the material bonds by which the corresponding constraint is effected cannot require any expenditure of energy ; since this imposition or removal does not change the position of any point of the system. Let there be now a system S i9 which without 27 is conser- vative, and let A and B be two configurations in which the condition U = Q is fulfilled ; then, as we have seen, the forces replacing £7 are conservative, and if they consume work in the motion from B to A, they produce work in that from A to B. Let the external work W bring Si from A to B subject to the constraint TJ. Let Q be the amount of potential energy thereby produced in the system, and E the work clone by the forces replacing U; then, the whole amount of work produced is W - Q + E. Now let this be used in bringing S 2 (precisely similar to S y ) from B to A without the constraint TJ, whereby Q is produced, and in doing such an amount of other work that Si may come to rest in the position B and S 2 in the position A. We may then without any expenditure of work impose the constraint TJ on S% and remove it from Si. Things are now in precisely the same state as at starting, and in the whole process, by an expenditure of work W, we have produced work whose amount is W + E. Hence in any motion of the system the work E done by the forces replac- ing the condition TJ = must be zero. 2 1) 402 Energy. As the amount of work done by these forces in an un- reversed motion cannot be influenced by the character of the other forces, but only by their amounts and directions, the work done by the forces replacing U = must under any circumstances be zero. 285. Equation of Energy in General. — If we have a system acted on by any forces external or internal, and sub- ject to any constraints or mutual connexions, the equation of energy assumes the form T-T + V- V = W. (5) T and V are the kinetic and potential energies in the initial position, T and V those in the position under con- sideration, and W the work done in going from the initial to the actual position by the external forces and by those internal forces which are not conservative or reversible in their character. As regards constraints and connexions, they may be di- vided into three classes. 1. Those producing forces whose work during any motion of the system is zero. Such con- nexions we have already considered ; they have no effect on the equation of energy. 2. Those which are capable of alter- ation under the action of external forces, and such that their alteration produces or consumes a corresponding amount of potential energy. The work done by the forces replacing these constraints and connexions is included in the expression Vq - V. 3. Resistances or connexions which introduce forces of a non-conservative character. Such are the friction of rough surfaces, the resistance of a medium, the forces deve- loped by the alteration of an extensible body which does less work in its recoil than the amount required to stretch it, &c. All such forces must appear as forces in the equation of energy, and the work done by them is included in W. 286. Virtual Velocities. — The principle of energy may, as we have seen, be expressed for dynamical purposes in the following form : — The work done on a system in any interval of time by external applied forces, diminished by the work consumed in the same time by the non- conservative forces of the system, is equal to the sum of the increments of the kinetic and potential energies. Virtual Velocities. 403 We have seen likewise that this principle holds good for a system subject to any invariable constraints or connexions internal or external as well as for a free system. We are now able to obtain the conditions which must be fulfilled in order that any system should be in equilibrium ; they can be expressed in a single statement, viz : — In order that any system should be in equilibrium, the ivork done by the applied forces in any possible infinitely small displace- ment, diminished by the increase of the potential energy of the system, must be cither negative or zero ; and, if this be true for every possible infinitely small displacement, the system is in equilibrium. The truth of this statement readily appears from the equation of energy. A position of equilibrium is one in which if the system be placed at rest it will remain at rest. Now the system will not remain at rest if there be any possible mode of displace- ment, in which the united action of the internal and external forces can produce a velocity in any of the points of the system. On the other hand, if the system move from rest in any manner, it will acquire a positive kinetic energy. _ Hence, if there be no possible way in which it can do this, its position must be one of equilibrium. In applying the principle of equilibrium we must regard the non-conservative forces of the system (if any) as applied forces, and introduce them with their proper signs. In the case of actual motion, forces of this kind always consume work, but in the case of virtual displacements this is not ne- cessarily the case ; e. g. suppose a heavy particle is placed on a rough inclined plane, and it is required to determine the condition of equilibrium. In this case we must consider the force of friction as acting upwards along the plane. If now we imagine a virtual displacement down the plane, friction will consume work ; but if we imagine a displacement up the plane, friction will produce work. In the case of actual motion, whether slipping take place up or down the plane, friction will consume work. Again it is to be observed, that if every possible set of displacements be also possible when reversed, the condition of 2D 2 404 Energy. equilibrium becomes simply that the total work done by all the forces internal and external be zero. In fact, if SPSp be negative and P remaining unaltered the sign of each §p be changed, SPSp becomes positive ; but this is inconsistent with the principle of equilibrium as stated above ; hence ^P^p must be zero. If we combine the principle of Virtual Velocities with D'Alembert's principle, we obtain the equation which embraces the whole theory of Kinetics, From this equation that of energy was deduced in Chapter IX. In the present chapter we have reversed this mode of procedure. 287. Equivalent Sets of Forces. — Two sets of forces acting on any material system are said to be equivalent when the motions produced by one set are identical with those produced by the other. If each of two sets of forces be capable of equilibrating the same third set, the two are equivalent. For let P be a force of the first set, Q one of the second, and R one of the set which each of the first two can equilibrate. Suppose the P set only to act. Introduce at the point where R would act two forces R and - R. This being done for each point of the system, the motion remains undisturbed. The system is now acted on by the three sets of forces P, P, and - R ; and, since the sets P and R are in equilibrium, the sets P and - R are equivalent. In like manner the sets Q and - R are equivalent. Hence the sets P and Q are equivalent. In moving a system from one given position to another, equi- valent sets of forces produce the same amount of work. The motion being the same whichever set of forces is in action, the intermediate positions of the system are at each instant the same ; consequently, since the two sets of forces are each capable of equilibrating the same set, we have %P$p = ^QBq at each instant. Hence the whole amount of work produced in one case is equal to that produced in the other. It can be shown in like manner that the work required to move a system from one given position to another, against the Examples. 405 action of any set of forces, is equal to that required to move it against the action of an equivalent set. 288. "Wrenches. — A wrench in Kinetics corresponds to a twist in Kinematics. If a rigid body be acted on by any forces, these forces can be reduced to a single force along with a couple whose plane is perpendicular to the direction of the force. Such a system is called a wrench about a screw, the axis of the screw being the line of direction of the force, and the pitch of the screw the line which is the quotient obtained by dividing the moment of the couple by the force. The magnitude of the force is called the intensity of the wrench. The wrench to which a set of forces acting on a rigid body is equivalent has been termed the canonical form of the set of forces. The canonical form of a set of forces is in general unique ; for, as may be easily seen, if two wrenches be equivalent, they must either be identical or else consist of equal couples in parallel planes. Examples. 1 . A particle of mass m moves with a simple harmonic motion ; determine its mean energy. If t and a be the periodic time and amplitude of the motion (Arts. 87, 88), and T the mean energy, 1 rrmv 2 , 7r 2 . T=~\ — - dt = m — a 2 . 2 T" r Jo 2. If the motion of the particle m he the resultant of any number of simple harmonic motions having different periods and amplitudes, find the mean value of the energy. If be an interval of time which is very great compared with the longest periodic time, -if* — -dt = W7T-2— . Z T" 3. Determine the mean energy of a system of vibrating particles. The rectangular components of the displacement of any particle are periodic functions of the time, and can therefore be expanded in a series of terms of the form • /2tt \ asm — t + a J . (i- _l yi jl. (<2 Hence, T= Tr-Xm — -. 406 Energy. 4. A rigid body is acted on by a couple whose moment is Pp ; determine the work done by the couple in any small motion of the body. If d9 be the angular displacement of the body round an axis perpendicular to the plane of the couple, the work done by the couple is Ppdd, see Art. 128. 5. Express the kinetic energy of a body having a fixed point in terms of the angles 0, cj>, $ (Art. 258), the body-axes being the principal axes at the fixed point. As an, «2, co3 are given in terms of 0, (f>, »//, Ex. 5, Art. 260, we have, Art. 263, 2T= (^sin 2 <|) + -Bcos 2 ^)0 2 +{(^cos 2 (/) + ^sin 2 (/))sin 3 0+C r cos 2 0} fr + C

+ 2 C$^ cos0. 6. If T be the kinetic energy of a body having a fixed point, and 2 ) = 2gm (z - z). Hence, if h be the initial height of the centre of the cylinder, and a> its angular velocity when it reaches the horizontal plane, or = ; — , x - Xo = — ■ ; (h — a) cot x. {m + m sin 2 i) a 4 + (m + m) k" 1 m + m 10. Show that the velocity v with which a fluid, under a uniform pressure p, escapes from a small orifice is given by the equation v 2 = 2gh, where h is the height of a column of the fluid which would produce the pressure p. Suppose a small mass m of fluid forced through an orifice, whose section is , M+m ( Mm ) [ fi 2 M 3 m 3 ) We have then h = y+x, 2Y-K=y 2 -—- If the whole system move as a rigid body, the angular velocity h, f(x) is positive, and therefore the biquadratic has no positive root greater than /* ; Examples. 409 but if x be positive and less than h, f(x) may be negative, and therefore the biquadratic may have two positive roots between and h. Asf'(x) = x-(ix — 37*), if the biquadratic have two real roots, one is greater than f A and the other less. The greater root makes /'(#)> and therefore — positive, and Fa minimum; d 2 Y the lesser root makes — - negative, and F a maximum. 13. Apply the preceding examples to determine the secular effects of tidal friction on the Earth-moon system, the moon being supposed to move in the plane of the equator. If the Earth's radius be denoted by 0, Cis approximately f Ma\ and M = XZm. Hence the unit of mass is f£, the unit of length 5-26*, and the unit of time 2 hours 41 minutes. Again, in the special units, the present value of r is 11 '454, and that of n is 0-704, whence x is 3-384 ; also h = 4-088. It is plain that for this value of h the biquadratic /(.r) = has two real roots. Theksser of these, xi, makes Fa maximum, and the greater, x 2 , makes Fa minimum. Again, f{x) is positive for values of x between and x x , negative for those between xi and xo, and positive for those greater than x 2 . As x is positive throughout, when f(x) is positive we have — ,>y, i.e. «>»; and — < y, i. e. w < w, when f(x) is negative. At present f(x) is negative, and therefore the present state of things corresponds to a value of x which lies between X\ and xi. We can now determine the effects of tidal friction. Since the friction resulting from the lunar tides constantly diminishes the sensible or mechanical energy of the Earth-moon system, Y must continually decrease (Art. 282;. Hence, as at present / (x) is negative, x must increase and y decrease until T reaches its minimum, after which the whole system will move as if rigidly connected. . . , It appears accordingly that the friction caused by the lunar tides diminishes the angular velocity of the Earth, i. e. increases the length of the day, and at the same time increases the Moon's distance and the length of the month ._ lhis process must go on till the day and month are of equal length, after which the lunar tides will cease. If at any past period the Moon moved as if rigidly con- nected with the Earth, this must have been when F was a maximum. Such a state of things was dynamically unstable, for the least disturbance of the rigidity of the motion would'have produced tides whose friction would have diminished the energy, and caused the system to depart farther from the configuration oi maximum energy. The departure from this configuration might have taken place in two ways, according as the Moon approached the Earth or receded from it. If the former event had occurred, the Moon's angular velocity in its orbit would have become greater than the angular velocity of the Earth's rotation, and the Moon must ultimately have fallen upon the Earth, as x must have de- creased continually along with F If on the other hand the Moon had receded the present state of things would have been reached. The value oi x which makes F a minimum lies between 4-073 and 4-074, and the corresponding value of n lies between 0-015 and 0-014. The ratios of the present value of n to these two values are 46-9 and 50-2. The present investigation would therefore lead us to conclude that, when the lunar tides cease and the day and month become equal, the length of the day will be between 46 and 50 times its present length. 410 Energy. Examples 11, 12, 13, and Example 3, Art. 213, are taken from a Paper by Professor G. H. Darwin, first published in the Proceedings of the Royal Society for 1879, and subsequently, with some alterations, in Thomson and Tait's Natural Philosophy, Part ii. In this Paper Mr. Darwin gives diagrams of the curves represented by the equations V = 2Y-K=F {x), &f = l, h = x + y, by means of which the results arrived at are exhibited to the eye. 14. A great number of smooth perfectly elastic particles are moving with great velocity in various directions within a rectangular parallelepiped, two of whose opposite faces are large compared with the others. If one of these faces be movable, determine the force required to keep it steady. Let u be the velocity of one of the particles whose mass is m, and

nu- cos 2 , describe a sphere of unit radius, and draw from its centre lines parallel to the directions of motion of the various particles at the beginning of the interval of time 0. Since the number of particles is very great and the direction of the motion of any one undeter- mined, we may assume that the energy of those particles whose directions of motion make an angle

and $ + clef) is to the whole surface. If The the total energy of the moving particles, we have then fTT 2 2»*w- cos 2 = T cos 2 tf> sin = - T. 2 TQ Hence M = . Now the required force J 1 must be such as to communi- 3 a cate to the movable face the momentum M in the time 0, and therefore 2 Td 2 T F6 = M=-—, arF=--. 3 a 3 a 15. A number of particles move as in the last example ; determine the pressure which they exert on the unit of area. Examples. 411 If S be the area of the movable face in the last example, and p the pressure 2 T of the particles on the unit of area, pS= F= - — . Hence, if v be the volume of 3 a 2 the parallelepiped, pv = - T. The results obtained in Ex. 14 and 15 are made use of to explain the pressure which a gas exerts against its envelope. The mode of investigation employed is due to Clausius. 16. Determine the mean kinetic energy of any system in stationary motion. A system is said to be in stationary motion when the coordinates and the velocities of its various points fluctuate within determinate finite limits. If we integrate x 2 dt by parts, we get $cb-dt = xx- jxxdt ; and similar equa- tions may be obtained corresponding to the other coordinates. Again, supposing each point of the system to be free, we have mx = X. Hence, if = t\ — to, 1 f'i 1 Tclt = — ^,m{x\xi + yiyi + z\i\ - (x x + y y + z z )} 6 Jt Zv -^ 2^(Xx+Y!/ + Zz)dt. If 6 be made sufficiently large, the first term on the right-hand side of this equation may be neglected, and we find that the mean value of T is equal to the mean value of -%2{Xv+ Yy + Zz). This latter quantity is termed by Clausius the virial of the system. Hence, the mean kinetic energy is equal to the virial. This theorem, and its application given in Ex. 17, 18, are due to Clausius, whose investigation will be found in the Philosophical Magazine for August, 1870. 17. If n be the virial of a system which is acted on by no external forces except a uniform pressure on its surface, prove that n = §pv--\ X S.r dzdx + jjzdxdy} = %pv. 1 fix Hence n = f pv — „ 2r<£ (r) dt. 6Jt 18. Determine the pressure of a gas in terms of its volume and the mean kinetic energy of translation of its molecules. A gas may be regarded as composed of a number of molecules which exert no action on each other except when in contact. If the gas be enclosed in an envelope, and its condition remain unaltered, its molecules must be in stationary motion. Hence, if T' be the mean value of that part of the kinetic energy which results from the velocities of the centres of inertia of the molecules, and n the corresponding virial, we have T' = U; but n = \pv (Ex. 17), since the time during which a particle is in contact with other particles is negligible compared with the interval between two such contacts, and therefore the other term of n may in this case be neglected. Accordingly pv — §T\ Section II. — The General Equations of Dynamics. 289. General Equations of Motion for any System. — The general equations of motion for any system are ob- tained in precisely the same manner as the general equations of equilibrium. If F= 0, G = 0, H= 0, &c, are the equations of condition representing the connexions and constraints, we have dF- dF„ dF. dF„ — 6x x + — - tii/i + —£% + -— &r 2 + &c. = 0. axi dy x dz x dx> ^ &* + &c. = 0, ~^x x + &o. = 0, &o. dx x dx x Multiply the first by A, the second by /u, the third by v, &c, and add to D'Alembert's Equation, and we obtain v d 2 x x .dF dG dH g \. - M . /1N Xi - nh W + X ^ + ^aV L +V a^ + &C ')^ + &G ' = °' W If there be n equations of condition we can assign to A, ju, v, &c, such values as to make the coefficients of the first n displacements in the above equation vanish. By means of General Equations of Motion. 413 these displacements we can satisfy the n equations $F = 0, S6r = 0, &c. The remaining* displacements are then entirely unrestricted, and their coefficients in (1) must therefore be each zero, and we have for the equations of motion ffix x _ .dF clG dE _ "| mi 7F =Xl + X d7 + fX d7 + v aV + &c ' (It tlXx LlJLi tlJbi dh,x _ A dF dG dH dt 2 dy x dy x dy x L -r^r = Z x + X i dF 1 dt dG dE p dz x r d% x dz x m 2 d 2 x 2 w &c, _ .dF dG dH X 2 + A — + fl-r— +V-j- + &C. dx % dx 2 dXi &c, &c. (2) From these equations we can obtain the Equation of Energy, if we multiply the first by dx Xi the second by dy x , &c, and add. In this manner we obtain (dF dF \ dT = 2 {Xdx + Ydy + Zdz) + X f — dx x + — dy x + &c. J + &c. Now, if the equation F = involve only the coordinates of the various points, dF _ dF - 7Z:7 n — - dx x + — c/yi + &c. = dF = 5 flfoi dy x and the condition expressed by the equation F = has no effect on the kinetic energy. This result was obtained from first principles in Art. 284, and by its means the Equation of Virtual Velocities in its usual form was deduced from the Equation of Energy. 290. Equation of Energy when Equations of Con- dition Involve the Time Explicitly. — If the equation 414 The General Equations of Dynamics. F=0 involve the time explicitly, the work done in any actual motion of the system by the forces capable of replacing the condition F = need not be zero. In a virtual displacement the work done by these forces must still be zero, because in such a displacement no lapse of time is supposed to take place. So far, therefore, as the equation of virtual velocities is concerned, t must be considered constant in the equation F= 0, and as in Art. 200 the virtual displacements must fulfil the condition dF rfF dF s dF. -t-COHi + -r-Oth + — OSi + -7— bx 2 + &c. = 0. dxi ay 1 dz v dx t The actual displacements on the other hand fulfil the condition dF. dF , dF dF j _ (dF\ M . — - dxi + -7- di/i + -— dzi + —- dx % + &c. + — - )dt = 0. dx x diji dz x dxo \dt J Hence in this case the Equation of Energy becomes dT = 2 (Xdx + Ydy + Zdz) - \ f^\ dt- ^ {^\ dt - &c. (3) 291. Similar Mechanical Systems. — Two systems are geometrically similar when each line of the one is equal to the corresponding line of the other multiplied by the same constant. Similar Mechanical systems are not only geometrically similar, but have also a similar distribution of mass, and a similar distribution of force, and work in a similar manner ; i. e. each mass of the one is equal to the corresponding mass of the other multiplied by a constant, each force of the one is equal to the corresponding force of the other multiplied by a constant ; and the systems are always geometrically similar at instants of time whose intervals from two fixed epochs are in a constant ratio. Let x be a coordinate of a point in the first system, m a mass, X a force, and t an interval of time ; and x', m\ X', t' Generalized Coordinates. 415 the corresponding quantities for the second system ; we have then the equations %' = Ix, ml = \xm, X' = AX, t' = vt. Hence, Sm'^S* + JfW+ ^§* J and 2 (X'&u' + F'S/ + Z'&O = A/s ( x ^ + F ^/ + zgs )« In order, therefore, that D'Alembert's equation should hold good for each system, we must have fil = \v~. This equation of condition may be put into another form by expressing v in terms of the ratio of the corresponding velocities in the two systems. If we denote this ratio by a, dx dx - . , dx I dx ,, « , j we have —7 = a — , but also, — - = - — ; therefore I = av, and do do do v do the equation of condition becomes XI = /ma*. If, as is generally the case, gravity be one of the moving forces in both systems, we must have A = ^ ; hence a 2 = /, or the velocity in each system must be proportional to the square root of its linear dimensions. 292. Generalized Coordinates. — If a system have n degrees of freedom its position is completely determined at each instant by the values of n independent variables, which may be termed coordinates, and be denoted by £1, £ 2 , ? 3 , . . . £». The Cartesian coordinates x, ?/, z of any point of the system are expressible in terms of these new coordinates, and are therefore functions of the n variables £1, ? 2 , &c., these latter being functions of the time. If we differentiate the equation x = /(?!, £ 2 , £ 3 > • • « ?n) with respect to the time we obtain , CtX A. CIX L, 11 X y. f A\ d£x a£ t d% n This equation shows that x is a function of the velocities fi, 416 The Genera! Equations of Dynamics. &c, and of the coordinates ? l5 &c, and is linear with respect to the velocities. From (4) we have dx dx dx dx /Jrx *"£ -S-3&* 8 - (5) dx Again, if we differentiate —, ■? with respect to t, we get dgi d dx d 2 x £ d 2 x £ d 2 x £ dttiZ = W? dfd& d^a%J n ; but by (4) this is the expression for the partial differential coefficient — ■ . Hence we have d dx dx d dx dx . f dtWi^dZi' dtd& = dQ {) Any set of n independent variables which completely deter- mine the position of a system may be taken as the generalized coordinates of the system. The number of these coordinates is fixed, bnt the actual coordinates are in general to a great extent arbitrary. 293. Kinetic Energy and Generalized Coordinates. — The kinetic energy T of any system in motion is given by the equation 2T = Hm(x 2 + if + z~) If we substitute for x, if &c, their values given by (4) and the corresponding equa- tions, we obtain a homogeneous quadratic function of the n velocities f i, § 2 , • • . ?», the coefficients of £ i 2 , ji ? 2 , &c., being functions of the coordinates £i> £ 2 , &c, and of the con- stants of the system. If we denote these coefficients by &i, 23£i2, &c, we have the equation 2T = H x & + %& + &c. + 23kf if, + 23£j£i£. + &c. (7) 294. Equations of Motion for Impulses. — If a system start from rest under the action of any set of impulses X, F, Z, &c, the initial velocities are determined from D'Alembert's equation by equating to zero the coefficient of Equations of Motion for Impulses. 417 each independent variation. Now, if ?i, £>, &c. be the generalized coordinates, where 2£i, S£ 2 , &c. are independent arbitrary variations. Hence, substituting for §#, Sy, &c. in D'Alembert's equation, we obtain as the equation of motion corresponding to the variation S£i, ( . dx . dy . dz\ „ / _ b jp 2 , &c. be the general- ized impulse components which would give its actual motion to the system starting from rest, these impulses p l9 &c. are determined by the equations dT dT dT Pl = l£> Pi = IF' Pn = 7T' ( 10) o?gi d& d% dT dT The differential coefficients -r-, — -, &c. are the gene- ralized components of momentum of the moving system. If a system in motion be acted on by any set of im- pulses whose generalized components are Hi, S 2 , &c, the changes of velocity produced by these impulses are given by the equations dT (dT\ dT (dTV „ . .„. fdTY where ( — - J , &c. correspond to the instant immediately dT before, and — =-, &c. to that immediately after the action of the impulses. Since the values of £ l5 &c. remain unaltered during the impulse, equations (11) may by (7) be written ft, (fe - £/) + £, (fc - £0 . . . . + ft. [tn - in = 9. ft* (f, - ?/) + ft* (& - 60 ■ . . . + ft. (I. - £„') = E 2 I &C &C. &C. y Kinetic Energy and Components of Momentum. 419 295. Kinetic Energy and Components of Momen- tum. — Since I 7 is a homogeneous quadratic function of ?x, £ 2 , &c, we have, by Euler's Theorem, • dT . dT 2T=% X ~ + f 3 i± + &c. -*&+!*&. • .^»5«. (13) dKi d& This equation may be written in the abbreviated form 2r=SQ4). (14) If we suppose the same system occupying the same posi- tion to have successively two different motions, in the first of which the velocity-components are f : , &c, and in the second, §/, &c, and if we put f/= £i + a 1? &c, and express the cor- responding values of T by T\ and Tj? , we have, by Taylor's Theorem, Tj, = Ti+Vpa + Ta, i.e. r^=r|+Sp(f'- |) + %_|). (15) If now we suppose a system to start from rest the values of certain components of velocity being prescribed, and if the system be set in motion by impulses such that there are no components of impulse except those corresponding to the prescribed velocities, the initial kinetic energy is a minimum. Let £i, &c, be the velocity-components of the initial motion produced in the manner described, then p l9 &c, are the impulse-components; and if any impulse-component p q be not zero, the corresponding velocity-component % q is pre- scribed. Let us now suppose the system to be set in motion in any other way, the prescribed velocity-components being the same as before, and let |/, &c, be the velocity- components of the new initial motion. We. have, then, 2p (?'- ?) = 0, since whenever £> is not zero, £' = %. Hence T% = T% + ^ (£'-£), and therefore T* is a minimum. This is Thomson's Theorem, Art. 199. Again, if fi, &c, p l9 &c. be the components of velocity and momentum of a system in any given position, and ?/, &c, p{ 9 &c. the corresponding quantities for a different motion of the same system in the same position, we have s (ptr> = s (/?). (i6) 2E 2 420 The General Equations of Dynamics. The truth of this equation appears readily by substituting fi + £/, &c. for |i, &c. in T%, and equating the two expres- sions which by Taylor's Theorem can thence be obtained. 296. Energy of Initial motion. — If we substitute % v dt for S£i, ^dt for S£ 2 , &c. in (9), we obtain for the initial energy T of a system starting from rest the equation 2r=S(S?). (17) Let us now suppose that on a system having £ x , &c. as its generalized coordinates constraints are imposed capable of being expressed as in Art. 284 by equations connecting the coordinates &, £ 2 , . . . . g n . The coordinates are then no longer independent, and if the system be set in motion by impulses 5Efi, &c, equations (8) no longer hold good, but (9) and (17) remain valid, f^ &o. being the velocity- components of the actual motion. Also T is the same function of fi, &c. as it was before the imposition of the constraints, the only difference being that certain relations hold good in the constrained motion connecting these velocity-components. In order to compare the initial kinetic energies of the system in the unconstrained and constrained motion, let &, &c. be the velocity-components corresponding to the former, and f/, &c. those corresponding to the latter, then by (17) we^have 22> = 2(Er) = S(i>|')> also 2Tj: = 2pt Substituting in (15), we obtain This proves Bertrand's Theorem, Art. 199. 297. Iiagrange's Equations of Motion. — We saw in Art. 294 that /. dx . dy . dz\ dT „_, V d^i d£ L d&J d% x ' V ' Lagrange s Equations of Motion. 421 if we differentiate each side of this equation with respect to the time, and substitute for — -r=-, &c. their values given Clt usi by (6), we obtain dx .. di/ d It 'z\ f . dx . dy . dz F + 2m [x -p + y -p£ + z -^ t,ij \ dt.x d% x dt,, d_dT t dtd%} dT t . „ / . dx . dt) . dz\ . -, . -■ but 2m x —r + z-k? + z -tt) is plainly , V d& J d& dl x ) * J dl x hence we have „ / dx .. dy .. dz \ d dT dT 2w [x -rr + y -£■ + z -? = — ■ -3- - -r— . (20) V rf& J d& dZj dtdl dl x v ' Now in D'Alembert's equation for continuous forces the coefficient of the independent variation §& is, dx . dy dz dl x J d& dl, d& d& dZi Hence, if we put _ dx T _ dy „ dz\ _ _ we have, as the equations of motion of any system, d_ (IT _ dT _ . dt dtx dti ~ * d_ dT_ dT dt 4* * >• (21) dt dt dl n ' " J The work which would be done by the forces of the system against the displacement §£i is - Si S?i, accordingly 35?i, &c. are the generalized components of force tending to alter the coordinates &, &o. It is to be observed that the 422 The General Equations of Dynamics. forces X, F, Z, &c. are not equivalent to the forces Hi, S 2 , &c, unless the variations $x, By, dz, &c, and the correspond- ing variations Sgi, S£ 2 , &c, result by orthogonal projection from the same possible displacements of the system. For a conservative system equations (21) become d_ dT It d% x d_ dT It d& dT dV_ d& + & r dv dh + dl, d_ dT dt d% n dT dV + — d%n d£ n u, ^_^_ + ^_ =0 1 ^ + ^-=o y J (22) Equations (21) and (22) were first given by Lagrange, and are known as Lagrange's equations of Motion in Generalized Coordinates. The proof given above for Lagrange's equations holds good even though the time appear explicitly in the equations which determine the Cartesian in terms of the Generalized coordinates. In this case x =/(?i, ? 2 , 5n, t), & c - * taen contains the additional term — ; but the equations dt dx dx d dx _ dx 4i = a%' G '' It S& ~ 7& c '' are still true, and therefore the proof of Lagrange's equations remains valid. If we put L = T - V, the function L is the difference between the kinetic and potential energies of the system, and is called Lagrange's Function. Equations (22) may now be written in the form d dL dL dt d% x d%i ^ _ £5 = 0. (23) dt d% n d%„ Effect of Constraints. 423 This form of the equation of motion is likewise due to Lagrange. 298. Deduction of the Equation of Energy. — If we multiply the first of equations (22) by fi, the second by £>, &c, and add, we get ±ld dT dT\ ^{j.dV\ n , nA , 2? — -r 1 + 2 ?— =0. 24 • dT Now 2T=2?-r-; and therefore 2 — = 2 ? — —r + ? -f > hence "•ej-s)-f-«(«f*'S)-f'« rfT ^/^rfT *dT since = 2 § — - + § — r Substituting in (24), we obtain (IT dV _ dt + dt " ' hence we have T+V=E. (26) 299. Effect of Constraints. — If a system having n degrees of freedom be subjected to additional constraints capable of being expressed by q equations connecting the coordinates of the form F= 0, G = 0, &c, we may either select a new system of n - q generalized coordinates, or else 424 The General Equations of Dynamics. retain the old system, and proceed according to the method of Art. 289. Equations (22) would then become d all dT dV .dF dG _ x dt dl l ~df 1 + d^ = X df l +fX d^ + &G ' d dT dT dV .dF dG a d_'dT_dT dV _ .dF dG_ dt d%n d% n d£ n d% n ** d% n In the case of impulses we may proceed in a similar manner, and still make use of equations (8) or (11), provided we introduce additional terms into S?i, &c. representing the impulses by which the constraints may be replaced. It is plain that both in the case of continuous and also in that of impulsive forces the terms in Lagrange's equations repre- senting the action of the constraints disappear from the equation of energy. Examples. 1. Determine in polar coordinates the equations of motion of a particle which moves freely in a fixed plane. Here T=\m (r 2 + r 2 6 2 ), whence d dT dT .. ., d dT dT d , . - — — — — = mr — mr9*, — -p- - — - = m-r (r 2 9), dt dr dr ' dt dd dd dt K h and the equations of motion are the same as those which would be given by (11) and (12), Art. 28. 2. Determine in polar co-ordinates the general equations of motion of a free particle. Here T = § m { r 2 + r 2 (0 2 + sin 2 9

2 J = Pr, m - {r 2 sin 2 9$) = Qr sin 9, Examples. 425 where i?, P, and Q are the components of the force acting on the particle, along the radius vector from the origin, perpendicular to the radius vector in the meridian of the particle, and at right angles to these two directions. 3. Prove Euler's equations for a body having a fixed point. The body-axes being the principal axes at the fixed point, the expression for Tin terms of 0, ) 0^ = *. If we substitute u>z for

aoo-2 and we have d (dT\ dTl dT — ( J — C02 + COl = 4> = dt \dcoz/ dan dwo = N. 5. Two particles m and m are connected by an inextensible string passing through a smooth hole at the edge of a smooth horizontal table on which m rests ; determine the equations of motion of the particles, and the tension of the string. Let r and 6 be the polar coordinates of m with respect to the hole as origin ; then IT = (m + m) r 3 + mr 2 2 , and the equations of motion are d (m + tri) r — mr 6~ = - m'g, — {mr 2 6) = 0. If t be the tension of the string, and h the value of mr 2 9, we have mr — mrd 2 = - r (Ex. 1), whence mm ( h 2 \ = — : — ; Iff+TZ)' m + m \ m i r a / 6. A smooth particle descends the upper edge of a thin vertical lamina which is capable of sliding freely down a smooth inclined plane with which 426 The General Equations of Dynamics. its whole lower ledge is in contact. If the plane of the lamina he perpendicular to the intersection of the inclined plane with the horizon, and the particle and lamina start from rest, determine their position at any time. Let x he the distance at any time of a point in the hase of the lamina from its initial position, | the distance which the particle has moved along the edge of the lamina, a the angle which this edge makes with the inclined plane, & the inclination of the latter, m the mass of the particle, and M that of the lamina. The kinetic energy of the lamina at any time is ^ Mx 2 , and that of the particle is \m { (x + £ cos a) 2 + £ 2 sin 2 a} . Hence 2T= (M + m) x- + m£ + 2mx £ cos a. Again, — V = Mgx sin /3 + mg { (x sin £ + | sin (a + j8) } , and therefore the equations of motion are (M + m) x + m'( cos a = (M + m) g sin j8, m (£+ x cos a) = mg sin (a + &), whence ( . m sin a cos a cos £ ) . j9 (M -4- m) sin a cos £ 2 J \ M+m sm ? - a ) a Jf + w sin 2 a 300. Ignoration of Coordinates. — If there be no force tending to alter one or more of the independent variables by which the position of a system is defined ; if moreover the expression for the kinetic energy of the system does not contain these variables, bnt only their differential coefficients ; and if the system start from rest ; then T may be expressed as a function of the other independent variables and their differential coefficients, and be treated as if these latter vari- ables completely denned the position of the system. Let & be one of the independent variables satisfying the conditions supposed ; then, as there is no force tending to alter &, dT ; and therefore -r- = constant ; d\ x also as the system starts from rest, and T is a homogeneous quadratic function of f 1, £ 2 . . . tm this constant must be zero ; dT hence — - = 0. In like manner, if £> be another variable d%x dT satisfying the same conditions, we have —r- = 0, and so on. dc,^ d dT dt df, Ignoration of Coordinates. 427 dT dT • • From the linear equations —r- = 0, -r = 0, &c, £1, £ 2 , &c, can be found in terms of the remaining differential coefficients 2^, . . . % n . Thus T becomes a function of if g . . . £ n , and of their differential coefficients, that is r = i?(5 fi ,? ?+1 ,&c., ?„&c). If now we regard f ? , £ ff+1 , &c. as completely defining the position of the system, Lagrange's equations are d dF dF_ It d$ q ~ « c " S *' &C * ; but these equations are true, for dF dT dT d& dT d& ' —r = —7- + — r — - + — - — ~ + &C, rf? f di t d& d% q di> di q clF _dT_ dT dt f^4i + &c . d% q d% q d% x d% q d% % d£ q whence, as — - = 0, — r = 0, &c, we have clF_dT clF_dT dtq d\q d%q ^ q The proposition proved above is given by Thomson and Tait {Natural Philosophy), and is the simplest case of what they have termed Ignoration of Coordinates. Examples. 1 . A particle descends from rest along one face of a smooth triangular prism which is supported by a smooth horizontal plane. The initial position of the particle lies in the vertical plane containing the centre of inertia of the prism and perpendicular to its edge ; determine the motion. Let x be the horizontal coordinate, in the vertical plane in which the particle moves, of the centre of inertia of the prism, M its mass, m that of the particle, 428 The General Equations of Dynamics. I the distance it has moved at any time along the face of the prism, and o the angle which this face makes with the horizontal plane ; then 2T=(M+ m) x 2 + mi? + 2mx\ cos a, V—— mgl sin a ; and the equations of motion are (If + m) x + mi cos a = 0, m\ + mx cos a = mg sin a. Hence, as the particle starts from rest, (M + m) x = ~mi, cos a, (M + m sin 2 o) f = (M + m) g sin a. The student will observe that if T were expressed by means of the first of these equations as a function of £ alone, and treated as such, the second equation would be obtained directly as Lagrange's equation. 2. In the preceding example, if the face of the prism down which the par- ticle descends be rough, determine the equations of motion. The force of friction tends merely to stop the relative motion of the particle and prism ; hence, F being this force, F8f= - juP5|, where P is the perpen- dicular pressure of the particle on the face of the prism. Now P = m(g cos a + x sin a), and therefore the equations of motion are (M + m) x + m\ cos a = 0, m\ + mx cos a = mg (sin a — ft cos a) — ixmx sin a. The latter of these equations can be reduced to the form £ cos X + x cos (a - A) = g sin (a - A), where tan x = [x. 3. A sphere, having no motion of rotation, and under the action of a force passing through its centre of inertia, moves through a liquid extending indefi- nitely in all directions on one side of an infinite plane : the liquid being origi- nally at rest, and not acted on by any force, determine the form of the equations of motion of the sphere. Let the origin be anywhere in the fixed plane, the axis of x being at right angles to that plane ; and let x, y, z be the coordinates of the centre of the sphere at any time, and £ a coordinate of any particle of the liquid, which may be defined as matter which is incompressible, devoid of resistance to change of shape, and incapable of exercising any friction against a surface with which it is in contact. dT If T be the kinetic energy of the whole system, we have —p = C, since there is no force acting on the liquid ; but as the liquid was originally at rest, and no impulse was imparted to it, C = 0. Hence_ T is a function of x, y, z, x, y, z. Again, the motion of the system at any instant could be pro- duced from rest by placing the sphere in its actual position, and giving it an impulse sufficient to impart to it its actual velocity, since the impulses which should be given to the liquid particles are zero (10), Art. 294. Hence, asthe initial circumstances are unaltered by changing the values of y and z, I 7 is a function of x, x, y, z. Again, a change in the sign of y or z can make no change Components of Momentum and Velocities. 429 in the value of T, which must therefore he of the form ^{Px 2 + Q (y 2 + i 3 )}, since the coefficients of xy, yz, zx must he zero. The equations of motion are then Qu +■ -f *y = 1, Qz + -^xz = ^, 4. Prove that a sphere projected through a liquid perpendicularly from an infinite plane boundary is at first accelerated, and then tends towards a con- stant velocity. Show also that if projected parallel to the boundary it moves as if it were attracted towards the boundary. Initial circumstances in Ex. 3 are altered in the same manner, whether we suppose introduced into the liquid a second bounding- plane parallel to the first, and between it and the sphere, or suppose the sphere placed initially nearer the original bounding plane. Hence a diminution of the initial value of x is equivalent to the introduction of additional geometrical constraints into the system. From this it follows by Bertrand's Theorem, Art. 296, that if x' < x, and P'x' = Px, the value of Px 2 must exceed that of P'x' 2 , and there- fore x < x, and P" > P, or P decreases as x increases. Similar reasoning can be applied to Q. If x be infinite, or the liquid unbounded in every direction, P and Q are constants. The statements made in the enunciation of this example follow then imme- diately from the equations of Ex. 3, by making X and Y zero. Examples 3 and 4 are taken, with some slight modifications, from Thomson and Tait {Natural Philosophy). 301. Components of Momentum and Velocities. — Equations (10), Art. 293, enable us to express the velocities Ei, &c. as linear functions of the components of momentum Pi, &c. If these values be substituted for Ei? &c. in T, as given by equation (7), a new expression for T is obtained which is a homogeneous quadratic function oip ly p 2 , . . . p n . We shall represent the two expressions for T by T* and T p . Equation (7), Art. 293 gives jT>, and the corresponding equation for T p is of the form T p = Pnpc + P22P2 2 + &c. + 2P 12 p 1 p 2 + &c. (28) In this equation P n , P 22 , &o. are functions of Ei> fe> &o. Thus Tz and T p are each functions of Ei, ?2, &c. ; but these coordinates, so far as they appear explicitly, do not enter in the same manner into the two expressions for T. Equation 430 The General Equations of Dynamics. (14) gives an expression for T which is symmetrical in £1 and pi, &c, and which becomes T$ or T p according as we express p h &c. in terms of % Vl &c, or &, &c. in terms of p„ &c. If we seek for — from equation (14) we obtain dpi 2-r-= gi + S^ — . (29) > Again, if we seek for — from (7) we have dT dTdt dT df, o v 4 ,o ft v = — — + — ; + &C. = 2^ • (OO) dpi d% x dp x f/? 2 dpx dp, dT Substituting this value for — in (29) we get n dT * dT dT t 2 -—=?! + — , whence — = \\\ dp x dp' dp x and as a similar result holds good for each component of momentum, we have dT g f^ t dT * The partial differential coefficients of T with respect to £ l5 &c. are different according as T is expressed by T^ or T p . dT If we seek for -~ from equation (14) or (7) we must m each case regard &, &c. as functions of #, ^ 2 , &c. ; &, ? 2 , &c. In this way we get from (14), *%-»%**%***- (32) Hamilton' '$ Equations of Motion. 431 and from (7) alTp dTi dTi dt dT t d% % — = — - + -4 — + —4 — + &c. d& d& rff, rff, d& rff, + Pi -Tg- + i? 2 -r=- + &c. (33) iff, J d& " <% Hence, by (32), * = _f + 2 _£, and therefore £ + I _ q We have then the system of equations dT p dT, clT p dT, dT p dT, It is plain that the reciprocal relations between compo- nents of velocity and momentum are analogous to the polar properties of curves and surfaces. 302. Hamilton's Equations of Motion. — If we put T p + V - U, we obtain a function 27" of Pi,pz, &c, £u ? 2 , &&., which represents the total energy kinetic and potential of the system, and whose value is constant. By the employment of U Lagrange's equations of motion may be expressed in another very symmetrical form due to Hamilton. -d /ian a l nnA d dT dpi dT dTp By (10), Art. 294, -^ = g, and by (34) - ^ = -g. Hence Lagrange's equations (22) become f + ^0,f + § = o,...% + ^-0. (35) dt d% x dt d%i dt d% n v ' Equations (35) have been termed The Equations of Motion of a system expressed in the Canonical Form. 432 The General Equations of Dynamics. It is easy to see that tlie equations which give the motion of the centre of inertia and the changes in the moments of momentum for any system are particular cases of equa- tions (35). Examples. 1 . In a moving system the total elementary change of momentum corre- sponding to one of the generalized coordinates is made up of two parts, one resulting from the forces acting on the system, the other from the previously dT existing motion. Show that — - dt expresses the latter, | being the generalized d\ coordinate. If », &c. be tha impulses which would give the existing velocities at any dT . (dTV , instant, — = p. At the next instant ( —v- ) =p . d£ \«| / From these equations it appears that the total elementary change of mo- mentum p' — p corresponding to £ is dT\' dT d dT J at. fdT\ ' dT d_dl Ui / «*£ or * 4 Now, by Lagrange's equations d dT , dT , — —dt = s.dt + — dt dt dt, di whence, as s.dt represents the change of momentum resulting from the applied dT forces, — dt must represent that due to the previous motion. d\ 2. Apply the method of the last example to determine the components of the centrifugal couple in the case of a body having a fixed point. Here IT - Aai 2 + Bootf + Cwi 1 . If now m, wz, a>3 be expressed in terms of e, , $ ; e, doo2 d

%. d(j> 3. If the Cartesian and generalized coordinates be connected by linear equations with constant coefficients, show that there are no terms in the equations of motion resulting from the previous motion. Calculus of Variations. 433 303. Calculus of Variations. — In the Calculus of Variations the form of the function which determines the dependent variable y in terms of the independent variable x is supposed to vary, and zs being the symbol of a given operation or set of operations, the fundamental problem of the Calculus is to determine the variation of zsy. If y =/(,r), a change whose magnitude is infinitely small in the function / (x) must be of the form i\p (x) , where i is an infinitely small constant. We have then By = ty (x). In consequence of y becoming f(x) + i\p{x), the differential co- m ■ 4. ^ \> d>[f \d>^ efficient — - becomes — - + i -—■. dx n dx n dx n Hence we have o -r-= = —j-r- U>b) dx n dx n If Q = i*»£-2i the variation £12 is the change in £2 in consequence of y changing from/(.r) to f(x) + i\p (a?). As the result of this change of y the function F becomes F + SF, where sw- dF x dFdSy dF d n $u ~dy J+ ~JW\ dx "" + d (d»y\ dx» ' \dx) and 12 becomes j Fdx + f SFdx. Hence we see that $Q = d$Fdx = fSFdx. (37) In the case of a definite integral whose limits are variable the complete variation is the sum of two parts, one resulting from the variation of the limits, the other from the variation of the expression under the integral sign. Hence if 12 = Fdx, and if D£2 be the complete variation of 12, we X have DQ = F"dx" - F'dx' + SFdx. (38) In general the complete variation Du of a dependent variable u is the sum of two parts, one resulting from a change of the independent variable x, the other from a 2F 434 The General Equations of Dynamics. change in the form of the relation connecting u with x. In the Calculus of Variations the symbol 8 is restricted to varia- tions of the latter kind. Hence, in general, Da = C ^dx + hi. (39) ax Examples. 1. A particle under the action of gravity is constrained to move from one given point A to another B along a smooth plane curve ; determine the nature of the curve so that the time of descent may be the least possible. The curve obviously lies in a vertical plane passing through the points A and B. Let the axes of x and y be a vertical and horizontal line in this plane, the positive direction of x being downwards, and let v be the velocity of the particle in any position, then, if the origin be properly selected, d s v 2 = 2y x, and therefore dt = . yflg* dy ax Hence, if a = j * J~^f dx > wnere we have to determine y as a function of x so that n may be a minimum, and therefore 5n = for all possible variations of y. Now 8n= p p ■ «r fc hence, integrating by parts, and neglecting the terms outside the integral sign , since y\ and yo are given, and therefore 5yi = dy = 0, we have p d_t P ) 5ijdx = , ho dx W'2gx(l+p~)/ but 8y being arbitrary, this equation cannot be true for all values of Sy, except ^ ,- * =0. dx N-lgx (1 + pf Integrating, we have p 1 = 2yc 2 x (1 + i^ 2 ). 1 • o dy If we put — -„ = a. and p = tan 0, we get x - a sm 2 0, — - = tan 0. 2gc~ dx Aeain — = — — = 2a sin'-0 ; hence we obtain, as the equations of the ° dO dx dd curve, x = a sin 2 tf, y = a (0 — sin 6 cos 0) + b, where a and b are arbitrary constants. The curve is therefore a cycloid {Differential Calculus, Art. 272). This problem is one of great interest in the history of Mathematics, as its proposal by John Bernoulli in 1696 led to the invention of the Calculus of Variations. Examples. 435 2. Prove that for any system of coplanar forces the curve of quickest descent is such that at each point the pressure on the curve due to the forces is equal to that due to the motion. Here n = j ,. dx ; hence, putting 5n = 0, have, after integrating hy parts, dx\v\/l+p 2 J v 2 dy T£ « A v L - i d /sinfl\ 1 dv It we put p = tan 0, this equation becomes — I ] 1 dsind sin (dv dv\ that is, — ( — + tan — v dx v- \dx dy J cos 9 dy ^sin0 sin (dv dv\ 1 dv — =0, COS f/y tf0 1 (dv . dv \ whence cos — = - ( — - sin — — cos . vr a dx ■ n (h J wu x ^ (dv dy dv dx\ jy ow cos = — , sin = — , and therefore v 2 — = v ( — — — 1 ; ds ds ds \dx ds dy ds J dd also — - = p, where p is the radius of curvature, and mv~ = 2 $(Xdx +Ydy) ; hence, substituting, we obtain mv~ dy dx — = X l — p ds ds which proves the theorem in question. The curve of quickest descent is called the Brachystochrone. The propo- sition here proved is a case of a more general theorem in the Calculus of Varia- tions, for the discussion of which the reader is referred to Jellett's Calculus of Variations, p. 140, or to the Encyclopaedia Britannica, vol. 24, p. 86. 3. Deduce Lagrange's equations of motion in generalized coordinates and the corresponding equations for impulses from D' Alembert's Principle by means of the Calculus of Variations. If x, y, z be the coordinates of any particle m, T is given by the equation T = 2/« {x 2 4- y- 4- z 1 ) ; but T can also be expressed as a function of the gene- ralized coordinates |i, &c., and velocities £i, &c. As these two expressions for T are always identical, so also are the expressions for jdTdt derived from them ; we have therefore f / dZx . dZy . ddz\ , f (dT dT dlh „ \ f 5 "> (• Tt + »* ' + -" * ) dt = j (se 5fl + W> ^ + &c ') "'■ If we integrate by parts each side of this equation, the terms remaining under the integral sign on one side must be equal to those remaining under that sign on the other, and a similar equality must hold good for the terms outside the integral sign at each limit. Hence we have Id dT dT\ Id dT dT\ dT' dT' and -t-t S£i' 4- -rs-r 5£ 2 ' 4- &c. = 2w (%' 5x' 4- y 5y' 4 z'5z'). «|i a| 2 2 F 2 436 The General Equations of Dynamics. Since the limits are arbitrary the latter equation may be written dT dT 7T 5 £i + -JT 5 £a + &c. = %m (x5x + y$y + zSz). «£i d\i If we now employ D'Alembert's Principle, the equations of motion are immediately obtained. 304. Iieast Action. — The integral J" 2 Tdt taken between two given configurations of a system is termed the Action of the system in passing from one of these configurations to the other. If we denote the action by A, we have the equation A = 2 Tdt, (40) where f and t" correspond to the initial and final configura- tions of the system. If v be the velocity, m the mass, and s the path of any particle of the system, it is plain that A may be expressed also by the equation A = S/rc vds = Sm (xdx + ydy + zdz), (41) where s and s" are in any individual motion the values of s for the particle m in the initial and final configurations. The Principle of Least Action asserts, that subject to the condition imposed by the equation of energy the mode in which a conservative system passes from one configuration to another is such that the action is a minimum. The equation of energy is T + V = E, where E is con- stant, and V a given function of the coordinates. This equation determines T as a function of the coordinates, but not v the velocity of an individual particle. Hence the value f s " of vds depends not only on the initial and final positions of the particle, but also on the relation which in any individual actual motion exists between v and s. If we consider the ex- pression for A given by (40) it is plain that the value of A depends on the equations which are supposed to determine the coordinates in terms of t in any individual motion of the system, and the Principle of Least Action asserts that in the actual motion of the system these equations are such as to render A a minimum. The student should observe that Least Action. 437 in (40) the limiting values of t are not given. In fact, when the initial and final configurations are given the correspond- ing values of t depend upon the actual motion of the system. To show that A is a minimum in the actual motion we must suppose the forms of the functions by which cc, &c, are expressed in terms of t to vary, and prove that the consequent variation of A is zero. We have then by (38) DA = 2T"dt" - 2T'dt' + j2STdt. Now ST + $V= 0, and therefore we get DA = 2T"dt" - 2T'dt' + J (ST - dV) dt; also, since 2 T = 'Em (dr + if + s 2 ) , we have $r= 2m (i£i + y§y + z$z), hence STdt . .d$x . dh/ . dSi . - + my J Sy + f^?+ miA &s J «. (43) If we integrate each term by parts, and substitute in the expression for DA, we obtain DA = 2T"dt"-2T'dt' + 2w(i"&*>" + jTW+ *"&") - 2f»@W + y V + s'&0 Now by D'Alembert's equation the part under the inte- gral sign must be zero, and therefore if the part outside that sign be likewise zero, we have DA = 0. But 2T'dt" + 2/rc {x" Ix" + y'ltf + z" Ss") = 2m{x" (x'dt" + &*>') + y" (fdt" + Bf) + z"\z"df + $z") J , and df'dt" + dx", &c. are by (39) the complete variations of %", &c, and therefore must each be zero, since x'\ y\ s", &c. are invariable, being the coordinates of the particles of the system in its final configuration, which is given. Hence, as similar results hold good for the other limit, we obtain DA = 0, and therefore may conclude that A is a minimum or a maximum. 438 The General Equations of Dynamics. If the potential energy of a system be given as a function of the generalized coordinates, the Principle of Least Action enables us to arrive at its equations of motion. To obtain the equations of motion in this manner we must seek to determine the generalized coordinates as functions of t in such a way as to make A a, minimum, subject to the condition that T + V = constant. This condition gives ST + SV= 0, and therefore if A be an indeterminate quantity we must have, when A is a minimum, DA+j\($T+SV)dt = 0. (44) In this equation the variations £|i, &c. may be regarded as independent and arbitrary, provided we can determine A so as to satisfy the equation T + V = constant. If we substitute ^ ^ + — S& + &c. for ST and d£ x dt d|i -rs- Sgi + &c. for SV in (44), we get, after integrating by parts, for the terms under the sign of integration, , Q V r/T .dV d (2 + A } — + A d& d& dt (2 + A) (IT dt S& + &o. <#. Hence, as the part under the integral sign must vanish independently of the terms outside that sign, and as <5£i, &o. are independent and arbitrary, we have the system of equa- tions /0 (dT d dT\ dV (2 + A) — ■ + A — \rf£i dt dlJ dKi 4i dt dT d\ _ ~\ (2 + A)(^ 1 !*? &c. dV (IT dX + A— -^-— =0 | d% 2 dt dt > (45) If we multiply the first of these equations by fi, the second by £>, &c. and add, we have Wit d_dT\t dt 4 / AS IT (46) Hamilton's Characteristic Function. 439 Hence, by (25) and (13), we obtain ,<> + K dT + \ dV 9T (/X -0 - [2 + 1) Tt +X lf' 2T dt-°' thatls ' rfF - 2— V It " 27a It = °- < 4 '> This equation becomes the same as the equation of con- dition T + V - constant, provided A = - (2 + A), or A = - 1. Equations (45) then become the same as Lagrange's Equa- tions (22). It is easy to see that if A =- 1, the terms outside the sign of integration in (44), after integrating by parts, vanish of themselves when the limiting values of £u £ 2 , &c are given. Some eminent mathematicians have deduced the equa- tions of motion from the Principle of Least Action in a strangely illogical manner. 305. Hamilton's Characteristic Function. — The motion of a given system having n degrees of freedom whose potential energy is a given function of the coordinates is completely determined if the initial values of the generalized coordinates and velocities be given. At any subsequent un- determined time t we have n equations connecting t with the corresponding values of the coordinates and the 2n quantities previously assigned. If t be eliminated from these equations n — 1 remain. Again, the kinetic and potential energies are at any time connected by the equation T + V = E, which gives another relation between the 2n assigned quantities. Hence we conclude, that if the initial values of the coordi- nates be given, and also their values at any subsequent undetermined time, along with the total energy E of the system, the motion is completely determined. It follows from what has been said that the action A of a system in passing from one configuration to another is a determinate function of the initial and final values of the coordinates and of the total energy. This function is called by Hamilton the Characteristic Function. Whenever it can be assigned it furnishes us with the first and second integrals of the equations of motion, as we proceed to show. 440 The General Equations of Dynamics. Suppose each of the initial and final coordinates, as well as the total energy of the system, to be slightly altered, then each coordinate, at any intermediate time, receives a corre- sponding variation, and so likewise does T, the kinetic energy of the system. Now A = 2 J Tdt, and therefore 8 A = J" 28TM ; but BT+SV= Whence $A=j(ST+$E-$V)dt. (48) If in this we substitute for jSTdt its value given by (42) and integrate by parts, we find, as in (43), that the part under the sign of integration must, in virtue of D'Alembert's equation, be zero. Hence $A must consist entirely of the terms outside the sign of integration. To ascertain what these are when T is expressed as a function of the generalized velocities and coordinates, we must put for ST in (48) the expression ^IT dTd 3 \d% d% dt Since SA as shown above consists entirely of the terms outside the sign of integration, if % l9 g 2 , &c, ?/, £ 2 ', &c, be the final and initial coordinates, we obtain thus 8A=(t-t')$E+~ S& + ^8g, + & ._f!*£sEi / + ^8&'+&o.\ dt d^ U/ dli J dT • Now DA = 2Tdt - 2T'dt' + BA, and 2T = S -4 ?, d% hence by (39) we get DA = {t- if) $E + pMi +P-M, + &o. - (pi'DE/ +P2D& + &o.) where ^1, &c. have the same meaning as in (10). Again, A being supposed to be expressed as a function of the initial and final coordinates and total energy of the system, we have _. dA y dA y B dA y , dA r , p dA ~ -, DA= — Dgi+— DL+&0.+ — D?i + — i>s2 +&c.+ — S-#. arbitrary, we get dA dA dA d& «-*•■ ' dl n ~ dA dA dA wr~ Pi > dE~ t t ' Hamilton' 's Characteristic Function. 441 Comparing the two expressions fori) A, and remembering that Z)£n Dt, 2 , &o. D?/, Df/, &c. and BE are independent and ft; (49) --*.'; (50) (51) Equations (49) and (51), if E be eliminated, furnish ex- pressions for |i, £ 2 , &c, in terms of the coordinates and the time, in other words, the first integrals of the equations of motion. Equations (50) and (51), if E be eliminated, enable us to express the coordinates themselves as functions of the time, and so furnish the second integrals of the equations of motion. In each case the initial coordinates £/, &c, and components of momentum pi, &c, are supposed to be given. It is to be observed that if we desire to have the first inte- grals in their usual form, in which the arbitrary constants are determined from the initial velocities, we must employ all the equations (49), (50), and (51), and eliminate £i, &c, as well as E. In the case of a set of free particles, equations (49) and (50) become dA . dA . dA . dA . Q /tox * = '"'*" fe = myi > 25 - " hZl > dT 2 = mH ' &0 - ; (52) dA ., dA ., dA ., dA . J2 = -"* - ^ = -»* . s> = -«•* . sj = - «* . &0 - («») The function ^1 satisfies certain partial differential equa- tions by which it may sometimes be determined. These equations are obtained thus : — Multiply the first of equations (49) by &, the second by £», &c, and add, and we have J A . rl A . ~ s. + 4- & + &o - = 2T= 2 c* - n- (s*) «S1 "S2 442 The General Equations of Dynamics. In like manner, from (50) we get rl A • rl A • ^ ^ + «£ & , + &c# = _ 2 r = 2 ( r - ^). (55) o%( dl % In equation (54) we must remember that £i, ? 2 , &c. are supposed to be expressed as functions otpi 9 p t9 &c, and thus, finally, as functions of clA clA ~o%? d& A similar remark holds good for (55). In the case of free unconnected particles, equations (54) and (55) take the simple forms, 2- m MQHfMS)]-^-' '• m 4PH£M»)'}- 3 <^ 157 > Examples. 1 . Find the characteristic function, and the initial and final integrals in the case of a hody falling vertically. Here there is only one coordinate, z the height of the hody from the ground. Since gravity tends to diminish z, the potential energy V = mgz> and E = T + mgz. We have, then, =(£)■-•<■-- >• =(»)->«»-^ where z is the initial height. If we attribute the negative sign to the square root in the first of these equations, we get, by integrating, 2,cj \ m j In this equation C is a function of z, and is to be determined from the second differential equation for A. Remembering that A must vanish when z = z\ we get finally Examples. 443 We have, then, dA \2{E-mgz) ., , dA J2(E ms=Pl = -d7 = - m V— iT - ' "" = Pl = - H = ~ m <~ mgz') dA I ( ( 2(E- mgz) \ \ _ / 2(E-mgz') \ § | ~d~E~g\ \ m / ~ \ w / i ' If we eliminate E and z from these three equations, and put z" = - v', we get the ordinary first integral of the equation of motion in which the initial velocity is supposed to he given. If we merely eliminate E between the last two of the above equations, and put z' = — v' , we get the ordinary final integral. t~ The resulting equations are £ = — {gt + v). z — — g — - v't + z'. The signs which we have attributed to the square roots correspond to the motion of a falling body projected vertically downwards. The results which hold good in the other cases of the motion of a body falling vertically are deduced from the general equations by giving the proper signs to the square roots. 2. A material particle is acted on by an attractive force passing through a fixed point, and varying directly as the distance ; find the characteristic function. Let m be the mass of the particle, and fir the magnitude of the force at the distance r, then dV 3 rr A*, o ~~ ~dx = ~ ftX ' 2 * + r ) ' Hence we have (a , +(S) , - tM, -^ + ^ > - w If we assume the equation (a) is satisfied, provided ci + c 2 = IE. (e) Since the differential equation to be satisfied bv 77 and -— is similar to (a), ' dx dij and since A must vanish when x = x' and y = y', we have 444 The General Equations of Dynamics. In this expression for A the constants c x and c% are subject to the condition ci + c% = 2E. In order that A should he expressed as a function of x, y, x\ y\ and JE, a second equation connecting c\ and c* with these quantities is required. This equation is, in fact, x A— — sin- 1 x'l—= sin- 1 y J sin -1 y'J- \ Again, dc\ + dc% - 0, since c\ +C2 = 2.E', and therefore we have — = — . aci aC2 Hence the required relation between c\ and c 2 must, in virtue of (c), be capable dd> deb _, „ ffo , ^ of being expressed in the form, -f- = - z . The expressions for — and — - are found most easily from (b). From these equations we have dA /- / c?L4 a/w — = Y m \/ ci — /j. x £ , whence Integrating, we have dA ,- [ x dx \m / U . . , Ifi \ In like manner dA \m I . , L . , , lju.\ . — =7 A - sin m/. - — sin l y \ - ) > dco 2 \^ V >^2 y >c 2 / , . + «& + gh = 0. (2) Now, for a small oscillation x must be small throughout the motion, and consequently s is a small quantity of the second order. 446 Small Oscillations. Hence, to the degree of approximation required, we have adz = x$x, and. az = xx ; therefore az = xx + or ; we may accordingly neglect z, and equation (2) becomes x + g - x\ Ix = 0, or x + 9 - x = 0. The integral of this equation is * = *sin(^ + X ), (3) as in Art. 102. In like manner, if any curve be taken instead of the circle, its equation, referred to the tangent and normal at its lowest point, may be written 2s = c % 2 + 2c Y xz + 2c 2 z 2 + &c. Accordingly, neglecting terms of a higher order than the second, we have gs = c&$x, and it is readily seen that z may 1 be neglected as before ; also observing that e = - {Biff. CaL, Art. 230), where p is the radius of curvature at the origin, we get immediately from (2), /*• sin f t ■IH- This shows that in all such cases the motion is represented by a simple harmonic function. 308. Oscillation on a Smooth Surface. — We shall next consider the case of a small oscillation, under gravity, on a smooth spherical surface. Taking the origin at the lowest point on the sphere, and the z axis vertical, the equation of the sphere is 2az = x 2 + f + z\ (4) Also, from D'Alembert's principle, x§x + i/hj + z$z + g§z = 0. (5) Oscillation on a Smooth Surface. 447 Here we may neglect z 2 and z as before, and thus we obtain immediately (* + f.)& + (# + j[ y )*-o. Hence x + — x= 0, j) + - y = ; c i ci accordingly we have x = m sin [t^ g - + x \ y = n sin [t^ + x . where m, n, ^ 1? X2 are arbitrary constants. These equations may also be written in the form x = a sin t I- + a cos t /- \a \a ,j = j5 sin we shall have the terms where jui = - Ai» Oscillation on a Smooth Surface. 449 The motion will then not be a small oscillation, as this expression will increase continually with t, unless in the exceptional case where hi = 0. Again, if R l and R 2 be the principal radii of curvature at 0, it is readily seen that A - 9 \ - g For let the ellipse ax 2 + 2hxy + by 2 = c be transformed to its axes, so that ax 2 + 2hxy + bif = a'X 2 + b'Y 2 ; then, since a + b = d + b', and ab - h 2 = a'b', the equation (ga - \){gb - A) - cfh 2 = becomes - {ga - A) {gb f - A) = 0. The roots of this equation are gaf and gtt ; but, as in Art. 307, we have «- = ! 6 '=i R\ H2 accordingly for a small oscillation both R x and R 2 must be positive, i.e. the surface must be convex towards the plane of xy. If Ai = A 2 , we have Ri = R 2i and the origin is a point of spherical curvature. In this case a small oscillation is the same as on the surface of a sphere, and is given by equations (6). Examples. 1 . A bar of mass m hanging freely from one extremity is slightly displaced ; determine its motion. Take two horizontal lines at right angles to each other passing through the point of suspension for axes of x and y. Let the small angular displacements of the bar at any time round each of these axes towards the other be 6 and

80), 450 Small Oscillations. thus we may neglect z, and have Sdmr- {6 56 + + ff-(p = 0, (■[v'V + Xji <|) = ^sin^v / ^+x2J. 2. Two balls connected by a horizontal bar, whose mass may be neglected, are suspended by two vertical cords of equal length. The bar receives a slight displacement of rotation round a vertical axis midway between the cords ; find the motion of the system. Let if/ be the angle which the bar makes with a horizontal line parallel to its initial position, 6 the inclination of one of the cords to the vertical (see figure in Ex. 4, Art. 244), I its length, b the distance from the middle point of the bar to one of the balls ; then x = bcosip = b{l-lf~), y = bi/, z = I (1 - \6 2 ) ; 5 2 „\ but » = ty; .-.z = l(l-±-rpA, and x' = - x, y' = -y then x, x, z, z' may be neglected, and equating to cipher the coefficient of 5\p in D'Alembert's equation, we have £ + |* = 0; therefore {&+*) where a and % are arbitrary constants. This shows that the period of vibration is the same as that of the pendulum whose length is I. 3. A heavy bar is suspended and displaced as in the preceding example ; investigate its motion. Let r be the distance of any point of the bar from its centre, and b the distance from its centre to the point of attachment of one of the cords ; then, as in the preceding example, J 8 \p JY 2 dm + g — i// jdm = ; therefore \f = a sin ( 7 -Jt * + X )> where J> 2 dm mk- Stable Equilibrium. 451 4. How must the bar in the preceding example be suspended in order that its vibrations should be isochronous with those of a ball hung by one of the supporting cords ? Ans. b = k. In the case of a homogeneous bar whose length is 2a V3 5. A uniform rod of mass m hangs from a horizontal pivot passing through one of its extremities. An inextensible string, whose weight is negligible, attached to the other extremity, passes through a smooth ring situated on the vertical line through the pivot at a distance below it equal to the length of the rod, and sustains a mass p. The rod being slightly displaced from its position of equilibrium, determine the motion. The equations of motion are % ma 2 9 = - mga sin 9 — 2a T, p z = T — pg, where a is half the length of the rod, and z the vertical coordinate of p. If z be measured from the position of p when the rod is vertical, z = 4a sin ±-9. Since 9 is always small, we may take sin 9 = 9 ; substituting for z and elimi- nating T, we have | a{m + 3p) 9 + nig (9 + — \ = 0. Hence the rod returns to its vertical position in a time l[ 4a (u 3p )} coa -i 2p \j 1 3g \ m J ) 2p + m9o where 0o is the initial value of 9. 309. Stable Equilibrium. — A position of stable equi- librium is one from which a system has no tendency to depart far if it be slightly disturbed. In a conservative system if the potential energy be a minimum the corresponding position is one of stable equi- librium ; as may be shown in the following manner : — From equation (4), Art. 282, we have T + V - V = T . Now since T = \ ^mv 2 it is always essentially positive ; also, V being the minimum potential energy, V - V is positive for all small values of the variables, and may therefore be reduced to a number of squares with positive coefficients. Therefore if T be small, each term both of T and of V - V must be small, and must always remain so. Hence, if the original disturbance be slight the system can never depart far from the position of equilibrium nor attain a high velo- city. The position is therefore one of stable equilibrium. 2 G 2 452 Small Oscillations. 310. Equations of Motion for an Oscillating Sys- tem. — In the following investigation of the small oscillations of a system ahout its position of equilibrium, it is assumed that the forces which act at the different points of the system are functions of the coordinates of those points, and that the constraints and mutual connexions can be expressed by means of equations between the coordinates. In virtue of these equations the coordinates of the points of the system are functions of n independent variables, and these again are at any time functions of their values in the position of equilibrium, and of the increments resulting from the disturbance from this position and subsequent motion. If the system perform small oscillations the increments of the variables are all small quantities, whose squares and higher powers may be neglected. Hence the equations of motion involve only the first powers of the variables and of their differential coefficients. In other words, they form a system of linear differential equations with constant coefficients. Let ai, a 2 , ... a n represent the values of the generalized coordinates in the position of equilibrium, and & c -> and substitute in the equations of motion, we obtain the n equations (/nX - ?ii) «i + (/12X - q u ) a t . . . + (/mX - ? 1M ) a n = \ (/12X- 0ia) 0] + (./ 22 X - g 22 ) 0, . . . + (/ 2M X - ^ a n = (/inX - qm) «l + (/anX - ? 2H ) flfe . . . + (/„„X - ^ nn ) «» = (19) These can be satisfied by the ratios of the n determinable constants « 1? a 29 . . . #», provided X is a root of the equation ,/nX - £11, ,/i 2 X - qu 9 . . . fmA - q xn /12X — ^12, / 22 A - § , 22> • • • y 2nX — q% n ,/i«X — g'm, ,/o M A y «»X — gv = 0. (20) The symmetrical determinant which enters into this equation we shall call A. It is usually termed the harmonic determinant of the motion. If the roots of the equation A = be all real and positive, and be denoted by A,, A 2 , . . . X», the complete values of Si, £2, &c., are given by the equations !i = /ci0nsin(*VAi+xi)+" : 2«i2sin(*VA.2 + X2) • • • + «««in sin (*VaI +%»)' I2 = «i«2i sin (t Vai + xi) + «2«22 sin (£ Va 2 + X2) • • • + «n«2n sin (ty/ A„ + X«) In = «1%1 Sin (^Ai + x0 + K 2«;«2 sin (*Va2 + X2) • • • + Knflnn sin (W\ n +Xn) t ,(21) Lemma in the Theory of Determinants. 455 where ki, %i? *2> X 2 > & c - are arbitrary constants, 2n in number, and On, a 2 i, . . . a nl satisfy the n linear equations for a lt a 2y . . . a n obtained by putting A x for A in (19) ; a l2 , a 22i . . . a n2 those obtained by putting A 2 for A ; and so on. If any root of the equation A = be real and negative, instead of Kia n sin (t */\ 1 + ^0, there will be in & a term of the form dn{icie u i^ + /ciV^'fTij, where fi x = - \ x ; and there will be corresponding terms in £ 2 , £ 3 , &c. In fact if we substitute klOl^IL for £i> KxOt^'v- for ? 2 j and so on in the equations of motion, we get a system of equations which differ from (19) in having - fi instead of A, and which can therefore be satisfied by «i : a 21 &c. pro- vided - fi be a root of the equation A = 0. Corresponding therefore to every real negative value of A there is a real positive value of /u. In this case, since % l9 £ 2 , &c. contain in general terms increasing without limit with the time, the motion cannot consist of small oscillations. If we suppose a u a 2 , . . . a n substituted for &, ? 2 , . . . % n in T, and for % u ? 2 , • • • S» in F, and denote the results of these substitutions by T' and V\ equations (19) may be written ^(Ar-n-o,^(xr-n-o,...^(Ar-n-o. (22) 312. Lemma in the Theory of Determinants. — If A be any determinant, and if the determinants obtained by erasing the first row and first column of A, the second row and second column, the first row and second column, the second row and first column, be denoted by A n , A 22 , - A 12 , - A 21 , and if also the determinant formed by erasing the first row and first column of A u be denoted by A im , then it is a well- known property of determinants that A n A 22 - An A 21 = AAi M . (23) For the convenience of the student we shall give here a proof of this theorem. 456 Small Oscillations. If we have the n linear equations a u x x + a 12 Xi + a x3 x :i . . . + a xn x n = y x «2i^i + a 22 x 2 + a^or s . . . + a 2n x n - y 2 a iX x x + a Z2 x 2 + a^x-i . . . + a m x n = y z a nx x x + a n2 x 2 + a nz x z . . . + a nn x n = y n (24) and solve for x x , &c, in terms of y u &c, we get another system of n equations, of which the first two are Ax x = Au^i + A 21 y 2 + &c, Ax 2 = A x2 y x + A 22 y 2 + &c. ; whence, eliminating y 2i we obtain between the n + 1 variables x n x *9 Pn Vzy - • • Vn the linear equation A(A 22 ^i - A 2 i^ 2 ) = (An A23 - A\ 2 A 2X )y x + &c. (25) Again we may obtain a linear equation between the same variables in another way, viz., by eliminating x 3 , a? 4 , . . . x n from the (n - 1) equations got from equations (24) by omitting the second. The result of this elimination is (26) (27) a xx x x + a x2 x 2 - y x , a xz , a u , . . . a ln a 3X x x + a 32 x 2 - y Sf a^, a u , . . . a 3n a n \X\ + a n2 x 2 — y n) # n3 , a n ^ . . . a nn which expanded becomes A 2 2#i - A 2V ^2 = A XX22 y x + &c, Since only one linear equation can exist between n + 1 vari- ables of which n are independent, (27), when multiplied by A, must be identical with (25). Hence we have AA1122 = A11A22- A12A21. In the case of the harmonic determinant, since it is symmetrical, we have A21 = Ai 2 , and therefore (23) becomes (28) Reality of the Roots of the Harmonic Determinant Equation. 457 313. Transformation of the Harmonic Determi- nant. — If we denote the quadratic function of n variables by P'and the function U. Again, when fi, | 2 , &c, are substituted for the variables in «9[it becomes the kinetic energy T of the system. Now, |i, | 2 , being generalized components of velocity, whatever small values be assigned to them, these values will belong to a possible motion of the system. Hence the quantic ^is positive for all real values of the variables, and may there- fore be transformed into the sum of n positive squares. If this transformation be effected we have 2^= m 2 + *•* + *• • • - + nn\ (29) 2© = s U Th 2 + 5 22 )]o 2 + 2s 12 ijin 2 + &c., (30) and the harmonic determinant is given then by the equation A— Sn, — «Si 2 , — §13, ... — Sm A= ~ 8 12 , X - S 22 , - S 23 , ... - S %n m ( 31 J ~~ SlMj ~" $2»j ~ $3», • • . A — S/j^ 314. Reality of the Roots of the Harmonic Deter- minant Equation. — If the first row and first column of the harmonic determinant be erased, and a similar process be applied to the determinant so obtained, and again to the determinant thus formed from it, and so on, we get a series of determinants beginning with the harmonic determinant 458 Small Oscillations, itself, whose degrees in X are n, n-1, n - 2, &c, and which in the present Article will be denoted by A»,A»_i, . . . Ai. It is to be observed that A n » A n -u A n -2 are identical with A, An, A 1122, and that Ai is simply X - s nn - If we place + 1 at the end of this series of determinants we obtain a set of (n + 1) quantities, such that when any one intermediate be- tween the first and the last vanishes, the two on each side of it take opposite signs. When A«_i (that is An) vanishes this appears from (28), and it is plain that a similar equation holds good for any three successive determinants in the series. Its last three terms are, A — S(»_i) (w-i)j — 5 (n-i)» , l, of which the first is negative when X - s nn = 0. If now we substitute + oo for X, each term in the series is positive, and if we substitute - go the terms are alter- nately positive and negative. Hence n variations of sign in the successive terms of the series have been gained in the pro- cess of diminishing X from + go to - go ; but, since when one of the intermediate terms vanishes no variation is lost or gained, a variation can be gained only by passing through a root of the equation A n = 0. In this way, therefore, n variations must have been gained. Hence the n roots of the equation A«, = are real, and a variation is gained in passing through each. From this last observation it follows that when A n first vanishes A«_i is positive, and that it must become negative before A n vanishes a second time, then again become positive before A» vanishes a third time, and so on. Hence the roots of the equation A»_i = separate those of A» = 0. In like manner the roots of the equation A«_ 3 = separate those of A„_i = 0, and so on. If we denote by t & and jB the quantics obtained, from & and @, by omitting all terms containing & the minor determinant A«_i belonging to A» in its most general form, as written in equation (20), is the discriminant of \ST- JB, and the special form of A«_i, considered in this Article, is the Stability of Motion. 459 discriminant of the same quantic after linear transformation. Hence the general and special forms of A»_i vanish for the same values of A, and we conclude that in general the roots of the equation A„_i = separate those of A» = 0. It is obvious that similar considerations apply in the case of A„_ 2 , &c The results in this Article might have been obtained directly for the determinants A n , A»_i, &c, in their most general form by using the conditions which must be fulfilled (Biff. Cede, p. 460) when the quantic ^is always positive. 315. Stability of the Motion. — If we make A zero in the series of determinants A», A«_i, &c. of Art. 314, we obtain a new series which may be denoted by (- l) n B m (- l) n_1 !)„_!, &c., where B n is the discriminant of @, and the remaining determinants, D„_u &c. are formed from B n by a process similar to that employed in obtaining the former series. It is clear, from Art. 314, that the number of positive roots of the equation A ?i = is equal to the number of vari- ations of sign in the successive terms of the series (- l) n B n , (-l^ZU, . . . - A, 1. Hence it follows that if B n , B n . x , &c. be all positive, the harmonic determinant equation has n positive roots. We conclude, therefore (Biff. Calc, p. 460), that in order that the roots of this equation should be all positive, the quantic @ must be positive for all values of the variables, and vice versa. Without assuming the truth of the conditions referred to, we may obtain the same result in another way by employing the following transformation : — We shall suppose that & and @ are of the form given by equations (29) and (30), and that the roots of the equations A = are all unequal. Apply a linear transformation which will change r\x + r)z -f y\,j + &o. into Z\ + ?2 2 + Ss 2 + &c., and at the same time reduce @ to its canonical form Pxfr + P.JV . . . +P»?» 2 . 460 Small Oscillations. In order to show that it is possible to do this by a real trans- formation, assume vi = viZ\ + m"?2 + »h'"?3 + &c. >?» = J?n'?i + W£a + »7n "?3 + &C. (32) where the ratios >?/ : 77/ : m : &c., are determined by the equa- tions *n»h' + SiM* + s ^h • • • + s mVn ~ t\\Vi 512*?/ + 522172' + S23»?/ . • • + S2n r ??/ = AilJ 2 Si«*?/ + S 2n r}2 + Sands' . . - + S nn r\n = AiTfo the ratios if/' : *j 2 " : 17/" : &c by the equations Snifi" + S l2 T)" . . . + SmVn = A 3 »7i" _•//.-" . ~ ft \ ft Si2*h + S22172 . • • + s 2 «r?rt = A 2 ?j2 «i»»)i" + s 2 „ij/' . . . + s„»ij»" = A 2 i?»" , (33) k (34) and so on. From equations (30) and (33) it follows that when \ x and A 2 are unequal, r tt . t rr , t ft , r // a /qk\ For A, (1,/ 1,/' + „/ „/' + &o.j = 1,/' (JjY + W (JJ + &c. /r/©v ,v«?@\" « > , , „ , „ o v = m ( -7- ) + m [~j~j + &c. = A 3 (»?i m + r? 2 *? 2 + &o.j. We shall now show that @ becomes of the required form. Stability of Motion. 461 r , . (KB , d<& ,d& ,d?/ 2 + &c) Z 1 + ( m V + ihV + &o.) Z% + &c. } It can be shown in a similar manner that ~jy- = A 2 (17 1" 3 + if*" 8 . . . + W 2 ) £ 3 , and so on. If then we assume, as is allowable, »7i + Vz + Vi ...+ Ti n .= L, rii + rjo + jj s • • • + »y» = 1, &0., we have the equations d<& dh v\\ + m Vz • • . + tin Tt\ n = 0, whilst the other still remains arbitrary. Hence the transfor- mation is complete, but one of the ratios which is determined in the case of unequal roots remains arbitrary in the case of equal. 464 Small Oscillations. The results obtained above for the determinants An, A 22 , &c. may be extended, as in Art. 314, to the first minors of A in its most general form. We may then assert, in general, that when A is a double root of the equation A = 0, the system of n linear equations (19) can be satisfied by (n - 2) of the quantities a l9 a 2 , . . . a n , the other two remaining arbitrary. The conditions to be fulfilled in the case of equal roots might have been deduced at once from the consideration that the roots of the equation An = separate those of A = 0, as shown in Art 314. If, on the other hand, some method different from that of Art. 314 be adopted to prove the reality of the roots of the equation A = 0, then the method of the present Article may be employed to investigate, as above, the case of equal roots, and also to show that the roots of the equation An = separate those of the equation A = 0. 317. General solution of the Differential Equa- tions in the case of Equal Roots. — When the roots of the equation A = are all unequal and positive, equations (21) may be written (40) |i = #n sin t VAi + flt'n cos t Vai + #12 sin t VA2 + #'12 cos t Va 2 + &0. I2 = #21 sin t Vai + rt'21 cos t Vai + o.2% sin t VA2 + #'22 cos t Va 2 + &c. &c. = &c. where the 2n constants a u , a' u , a 12 , a m &c, in the expression for £1 are all arbitrary, and the corresponding constants in g 2 , &c. may be found in terms of these arbitrary constants by the solution of linear equations, the equations connecting a n , #21, «3i? • • • a n\ being the same as those connecting a' Ui a 21, ft 3i» • • • (i m.' If now two roots Ai and A 2 of the equation A = become equal, equations (40) are reduced to the form £1 = «n sin t\/\i + «'n cos ^Ai + #13 sin t\/\3 + a n cos t\/\z + &c.\ £2 = #21 sin t\/^i + a 21 cos<-\/ai + #23 sin t\/te + a'23 cos t V A3 + &c. \.. (41) In - «n\ sin t \/x\ + a'ni cos ty'xi + a»3 sin ty A3 + #'»3 cos t v A3 + &c./ Principal Coordinates § Directions of Harmonic Vibration. 465 In this case there are only 2(^-1) arbitrary constants in gij but since the system of n linear equations corresponding to A t can (Art. 316) be satisfied by (n - 2) of the unknown quan- tities, the other two remaining arbitrary, we may in the present case, in addition to the (2n - 2) constants in £ l5 con- sider a 2l and a- n ' also as arbitrary. We thus have still 2n arbitrary constants altogether, and the solution of the diffe- rential equations (18) is therefore complete. A particular case of this has been already considered in Art. 308. It is easy to see that we may still, if we please, express the values of ?!, &c. by equations (21), but when A x = A 2 the constants tf 2 i and a 22 are arbitrary, as well as K X a ll9 K 2 a l2y xu and x^ and in terms of these six we can express the four arbitrary constants which belong to the solution of the differential equations. If there be several distinct double roots similar considera- tions apply to each of them, and in general, corresponding to each doable factor of A there are four arbitrary constants in the solution of the differential equations. The preceding investigation can be readily extended to the case in which the equation A = has r equal roots. In this case 2r constants a n , a 21 , . . . a rl , a n ', a 2l ', . . . a r { are arbitrary, and the n linear equations corresponding to the multiple root, which in general determine (n - 1) quantities in terms of the remaining one, are equivalent to only (n - r) independent equations. In fact, from what has been proved above, it appears that every double root of the equation A = must be a root of An = 0. Hence if the former equation have r equal roots the latter must have (r - 1). Again, it is plain that A n is related to An 22 in the same way in which A is related to An, and so on. "We may therefore conclude that if the equation A = have r roots equal to A x , then (r - 1) successive minors of A must vanish for that value of A. 318. Principal Coordinates and Directions of Harmonic Vibration. — Since in the present case the equations are linear which connect different sets of co- ordinates, the generalized components of velocity are ex- pressed in terms of each other by the same equations as those which connect the corresponding coordinates. Hence 2H 466 Small Oscillations. the transformation of coordinates by which 2& becomes Si 2 + V . . . + In, reduces 2 T to the form ^ + £ 2 3 . . . &. Now, Art. 315, 2© is in this case of the form A^ 2 + X 2 ? 2 2 . . . + X„?» 2 , and therefore by the solution of the differential equations for this particular set of generalized coordinates we have Ci = h sin *(Va! + xi), & = *a sin (' Va 8 + %*), . . . f» = *« sin (* V\^ + X »)> ( 42 ) where Xi, &c. are the roots of the equation A = 0, and hi, k 2 , . . . k m xi» X2» • • • X» are arbitrary constants, 2w in number. The coordinates f u J 2 , &c. are called the Principal Co- ordinates of the oscillating system. The Cartesian coordinates a?, y, s of any point of the system are given in terms of the principal coordinates by equations of the form x = x + A£i + A£ 2 . . . + A n Z n \ y = y. + B£i + B& . .. + £„£»{, z = s + C\£i + C 2 ? 2 . . . + i£«» = -77-, &c ; hence we have dxpi (49) Effect of Increase of Inertia. 469 In a precisely similar manner we can show that @ = @^r- + e,^, 2 . . . + @„^ n s = \i^i 8 + x a #^ a a . • • + x«#^v (so) If we select the constants a n , a 12 , a 1? „ . . . a ln so as to satisfy the equations ft = 1, & = 1, . . . #» = 1, the simple harmonic functions ip l9 ip 2 , &c. express the values of the principal coordinates of the system. When the harmonic determinant equation has equal roots the orthogonal transformation which reduces (5 to its canoni- cal form though valid is no longer determinate Art. (316), and there are an indefinite numher of sets of principal coordi- nates. 319. Effect of Increase of Inertia.— If the mass or inertia of any part of a moving system be increased, the expression for the kinetic energy receives thereby the addition of one or more terms of the form vQ\ where v is a positive constant, and 6 is a linear function of the generalized com- ponents of velocity. The coordinates may be transformed in such a way as to make the linear functions 0, &c. identical with an equal number of the generalized coordinates \ X9 &c. If the forces acting on the system remain unaltered, and if there be only one additional term in the expression for the kinetic energy, the harmonic determinant A' of the system in which there has been an increase of mass or inertia, is given then by the equation X(/ n + v) - q ll9 X/12 - qu • X/l2 - 012 A' = A + v\ An, where A is the harmonic determinant of the original system. If the original position be one of stable equilibrium all the roots Xi, . . . \ n of the equation A = are positive, and are separated by the roots fii 9 . . . fi n -\ of the equation An = 0. Hence A r is positive for X = Xi, negative for X = /ui, negative for X = X 2 , positive for X = ju 2 , and so on. Consequently the roots of the equation A' = are each less than the correspond- ing root of the equation A = 0, but are all positive and are separated by the roots of the equation An = 0. 470 Small Oscillations. It follows from what has been said, that when the forces remain unaltered an increase of mass increases the several periods of vibration. If the generalized coordinate or & were rendered invariable the system would have only [n - 1) degrees of freedom, and the harmonic determinant would become An. Hence no root of the equation A = is diminished by an increase of inertia as much as it would be by rendering the corresponding coordinate invariable. It follows that if any period of oscillation belong to a system both before and after a certain coordinate has been rendered invariable this period belongs also to the system when the mass corresponding to this coordinate is increased. The substance of this Article is taken from Eouth's Rigid Dynamics. 320. Energy of an Oscillating System. — If we put t ^/Ai + Xl = 0U t \/^2 + X2 = 02, &C, and substitute in T the values of £i, £ 2 , &c. obtained by dif- ferentiating equations (42) we have 2T = XJcS cos 2 0! + A 2 & 2 2 cos 2 2 + &c. (51) Again, substituting in V the values of &, £ 2 , &c we have 2 V = 2 Vo + Xifc" sin 2 fa + A 2 & 2 2 sin 2

0* = #0 sin (t Va + x), &c. where A- and x are arbitrary constants, we get to determine a, )8, 7, &c, and \ the equations (?«i + w 2 + &c.)(aiA-5')o + (w2 + W3 + &c.)a2A)8+ (w3 + &c.)a 3 A7+&c. = 0, (W2 + W3 +&c.)aiAa+ (»»2 + »«3 + &c.)(«2A.-5')i8+ (mz + &c.) a 3 \y + &c. = 0. (flio + «2)8 + «37 + &c. + fl n w) A - ga> = 0. This problem can also be treated by the general method of Art. 310. For, since the vertical motion of each ball is very small in comparison with its hori- zontal motion, the velocities si, Z2 f &c. may be neglected ; and we readily find 2T- mi <7r 0r + w 2 (tfi 0i + a 2 2 ) 2 + m z (a x 0i + a 2 02 + «3 03) 2 . . . + m» (ai 01 + « 2 02 + • • • + d n 0„) 2 . Also, if the potential energy be estimated from the position of equilibiium of the system, 2 V = mi ga\ 0r + W20 («i Or + «2 02 2 ) + . . . + »M ( a i #i 2 + #2 02 2 + • • • + «« 0n 2 ) • The preceding differential equations immediately follow from these equa- tions by the method of Art. 310. 472 Small Oscillations. 2. The system of balls suspended as in the last example are displaced in different vertical planes. In this case, 0i and cpi being the angular displacements of «i towards the axes of y and x, B 2 and

are small, and the initial position is one of equilibrium (Art. 310), V = Vo + | (?110 3 + + 2^130^ + 2^23^)- Again, neglecting small quantities of the second order, «i = 0, u>z = <£, «3 = rp ; and therefore (Art. 263) T=i(^0 2 + £(p 2 + Cty 2 ), neglecting small quantities of the third order. Hence equations (18) become Ad + q n 6 + Qi2

+ 533^ = 0. Assuming = Jc a sin (* Va + x)i

^ = &7 sin (* l 7 * + x ), we have, for the detennination of a, £, 7, A, the equations AXa = g-no + 512)8 + ?i37> -#Aj8 = £i 2 a + 522)8 + 5-237, C\y = qua + 523)8 + 5337- If 01, 02, &c. be the values of a, &c. corresponding to Ai and A 2 , two of the roots of the cubic for A, it is easy to see that (Ai - A 2 ) {Aaiaz + Bfafa + C7172) = ; hence Actio.* + Bfiifiz + C7172 = 0, and therefore also 02 (51101 + 512)81 + 51371) + )82 (5i2«i + 522^1 + 52371) + 72 (51301 + 523)81 + 5337O = °- Accordingly the lines whose direction cosines are proportional to 01, £1, 71 ; 02, /32, 72; 03, £3, 73 ; are conjugate diameters of the momental ellipsoid, and likewise of the quadric E, whose equation referred to the principal axes of the body at the fixed point is 511 #- + 522^ + 533Z 2 + 2512 xy + 2qi 3 xz + 2q 2 *yz = K. Since the initial position is one of stable equilibrium, E must be an ellipsoid (Art. 315). Examples. 475 An angular displacement Q x - Q' 2 . Then the source of heat at t x remains as before, whilst the source of heat at t% has received Determination of Carnofs Function. 487 Q : and given out Q 2 units of heat. Moreover, an amount of work represented by Q 2 - Q 2 has been accomplished. The result of the whole process is that work has been done by means of heat obtained from the coldest body in the system. As this result is opposed to the second fundamental Principle (Art. 328), we conclude that Q 2 is the same for all bodies, and is therefore simply a function of Q lt t u and t 2 . From what has been now proved it follows that if we sup- pose the curve A l B l divided into n parts, for each of which Q x is the same, and adiabatics drawn through the points of section, the corresponding values of Q 2 are equal. Hence it is easy to see that Q 2 becomes n Q 2 if Q y become n Qi, and therefore that -=r must be independent of Q x ; accordingly, we have | =/(*., t t ). (17) Again, if W be the amount of heat converted into work in the process, we get, from (17), J-l-/ft*<* (18) 330. Determination of Carnot's Function. — In order to determine the function/ we have merely to select a body for which the isothermal and adiabatic curves are known. Let us then select a perfect gas. In this case, by (13), JQ 1 = RT X log 5»a, JQ 2 = RT 2 log &. Again, as the points A x and A 2 lie on the same adiabatic, by (16) we have fcf^.and likewise tef-£; therefore — = — -, and 7r = — . 488 Thermodynamics. Hence, whatever be the body employed, we obtain the equation ft Q 2 T, (19) 331. Extension of Carnot's Cycle. — If heat imparted to a body be regarded as positive, and heat given out by the body as negative, (19) may be written r A + T 2 ■^ + ^ = 0. (20) If we now suppose a reversible cyclical process represented by any number of isothermals and adiabatics, each isothermal being followed by an adiabatic, and if Q be the number of units of heat imparted to the body at the temperature T, Q we have the equation 2^=0. In order to prove this, let us first suppose a cycle in which there are three isothermals, A^Bi, B 2 C 2 , and A z C z , cor- responding to the temperatures T x , T 2 , and T z . Produce the adiabatic B X B 2 to B z , then Q z =q z + q*, where q z corresponds to B Z A 3 , and q z to C Z B Z . Now by (20), Qi ft n , Qz q' ft - + - = 0, axui- + - = from which, by addition, we have Qi 0* Qz r, + r 8 + t z o. This result "may be extended in a similar manner to a cycle containing four isothermals, and so on. Hence, in general, 4=o. (21) Entropy. 489 Again (21) holds good for every reversible cyclical process, whatever be the nature of the curves by which it is repre- sented. This appears from the consideration that two infinitely near points A and B on any curve can be connected by the element of an isothermal followed by that of an adiabatic, and that the area bounded by these elements, the ordinates of A and B, and the axis of abscissas, differs only by an infinitely small quantity of the second order from the area of which the arc AB is the boundary. For every reversible cyclical process, however effected, we have, then, the equation jf = o. (22) 332. Entropy. — If a body pass from any one state to any other, we may suppose the change of state effected by means of a reversible transformation ; and, whatever this pro- cess be, -=■ between the limits corresponding to the two states must have the same value, since the cycle may be completed [clQ by a definite invariable transformation. Hence — depends only on the state of the body, and is independent of the mode (supposed reversible) by which the body is brought into this state. If we put — = , the quantity

are supposed to be expressed in mechanical units. 490 Thermodynamics. 333. Energy and Entropy. — For every reversible transformation in which the external work done by the body is due to its own expansion we have, if Q be expressed in work units, the two equations dQ = dU + pdv) (24) dQ = Td ) The energy U and the entropy <£ are functions of the independent variables on which the state of the body depends, and dU and dip are therefore perfect differentials ; Q depends not merely on the state of the body but also on the mode in which it has been brought into that state ; hence dQ is not a perfect differential. The limits of the quantities , , &c. are expressible in terms of the independent Av Ap variables and the differential coefficients of JJ and v. They are therefore functions of the two independent variables which determine the state of the body, but are not differential coeffl- £Q cients. They may be written -^, &c. cv Again from equations (24), we have dU= Td

, p; T, p ; T, v ; and v, p ; and express in each case the condition that dU should be a perfect diffe- rential, we obtain a system of equations which hold good in any reversible transformation in which the external work done by a body is due to its expansion against the pressure on its surface, and which are as follows : — dvh~ WA' WA"WV VpJt" \dT)p\ .... d±\ = (dp\ (d_T\ (dj\ _ (clT\ (d£\ m dv) T \dT)J \dpJ v \dvj P \dv) p \dp) v J Elasticity and Expansion. 491 Briot remarks that from the first of these equations the three succeeding can be obtained by interchanging p and i\ or T and but d * = {d- v ) P dv+ [dp) v d *> /#\ ldT\ and therefore 1&L = - { ±) . In like manner iftil = - (±) ; \dp) v \dv) p idp\ hence £ = A*l2± = f» (Art . 334) . \dv J t Examples. 493 4. Prove that dQ = c p dT - cvTdp. "- ®."*(S),* "*" + '(8)r* but by (26) -we have (— ) =— ( t^,) , an d hence by substitution we obtain from (28) the required result. 5. Assuming that the square of the velocity of the propagation of sound is proportional to the elasticity of the medium divided by its density, show that in a gas the velocity of sound varies as \ZkBT. Since the compression of the air during the passage of a wave of sound is very sudden, the compression may be regarded as adiabatic. Hence the velocity of sound varies as ^ E^v, but E$ — JcE T (Ex. 3), and E T = P, therefore, &c. By means of the results obtained in this Example and in Ex. 1, Art. 324, if the velocity of sound be determined by experiment, C p and C v can be calculated. Conversely, if C p be known by experiment, C v can be found from the velocity of sound, and hence the value of / can be determined. 6. Show that bodies which expand by heating are heated by compression ; those which contract by heating are cooled by compression ; and, if the tempera- ture be maintained constant, determine the rate at which heat is given out or absorbed according as the pressure is increased. If Q be the heat required to keep the temperature constant, the rate of ab- sorption is ( — J ; but (S)'-*(2),--*S),- : -- E *-«-»>■ Hence 8Q is negative if e be positive, and conversely. 7. Prove that in water not far from its maximum density the rise of tem- perature produced by an increase of pressure is given approximately by the formula, 2950000 *' where t is expressed in degrees centigrade, and p in atmospheres. If vo be the volume of the unit mass of water at 4°, when the density is a maximum, the empirical formula v = vq f 1 + ) represents, according to Kopp and Tait, the results of numerous experiments. From this formula we have approximately e = --^qqq- Hence, assuming the pressure of the atmosphere to be 1033 grammes on the square centimetre, we obtain the required result. 8. If the internal energy of a body be a function of its temperature alone determine the relation which must exist between v, p, and T. 494 Thermodynamics. In this case (25) becomes Td

= adT + (b + p) dv, whence b + p = Tf(v). If by means of this last equation we express dv in terms of dp and dT, we have dQ = { a -7) iT +h- Now *-Gf), P and therefore, if n be the constant value of c p , we obtain a-'— = n. From this we have [a - n) ■£■ = dv, and integrating we get (v + C) f= (n - a), where is the constant of integration. Hence we have as the required relation (b+p){v+ C) = {n-a)T. 10. If the specific heats of a body at constant pressure and at constant volume be each constant, show that the energy is a linear function of the volume and absolute temperature. ~Letc v = m, c P = n, then Ijj,) = % and therefore U = mT + f (v). Ah0C *={§) P +p (is),- whence " - •" + (/ ' +p) (%)„■ (a) Again, from (25) we have Td

, _ ^ = / ( C v \og-+ C p log -) =J(C P - C v ) log - = R log -. Since vz >v\, the entropy is increased by the supposed transformation. This transformation, it should be observed, is non-reversible, and therefore not adia- batic, though no heat is lost or gained. ' 3. A vertical cylinder, whose horizontal section is S, is filled with gas at the atmospheric pressure p\ and temperature 1\, and closed by a piston on which is placed a weight w which pushes it down. Supposing no external heatto pass into or out of the gas, determine the temperature when equilibrium is established. The transformation here is non- reversible, since the external pressure ex- ceeds that due to the state of the body by a finite amount. Since no heat is lost or gained the external work done on the gas must be equal to the change of energy. Let pz, Ti, V2 be the final pressure, temperature, and specific volume, the w initial specific volume being vi, then pi = p\ + -, and the work done on the unit of mass is pz {v\ - vz). Hence from (10) we have c„ (r 2 - Ti) =p 2 {vi - vz) = P2V1 - R T 2 , and therefore {fin + R) T 2 = c v T x 4 (px + ^\ vi, 4. In Ex. 3 determine the increase of the entropy of a unit mass of the gas. Tz „ ,. , vi) Tz = c p T\ + — v\, which determines Tz increase of the Am. $, then T 2 = Ti (I - ufl)-\ also, B* 5, T 2 '=T ia v = T, jl + * log (1 + jB) + ^ log^l + 0) + &c. but j8 > log (1 + j8), for e* 3 > 1 + = e l0 §( 1+ |S>. Hence each term of the series for T 2 is greater than the corresponding term of the series for T 2 , and as all the terms are positive, T 2 > T/. The increase of entropy in the gas compressed, as in Ex. 3, might be expressed in the form p 2 , and let 1 + = — = a, 1 = va, Vipi p\ then o = -~ 1 {alog(l-y)-log(l-yo)}; and since a > 1 , we readily see that the quantity inside the bracket is positive. 336. Absolute Scale of Temperature. — The result obtained in Art. 330 may be arrived at by a different method independent of the properties of any particular substance. We have seen in Art. 329 that if Q be the heat drawn from 2 K 498 Thermodynamics. the source, and W the heat converted into work in Carnot's cycle, — i s a function of the extreme temperatures only, and is independent of the suhstance employed. In order, then, to construct a scale of temperature independent of any parti- cular body we may proceed as follows : — Draw the isothermal AB of a sub- stance chosen at random, corresponding to any arbitrary temperature, which may be indicated by T, and draw the adiabatics AA and BR corresponding to the con- dition of the body before and after a certain arbitrary amount of heat Q has been imparted to it. Draw another isothermal at a tem- perature T' less than T, so that the area ABB' A' may be of given magnitude or correspond to a given amount of heat w. Now draw a series of isothermals T" , T" ', &c, at intervals such that ABBA = AB'B" A' = A'B"B!"A" = &c. ; then \IT-T be the unit of temperature, T- T" is two units, T- T" three units, &c. Since T, Q, and w are fixed quantities, and W correspond- ing to T^ is nw, Equation (18) shows that two bodies are at the same temperature if each indicates in the manner described n degrees of temperature below T. This method of estimat- ing temperature is, therefore, independent of the body em- Again, if T be any temperature lower than T estimated in this manner, and W the heat converted into work in the cor- responding cyclical process, we have W = (T- T') w, and in like manner for another temperature T" lower than T we have W"={T-T")w. If we now suppose a cyclical process between the tem- peratures T and T", indicated by the points A', B% B" , A\ the heat converted into work is W" - W\ and we get JT»- W'={T'-T") w (31) Absolute Zero. 499 Again, the heat Q / drawn from the source at T f , is equal to that given to the condenser in the process in which T and T' are the extreme temperatures ; hence Q . q = W' = (T- T) w, that is, Q' = Q - (T- T') w. (32) 337. Efficiency of a Heat Engine. — A system work- ing in the manner required by Camot's cycle may be termed a reversible heat engine, and the ratio of the heat converted into work to the heat drawn from the source is called the efficiency of the engine. It appears by the reasoning of Art. 329 that the extreme temperatures being given, the efficiency of a non-reversible engine cannot exceed that of a reversible, and that the efficiency of all reversible engines is the same. 338. Absolute Zero. — From Art. 337 it appears that the efficiency of a reversible engine working between the temperatures T and T" is j- f . By (31) and (32) this H rpr _ /xt// becomes -. -^ • r \ T -l) As T" decreases, the efficiency increases, but the limit which it can never exceed is unity, since the mechanical work done by an engine can never exceed the equivalent of the heat drawn from the source. Hence, if we make the effi- ciency unity, we obtain for T" the smallest possible value, which is T This temperature T" , since it is the lowest which can be attained by any body, must be the absolute zero. Hence '-!-"• - 14 The expression for the efficiency of a reversible engine working between any two temperatures T r and T" becomes 2 K 2 500 Thermodynamics. rpr _ rprr then — ^7 — , and for the cyclical process described in Art. Q - Qo T —To 329 we have — -jz — - = — ^= — -. Carnot's function has thus H\ J- i been determined independently of the properties of any particular substance. Again, this mode of determining Carnot's function shows that the existence of an absolute zero of temperature, sug- gested and rendered probable by the known properties of what are called permanent gases, follows necessarily from the two fundamental Principles of Thermodynamics. The experiments of Joule and Thomson have shown that the absolute zero is 273*7 below zero on the Centigrade scale, or 460*66 below zero on the Fahrenheit. This is very nearly the same result as that of Article 324. Examples. 1. The entropy

= — - -f B — . Hence -dU must be a perfect differential, whence U= F(T). 2. Gas is made to pass uniformly through a tube in which a porous plug, such as cotton-wool, is placed. No heat is permitted to leave the gas or enter it from any external source ; determine the connexion between the variations of pressure and temperature caused by the plug. Since the density of the gas at any particular cross section of the tube does not vary during the experiment, equal masses of gas pass through each section in the same time, or the velocity of the unit of mass is constant. Again, any energy which is lost by frictien is restored as heat. We are therefore entitled to assume that any change in the energy of the gas as it passes through diffe- rent parts of the tube is due to the work done on it or to the work which it Suppose two cross sections A and B of the tube, one on each side of the plug, the pressures at which are p\ and p 2 . As a small quantity dm of gas passes A the pressure driving it forward does work on it whose amount is pividm. At the same time dm does work on the next layer of gas which is equal to the work done on dm when passing the section consecutive to A. Thus, in going from A to B tbe work done by dm and the work done on dm compen- sate each other, with the exception of pwidm done on dm, and p 2 v 2 dm done by dm. In other words, in the passage from A to B the whole external work done by dm is (p 2 v 2 - pwi) dm, and therefore, since no heat is lost or gained, we have U 2 - U\ + P2V2. - p\V\ = 0. Now U=jTd(j>- J pdv =$Td in consequence enter a source at To, the loss of available energy is (T, - To) | - (21 - Jl) | 2 , or Q t - ft- T (| - -|). 508 Thermodynamics. 341. Dissipation of Energy. — If the transference of heat from a source at T y to a source at T 2 take place through the medium of a reversible engine undergoing a cyclical process, ~ ■ - -£ is zero, and the loss of available energy is Q\ - Q 2 , which is the same as the work done. Thus the un- compensated loss of available energy is zero. In the case of an engine undergoing a non-reversible cyclical process, Q l - Q z cannot be greater, and is usually less, than (2\ - T t ) % (Art. 337), or ~ - % has a negative value 1 \ J- 1 Jj which may be denoted by - N. In this case the uncompen- sated loss of available energy is T N. By a method similar to that employed in Art. 331 this result can be extended to every non-reversible cyclical pro- cess. In this case, if Q be the heat which enters the Q . engine at the temperature T, the quantity S -~ is negative, and the uncompensated loss of available energy is - T E -=. To prove this, we have only to substitute for the actual process A a process B in which the cycles corresponding to each pair of temperatures are completed by reversible trans- formations, each of which is accomplished first in one direc- tion, then in the opposite. As these transformations are passed through in both directions, the value of 2 — and of the un- compensated loss of available energy is the same for A as for B ; but 2 -= f or B is the sum of the values of S -j, corre- sponding to the small cycles, since the remaining part of B forms one reversible cycle. Hence we obtain the required results. The uncompensated loss of available energy is called the Dissipation of Energy. From the present and preceding Articles it appears that this dissipation takes place whenever heat passes without the performance of work from a body at a higher to a body at a lower temperature, and also, in general, in non-reversible Increase of Entropy. 509 cyclical processes. A strictly reversible process cannot be realized in nature, since the absence of friction and the perfect equality of internal and external pressures and temperatures cannot be attained. Hence we may conclude, that in natural processes there is, in general, an incessant dissipation of energy. There is one class of irreversible transformations in which, according to Mr. Parker (Philosophical Magazine, June, 1888), there is no dissipation of energy. Mr. Parker in the Article referred to defines an equilibrium path to be one at every point of which the system is in equilibrium. The path corresponding to a reversible trans- formation is always an equilibrium path, but an equilibrium path is not necessarily reversible. As a result of experiments on the solubility of various substances, Mr. Parker has been led to adopt the conclusion that in an irreversible equilibrium cycle there is no dissipation of energy. It is to be observed that the theory of dissipation depends on the assumption of a certain temperature as the lowest which is available. If the lowest available temperature were absolute zero there would be no dissipation of energy. 342. Increase of Entropy. — If an element of heat rlQ pass from a body A, whose temperature is T i9 to another body B at a lower temperature T 2 , and if we suppose the volumes of A and B to remain constant, the entropy of A is diminished by 7=-, and that of B increased by — , and as 1\ 1% T x > To, the whole entropy of A and B is increased. Again, in a cyclical process, if we suppose the source A and the condenser B to remain at constant volume, in which case their temperatures will of course vary, 2 ~ is the loss of entropy by A, and 2 -^ the gain of entropy by B. Hence the entropy of the whole system is increased by the quantity 2 ( ^ - -^ ). In a reversible process this quan- tity is zero, but in a non-reversible process it has in general a positive value N. 510 Thermodynamics. "We have supposed A and B to remain at constant volume ; but if this be not the case, the results obtained still hold good, provided the transformation applied to each of these bodies is reversible when each body is considered alone. Under these circumstances the uncompensated loss of available energy in a non -reversible cyclical process is equal to the product of the limiting temperature and the increase of the entropy of the system. Since, according to Mr. Parker, there is no dissipation of energy in an equilibrium cycle even though it be irreversible, in such a cycle the entropy of the whole system is constant. Again, it would appear that the definition of entropy in Art. 332 is unnecessarily restricted, and that entropy may — along any equilibrium path. It would seem that the result of Mr. Parker's experi- ments might have been anticipated. For, when a system under- goes a transformation corresponding to an equilibrium path, the irreversibility of the transformation for the whole system can result only from the way in which heat is communicated to or leaves the system, or on the mode in which it passes from one part of the system to another part. We may therefore suppose the system divided into portions for each of which taken separately a reversible path may be assigned coinciding with the actual equilibrium path. If Q if Q 2 , &c. be the quantities of heat which at any stage of the transfor- mation have passed into these portions, U l9 U 2 , &c their energies, v l9 i\, &c. their volumes, p l9 p 2 , &c. their pressures, T l9 1\, &c. their temperatures, and 1? <£ 2 , &c. their entropies, we have dQ x = dUi + pidi\ = T x dfr, since the path coincides with a reversible path. In like manner dQ 2 = dU 2 + p 2 dv 2 = T 2 d 2 , dQ 3 = dU 3 +p 3 dv ?J = T 3 dfa, &c. Now, since the whole system is in equilibrium, r l= T 2 = T 3 = &c. = T, pi=p 2 =p 3 = &c. =p. Path of Lead Heat. 511 Hence, if be the entropy of the entire system, and Q the quantity of heat imparted to it, p dQx + dQv+ &g. dQ d$ = dfa + dfa + &c. = -= = -jr, and therefore so far as the relation between heat imparted and entropy is concerned, the whole transformation may be treated as if it were reversible. We may conclude from what has been said, that natural processes have a tendency to increase entropy, or, as stated by Clausius, the entropy of the universe tends to become a maximum. 343. Path of Least Bleat. — Let us suppose that a body, whose entropy is fa, passes from the state A to the state B in which its entropy is fa, less than fa. If Q be the heat given out by the body when at the temperature Q T, and if S denote the value of S — for the whole process, S cannot be less than fa - fa. To prove this, first suppose the transformation reversible, then 8= fa- fa. Next suppose the transformation non-reversible, and let the cycle be com- pleted by a reversible process which brings the body from B to A. The value of 2 -p=, for the cycle is then S- (fa- fa), and this must be positive (Art. 340) ; hence 8 > fa - fa. Let us now consider by what path a body, whose tempe- rature can never be less than T 0f should pass from the state A to the state B at T , fa, so that the heat given out in the passage should be a minimum, no heat being supplied to the body from any external source. Let H be the heat given out ; then for a non-reversible transformation, since T > T , and since any element of heat which enters the body at T must have previously passed out of it at a temperature higher than T, we must have H > T V) S > T ((pi - fa). For a reversible transformation Tdfa which is least when T = T . The least value of tf>0 H is therefore T (fa - fa). Hence the path consists of an adiabatic at the entropy fa from T x to T , and an isothermal 512 Thermodynamics. at To from fa to fa. Since JTi - Z7"o = TF+ #, where TFis the work done by the body during the transformation, when H is least W is greatest, and the maximum work which a body can perform under the circumstances supposed is Ui-U Q -TAfa-fa). Examples. 1. Prove that the available energy of any system of bodies is J T where T\ is the initial temperature of mi, and c\ its specific heat at constant volume. 2. If the system in Ex. 1 be enclosed in an envelope impermeable by heat, show that To is determined by the equation >r i dT I,.*-?- - The actual work performed by the system during the transformation in which all its parts are brought to the temperature To is ] T P CidT; J T but, if the transformation be that in which the greatest possible work is done, this work must be equal to the available energy, and therefore 2mi When the limiting temperature To is determined from within, as in this ex- ample, or, in other words, when one part of the system acts as condenser to another part, the available energy is called by Thomson the Internal Thermo- dynamic Motivity. When To is independent of the system, i.e. when heat can pass out of the system to an external condenser, the available energy may be termed the External Thermodynamic Motivity. In this case To must be as- 3. If a system consist of two equal masses of the same substance whose specific heat is constant, show that the limiting temperature of the internal thermodynamic motivity is \ ' T1T2, where Ti and T 2 are the initial temperatures of the two masses. 4. In the preceding example prove that the thermodynamic motivity of the system is me (V ~Ti - Vl^) 2 . Examples. 513 5. If the entropy of a substance be increased, its energy remaining constant, prove that the work which can be obtained by a transformation to a given state is diminished. 6. A unit mass of gas, whose volume is v\, is allowed to expand into a per- fectly empty vessel, whereby its volume becomes v% • show that its capability of doing work is diminished by the quantity TaR log—. V\ 7. Determine a transformation by which, without the transference of any heat, gas at p\V\ may be brought by the application of the smallest possible amount of external work to p>2V% ; where pi > p\, vi > V\. Since »2 > »i the gas must expand, and since no heat is given it must ex- pand by its own energy. It will do this with the smallest possible expenditure of energy by expanding into a vacuum. If TT% be the energy corresponding to P2V2, the smallest amount of external work capable of changing the energy from U\ to Ui is Z7o — U\, and in order that no more than this should be required the compression must, by (25), be adiabatic. Hence let the gas expand into a vacuum till its volume become v, and then let it be compressed adiabatically till its volume become v%. In order to determine v, let T\ and To be the tem- peratures belonging to the initial and final state ; then, by (16), T\v k ~ x = T2V2*" 1 , whence ■0-- 2 L 514 Miscellaneous Examples. Miscellaneous Examples. 1. If two points fixed in a lamina slide upon two intersecting straight lines, and if one point be made to oscillate backwards and forwards so as to have always the same velocity, the ellipse described by any fixed point of the lamina will be described under acceleration which is fixed in direction. 2. A material point of given mass moves freely under the action of a central force of given absolute intensity, varying inversely as the square of the distance ; given the initial circumstances of projection, determine the major axis, eccen- tricity, and line of apsides of the orbit it describes. 3. The extremities of a uniform rectilinear bar move on the circumference of a smooth vertical circle ; find its period of oscillation under the action of gravity consequent on a small displacement from its position of stable equi- librium. 4. A circular plate, revolving round its centre in a vertical plane, becomes suddenly attached at its lowest point to a heavy particle previously at rest ; re- quired the mass of the particle in order that, at the end of a semi-revolution, the system may be brought to rest under the action of gravity. 5. A uniform beam is supported symmetrically on two props ; find where they should be placed in order that if one of them be removed the instantaneous pressure on the other may be the same as the statical pressure. 6. A circular board lies upon a smooth table ; in the board is cut a circular groove along which a molecule is projected with a given velocity ; determine the pressure against the side of the groove. 7. A straight rod which passes through a small fixed ring is in motion in a horizontal plane ; determine the motion of its centre of gravity. 8. A lamina unacted on by any force is projected in its own plane ; prove that its space centrode is a straight line, and its body centrode a circle. 9. A sphere, rotating about a horizontal axis through its centre of gravity, falls vertically ; prove that its space centrode is a parabola, and its body cen- trode a spiral of Archimedes. 10. Given the motion of one point in a body and also its space centrode, find its body centrode. 11. A small ring slides down a rough rod from a given point to a given right line ; find the direction of the rod so that the time of descent may be a minimum. (a) Find the limits of the coefficient of friction for which the required posi- tion is vertical. Miscellaneous Examples. 515 12. A material particle, attached to a fixed point by an inelastic string, is allowed to descend a smooth inclined inelastic plane, starting without initial velocity from the foot of the perpendicular from the fixed point on the plane. Describe the subsequent motion, and show that the total length of the path described by the particle on the plane before it comes to rest is \sm sin/3 cos 2 /3 l + cos'-jS where I is the length of the string, and j3 is the angle which, when stretched, it makes with the perpendicular. 13. A homogeneous sphere rolls down the concave surface of a rough semi- circle, the axis of which is vertical ; find its velocity and entire pressure against the semicircle in any position. 14. Two balls of different masses, moving in the same right line with diffe- rent velocities, become suddenly connected by a weightless inextensible rod : given all particulars, required, in magnitude and direction, the initial strain on the rod. 15. A material particle, constrained to oscillate without friction in a curve tautochronous with respect to any point under the action of any force, being supposed retarded throughout its motion by a resistance to its velocity of con- stant intensity ; determine the law of diminution of its several successive arcs of vibration. 16. The resisting, in the preceding, being supposed small compared with the moving force ; show that, if the friction vary as any function of the velocity, its effect will be ultimately inappreciable on the time of description of any com- plete arc of vibration of the particle. 17. A rigid body, revolving round a fixed axis, strikes perpendicularly against a fixed obstacle; required the height through which the same body should fall vertically, without rotation, so as to strike against the obstacle with the same force of percussion. 18. A rigid body connected with a fixed point by an inextensible cord, is in constrained equilibrium under the action of a force passing through its centre of inertia ; all the other restraints being supposed suddenly removed, required the initial stress on the cord. 19. A sphere, rolling without sliding on a rough horizontal plane, is acted on by a central force, varying inversely as the square of the distance, emanat- ing from a fixed point in the parallel plane passing through its centre. Show that it describes a focal conic round the centre of force ; and determine the initial velocity for which the motion is parabolic. 20. A rigid body, being set in motion by a single impulsive force, show that all axes of initial pure rotation, corresponding to different directions of the per- cussion, envelope a quadric cone, diverging from the centre of inertia, and touching the three central principal planes of the body. 2L 2 516 Miscellaneous Examples. 21. A rigid body, having two fixed points, is set m motion by an impulsive force ; determine in magnitude and direction the initial percussions at the points perpendicular to their line of connexion. 22. Two material particles, resting on a rough inclined plane, and connected by a slight flexible cord, passing without friction through a small ring attached to a fixed point on the plane, are in equilibrium under the action of gravity ; the inclination of the plane being supposed gradually increased, or its roughness ness gradually diminished, determine the nature of the initial motion of the particles. 23. Two material particles, moving without friction in two non-intersecting rectilinear tubes of indefinite length, attract each other with a force varying directly as their distance asunder ; determine completely their motion. 24. In the general displacement of a solid from one given position to another, find, by geometrical construction, the twist by which the body can be brought from the former to the latter position. — (Prof. Crofton, London Mathematical Society, 1874.) Let A be any point of the solid in its first position, B the new position of the same point ; again, let C be the new position of the point which was origi- nally at B, and D the new position of that point originally at C; then, to find the required twist, bisect the angles ABC and BCD by the lines BBTand CK; find EKthe shortest distance between these bisectors. The body can be brought from the first to the second position by a translation HK, and a rotation round HK through an angle which is equal to that between BIT and CK. 25. Calculate, in C. G. S. units, the mutual attraction of two units of mass at the unit distance apart, according to the law of gravitation. Let 7 denote the quantity in question ; then the attraction of the earth on a unit of mass at its surface is %irypB, where p is earth's mean density, and It is its radius. Hence we have g = iirpylt. Now, in the system of units adopted, we have g = 981, and irlt = 2 x 10 9 . Thence, assuming p = 5-67, we get I = ? x 551 x 10 7 = — x 10 7 = 15,410,000, approximately ; 7 3 981 109 ••• ? = 15,410,000 d r neS ' 26. A body is rotating about a fixed point. Express the element of the curve described by the instantaneous axis on a sphere fixed in the body in terms of the angular velocities round the body-axes. Let the instantaneous axis at any time make angles A, /j., v with the body- axes ; let the spherical surface be intersected by the two consecutive positions of the instantaneous axis in J and /' ; let OJand OI" represent the correspond- ing magnitudes « and w + rfwof the angular velocity. Then the projections of II" on the body-axes are proportional to du\, da2, decs, and II" is propor- tional to du. Miscellaneous Example 517 Now II" 2 = IT 2 + OFdy* ; hence dwi 2 + do>2 2 + door = dot 2 + ordty 2 , and dip 2 = — (^a>i 2 + dcez 2 + ^a>3 2 - d) — n sin , The values of the cosines of the remaining angles can now be written down from symmetry. If we put v = cos 7f , fj. = m sin § ', and its direction cosines I', m', ri ; <£>' -

i sm 2 ^P — ^— , — g= m 1 - J 2 ) cot J tf>, eft smid> * Again 2 — = wi sin ^P sm PP— r— r — = «i (w - fo» cot f 0), o)i sin AP sin CP sini(/> cos CPP BOO. ftp \i, «2, a>3, round three rectangular axes OA, OB, 00 fixed in the body. Determine the differential coefficients of Rodrigues' coordinates with respect to the time. By means of the last example we can write down the changes produced on , I, m, n by each of the rotations co\dt, w^dt, oizdt. Adding, and dividing by dt, we get 2 — = — (aon + W3«J 4- COt|(J>{a>i - l(lw\ + w«2 + nooz)}, dt 2 — = - wil + w\n + eoth2 1 + cot |0{«3 - n(lu)\ + ma>2 + na>z)}, dt d

3 ; dt whence, also, we obtain n dv n d\ 2 — = - <)>l\ - W2H - WiV, Z— = CCIV — C02V + 003 fJ., dt dt du . dv 2 = OJOV — CCZ\ + OJ\V, 2 — = 0>3U - Wljll + o>oA, «£ at where v, A, /*, v have the same meaning as before. 30. A rigid body is moving in any manner ; one point is suddenly arrested ; determine the impulse exerted on the body. Let u, v, w be the components of the velocity of the point immediately before it is arrested, x, y, z its coordinates, and X, Y, Zthe components of the impulse, the axes being the principal axes of the body at the centre of inertia, then X is given by the equation - [^- + A(B + C)x> + B(C+A)y°- + C(A + J5)z 2 + mir^X = {ABC + ®l[A{B + C)x i + BC(y 2 + z 2 ) + BtIt*sP\}« + Tl{AB + Wilr°~)xyv + Wl(AC + 2)llr 2 )xzw, where i" is the moment of inertia of the body round the line joining the arrested point to the centre of inertia, r the distance between these points, and A, B, C, the principal moments of inertia of the body. 31. A sphere is projected in any way along an imperfectly rough inclined plane. Investigate the motion. (This investigation, with some slight modifications, is taken from Routh, Rigid Dynamics.) 520 Miscellaneous Examples, Here the equations of motion are Mx = X 4- Mg sin i, My = Y, § Mr-wi = rY, %Mr 2 d>o = - rX, whence, eliminating X and Y, we obtain, on integrating, z + %ri = & -frHi, where a, )8, Hi, and Ho are the initial values of x, y, «i, and «2. Again, if u be the velocity at any instant of that point of the sphere which is in contact with the plane, and 9 the angle which its direction makes with the axis of x, u cos 9 = x - r«2, u sin 9 = y + ru\. Differentiating, substituting for x, &c, from the equations of motion, put- ting for X and Y the values which they take as long as there is slipping, viz., - [xMg cos i cos 9 and — fiMg cos i sin 9, and solving the resulting equations for u and u9, we have u = g sin % cos 9 — ^fxg cos i, u9 = — g sin i sin 9. Hence, if |/a cot i = n, we obtain, by integration, u sin 9 = K\ (tan J0)". Substituting the value given by this equation for u in the equation for u9, and integrating, we have (tan|fl)» +1 (tanp)"' 1 = ^ _ tysini ^ w + 1 n - 1 2 -fiT 2 ' A^ is determined from the initial value of 9, and K\ from the initial values of 9 and w. These latter are given by the equations u Q cos 0o = a — rfc, u sin 9q = £ + rfli ; then «* and 9 being known, #, y, o>i, and &>2 can be determined. If n or f fx cot i > 1, u and become continually less until they vanish together. Pure rolling then begins at a time t , which is given by the equation to= - — r— :• After pure rolling begins the values of x, y, u\, and wz, at any time, can be obtained from the combination of the equations of motion with the equations x — r«2 = 0, y + rw\ = 0. If n < 1, 9, though constantly approaching zero, as appears from the expression for u9, will not vanish in any finite time, and u tends to increase without limit. Miscellaneous Exercises. 521 If u = 0, the problem is at starting reduced to that of Ex. 3, Art. 278. The force of friction requisite for pure rolling is then §■ Mg sin i. Hence, if f Mg sin i < n'Mg cos i, or J y! cot i > 1 , where yJ is the coefficient of statical friction, pure rolling will commence and continue. If %/a cot i < 1, slipping will begin at once and never cease. 32. A body rests with a plane face on an imperfectly rough horizontal plane. The centre of inertia of the body is vertically over the centre of inertia of the face and very near it, the connecting line being a principal axis at the former point. The form of the face is such, that its radii of gyration about all lines in it passing through its centre of inertia are equal. The body is projected with an initial velocity of translation U, and an initial very small angular velocity fl round a vertical axis through its centre of inertia : determine the motion. Take the initial direction of translation, and a horizontal line at right angles thereto for axes of x and y. Let u and v be the components of the velocity of the centre of inertia of the body at any time, and w the angular velocity. Then, x and y being the coordinates of any point of the body, and £ and 77 its coordinates referred to parallel axes through the centre of inertia, clx dy - = u-v», #=? + *>. If F be the magnitude of the whole force of friction at any point, its com- ponents X and Y are given by the equations X = - F H ~ ^ — = -F a v V{^ + 77o>) 2 +(v + !a>) 2 } V + £co --■F—T-* 2-P- since 0, £a>, and rjca are small compared with u. Again, if S be the area of the plane face, the magnitude of the normal re- action of the horizontal plane on an element of the face is equal to (£, ij) dS, whence F = n

(|, r,) = R, G y =tia$ItdS-fl%dS; therefore J B£dS = iimga. Assume R = K+ eA, where iT and e are constants, then fimya = K j £dS + e f Af«fe, but f&S = ; therefore 6 must be small ; also mg = KS + e J AdS. 522 Miscellaneous Exercises. Again, &» = J" Bi\dS - jua f V -^- KdS ; and, since the second member of G x is zero, q.p.,vre have J" iSrj^/S = 0. Hence the resultant normal reaction passes through a, point on the axis of x. To determine the motion of the centre of inertia, m— = 2X = - n J" BdS = - \tmg ; therefore w= JT-figt. dv v f f v Again »» — = 2F = - ^ - \ RdS - n - \ R£dS = - \nmg -, q.p. ° dt u] u J w hence = cu ; and since » = when u= U, c = 0, therefore v = 0. To find the angular velocity, mk 2 3F--^J«"--i = ft (77)** • INDEX. Absolute, units, 54, 126. force in central orbit, 175. force suddenly changed, 179. zero of temperature, 481, 499. Acceleration, uniform, 12. variable, 15. total, 17. tangential and normal, 17. angular, 19. areal, 21. Acceleration-centre, 267, 340. Acceleration of rotation, 327. Action and reaction, 58. Adiabatic curve, 483. Airy, on Earth's density, 107. Ampere's Cinematique, 5. Angular velocity, 19, 95. of a body, 95. Apsides, 190. Apsidal angle, 191, 215. Areas, uniform description of, 164. accelerations of, 244. for principal plane, 245. Attraction, law of, 92, 130, 147. Atwood's machine, 60, 64, 138. Axes, relation between rotations round space and body, 327, 331. Ball. Sir R. S., referred to, 59, 334, 338, 407, 474. Ballistic pendulum, 271. Bertrand, on closed orbits, 203. theorem of, 231, 420. Billiards, problem in, 387. Body axes, 330. motion referred to, 385. Breaking weight of elastic string, 158. Bresse, on acceleration, 268. Bonnet's theorem, 208. Bordoni, 82. Brachystochrone, 435. Calculus of variations, 433. Canonical form of equations of mo- tion, 431. Carnot, S., cycle of, 486. extended, 488. determination of function of, 487, 500. Central forces, 90, 147, 164. potential of, 129. Centres of oscillation and percussion, 276, 277. Centre of inertia, 76. of oscillation, 142. motion of, 241. motion relative to, 242. Centrifugal and centripetal force, 88. acceleration, 89. force at Earth's equator, 91. Centrifugal force, resultant for ro- tating body, 96. in pendulum, 118. Centrifugal couple, 369. axis of, 370, 374. Centrodes, 261. Change of state of a body, 501. Circle of inflexions, 268. Circular, motion, 84. orbits, 90. orbits approximately, 194. Clausius, on energy of a gas, 410. on second fundamental principle in thermodynamics, 485. on entropy, 489. on saturated steam, 505. Coaxal circles, property of, 120. Coefficient of restitution, 67. Collision, of spheres, direct, 66. effect on energy, 235. oblique, 70. of smooth bodies, 379. of rough bodies, 3S0. 524 Index. Compound pendulum, 141. Compression, force of, 67. Composition of velocities, 7, 257. of rotations, 317, 321. of twists, 334. Cone, employed graphically in rota- tion, 328. Conical pendulum, 115, 224. Conservative system of forces, 129, 233, 397. Constrained motion, 206, 241, 247. Coulomb, on dynamical friction, 63. Couple, of rolling friction, 311, 314. of twisting friction, 311. tending to break moving rod, 303. Curtis, 225. Cycle, Carnot's, 486. Cycloid, tautochronism of, 115. is curve of quickest descent, 433. Cylindroid, 338, 339. D'Alembert's principle, 59, 227, 228, 417. applied to small oscillations, 445. Darwin, on friction of tidal action, 408. Degrees of freedom, 254, 269. Disturbing forces in focal orbit, 188. Dyne, 54, 126. Earth, atttaetion of, 151. mean density of, 107. Efficiency of agents, 126. of a heat engine, 499. Elasticity, 67, 302. in collision, 334, 383. Elastic strings, 155. Elasticity and expansion of a sub- stance, 491. Ellipsoid momental, 348. graphical use of, 348. Ellipsoid, of gyration, 349, 360, 371. of equal energy, 474. potential, 474. conjugate, 363. Energy, 59, 133, 396. . potential and kinetic, defined, 133. measure of kinetic, 133. equation of, 136, 396, 402. in thermodynamics, 478, 485. conservation of, 397. of initial motion, 420. of an oscillating system, 470. Entropy, 489. Erg, 126. Euler, equations of rotation, 354, 368, 425. for impulses, 346. Focal orbit, 173. velocity in, 177. constructed, 178. Force, function, 398. measure of, 53. absolute unit of, 54. gravitation unit of, 54. Forces of inertia, 59, 227. Fly-wheel, energy of, 137. Free motion of a body, 320. Freedom, degrees of, 254, 269. Friction, laws of dynamical, 50, 63, 296. work expended on, in pivot, 132. rolling, 311. twisting, 311. impulsive, 308. Gauss, absolute unit of force, 54. Generalized coordinates, 415, 452. equations of motion, 421. impulse components, 418. Geometrical representation of rota- tion, 321. Goodeve, 265. Gravitation, units, 54, 126. law of, 176. verified, 93. Gravity, acceleration due to, 29. variation of, 30. affected by Earth's rotation, 92. determined by pendulum, 102, 143. Greenhill, 83. Gyration, radius of, 141. ellipsoid of, 349. Hamilton's equation of motion, 431. characteristic function, 439. ; Harmonic motion, simple, 85. elliptic, 86. Harmonic determinant, 454. real roots of, 457. case of equal roots, 462. Haughton, on Earth's density, 108. Index. 525 Heat, mechanical equivalent of, 477. specific, 479. latent, of liquidity, 502. of vaporization, 503. of expansion, 491. Height, due to velocity, 30, 40. Helmholtz, 399. Herpolhode, 372, 378. Herschel, on disturbing forces, 188. Him, 505. Hodgkinson, on laws of restitution, 68. Hodograph, 19. application to focal orbit, 181, 182. Hooke's law, 137, 155. Huygens, on pendulum, 104. Ignoration of coordinates, 426. Impact and collision of spheres, 66, 381. of bodies generally, 280, 288, 379. Impulse, measure of, 56. in D'Alembert's principle, 228, 287. exerted on a fixed point, in rota- tion, 367. maximum, 281. Increase of inertia in an oscillating system, 469. Indicator diagram, 483. Inertia, law of, 25, 76. forces of, 59, 227. Initial tensions, 293, 356. Instantaneous centre, 261. screw, 334. Irreversible transformations, 495. Isentropic curve, 483. Isochronism of pendulum, 102. Isothermal curve, 483. for a perfect gas, 484. Jacobi, on motion in vertical circle, 121. Jellett, 50, 386, 394. Joule, on mechanical equivalent of heat, 477. Kater, on determination of force of gravity, 144. Kepler's laws, 91, 175. modification of third law, 184. Kilogrammetre, 126. Kinematics, 5, 254. Kinetics, 5, 27, 268. Kinetic energy, 133, 416, 419, 453. Lagrange, 210, 462. on spherical pendulum, 216. on small oscillations, 453. generalized coordinates, 421, 435. generalized equations for im- pulses, 417. Lambert's theorem, 183. Laplace, referred to, 332. Latent heat, of liquidity, 502. of vaporization, 503. of expansion, 491. Laws of motion : see Newton. Least action, 436. Line of quickst descent, 36. M'Cullagh, on rotation, 361, 377. Mass, 32. of Sun, 186. Mean value employed, 87. Mean energy in vibration, 411. Mechanical equivalent of heat, 477. Metric units, 23. Minchin, referred to, 50, 107, 163, 265, 272, 341, 343. Moment of inertia, 137. Momental ellipsoid, 348, 370. Momentum, 53. estimated in any direction, 74. conservation of, 75, 248. moments of, 243, 246, 273, 286. axis, 360, 373. Morin's apparatus, 46, 309. on impulsive friction, 309. Motion, first law, 25. second law, 26. third law, 58. on an inclined plane, 34, 46, 51. parabolic 39. of a particle, general equations of, 57. of a variable mass, 57. in a vertical circle, 99. on a fixed curve, 206. on a fixed surface, 211. of body round fixed axis, 255. round a fixed point, 353, 357. of solid of revolution, 389. Moving axes, 22. 526 Index. Newton, fluxion notation, 4, 9, 16. referred to, 76, 153, 176. laws of motion, 25, 26, 58. movable orbits, 196. central orbits, 166, 171. on coefficient of restitution, 68. on resistance of medium, 219. Orbits, central, 160. movable, 196. Oscillation of a simple pendulum, small, 101. in general, 108. period unaffected by resistance of air, 123. centre of, 142, 277. Oscillations, small, 445. Parabolic motion, 39, 72, 80. Parker, on equilibrium path, 509. Peaucellier's cell, 265. Pendulum, simple, 100. compound, 102, 141. conical, 115. spherical, 212. ballistic, 271. Percussion, centre of, 276. Perfect gas, 481, 484. Periodic time in central orbit, 161, 175. Planetary perturbations, 185. Poinsot, 378. Pole of rotation, 317. Polhode, 372. Poncelet, referred to, 159. Potential, 130, 135. energy, 133, 398. Poundal 54, 126. Principal axis, property of, in uniform rotation, 97. rotation round, 275. Principal moments, couple of, 347. Principal plane, 245. Projectile, parabolic path of, 39. Pure rolling, friction in, 234. Quickest descent, line of, 36. Range of a projectile, 41. Rankine, on steam, 505. Eebound from a plane, 69. Rectilinear motion, 25, 147. in resisting medium, 219. Relative motion, 6, 10. Resistance, of air, 48. does not affect pendulum period, 123. see Friction. Resisting medium, motion in, 219. Restitution, forces of, 6Q . coefficient of, 67. Reversible transformations in heat, 483. Richer, observed retardation of pen- dulum, 104. Rigid body, 240, 258. equations of motion of, 240. complete motion of, 329. Rodrigues, on screw motion, 336. coordinates of, 517. Rolling, pure, 261. Rolling friction : see Couple of. Rotation, velocity in, 226. acceleration in, 327. of a rigid body, 94. of a plane lamina, 95. energy of, 137. motion of, 318. instantaneous centre of, 261. Rotations, composition of, 317, 321. Routb, referred to, 144, 307, 309, 366, 391, 462, 468. on conjugate ellipsoid, 363. on equal factors of harmonic- determinant, 462. Salmon, referred to, 378. Schell, referred to, 279, 343. Screw, axis and pitch of, 333. of resultant twist,' 337. Seconds pendulum, 102. length of, 123. Similar mechanical systems, 414. Small oscillation of simple pendulum, 101, 445. Small oscillations in general, 445, 452. Source and condensor in Camot's cycle, 486. Space-axes, 330. Sphere used graphically in rotation, 317. Stability of motion of. small oscilla- tions, 451. Statical measure of force, 32. TTTVTJT 7 DAY USE RETURN TO DESK FROM WHICH BORROWED 3 LIBRA This publication is due on the LAST DATE stamped below. -QCL-ir&3 —fJCT ^^ u kn?w W%$f' vgpzrw F|p-'62 UlTTTlW oftifci^f? [JUL gQ 1071 — RB 17-60m-12,*57 (703sl0)4188 General Library University of California Berkeley N 6G54*>G UNIVERSITY OF CALIFORNIA LIBRARY