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AN ELEMENTARY TREATMENT OF THE THEORY OF SPINNING TOPS AND GYROSCOPIC MOTION

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■HI AN ELEMENTARY TREATMENT '^H OF THE THEORY OF SPIN]SriNG TOPS AND GYROSCOPIC MOTION BY HAEOLD CRABTREE, M.A. FORMERLY SCHOLAR OF PEMBROKE COLLEGE, CAMBRIDGE ASSISTANT MASTER AT CHARTERHOUSE WITH ILLUSTRATIONS LONGMANS, GREEN, AND CO, 39 PATERNOSTER ROW, LONDON NEW YORK, BOMBAY, AND CALCUTTA 1909 All rights reserved

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GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBEBX JIACXEaOSE AND CO. LTD. ..^w THE LIBRARY BRISHAM YOUNG UNIVERSITY. PROVO, UTAH _^

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TO MY OLD COACH AND FRIEND DR, W. H. BESANT, F.R.S. {Senior Wrangler, 1850) Tub last of all your pupils I, the last Of a long line, and some far-known to-day. For vie no fame : enough, the humbler way. To follow you who pioneered the past. And hoiv we loved your hook ! wherein you cast Your spells on moving ivorlds or meaner clay : — The ^^jet of sand,'^ the *^ falling chains,'^ the sway And poise of planet-spheres, dim, distant, vast. You tell me now the loheels of Life run sloio, And the chains loosen. Sinks the lessening sand While, as you rest, comes the calm evening glow, ^^ Cometh the night" — your motto. So you stand To loatch the day depart; yet, Master, know Still hums your torch, passed on from hand to hand. H. C. 1909.

PEEFACE. The object of this book is to bring within the range of the abler Mathematicians at our Public Schools and of First Year undergraduates at the Universities, a subject which has hitherto been considered too difficult for any but the more advanced students in Mathematics, while even they have in many cases failed to derive more pleasvire from the vstudy of spinning tops than is contained in submitting the problem to the action of a complicated piece of Mathematical machinery which automatically, though unintelligently, turns out the correct result. In this attempt to present an elementary, and at the same time a scientific view of the subject, I have expanded several of the suggestive ideas put forward in Dynmnics of Rotation by Professor Worthington, to whom is due a large debt of thanks for much liberal criticism and assistance in the earlier chapters, and for permission to borrow his article on Brennan's Monorail. In my other critics I have been similarly fortunate. Dr. Besant, Fellow of St. John's College, Cambridge, read and commented on several chapters in their first edition of manuscript ; and in dedicating this volume to him I have endeavoured to express the interest in the subject, and the love of it, which I owe to his inspiring teaching. Professor Hopkinson of the Engineering Laboratory at Cambridge has also read certain portions, while much assistance has been rendered by my friend and late colleague Mr. W. M. Page, Fellow of King's College, Cambridge, with whom I have spent many pleasant hours discussing various points. Lastly, the whole work has been carefully criticised by Mr. G. Bistwistle, Fellow and Principal Mathematical Lecturer of Pembroke College, Cambridge, to whom my sincerest thanks are due for much time devoted to the book and for many valuable suggestions.

viii PREFACE Considerable stress has been laid on Dimensions, particularly in the earlier pages, where quantities are equated which to the beginner appear at first sight to bear no relation to one another ; and a series of " Questions " has been interspersed throughout, the answers to which, though requiring no calculation, ensure an intelligent grasp of the principles involved, and may induce the reader to think for himself of further side issues. It has been found convenient to distinguish the axis of symmetry of a top by the term axle, this being in general the only axis which is not passing through the material of the top, and whose motion involves a corresponding motion on the part of the top. Particular care has been taken to preserve the elementary character of the first four chapters, which should be intelligible to any reader conversant with the principles of momentum and energy as applied to Particle Dynamics, while at the same time they embrace all that is necessary for understanding the motion of a spinning top, and the practical applications to the steering of a torpedo, the steadying of ships, and the monorail, which are given in Chapter V. Equations of motion, both steady and general, and a graphical representation of the paths described in space by the head of the top have been obtained directly from the principles of momentum, energy, and gyroscopic resistance. The latter has also been employed to establish the general equations of motion referred to moving axes, which are discussed subsequently and have been applied to the advanced part of the subject, including the oscillations of the sleeping top. For great assistance in reading the proof sheets and for many valuable suggestions while going to Press, I have to thank my colleague Mr. W. A. Nayler; and also Mr. W. H. C. Romanis, now Minor Scholar of Trinity College, Cambridge, with whom I have conducted many experiments at Charterhouse, and to whom several of the earlier examples are due. I also wish to acknowledge the courtesy and kindness of Messrs. Longmans, Green & Co., during the preparation of this volume. Extreme care has been taken both in the arrangement and expression of ideas, but doubtless deficiencies yet remain, and any corrections or criticisms will be gratefully acknowledged. HAROLD CRABTREE. Charterhouse, Feb. 1909.

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CONTENTS. INTRODUCTORY CHAPTER. PAGES Behaviour of Ordinary Spinning Tops 1-5 Tops containing Liquid 5^ ^ Celts ..-- 7 The Gyroscopic Top - - - - 8 The Gyroscope . - - - 9_12 CHAPTEK I. ROTATION ABOUT A FIXED AXIS. ARTS. 1-7. Angular velocity and acceleration . - _ _ 13-16 8. Law of inertia _ _ _ i§ 9, 10. Moment of momentum or angular momentum. Moment of inertia - - - - - - - 17, 18 11. Moments of inertia for a thin rod and solid disc - 18-20 12-14. Change of angular momentum 20, 21 15. Kinetic energy 21, 22 16. Work done by a couple 22 17. Analogies between linear and rotational motion - - 23 18. Table of moments of inertia 23, 24 19. Illustrative examples 24-28 Examples for solution. Diagrams of certain tops and of the mechanism for steering a torpedo - - 28-31 CHAPTER 11. REPRESENTATION OF ANGULAR VELOCITY. PRECESSION. 20. Angular velocity a vector quantity . . . . 32 21, 22. The physical meaning of composition of velocities - 32, 33 23. Simultaneous rotations ------- 33 24-27. Parallelogram of angular velocities and of angular accelerations ..----.- 34^ 35 28. Definition of angular velocity about a moving line - 35 29-31. Representation of angular momentum - - - - 36

CONTENTS ARTS. PAGES 32. Distinction between axis and axle _ . - - 37 33. Definitions : Gyrostat, Gyroscope, Spin of a gyroscope 37 34, 35. Motion of a gyroscope with its axle horizontal - - 37, 38 36-40. Precession 38-40 41. Analogy in linear motion --..-. 40 42. Diabolo 40, 41 Examples 42 43. Independence of translation and rotation - - - 42, 43 44. Illustrations of precession 43, 44 45. Gyroscopic resistance - - 44, 45 Examples 45, 46 CHAPTER III. DISCUSSION OF THE PHENOMENA DESCRIBED IN THE INTRODUCTORY CHAPTER. 47. Rise and fall of a top 47-49 48-50. Behaviour of a top containing liquid - - - - 50, 51 51. The gyroscopic top 51, 52 52. Efi'ect of reactions in the mounting of a gyroscope - 52 54. Swerving of a sliced golf ball. Drifting of a rifle bullet 53, 54 55. Explanation of the behaviour of Celts- . - - 54 CHAPTER IV. THE STARTING OF PRECESSION. OSCILLATIONS OF THE GYROSCOPE. 56. Conservation of angular momentum. Illustrations - 55-57 57. Summary of results 57 58. Increase of precession in a dying top - - - - 57, 58 59. Inertia - - - 58, 59 60, 61. Dipping of a gyroscope at the start of precession - 59-61 62, 63. Oscillations - - - - 61, 62 64. Summary of results 63 65. Principle of energy applied to a rising top - - - 63 66. Principle of energy applied to a dying top - - - 63, 64 67. Energy lost in heat. Electrical analogy - - - 64, 65 CHAPTER V. PRACTICAL APPLICATIONS. 68. The steering of a torpedo ------ 66 69. Herr Schlick's method of steadying vessels at sea - 67, 68

CONTENTS xi ARTS. PAGES 70. Extract fiom papei' read by Herr Schlick before the Institution of Naval Architects . . . . 68-70 71. Apparatus fitted to the R.M.S. "Lochiel" - - - 70 72, 73. Brennan's Monorail 70-73 73. Extract from the Times Engineering Supplement - 73-77 74. Differences between car and model . . . . 77-78 CHAPTER VI. STEADY MOTION OF A TOP. 77, 78. Rotation about one axis involves angular momentum about two other axes -.-..- 79^ 80 79. Particular case of a solid of revolution - - - 80, 81 80. Moving axle 81 81. Moving axis 82 82. The velocities and momenta of a spinning top in steady motion 82, 83 83. Velocity axis . - - 83 84. Momentum axis S3, 84 85. Rotation of momentum axis ------ 84, 85 86. Steady motion on a fine fixed point - - - - 85-87 88, 89. Equations deduced from gyroscopic resistance - - 87, 88 90. Motion equivalent to that of one cone rolling on another 88, 89 91. Application of the hodograph to angular momentum - 89 92. Deduction of the equations of motion - - - - 89, 90 93. Solid of revolution, or top with blunt peg, spinning on a rough horizontal plane 90, 91 94. Steady motion on a surface of revolution - - - 91 95. Systems of gyrostats ------- 91-93 96. Rolling cone 93, 94 Examples 95, 96 CHAPTER VII. GENERAL MOTION OF A TOP. 97. The angular momenta of a top spinning on a fine fixed point .....--- 97^ 98 98. Motion when the centre of gravity has no initial velocity 98 99. The limiting values of oscillation 98, 99 100. The values of the azimuthal velocity at these limits - 99, 100 101,102. Least possible values for the spin of a top- - - 100, 101 103. Change in direction of the azimuthal velocity - - 102 104. Oscillations deduced from gyroscopic resistance - - 102, 103

xii CONTENTS ARTS. PAGES 105. Value of the resisting torque 103 106. Equation of motion in the azimuthal plane - - 103, 104 107. Motion when centre of gravity has an initial velocity 104, 105 108. Paths in space of the head of the top - - - 105, 106 109,110. Motion of solid of revolution - - - - - 106-108 CHAPTER VIII. MOVING AXES. 111-114. Linear velocities in two dimensions . . - - 109-111 115,116. Accelerations 111,112 118-120. Linear velocities and accelerations in three dimensions 112, 113 .121, 122. Angular velocities and momenta - - - - - 113, 114 123. General equations of motion 114 124. Euler's dynamical equations - - - - - - 114 125. General equations deduced from gyroscopic resistance 114, 115 126. Art. 34 a particular case of general equations - • 115 127-129. Applications of moving axes 115-118 CHAPTER IX. STABILITY OF ROTATION. PERIODS OF OSCILLATION. 131. Rotation is stable about axes of greatest and least moment of inertia- - - - - - - - 119, 120 132. The Diabolo which will not spin 120, 121 133. A simplified case of Schlick's problem - . - - 121-123 134. Oscillations of a spinning top about position of steady motion 123, 124 135-138. Oscillations of the sleeping top. Integration of the equations 124-127 139. Path in space of the head of the top - - - - 127, 128 140. Periods of normal vibrations - - - - - 128, 129 141. Geometrical interpretation 129, 130 142. Value of the azimuthal angle at any instant - - 130 Miscellaneous Examples 130-133 Appendix 134, 135 Answers to Examples- --.-.. 136-138 Answers to Questions 139, 140 Plates : I. Celts, Figs, xi.-xiii. II. Photograph of Cat righting itself in mid-aii-. III. Schlick's Apparatus for Steadying Ships. Model exhibited before the Royal Society in 1907 - - at end

INTEODUCTOEY CHAPTER. There are few of us who as boys have not been mterested in Spinning Tops ; but the day soon arrives when we become too old or too proud to spin them, and most of us from that time never give them a thought again. A few years later, perhaps, we become interested in some of the numerous inventions of modern days, or even look forward to exploring advanced regions of Mathematics and Science ; but our tops have long since been forgotten. And yet, what surprising possibilities of knowledge and power have been put aside with our neglected playthings. Who would have thought that in them lay concealed the secret of steering a torpedo, of steadying ships, or of travelling with security in a single car on a tight-rope at the rate of 130 miles an hour ? Last, but not least, who would have thought that on such motion as theirs depends the foundation of the most astounding theory of Modern Science : that the so-called solid matter, with which we come in contact every day, is not really solid after all, but composed of a vast infinitude of particles whirling round one another at inconceivable speeds like planets in the Heavens. Professor Perry, in his most fascinating book entitled Spinning Tops,^ reminds us that an ordinary flexible metal chain or india-rubber band, which if left to itself would fall helplessly on the floor, will, if given a high speed of rotation on a revolving drum and then slipped off the drum, bounce along like an ordinary solid hoop ; that a circular disc of paper if made to revolve very fast about an axis perpendicular to its plane will behave as if solid, and if struck with a stick make a loud resounding noise :t while, if rings of smoke are blown in a certain manner and made to collide, they will bounce off each other as if made of india-rubber. These instances begin to give *The Romance of Science Series. S.P.C.K., price 2s. 6d. t A comparatively small rate of rotation will enable a circular piece of ordinary writing paper to act as a circular saw, su;fficiently powerful to cut another piece of paper right through. It is not difficult to see the reason for this quasi-rigidity in the case of the flexible chain or the revolving paper disc. If we consider an element AB of the A

2 INTRODUCTORY CHAPTER US some faint idea of how what appears to be solid may in reality be made up of minute loose particles revolving round each other at an enormous rate : and it is interestino: to think that nearly 2,000 years before this theory was formally stated, the Roman poet Lucretius should have written six books of majestic verse, one entirely, and the others partly, in support of a theory extremely similar to this, propounded by the Greek philosopher Democritus 400 years previously. Let us consider for a few minutes the behaviour of an ordinary spinning top. It is full of surprising contradictions. In the first place, to take the most obvious of all, we cannot balance it on its peg ; but give it a spin and it will stay balanced for a long time. We have known this all our lives; but few people can explain the reason, except to say " because it's spinning " — which begs the question. Again, supposing we spin it with its axis vertical and then give it a knock, it will go round the table in a slanting position ; but if we spin it slanting to begin with, it will almost immediately stand upright. And once again, although it is the friction of the table and the resistance of the air which eventually bring a spinning top to rest, yet if we take a very smooth surface, such as a glass plate, we find that many tops will not spin on it at all. The childish delight which we felt in watching our tops spinning remains with many of us a vivid recollection to this day. Some tops would buzz about busily before settling down to a regular motion ; others would be steady and stately from the first. Some would " go to sleep " almost at once, and, if disturbed, would only show signs of life for a short time before going to sleep again : these when they " died " would die very suddenly. Others when disturbed would spin about in a slanting position for a long time, particularly those with rather a long "leg"; these took a long time to "die." It is within everybody's experience that if a top is spun so that the foot traces out (approximately) a circle on the table, this circle will be described in the direction of rotation of the chain we see that it is acted on (in addition to its weight) by two tensions, each of very great magnitude, since they act at a very small angle to each other and yet supply the normal force necessary for the circular motion. Hence any force which deflects AB from its circular position will have to be extremely large to overcome the resolutes of these tensions which would be immediately called into play. Similarly in the case of the disc, any small elemental area is under the action of two very large tensions, resolutes of which are immediately called into play when a lateral blow is given to the disc.

INTRODUCTORY CHAPTER top; in fact the whole top rolls round the circle leaning towards the centre of the circle; but the top in the accompanying hgures Fig. I. Fig. II. will behave in exactly the opposite way. Here the string is fixed to the head by a swivel, and to spin the top the string Fig. III. (6). Fig. III. (a). is pulled tight down to the toe and wound round, starting from the toe as for spinning an ordinary peg-top (Fig. ii.). The top

INTRODUCTORY CHAPTER is then thrown down, when of course, instead of falling to the ground, it remains suspended from its head and spins, but instead of its toe (and its axis) moving round in the direction of the top's spin viewed, say, from above, it goes in the contrary direction (Fig. iii. a). If the string is lowered so that the peg can touch the table (or floor) the toe of the top will immediately reverse its direction of motion (Fig. ill. h) ; nor will the slightest jerk or eflbrt be visible. The top will probably begin to rise to a more vertical position. If this motion is allowed to continue till the top begins to " die " and the axis becomes rather more horizontal, when it is lifted off the table by the string the immediate reversal of the direction of motion of the toe will become very obvious. The effect is a very pretty one, and it is interesting to note in passing that the Earth is a top of this description, i.e. one whose axis revolves (with a motion called " precession ") in the opposite direction to the spin of rotation. The particular top represented in Figs. IV. (a) and IV. (b) is an ordinary whiptop through which has been inserted a metal spindle, which is capable of sliding movement through the top; so that either a very long or a very short leg can be used and the main body of the top can be either high up or low down. At the "foot" either a blunt or a fine peg can be screwed in, and it is found that, whether a long or short leg is used, the top will always rise more quickly from the slant position with the blunt peg. Any top will also rise more quickly on a rough surface than on a smooth one. If care is taken at the start to spin this top in a direction which should tighten the screw at the foot, it will be found when the top is "dead" that the screw is quite loose; but if the reverse spin be originally given this screw will end up tight. This top, if on its fine peg, maintains a steady motion for a long time when it is so nearly horizontal that the body nearly scrapes the table. Fio. IV. ((0— Blunt peg. Fig. IV, (6).— Fine peg.

INTRODUCTORY CHAPTER 5 When the blunt peg is used this (practically) horizontal motion does not continue so long : and, in general, a top with a fine point or with a long leg will spin at a greater angle to the vertical than one with either a blunt point or short leg. A loaded sphere when spun on a rough surface also presents a curious contradiction. Fig. V. (rt). Pig. V. (6). If, for example, a small hole is made in the side of a croquet ball and filled up with lead, when placed on a table the ball will settle down to the position where the lead touches the table (Fig. v. a). But if a really good spin be given to the ball the loaded part will persistently rise, as indicated in Fig. v. (h), and if the table is very rough it will get to the position where it is at the highest point of the sphere. Many of the tops we have been accustomed to spin are hollow. Has it ever occurred to the reader to ask what would be the efTect of filling a tin top with water, and making it watertight ? The answer that occurs to most minds at once is probably that it would spin much better; it is heavier, and Fig. VT.— Top full. Fig. VII.— Top empty. there is more of it generally. Let the experiment be made. Figs. VI., VII. represent two tops which are in every particular the same, except that the left hand one is full of water hermetically sealed. The empty one, if spun in the ordinary way, will

INTRODUCTORY CHAPTER continue to spin in an upright position ; the other one will lie down on its side at once, and spin violently lying at full length on the table. Some such tops are a little uncertain which to do. That in Fig. viii. has been constructed so that the head can be unscrewed and water poured in. If empty it spins very well, whether a big or little spin be originally given to it. When it is full of water a little spin will only result in the top falling to the ground ; a good spin will keep it upright in spite of the water. Such tops can be readily constructed out J) of small tins or similar receptacles capable of being soldered and made water-tight. Figs. IX. and x. represent two hollow china eggs, exactly similar, with the exception that one has been filled with water and the hole stopped up with sealing wax. If they are laid down on the table, and a spin is given to each of them about a vertical axis, they will behave in entirely different ways. The empty one jumps up briskly on its end and continues to spin in that position for a long time ; the other will spin slowly on its side for a short time and continue this uninterestino; motion till it stops : or if stopped prematurely by laying a finger on it, will begin to spin again on removing the finger. Fig. viii. -K Fig. IX.— Egg full. Fig. X.— Egg empty. ^ A real egg, if unboiled, and especially one which has had its yolk thoroughly shaken up into liquid form, will behave m precisely the same way; but a hard boiled egg will spin up on its end at once (especially if the table is rough) and continue spinning for a long time. A similar phenomenon can also be observed with acorns at a time when they are lying thick m the roads. If they are kicked, in any way whatever, they almost invariably skid along for a little way and then

INTRODUCTORY CHAPTER 7 rise up on their longest axis, in which position they remain spinning violently, though otherwise stationary on the road. What seems at the present time to be a favourite form of whiptop with children in the streets is an instance of the same thing. It is, roughly, in the shape of a large acorn with a thick cap to it. In whatever position it is first put down on the road violent lashing will always make it rise on to its longest axis, and continued lashing causes it to continue to spin in that position. Celts. Figs. XI., XII., XIII. (see Plate i.) represent three celts, stone implements used in times past by primitive man for cutting or cleaving, and now discovered from time to time in ancient barrows (Latin : Celtis, a chisel). They provide typical instances of the curious phenomena exhibited by all smooth celts when spun on a smooth horizontal surface (such as a sheet of glass). Let us perform the same experiment on each in turn and note the results. In each case, the stone being considered at rest, GAB is a horizontal section (very roughly elliptical) through the centre of gravity G, and GA, GB, GO are mutually at right angles. Fig. XI. (See end of book.) (a) If a vertical downward tap be applied at A the stone will rock about GB, and if the tap be applied at B the rocking will be about GA. (b) If spun about the vertical GG the stone will continue to spin steadily about GO until brought to rest by friction. These results are what we should ordinarily expect. But now let us consider the other two celts. Fig. XII. (a) When a vertical downward tap is applied through A the stone oscillates and then begins, almost immediately, to rotate from A to B ; but if the tap be applied at B, the stone, after a moment's oscillation, begins to rotate from B to A. (b) If spun in either direction it will begin to oscillate and gradually reverse its direction of spin. Fig. XIII. (a) A vertical downward tap through A or B sets the stone rotating in the direction from A to B. (b) If spun in the direction J. to .B it will continue to spin steadily in that direction ; but if spun in the direction B to A the stone will oscillate violently and eventually reverse its direction of spin, i.e. will spin in the direction A to B.

INTRODUCTORY CHAPTER The "Spider tops," which are frequently sold in the streets of London, consist of a heavy little disc mounted on a spindle (Fig. XIV.). When the disc has been set spinning a small curved piece of metal is placed to touch the toe, and at once begins to slide round it, first the side (a) in the figure, and then the side (h), the motion continuing backwards and forwards till the top comes to rest. The fact is that the toe is magnetic, and this being the case it is easy to see that the rolling of the toe on the side of the metal produces the motion. But Figs. XV. (a) and (b) illustrate a top, whose spindle behaves in exactly the same manner as the toe of the spider top, and yet is in no way magnetic. The action is purely mechanical, as we shall explain in later pages. It is an attractive top to watch, especially as it rushes round the corner when it comes to the end of the coil.^ Fig. XIV, Fig. XV. {a). Fig. XV. {b). But the most interesting top of all is undoubtedly the ordinary gyroscope. That depicted in Figs. xvi. to XXI., although merely sold as a toy, is nevertheless capable of illustrating the gyroscopic phenomena which have been so much made use of in modern mechanical invention. If the gyroscope is spun as in Fig. xvii. the surface on which it is standing should be as smooth as possible. See page 49, question 14. *This top, which is known as a "Gyroscopic Top," can be purchased of Newton & Co., Scientific Instrument Makers, 3 Fleet Street, London, E.G., price, £1. 68.

Fig. XXI.

10 INTRODUCTORY CHAPTER Scientifically constructed Gyroscope. Fig. xxii. represents a scientifically made instrument.* The wheel AB which has (for its size) a very heavy rim, is free to spin about its axle XX\ while the frame in which this axle is mounted can turn about the axis YY\ Finally, the frame in which YY' is mounted can turn about the vertical pedestal ZZ\ but at Z there is a screw which can be tightened to prevent this turning. X'.,_ T' .^Y' ^X Fio. XXII. Experiment 1. Suppose now that the wheel is free to turn about all three axes, and that friction is so slight as to be negligible. Let a good spin be given to the wheel in the direction marked, and let a weight be suspended at X. It will be found that, instead of the wheel tilting about Y'Y, it will turn about ZZ' in the direction indicated. But if the screw at Z is tightened the revolving wheel will at once turn over about TY, If, when the wheel is free to turn about all three axes, we apply at X a force which is more easily removed than the weight, such as the pressure of a finger, it will be found tliat, when the rate of spin is small, the wheel will tilt about Y'Y appreciably, and on the finger being removed the whole system will oscillate violently. If, however, the rate of spin is large, only a very small tilting, if any, will be appreciable, while on the removal of the finger the violent oscillations observed previously are now hardly more than a " shiver " of the axle. ^ The gyroscope here depicted is known as Wheatstone's Compound, and can be procured of Newton & Co. , at the cost of 3 guineas.

INTRODUCTORY CHAPTER 11 Experiment 2. If the wheel is taken out of its bearings at Y, Y', and the framework held in the hands at X, X\ a violent attempt to tilt it about Y'Y will, unless proper care is taken, result in the wheel wriggling out of the hands altogether — for the same reason that when properly mounted just now it turned about ZZ'. It is clear that an opposite direction of spin or any attempt to tilt in the opposite direction, produces an opposite direction of turning about ZZ'. Experiment 3. Let us now take the gyroscope when the wheel is not spinning, and its axle XX' is inclined at an angle to the vertical as in Fig. xxiii., the screw at Z not being clamped. If the pedestal ZZ' is held in the hand and slowly swung round, it will be found that XX' turns round about the vertical and points in the same direction as the hand. But let a spin be given to the wheel, and it will be seen at once that, though the hand revolves round the body, XX' remains always pointing in the same direction in which it pointed originally* This stability of the axle of the gyroscope may be employed to prove the rotation of the earth. For suppose the spin is maintained during a period of several hours by means of electric power, and that the gyroscope is set down in a room with its axle pointing to some particular object in the room. After a few hours it will be found that the axle no longer points to that object ; showing, not that the axle has changed its position in space, but that the room is in a different position in space owing to the rotation of the earth. Thus, if the axis is placed horizontally pointing due east, six hours later it will be pointing south (and upwards), while after another six hours it will be pointing due west (and horizontally). Experiment 4. Now, while the wheel is still spinning, let the turning movement be given to the hand when the screw at Z has been made fast. The gyroscope at once sets itself with the Fig. xxiii. * The friction may not be quite negligible though very nearly so. The frictional couple at Z is probably sufficiently large to turn the frame YY' round when the wheel is not spinning, but totally inadequate when it is spinning. So much as is appreciable tends to turn it about Y'Y, as will be seen later.

12 INTRODUCTORY CHAPTER axis XX' vertical as in Fig. xxiv., and when once this position has been gained a further swing of the hand in the same direction finds it in stable equilibrium : but if we check or reverse the swing we find that the equilibrium is unstable, and if the checking or reversal be sudden, the gyroscope will turn a complete somersault ! Fig. XXV, Fig. xxY. represents the same gyroscope taken to pieces and put together in another form. Here the weight of the instrument itself, when no extra weight is suspended, instead of tilting the gyroscope over, causes it to turn about ZZ' as before ; and it will be found that if the rate of turning be hurried by the application of a horizontal force to the frame of the gyroscope, then the centre of gravity of the instrument will at once begin to rise. The following pages will be devoted to discussing the phenomena described in this chapter, and to explaining why the gyroscope appears to defy all the recognized laws of gravity. The same principles account for an elongated bullet, when fired from a rifle, always " drifting " to the left-hand side, as also in part for the gyrations of a boomerang, but not in general for the swerving of a sphere, as for instance a cricket ball or a " sliced " golf ball. In order to get a clear idea of the subject it will first be necessary to master a few elementary principles in Dynamics which are discussed in the first two chapters of this book.

CHAPTER I. ROTATION ABOUT A FIXED AXIS. 1. Definition of angular velocity. If a point P is moving relaitively to a point 0, but not along the line OP, it is said to have angular velocity about 0. The angular velocit}^ is measured by the rate at which the line OP is describing an angle. For instance (Fig. 1), if 0 is the centre of a circle round which a point P is moving uniformly with velocity v, having started from the position A, then the rate at which the angle POA is being described is the angular velocity, co say, of the particle P about 0. Fig. 1, In this case if r is the radius of the circle, and v units of length are traversed in a unit of time, say one sec, then V O) = - radians per second. r It is clear that when we speak of the angular velocity of a point (P) about a point (0) we really mean the rate at which a line (OP) is revolving. As a further illustration (Fig. 2), suppose P is moving in a straight line with velocity v, in the direction AB, and that 0 is a fixed point. Let P be the position at the time t, and P^ at the time t-]-Sf. In the small time St P describes an angle POP-^ about 0, and if P^N is perpendicular to OP, the circular measure of this angle is P.N PP. sin 0 vSt sin 0 .^ii^d a\ -OP^^V~0-p-^-0^' (OPi^ = f?)

14 ROTATION ABOUT A FIXED AXIS V sm u .'. the rate at which this angle is described is (jo='-jyp~' .'. the angular velocity of P about 0 is . 2. The angular velocity of a line in a plane is the rate of increase of the angle it makes with some fixed line in that plane. For instance, if a stick is so thrown that its motion is in a vertical plane, its angular velocity is the rate of increase of the angle it makes with some vertical line in the plane of motion (Fig. 3). Notice that we speak of the angular velocity of a point ahotU a point, but of the angular velocity " of a line," simply — not " about " anything. ^^^ Fig. 3. Fio. 4. 3. Angular velocity of a plane. The angular velocity of a plane is the rate of increase of the angle it makes with any plane fixed in space. For instance, in Fig. 4, if 0 is at any instant the angle between the fixed and revolving planes, the angular velocity, . do a, say, is ^. 4. Rigid Body. By a rigid body or system we mean one in which the particles composing the body or system never alter their positions relative to each other. An ordinary walking stick is an approximate instance of a rigid body : an umbrella is not. 5. Angular velocity of a rigid body. The angular velocity of a rigid body is the angular velocity of any plane fixed in the body. In the accompanying figure the body is represented by a cylinder. If 0 is at any instant the angle between the two planes, then the angular velocity of the cylinder is do ""^di'

ROTATION ABOUT A FIXED AXIS 15 If the plane YOX were not fixed in space, but revolving do about OY, then w, if measured by -j^, would be the angular velocity of the body relative to the plane YOX and not "the angular velocity of the body." Angular acceleration is the rate of increase of angular velocity. Fig, 6. Torque. A force, or system of forces, capable of turning a body about any axis is said to be, or to have, a torque about that axis. It is measured by the moment of the forces about the axis. 7. Uniform angular acceleration. If a rigid body is rotating about a fixed axis with tvniforon angular acceleration a, then after a time t it is clear that, supposing the initial velocity to have been co, the final velocity Q = (jo-\-at (1) It is also clear to those familiar with the elementary formulae of particle dynamics, that if in time t, the rigid body under the same conditions turns ' through an angle 0, the average velocity at which this angle is described is Q + (jO . ^ 0 + co , O) + a^ + a) ^ =(^t-\-\at\ .(2) Combining equations (1) and (2) as in particle dynamics we obtain Q2 = a)2 + 2aa (3) These formulae are exactly analogous to the similar ones relating to the rectilinear motion of a particle under uniform rectilinear acceleration ; for they are obtained from exactly

16 ROTATION ABOUT A FIXED AXIS similar considerations ; but the student may feel at first that he is dealing with a different kind of velocity, and a difTerent {i.e. angular) measurement of space. In order, therefore, to obtain a clearer realisation of the quantities employed, the dimensions of the equations should be considered, this being a most important method of checking algebraical expressions. Dimensions. (1) i1 = iio-\-at\ 1 1.1 r,x[n [rr=[T]-^+[Tr. (2) 0 = wj^+Jaf2; (3) Q-^ = a)2 + 2ae; [r]-=[r]-H[T]-^ EXAMPLES. 1. A Avheel acquires a v^elocity of 100 racl./sec. in 16 sees, under a uniform angular acceleration, having started from rest. Find the acceleration. 2. Through what angle has it revolved in the 16 sees. ? What angle will it turn through in the next 16 sees. ? 3. A wheel is given an initial velocity of 1,000 reA^olutions per sec, after which it is acted on by a uniform retardation which in 3 sees, reduces its velocity by 12 revolutions a second. How many revolutions will it have made fi-om the start when it is making 20 revolutions a second ? 8. Inertia. Newton's First Law of Motion asserts that, Every body will continue in its state of rest or of uniform motion in a straight line, except in so far as it is compelled by impressed forces to change that state ; which is equivalent to saying that a body has no power of itself to change its state of rest or of uniform motion in a straight line. This statement is sometimes called the " Law of Inertia," " inertia " being regarded as a property of mass (Latin : iners). Newton's Third Law of Motion asserts that. To every action there is an equcd and opposite reaction ; and thus the existence of a force implies of necessity some resistance against which the force is applied. When a force is applied to a body at rest the resistance offered by the body is due to the inertia of its mass. If the body had no mass no force could be exerted on it. So also, if we attempt to alter the existing motion of a body, a resistance is experienced. This co-existence of action and reaction is described by the term stress ; and the resistance which a body offers by reason of its mass is sometimes called the force of inertia of the body, meaning the resistance due to inertia.

ROTATION ABOUT A FIXED AXIS 17 9. Let us now imagine a rigid body to be rotating about a fixed axis AB with angular velocity co, as for instance a gate swinging on its hinges, Fig. 6. Any given particle, such as P, in the body is moving with a definite linear velocity f, perpendicular to AB, which is represented by rco, r being the length of the perpendicular from the particle in question upon the axis. Therefore if ni is the mass of the particle, its linear momentum is onv or mrco : and if we take the " moment " of this momentum about AB, as we take the " moment " of fig. 6. a force, then the "moment of momentum " of the particle about AB is "iiiv . r = nfiTci) . r = mr'^uo ; and the '' moment of momentum " of the whole body is ^mr^w, or (jd^mr^ since w is the same for all the particles. Now 2'?7ir^ is a quantity dependent in each case on the configuration of the particles forming the body, and on the position of the axis AB ; and it is clear that for any particular body and axis an expression Mk^ can be found equal to S^nr^, where M is the mass of the whole body and k some length which is constant for the given body and axis. This expression is termed the "moment of inertia" of the body (see Art. 10) about the axis in question, and is frequently denoted by /. It is a most important quantity whenever the motion of a rotating body is considered ; for the rate of change of '^mr^oa or Mk^co, i.e. the rate of change of the moment of the momentum, represents, as we shall see later, the turning effect about the axis in question of the external acting forces, and therefore enables us to determine the motion which these forces cause. The length k is called the radius of gyration of the body about the axis. The moment of the momentum of a body about an axis, i.e. Mk'^w, is frequently called the angular Tnomentu'm of the body about that axis. QUESTIONS. [1.] What are the dimensions of (a) Moment of inertia ? (/5) Angular momentum ? (y) The moment of a force, i.e. a " torque " ? [2.] A heavy iron door weighs a cwt. Taking its radius of gyration as 2 ft., what is its moment of inertia about its hinges ? State clearly the units you employ. B

18 ROTATION ABOUT A FIXED AXIS 10. The phrase " Tno'tnent of inertia " means in the first instance " the importance of the inertia " ; and its significance will be easily grasped if we consider the obvious importance in the motion of some rigid bodies of the configuration or arrangement of the particles which form them. A golfer might conceivably possess a " putter " of the same total mass as his " driver " ; but the difference in their usefulness for conveying force to the ball is clear at once. This is of course due to the different configuration of particles in the two clubs. Similarly, a cricketer "takes the long handle" when hard hitting is his primary object. Or again, suppose we have two wheels, of the same total mass, rotating about fixed axes with the same angular velocity, the only difference in them being that one has most of its mass concentrated near the centre, the other at the rim. It will be found that the one with the heavy rim requires a much greater force to stop its rotation than the one with a heavy centre and light rim. The devices for finding the value of different moments of inertia, either without or with the aid of the Calculus are discussed fully in various text books. The following two elementary examples will illustrate the method when the Calculus is employed. 11. To find the moTnent of inertia of a thin rod of mass M and length I, about an axis which passes through one end of the rod and is perpen- a' dicular to its length. Let OA be the rod of which PQ is a small element dx, at a distance OP ( = x) from the „ ' ^ axis, l*ig. 7. Let y(x = mass per unit length of the rod. Then the moment of inertia of the whole rod = lljULdx. x^ PQ dx.x^ 10 "LTJo k^ = w, I being the length of the rod.

ROTATION ABOUT A FIXED AXIS 19 From the definition of moment of inertia as being the value of 2mr^, it is clear that the moment of inertia for any body is the sum of the moments of inertia of its component parts, and, therefore, if 0 were the middle point of a rod A' A, the moment of inertia about the central perpendicular axis would be 2—^ {M' being the mass of a length I which is in this case half the rod), Mh^ or M being the whole mass and h half the length of the rod. To find the Tnoment of inertia of a solid disc of mass M, radius a, and thickness 6, about its central perpendicular axis. Let us consider first a thin circular section ABC of the disc (Fig. 8), the whole mass of which is dM, and let /x represent the mass of this section per unit area. Divide the section further into thin rings, one of which is represented in the figure of radius r and thickness dr. Fig. 8. The mass of this ring is fxlirrdr. Every particle of the ring is at a distance r from the axis ; .". the moment of inertia of the ring is yw27rr dr . r^, or ^jjurr^dr ; .*. the moment of inertia of the thin section ABC f = I 2/uL7rr^dr = 2[^ dM , where dM is the mass of the thin circular section.

20 ROTATION ABOUT A FIXED AXIS Now if we consider the (thick) disc to be made up of an infinite number of thin sections exactly similar to ABC, each dM a^ having a moment of inertia — ^ — about the axis in question, then for the whole solid disc made up of an infinite number of particles each of mass m, the moment of inertia ^tnr^ is equal to I dM -^ = Ma" and 1 = -^, i.e. the radius of gyration is — t= [3.] How has the thickness h come in ? [4.] What is the angular momentum {MBim) of a grindstone about its axis, in pound-foot-radian-second units, if its mass is 20 lbs., radius 1 ft., and it is making two revolutions a second ? A list of moments of inertia for some simple solids will be found at the end of this chapter. 12. Rate of change of angular momentum. It was stated in Art. 9 that the rate of change of the angular momentum (Jw) of a rigid body rotating about a fixed axis represents the turning efiect about the axis in question of the external acting forces : i.e. the moment of the forces about the axis. Let us consider a body turning about a fixed axis (through 0 perpendicular to the plane of the paper in Fig. 9), under forces whose moment about the axis is K, with angular velocity w, and angular acceleration a. Let P be any particle of mass m, Fig. 9. and the forces acting on it be as shown in Fig. 9, X, Y being components of the external forces and X\ Y' components of the reactions with neighbouring particles. The resulting acceleration of P will have components rco^ and ra as marked. Then, by Newton's second law, F+ Y' = mva ; ... {Y-\-Y')r = mr''-a.

ROTATION ABOUT A FIXED AXIS 21 Taking then the motion of all the particles, since a is the same for all the particles, = Ia. But since the internal reactions are equal and opposite, their moment about the axis is zero. .'. 2FV is zero ; .'. ^Yr = Ia. But ^Yr represents the moment of the external forces about the axis, i.e. K; ' K=Ia that is, about any fixed axis, the moment of the external forces is equal to the rate of change of the angular momentum. Dimensions. ^^^M^ •[L] = [M][LJ . ^ 13. The student should notice that as in linear motion we have the equation force = inertia X acceleration, so in rotational motion we have moment of force = moment of inertia X angular acceleration. A similar analogy offorin can be seen in all equations relating to rotational motion. 14. Change of angular momentum. From the above equation we see that if a and K are constant, Kt=-Ia.t where co becomes co^ after time t Further, this is true when K and a vary, if t is so small that they may be regarded as constant during the interval. If now K be made infinite and t indefinitely small in such a way that Kt is finite and equal to K\ we have an impulsive couple of moment K' and that is, about any fixed axis, the moment of the external impulse is equal to the change of the angular momentum. [5.] What is the analogous equation in linear motion ? 15. Kinetic Energy. // a body is rotating with angular velocity w about a fixed axis its kinetic energy is \I(jo^. For, using the notation of Art. 12, the linear velocity v of P is ro).

22 ROTATION ABOUT A FIXED AXIS .'. kinetic energy of particle m =^7nv'^ = \nfir^(f)^. „ „ whole body = 2 JmrW CO ^ 9 = IM1^^\ or J/o)^. Dimensions. K.E. = J /co^, or W[-^]' FTP ' ^.6. same dimensions as force X distance, which is work done. [6.] What is the analogous equation in linear motion ? 16. Work done by a couple. // a constant couple K turns a rigid body about a fixed axis through an angle 0, then the work done by the couple on the body is KO. For, in the small element of time St, let the body turn through a small angle SO. Then P (see Fig. 9) is displaced through a distance rS6 along the direction of the force F+ Y\ .'. The work done on the particle P in time St = (Y+Y')rSe. .*. The work done on the body in time St = 2(Y+Y')rS0 = Se^(Y-hY')r = Sei:Yr, since SFV = 0 =se.K, where K is the moment of the couple. Hence, the couple K being constant, the work done by it during the whole time is given by KO. Since the work done on any body is equal to the change in kinetic energy, we have ir^ = iJ«-co2), if CO changes to coy This also follows from the equations K = Ia, Art. 12, and Wi2^ft)2 + 2a0, Art. 7, V. ^ ^ 20K whence m-^^ — w"^ = — ^ or KO = iI(co^^-w^). Dimensions. i^Ml [X] = [M] [Lf . ^, . [7.] What is the analogous equation in linear motion ?

ROTATION ABOUT A FIXED AXIS 23 17. Summary of preceding results. Linear Motion. Rotational Motion. Inertia or mass = i/. Moment of inertia =Mk^ = L Momentum =Mv. Moment of Momentum' or ' = Id). Angular Momentum J Force = Mf. Moment of force =/a. Impulse Kinetic energy Moment of impulse = /(w^ — w). Kinetic energy =^Ioi^. Work done by constant force P moving its point of application through distance 5 = Fs = ^M(v^^-v^). Work done by constant couple K in turning body through angle $ = /f(9 = 4/(a)i2-(o2). 18. Table of moments of inertia for some simple solids of common occurrence. Most of the following results (the exceptions are noted by an asterisk) are included in a rule enunciated originally by the late Dr. Routh, and known as "Routh's Rule." "The moment of inertia of a solid body about an axis of sum of squares of perpendicular semi-axes symmetry = mass X o^ a k ' the denominator being 3, 4, or 5 according as the body is rectangular, elliptical, or ellipsoidal." It should be remarked that these perpendicular axes are axes of symmetry, and that a circle and sphere are special cases of an ellipse and ellipsoid respectively. The results can be obtained by simple integration similar to that indicated in the earlier part of the chapter. Except where otherwise stated the axes core taken through the centre of gravity. (1) Thin straight rod — about an axis perpendicular to its length : * Through one end, -— , where I is the length. Through the centre of gravity, half-length. Mh^ , where h is the (2) Thin rectangular lamina, sides 2a, 26 — about an axis : Perpendicular to plane, M . — ^^. Parallel to side 2a, 3 (3) Rectangular parallelopiped sides 2(X, 26, 2c : 62 + c^ Perpendicular to 26, 2c, M. 3

24 ROTATION ABOUT A FIXED AXIS (4) Circular wire, radius a : * Perpendicular to plane, Ma^. * Diameter, ^— — . 2 (5) Circular disc, radius a : Perpendicular to plane, — — . JJiameter, — ; — . 4 (6) Circular cylinder, radius a : Central axis, . 2 (7) Elliptical disc, axes 2a, 26 : Axis 2a, ——-. 4 Axis 2b, ^If. 4 Perpendicular to plane, M . — ^^ — . (8) Spherical shell, radius a : * T^. , 2Ma^ * Diameter, — - — . o (9) Sphere: y.. , 2Ma^ Diameter, — -— . 5 (10) Ellipsoid, axes 2a, 26, 2c: Axis 2a, M . — ^^— , and similarly for other axes. (11) * Right cone, height h, radius of base a: 3a-^ Axis, M. ^^ Perpendicular to axis, ^^ 'WK\Gi?-\--r\ 20 \ 4/ 19. We will now proceed to the discussion of some problems in connection with the angular momentum and kinetic energy of a rigid body rotating about a fixed axis, on the assumption that the general principles of work and energy are understood. ILLUSTEATIVE EXAMPLES. 1. A spinning top weighing 8 oz. is set rotating with a velocity of 5 revolutions a second. Taking the radius of gyration as 2 ins., calculate (i) its angular momentum ; (ii) its kinetic energy.

ROTATION ABOUT A FIXED AXIS 25 (i) Angular momentum = /w. Here w = 5 . 27r = IOtt radians a second. Angular momentum = /(o 5 = ir7;7r Ib.-ft.^/seC. (ii) Kinetic energy = -/w2 ^I'l-ie^^^''' [7r2 = 10approx.] 250 125 _ „^ ,, = -gg- = -^ = 7 f t.-pdls. approx. 2. Round the spindle of a top ^ in. in diameter, is wound a thin string 3 ft. in length which is pulled off by a uniform force in 2 sees. (i) Find the angular velocity after that time. (ii) If the angular velocity is IStt rad./sec, how long did it take to pull off the string ? (i) The point where the string is held passes over 3 ft. in 2 sees. .'. its average velocity = 1 ft. /sec. .*. since the pulling force is uniform its final velocity = 3 f t./sec. (Art. 7), and this is the final velocity of any point on the spindle's rim. Hence, since the spindle is j in. in radius or ^^ ft., the final angular velocity of the spindle or of the top = 3-^4^3 = 144 rad./sec. (ii) Let t sees, be the unknown time. Then as before, the final velocity of a point on the string or on the rim of the spindle = 2. 1 f t./sec. The final angular velocity of top therefore = — -48^^^^-/^''But this also = 15:7 rad./sec. 6 . 48 , ^ .'. — ^ = 157r; 6 . 48 96 • t = — — = — sees. IStt 57r 3. A heavy top weighing 1^ lbs., whose radius of gyration may be taken as 4 ins., is revolving at the rate of 450 turns a minute. (i) What energy is expended in stopping it ? (ii) If a frictional force equal to — lbs. wt. be applied to the rim at a distance of 6 ins. from the axis, how soon will the top be brought to rest, supposing the axis to be fixed vertically all the time ? (iii) How many turns will it have made since the force was applied ? (i)Here I=l{±J or ^ Ib.-ft, 450. 27r , , ,0) = ——- — rad./sec. = IStt. bO

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26 ROTATION ABOUT A FIXED AXIS The kinetic energy of the top (Avhich must require an equal amount of energy to stop it) = ^ . ^ . 1 5'^ TT^ [7r2 = 10 approx.] = 187ift.-pdls. (ii) We must first find the angular retardation produced. Denoting this by a rad./sec.^ we have rate of change of angular momentum = /a = -a Ib.-ft.-rad.-sec. units. 6 5;r 1 Moment of applied torque = ^^ . 32 . - units. Hence, S'^^.H"' .•. required time = a ^1573= 3 sees. Or we may proceed directly. Change of angular momentum in time t = /a> = ^157r units. Moment of applied torque K^ 577 "96* -•i units. Kt = = /a) ; /IStt . bTT 16 L — 6 ' 96 • = 3 sees, as before. (iii) Since the final angular velocity = 0, and initial „ „ =157r rad./sec. ; .-. average „ „ =-k~ rad./sec. — ^-f27r revolutions/sec. .*. in 3 sees, top will have made — or llj revolutions. 15 4 ' 45 4 Or we may use the equation KQ = \I{i3i^ - (o^). For from (i) Kinetic energy = 187^ ft.-pdls. 57r 1 Applied couple /f =^ x 32 x - ft.-pdls.

ROTATION ABOUT A FIXED AXIS 27 .*. if 0 be the angle through which G moves in destroying the kinetic energy, — 32 i 96 • '^^ • 2 Hr.32.i.^=187j. n 375.3 ,. u= — radians OTT revolutions = ^ (taking 7r2 = 10) = llj as before. 4. If the mass of the top in example (2) be 8 oz,, its radius of gyration about its axis of symmetry 2 ins., and the pulling force be 10 lbs. weight throughout, how many revolutions a second will it be making when all the string is unwound ? The work done = 10 . 3 . 32 ft.-pdls. If to = the angular velocity generated, in radians per sec, kinetic energy = -/w2 22V6 Equating the K.E. to the work done, we get 0)2 = 10. 3. 32. 4. 36; .-. a> = 96N/T5; .'. number of revolutions per sec. 0)2. 48\/l5 TT Notice that in this case the size of the spindle does not affect the problem. If the spindle were larger the string would be unwound in fewer turns, hut in a shorter time. The force would act at a longer arm, and the power or rate of work would be greater, but the actual work done, or kinetic energy generated, would be the same. For the same reason the string need not be "thin." 5. To a wheel and axle of mass 48 lbs. and radius of gyration 6 ins. is attached a weight of 13 lbs. by means of a rope wound round the axle. Taking the radius of the latter as 4 ins., and neglecting the weight of the rope, find the velocity of the 13 lbs. weight after it has descended 52 ft. We shall equate the work done by gravity on the whole system to the kinetic energy of the system. Let v ft. /sec. be the required velocity of the weight : then the corresponding angular velocity of the wheel is v-f ^ = 3v rad./sec. o The K.E. of the whole system = |.48.§,(3.)2 + ll3..2 = -.121 ft.-pdls. 2 ^ Work done by gravity on the system = 13. 32. 52 ft.-pdls.

28 ROTATION ABOUT A FIXED AXIS 191 Hence, ^i;^ = 13. 32 . 52, 2 „ 2.13.32.52 "^= 121 ' 26.8 = 19 ft./sec. nearly. 6. The following example involves the use of the Calculus. Find the energy communicated to a top when it is set in motion by a string 1^ yds. in length, being pulled with a force which varies as the length of the string already unwound. Suppose that .v ft. have been unwound at any moment. Then the pull at that moment = A^ units of force. The whole work done or energy communicated when 1^ yds. are unwound = X I xdx Jo 2 81A units of energy Here neither the size of the spindle nor the radius of gyration make any difference. EXAMPLES FOR SOLUTION. 1. A spinning top weighing 6 oz. makes 300 revolutions a minute. Taking the radius of gyration about the axis of revolution (supposed vertical) as 1^ ins., calculate (i) its angular momentum ; (ii) its energy. 2. It is said that a Diabolo spool in full rotation spins 2000 turns a minute. Taking the mass as 3^ oz. and the radius of gyration as ^ in., find its angular momentum and how much energy has been expended on it, assuming that half has been wasted. 3. The weights of the minute, hour, and seconds hands of a watch are as 15 : 10 : 1. Compare their angular momenta, their lengths being as 3 : 2 : 1, assuming that the minute and hour hands revolve about one end, but the seconds hand about its middle point. (For the value of k, see Art. 18.) 4. A door a ft. wide is shutting with angular velocity w rad./sec. If it comes to rest in y^j sec, find the shock on the door post, the mass of the door being i/lbs. and its radius of gyration k ft. 5. A door weighing 40 lbs. is rotating about its hinges, when its edge, 3 ft. 6 ins. away from the hinges, and moving at 30 mi./hr., meets an immovable object. What is the measure of the shock experienced by that object during j of a second ? Radius of gyration 2^ ft. 6. Given that the top in Fig. ii. weighs 3^ oz., that the string is li yds. long, and the average radius of the conical body is 1 in., calculate the angular momentum and the energy due to rotation when the top falls from rest so as to unwind in 1^ sees. Take the radius of gyration as f inch. 7. If the top in the preceding question is making 360 turns a minute just after falling, how long did it take to fall ?

ROTATION ABOUT A FIXED AXIS 29 8. The following method of spinning a top is frequently employed. A rod AB fitted with a spiral thread can be pressed vertically downwards along the axis of symmetry of the top, thus causing the top to rotate. The rod can be raised up and Dressed down again, and so on (Fig. 10). Find the energy communicated to the top after pressing down five times with a force of 6 lbs., the rod BC being 6 ins. If the top weighs 5 oz. and the radius of gyration about its axis is 3 ins., what is its angular velocity after the five downward strokes ? 9. In the last question how many times must the sleeve be pressed down with a force of 5 poundals to give the top an angular velocity of 32 radians a second ? 10. The steering of a torpedo is effected by means of a gyroscope mounted in gimbals, which in the accompanying figure are not shown. The principle employed is the maintenance of its direction in space of the axle A C of the wheel EF, when once the latter has been set spinning, and the gimbals are unlocked to allow of free motion in every direction. The figure illustrates the method of spinning the gyroscope. The gimbals (not shown in the figure) are first locked so that the axle becomes mounted Fia. 10. End view of axle atAB Fio. 11. in fixed bearings. While it is in this position the teeth at the base of the segmental cone engage in the grooves of the axle at the end AB. The cone is attached to a spring S^ which is automatically released as the torpedo passes through the " impulse tube," The released spring swings the cone round in the direction indicated, thus communicating rotation to the axle of the gyroscope. When the teeth at T' arrive at the axle AB, i.e. after I of a revolution of the cone, the rotation of the latter is checked almost instantaneously, by mechanism not here shown, and the gimbals are simultaneously unlocked so that the gyroscope is free to move in any direction. The weights of the wheel (with axle) and cone are 1 lb. 12 oz. and 1 lb. 2 oz. respectively, and their radii of gyration Ij ins. and 2 ins. ; the diameter of the cone at the base is 5 ins., and of the section AB of the axle f in. If the gyroscope starts spinning at 2400 revolutions a tuinute, determine ^ (i) the energy communicated by the spring ; '"■[■■■ ■%:.

30 ROTATION ABOUT A FIXED AXIS (ii) the average pressure between the teeth of the cone and those of the axle. (iii) If the wheel continues spinning for 30 minutes, find the mean resisting couple due to air and friction."^ 11. A small top weighing 3 oz., is so constructed that on its head is a series of small slanting flanges arranged like the sails of a windmill. The object of these is that the spinning of the top may be prolonged by blowing down on the flanges. Supposing that it is given an initial spin of 7 revolutions a sec, in the direction in which blowing assists, find the value of the couple due to friction at the toe and the resistance of the air when it is kept spinning uniformly under the following conditions : There are 7i flanges each of area A sq. ins. inclined to the vertical at an angle of 0°, the blowing pressure is P lbs. to the square in,, and the distance of the centre of pressure of each flange from the axis = ^ inches. Supposing the initial spin is given in the wrong {i.e. opposite) direction, and the same blowing force applied, how soon will it be brought to rest, taking into account the resistance of air and friction, assuming their mean value to be half of that in the previous part of the question, and that the radius of gyration of the top is k ins. ? 12. Fig. 12 represents a top which is spun in the following manner. The spindle projecting from the head of the top works freely in a little cap to be held in one hand, while the other pulls a short string of about 4 ins. length, one end of which is fixed in the spindle. The string is pulled out, allowed to slack and wind again on the spindle in the reverse direction. The process is repeated, and after two or three pulls the top seems to have its maximum rotation, for it starts humming. Why should the rotation be greater after, say, three pulls than after one pull ? Fig. 12. Fig. 13. 13. In another top (Fig. 13) the spindle projects from its mounting, which can be held without the top's rotation being affected ; one end of the string is looped once round the spindle and this end is pulled till the whole string has been pulled over the spindle. Taking the tension in one part of the string to be e^'^i^ x the tension in the other part, where /x is the coefficient of friction and 27r the angle through which the string is wrapped, how does this method of spinning with a string compare with the more ordinary one? * Further information on the steering of a torpedo is given in Chapter V. For the information there contained, and for the above approximate measurements, the author is indebted to the Admiraltj'- authorities and to the courtesy of the Assistant Superintendent of H.M. Torpedo Factory, Woolwich Arsenal.

ROTATION ABOUT A FIXED AXIS 31 14. A bucket weighing 2 lbs. hangs by a rope wound round a wheel and axle, mass 8 lbs., whose radius of gyration may be taken as 15 ins. If the diameter of the axle is 6 ins., find the angular momentum of the wheel and axle after the bucket has attained a velocity of 20 ft./sec. Express also the kinetic energy of the whole system in this case. What is the angular velocity of the wheel when 25 ft. of rope have been unwound, and how long has the unwinding taken ? It is clear that the whole system moves quicker and quicker under the action of the (constant) weight of the bucket. So when a top is set spinning by means of a string, unless the hand holding the string moves quicker and quicker the string will not remain taut. The accelerated motion of the hand does not necessarily mean an increasing pull. 15. In the above question, supposing the bucket to contain a mass of 40 lbs., find the angular momentum of the wheel and axle when the bucket has fallen 36 ft. from rest, and the kinetic energy of the whole system. For how many seconds after this must a frictional force of 6j lbs. weight be applied to the rim of the axle to bring it to rest, assuming that the rope has become slack ? 16. A wheel and axle of mass 6 lbs. is mounted on a table with its axis vertical, and is made to rotate by the descending of a weight of 2 lbs. attached to a string passing over a pulley at the edge of the table, the other end being wound round the axle of the wheel. The weight falls 12 ft. in 3 sees. ; find the radius of gyration for the wheel and axle, if the radius of the axle is 1 in. 17. In experiments recently conducted on the See-har for steadying ships at sea, a big flywheel was mounted amidships with its axle vertical. The wheel, which was in the form of a solid disc, was approximately 1 yd. in diameter, weighed (without the spindle) 1100 lbs., and was made to rotate at 1600 revolutions a minute. Neglecting the spindle, find what horse power would generate the velocity in 86f sees. If steam is shut off and the flywheel comes to rest after 2750 revolutions, what is the value of the frictional couple due to the resistance of the air and the bearings ? 18. Find the energy communicated to the top in example (1) by winding it up with a string a yard long and pulling (i) with a constant force, F lbs. ; (ii) with a force which varies as the length of string (a) already used, (^) not yet used ; Comment on the similarity of answer to (a) and (^). (iii) with a force varying as the square of the length already used ; (iv) varying as the nth. power of the length yet to be used. How would you find the angular velocity in each case ?

CHAPTER 11. REPRESENTATION OF ANGULAR VELOCITY. PRECESSION. 20. If a body have a point fixed, then the angular velocity or rotation of the body has three essential properties. For : (1) It is about a definite axis. (2) It is in a definite direction about that axis. (3) It is of definite magnitude. These three properties can be represented geometrically, for : (1) Through a fixed point, 0 say (to represent the fixed point in the body), can be drawn a straight line OA to represent the axis of rotation. (2) The direction of rotation about this axis can be represented by considering as positive the rotation of a a' J 0 y A Fio. 14. left-handed screw as one looks along the axis. For instance, in the adjoining figure the same rotation is + 0) about OA or A' A or — w about OA' or AA'. (3) Finally, we can mark oflf the length OA to represent on any scale we like, the magnitude of the rotation. Hence it follows that angular velocity, like linear velocity, is a vector quantity. 21. On the physical meaning of composition of velocities. Relative and total velocity. When we say that a particle P simultaneously possesses two velocities u and v in two directions, OA, OB inclined to one another at an angle a, we do not mean that it is moving in space in two directions at once. That is impossible, for it moves with a definite velocity in a definite direction in space. The physical meaning of the "two simultaneous velocities" will be best understood by supposing (Fig. 15)

REPRESENTATION OF ANGULAR VELOCITY 33 that we have a thin wire rod, OA, on which a small bead P is moving with velocity it relative to the end 0 of the rod from which it started, while the rod moves with velocity v in space, keeping parallel to itself, in the direction OB. The total velocity In space is then along a definite straight line OB passing through the original position of 0 in space. The same total velocity would have been obtained by taking the velocity v of the particle relative to a rod OB, while OB is moved parallel to itself in the direction OA with a velocity n. 22. It will be noticed that at any instant the toted velocity of F in the direction O'A' in space is equal to that relative to O' + thatofO' =«+^cos«, which, if a = 90° (namely if OA, OB are at right angles to each other) =u-hO. Hence in this case the total velocity of P in the direction O'A' in space, is the same as the velocity relative to the rod ; so that, when we compound two velocities at right angles, each component is the total velocity of the particle in the direction considered. Similarly, if a particle possesses three simultaneous velocities u, v, w not in one plane, we must further suppose the whole frame AOB to move in space with velocity w keeping parallel to itself, in a direction OC not in the plane of the paper; and if the three directions OA, OB, OC, are mutually at right angles, each of the components is the total velocity of the particle in the direction considered. 23. We will now extend the above observations to the case of angular velocities or rotations. Simultaneous Rotations. Definition. If a rigid body is rotating with angular velocity coi about some axis OA, fixed in the body, relative to a frame in which that axis is fixed, while the frame rotates about some fixed axis OB with angular velocity 002, then the total angular velocity of the body is said to be compounded of the two simultaneous rotations cd;^, w^ about the axes OA, OB. c

34 REPRESENTATION OF ANGULAR VELOCITY It is clear that the frame mentioned might rotate with a velocity w^ relative to a second framne which had a total angular velocity Wg , so that the definition can be extended to any number of simultaneous rotations. 24. Parallelogram of angular velocities. // a rigid body, of which one point 0 is fixed, has two simultaneous rotations ft)i, 0)2 about axes OA, OB, and if w^, Wo cere rejjresented in direction and magnitude by OA, OB, respectively, then the resulting rotation, w, is represented in direction and magnitude by OC, the diagonal of the parallelogram of which OA, OB, form two sides. Fig. 16. Let OA, OB (Fig. 16), represent the two rotations co-^, Wg about OA, OB, respectively, the direction of rotation in each case being positive viewed from 0, i.e. that of a left-handed screw. Let us suppose that the angular velocity w^ is relative to the frame AOB, whilst co.2 is the angular velocity of the frame itself. Complete the figure as shown. Let (7 be a point in the body passing through the plane of the frame AOB and coinciding for the instant with a point C-^^ in the plane of the frame. Then the linear velocity of the point C is (away from the reader and perpendicular to the paper) = that relative to (7^ + that of 0^ = h.OA.CM-k.OB.C^N, where h is a constant depending on the scale chosen, = /c. (2a ^0(7- 2a 50(7) zero. .*. the point C is at rest (as also the point 0), and therefore the body is rotating about OC, and OG represents the direction of the axis of rotation.

REPRESENTATION OF ANGULAR VELOCITY 35 Again, the linear velocity of a particle at ^ is : CO . AL and also = ay^AN' = k.OB,AN' = k.2A0BC ^=k.OC.AL; .'. M = k.OO, i.e. OC represents the magnitude of the resulting rotation oj. Similarly, any number of simultaneous rotations may be combined by the parallelogram law. The order in which the rotations are combined does not affect the final result. So also any single rotation can be resolved into any number of component rotations. It follows that angular velocities, like linear, are subject to the parallelogram law. 25. If 0)2 had been in the reverse direction originally (represented by the dotted line) we should have most easily obtained our resultant by drawing OB in the opposite direction. The [B] resultant o) would in this case / have been represented by the / diagonal OG as shown in Fig. / 17. / 26. In practice it is sometimes more convenient to represent rotations right-handed instead of left-handed. The resultant oo of any two rotations w-^, Wg will be best found by always drawing OA, OB so that w^, (jo.2 are in the fio. it. same direction (whether positive or negative) viewed along OA, OB ; co will then be in this direction viewed along OC, where OC lies between OA and OB. 27. The parallelogram of angular accelerations follows at once as a corollary from tliat of angular velocities. 28. Definition. By angular velocity about a line which is moving, we mean the total angular velocity about the fixed line in space with which the moving line happens to be coinciding at the instant under consideration. If in Fig. 16 we consider the angular velocity ^i about OA to be relative to the frame, which rotates with velocity ^2 about OB, it is clear (since angular velocities, like linear, are vector quantities) that the angular velocity of the frame about the line OA of the frame is 0^2 cos a, where the angle BOA = a.

36 REPRESENTATION OF ANGULAR MOMENTUM Hence the total angular velocity of the body about OA, i.e. about the line fixed in space with which OA happens at the instant to be coinciding =0)1 + 0)2 cos a = 0)1 if a = 90°. Thus, if we compound two angular velocities about axes at right angles to each other, each component is a total velocity — not a velocity relative to some moving plane or frame. So too with three rotations about axes mutually at right angles. Conversely, if we resolve a single rotation into three components mutually at right angles, each component is a total velocity, and each is independent of the other two. The importance of the above will be seen more clearly in Chapter VI. 29. Representation of angular momentum. If we have a wheel rotating about its axle H'OH with angular momentum ^ Iw, say, then using the same conventions as when representing angular velocity, we can take the line OH to represent this angular momentum. Therefore, angular momentum is a vector quantity, and hence it follows that angular momentum, like angular velocity, obeys the parallelogram law. For instance : if OA (Fig. 16) be taken to represent the component angular momentum JiO)i of a body about OA, OB to represent the component I^w^ about OB, then the resultant angular momentum Im of the body is about OC, the diagonal of the parallelogram, and is represented by OC, ft)i and 0)2 being defined in the same way as in Art. 24. Fig. 18, 30. It should be noted that the resultant axis of total angular momentum of a body is not necessarily the same as the resultant axis of total angular velocity : for I^w^ , I^w^ are not proportional to o)i, 0)2 respectively, unless 1^ = 1^. 31. Referring now to Fig. 18, we see that if OE is a line making an angle a with the axle OH, then, OH being taken to represent Iw about the axle, OH cos a, i.e. OK, represents the resolute about OE, where HKO is a right angle. We might also have taken the area, A say, of the wheel's disc to represent /o), in which case A . cos a represents the resolute about the axis OE. In other words the resolute would be represented by the area of the projection of the disc as seen from a point E on OE. This method of estimating the resolute about any axis of the angular momentum of a rotating wheel is sometimes convenient.

MOTION OF A GYROSCOPE 37 32. Distinction between axis and axle. The axis of symmetry of a solid of revolution we shall in general call its axle, as distinguished from any other axis in the body about which it may be rotating or have a component rotation. It should be noticed that the axle in this sense is a geometrical straight line — not a thick spindle relative to which, or fixed to which, the body rotates. 33. Definitions. Gyrostat. A solid of revolution which is capable of rotation about a straight rod coincident with the axis of the solid is frequently called a gyrostat. Gyroscope. A gyroscope is an instrument in which a gyrostat is mounted in a frame. The most common form of a gyroscope is that in which the frame is so mounted that the gyrostat can turn in any direction. See Figs. xxii. sqq. We shall, however, in general refer to the spinning wheel of the instrument as " the gyroscope" Spin of a gyroscope. By the spin of a gyroscope or top about its axle we mean the total angular velocity about the line fixed in space with which the axle happens to be coinciding at the instant in question. It is not the angular velocity relative to the frame. F< z Q An^utoir momen,tu.m. v j^ A' Fig. 19, Fia. 20. 34. Motion of a gyroscope with its axle horizontal. By means of the parallelogram of angular momenta we can discuss the motion of a gyroscope which is moving, under the action of gravity only, about the fixed vertical through its point of support with its axle horizontal as in Fig. 19.

38 PRECESSION We will consider the frame to be so light that we are only concerned with the motion of the wheel itself. Let OX, OY, OZ (Fig. 20) be three mutually perpendicular lines, OZ being vertical. Suppose the gyroscope rotating with angular velocity o) about OX, its axle, whilst the latter rotates about OZ with angular velocity Q. The component angular momenta of the gyroscope are Iw and I'il where /, /' are the moments of inertia about OX and OZ. Let OA, OG represent these components in magnitude and direction. After time St the angular momenta are lo) and /'Q, about OA' and OG, where the angle AOA' = mt Then if A'B be parallel to OA, OB represents the change of angular momentum in time St, for if compounded with OA and OG it yields OA' and OG. But OB = A A' = OAx (QSt) = IwQSt ; .'. the change of angular momentum in time St is about OB and = IcoQSt ; . • . the rate of change of angular momentum is about OB and But this requires for its production a torque IcoQ about OB. Thus the rotation of the axle about the vertical requires for its maintenance the application of a torque K, say, perpendicular to the vertical, and to the axle of the gyroscope, such that 35. In the gyroscope of Fig. 19, the necessary torque is supplied by the weight Mg of the system, and the equal and opposite vertical reaction at the point of support, which together have a moment Mga, where a is the distance of the centre of gravity from the point of support. Hence, in this case, Mga = IcoQ. Dimensions. Mga = ^-^^^^— . [Z]. It should be noticed that the centre of gravity of the system describes a horizontal circle with uniform angular velocity Q under the action of the horizontal reaction at the point of support. (See Art. 43.) 36. Precession. The rotation of the axle OX of the gyroscope, by a torque about 0 Y, round the third perpendicular axis OZ we shall call a precessional motion, and the axle will be said to precess about OZ, the axis of precession. The equations of steady motion of a gyroscope or top with its axle inclined to the vertical are given in Chapter VI.

PRECESSION 39 37. The preceding results may be stated more generally as follows, without any reference to vertical or horizontal (see Art. 85): If a body which has angular momentum /w about an axis OX be under the action of a torque K about a perpendicular axis OY, then the angular moTnentum tvill be rotated about the third perpendicular axis OZ with angular velocity Q determined by the equation jr _ j c^ 38. Rule for direction of precession. It will be clear from the above discussion that : If the rotated angular momentum and the torque axis are drawn in the same sense, then the angular momentum sets itself towards the torque axis. 39. It should be noticed that although the gyroscope in Fig. 19 is rotating about the vertical, yet the torque does not create angular momentum about the vertical. It creates angular momentum about its oivn horizontal axis, which combining, as fast as created, with the angular momentum about the axle, causes the axle to take up successive positions in the plane XY. We have already said (Art. 34) that our investigations refer to the maintenance of an existing motion : otherwise our equation K = I(joQ involves the paradox that if (0 = 0, i.e. if the body is not spinning, then 1^2, the precessional velocity, is infinite ! This is, of course, not true, but it is true that a wheel spinning with a small velocity precesses with a large velocity, and vice versa. A full discussion of all that occurs when precession is being started, is given in Chapter IV., where it is shown that oscillations are set up which are quickly destroyed by friction, and in many cases are scarcely visible. In instances given before Chapter IV., where, strictly speaking, the starting of precession is in part involved, the preliminary phenomena will either be assumed or neglected. [8.] Explain what effect the couple due to the action of gravity on a Diabolo spool will have, if the string is not quite under the centre of gravity, (i) when the spool is not spinning ; (ii) when it is. 40. It is clear that the investigation of Art. 34 is equally true when K varies and consequently Q as well. For let 0 be the angle which the axle makes at any instant with its initial position. Then Se = AOA' and AA' = IwSe. Hence the increase of angular momentum in time St is about OB and is equal to IwSO : namely, the rate of change of angular momentum is equal to dt

40 PRECESSION It follows that, if w is constant, 10) J or the precessional angle swept out during any time is a measure of the time-integral of the applied couple. 41. Analogy in linear motion. The application of a torque to maintain the steady rotation of the axle of a spinning body about a perpendicular line has its analogy in linear motion, where a force is needed to maintain the steady rotation of the line of motion of a particle moving uniformly in a circle. For let a particle P, of mass on, move with constant velocity V round a horizontal circle whose centre is 0, towards which is a constant force F. Let P,Z„ P,Z„ F,Z, (Fig. 21) be successive positions of the vertical through successive positions of P. Then the effect of applying the force F perpendicular to the line of the momentum imiv of the particle, is to turn the line of momentum at a constant rate about an axis perpendicular to F and itself. In this case, if the rate at which the line of momentum is being turned be ^2, then Q is equal to the angular velocity of V the particle about 0, i.e. 0 = -, and we have Fig. 21. F: onv = mv . il. 42. Diabolo. The rule given above (Art. 38) for the direction of precession will at once explain the reason for the instructions (variously worded) which are given to beginners learning to spin a Diabolo spool. Let us suppose that the Diabolo is being spun right-handed as in the illustration (Fig. 22), when the end farther away from the performer begins to dip down, and the nearer end to rise. To get the spool again horizontal it is necessary for the performer to pull back his right hand and (in imagination) press down the rising end of the spool with what may be called the "working" end of the string. It will thus be seen that a torque is created about an axis perpendicular to the axle of the spool, and since the axle tends to set itself towards the torque-axis (both being drawn in the same sense), the

PRECESSION 41 lower end of the spool rises, as illustrated in Fig. 23. The same principle holds if the end next the performer is dipping ; also if the spool is being spun left-handed ; and a general rule Aa:le Axle. ^ Fig. 22. could be formulated to the exclusion of all ideas of left- and right-handed by saying that the working end of the string must be moved so as (in imagination) to press down the rising end. Let us now suppose that it is required to make the Diabolo turn round to the right of the performer when it is being spun as in Fig. 22. It is clear that a torque applied by the strings about a vertica,! axis to the right, as viewed from above, will result in the near end dipping. A torque must be applied about a hm^izontal axis, and taking the rotation of Fig. 22, this torque _ _ _ _ "f ri^^Jl^.^i^ can be applied in the right 'sense" f _— — -"^"TV^''^Io. by the performer dragging the spool '^ Jixl^ ^"^ ^^ slightly towards him, or (without fig. 23. moving the spool) by changing the point of support so as to make it slightl}' nearer to himself. If the spool is dragged from the under side it is advisable to extend the two strings at a large angle, as otherwise the dragging action will tend to depress the end nearest to the performer and thus produce the opposite effect to that aimed at. [9.] If the Diabolo is being spun left-handed, what is the effect caused by the performer (i) drawing in his right hand 1 (ii) drawing in both hands ? (iii) drawing in his left hand when the end near him begins to dip 1 A discussion is given in Chapter IX., on will not spin. Diabolo which [10.] In a motor-car the fly-wheels are mounted so that their axles are parallel to the direction and motion of the car. Supposing the fly-wheels

42 PRECESSION to revolve right-handed when viewed from the back of the car, what is their effect on the car as it turns a corner (a) to the right, (6) to the left ? [11.] (a) When a paddle-boat is steaming ahead, what is the gyroscopic effect of paddle-wheels when a wave strikes the ship and gives it a list to starboard ? (6) What is the effect of steering to starboard ? EXAMPLES. 1. In the equation /i'=/wi2, if w is 20 radians a second, 12 = 5 radians a second, /=50 Ib.-ft.^, what is the value of K in ft.-lbs. ? 2. The moment of inertia of a heavy wheel about its axle is 15,000 Ib.-ft.^. If it is making 3,000 revolutions a minute, what will be the velocity of precession when a couple of 12,000 ft.-pdls. is applied to the axle of the wheel '^ 3. Find the length of the axle of the wheel in Fig. xxii., supposing its moment of inertia to be 3,000 gram.-cm.^, and that, when spinning at the rate of 50 turns a second, a weight of 60 gr. suspended from X causes it to precess at 3 turns a minute. 4. Find the rate of spin of a gyroscope, taking the length of its axle as 12 cm., the moment of inertia 3,500 gr.-cm.^, the rate of precession 3 turns in 70 sees., and the suspended weight as 75 grms. 5. A heavy wheel, mass 50 lbs., in the form of a disc, is rotating about its axle with velocity 16 radians a second, and precessing at the rate of 4 radians a second under the influence of a couple of 100 ft.-lbs. Find the radius of the wheel. 43. Independence of translation and rotation. It is proved in works on rigid dynamics that any motion of a rigid body can be completely represented by a translational motion of its centre of mass G combined with a rotational motion about some axis through G; and further, that these two motions are independent of each other and produced independently by the acting forces. Hence, in considering the motion of any rigid body, whether some point in it is fixed or not, we may discuss the rotation of the body as if G were fixed, and the translation of the bod}^ as if it were a single particle, the entire mass being concentrated at (?. The truth of this can be illustrated by taking an ordinary walking stick and marking the position of its centre of mass with a piece of chalk. If the stick is then suspended loosely from the fingers of one hand while a blow is given to the stick, it will be found that, whenever the blow is struck so as to pass through the centre of mass, the stick flies out of the hand with a motion of translation only, that is to say, it has no rotation. If, on the other hand, the blow is struck through any other point than the centre of mass, then it will be observed that not only does the centre of mass have a translational motion, but the stick also rotates about this (moving) centre of mass.

PRECESSION 43 A familiar illustration of this independence of translation and rotation can be given by tilting back a chair and then letting it fall forward again. The chair not only swings forward to its original position, but also moves forivard along the carjjet : for, as it swings forward, its centre of gravity has a forward translational motion, and, before the chair can come to rest, this motion has to be stopped (by the friction on the carpet), independently of the rotation. The rotation is, of course, stopped by the moment of the vertical reaction of the floor on the two front legs about the line joining the points of contact of the two hind legs with the floor. 44. Further illustrations of precession. Hoop. If a wheel, or an ordinary hoop, is rolling along the ground it has an angular momentum about its central perpendicular axis or axle. If the hoop begins to move in the slightest degree out of the vertical plane, the external forces acting on it (i.e. the reaction of the ground and the weight of the hoop) tend to tilt over the axle, with the result that it " precesses " about an axis very nearly vertical, and its path becomes a curve instead of a straight line. Engine wheels. The tilting moment may be considerable without the wheel leaning over at a large angle to the vertical, as in the case of wheels of a railway engine where the reaction of the rails on the wheels is very large. Let us suppose the engine turning to the right, looking in the direction in which the train is going. To make each wheel precess to the right, a couple must be applied about a horizontal axis (through its centre of gravity, say) parallel to the direction of motion of the train ; and this couple must, as we have seen, be in a direction tending to lift up the left-hand end of the axle and to depress the right-hand end. Therefore, owing to the rotation of the wheels (apart from any motion of translation), it is necessary to raise the left rail slightly in rounding the curve. If this were not raised the whole turning of the train would be effected by the lateral pressure on the left-hand flanges, which (in addition to considerable wear to rails and wheels) would tend to make the wheels precess to the left, and would increase the risk of their "jumping the metals." Gyroscopic action in a bicycle. The wheels of an ordinary bicycle have rims so light compared with the rest of the machine that their angular momentum does not play a preponderating part in the motion of the whole machine. At times, however, it is quite an appreciable quantity, as can be easily realised by taking the front wheel out of its bearings, holding the axle in the hands and attempting to turn the axle after a considerable spin has been given to the wheel. There is a certain amount of gyroscopic action brought into play when the rider, finding himself falling over, say to the

44 GYROSCOPIC RESISTANCE right, gives the handle bar a sharp turn to the right. In this case we shall see, if we consider the directions of the rotations in question, that this twisting of the handle bar causes the front wheel to precess about a horizontal axis into a more vertical position, i.e. to right itself. Simultaneously with the application of the couple to the front wheel by the man there is a counter couple applied to the man, and therefore to the back wheel, which causes it to turn at a greater inclination to the vertical, and so to increase the curvature of the path of the bicycle. The preponderating cause, however, of the bicycle righting itself is to be found in the linear momentum of the whole machine and of the rider, for by far the greater part of the motion is not rotational. When the front wheel is turned at an angle to the back wheel, the linear momentum of the rear part being checked by the front part brings a force to bear on the latter, one component of which drives it along its path, while the other, perpendicular to the wheel, raises it to a more vertical position, and consequently the whole cycle proceeds in a more vertical position. It is clear that if the brake be applied to the front wheel and not to the back, the bicycle will tend to skid more than if the linear momentum of the rear part is checked while the front wheel is free to move along its path. Motor cycle. In a motor bicycle there is a heavy fly-wheel rotating rapidly in the plane of the road wheels, so that as the cycle rounds the curve the fly-wheel is constrained to precess, and therefore, on account of this fly-wheel alone, a couple IwQ must be applied to cause this precession, apart from the other forces required to turn the rest of the machine. In other words, the rider must lean over more than if the fly-wheel were not there, or were not rotating. In practice this wheel rotates in the same direction as the road wheels. This necessity for leaning over an extra amount accounts in part for motor bicycles being more liable to skid when going round a corner than ordinary bicycles. 45. Gyroscopic resistance. We have seen that in order to turn the axle of a rotating wheel about a perpendicular axis, it is necessary to apply a couple K, about a third axis, perpendicular to the other two, whose measure (employing the above notation) is /coQ. Thus the resistance which the axle of a rotating wheel offers to being turned about the axis of the applied couple K is measured by ImQ, and this expression is frequently called the gyroscopic couple, or gyroscopic resistance. Hence, the only difl(erence in discussing the motion of a rotating body, as compared with that of the same body not rotating, is that we must include the gyroscopic couple among the resisting forces.

GYROSCOPIC RESISTANCE 45 As a further illustration of the gyroscopic couple, it will be noticed that if we try to turn the axle of a wheel which is not rotating we meet with no resistance beyond that of the inertia of the body, i.e, co = 0, and the gyroscopic couple /wQ becomes zero. So in the Introductory Chapter, when the gyroscope (Fig. xxii.) is prevented by the clamp at Z from turning about the vertical, i.e. when Q = 0, the gyroscopic couple is again zero and the body offers no resistance to being turned about YY' , except that of inertia. Hence we see that a rotating wheel will offer no gyroscopic resistance unless it is free to precess. Again, when we apply the equation Mga = Ia)Q, in Art. 35, to obtain the equation of steady motion of the gyroscope, we are expressing the fact that the gyroscopic couple IwQ is balancing the couple Mga due to gravity. [12.] What corresponds to the gyroscopic couple in uniform circular motion ? If we neglect the inclination to the vertical of a wheel as it rolls round a curved path, the expression for the gyroscopic couple simplifies. For if v is the rate at which the wheel is travelling, r its radius, R the radius of the curve, r it and ift)12 = -y5 — » K .r k being the radius of gyration. The investigation when the inclination of the wheel to the vertical is taken into account, will be found in Chapter VI. FURTHER EXAMPLES. 1. A metal disc of radius 1 ft. and mass 2 lbs., is made to roll uniformly round a curve of radius 12 ft. The gyroscopic couple is 5 ft. -lbs. Neglecting the inclination of the disc to the vertical, find the rate of rolling. 2. An engine on six wheels, each of radius r, is rounding a curve, radius R. If V is the speed of the train, M the mass of each wheel, k the radius of gyration, and 2A the width of the gauge, show that the gyroscopic couple due to the wheels is Owing to their rotation, which wheels tend to come off the rails V 3. If the wheel of the gyroscope in Fig. xxv. is a solid disc spinning with velocity cu, and precessing with velocity 12, find its radius if 2a = the length of the axle. 4. The propeller shaft of a Torpedo-Boat Destroyer is a cylinder of radius a ft., and weight M lbs. When the shaft is revolving with an angular velocity w the speed of the ship is V ft./sec, and the radius of

46 GYROSCOPIC RESISTANCE the turning circle is R ft. Find the torque exerted by the bearings of the shaft. 5. In the preceding question, taking a = 4 ins., 3/"= 12 tons, F=:35 knots, (0 = 750 re vohitions per minute, /? = 605 yds., estimate the torque in ft.-lbs. These values are taken from H.M. T.B.D. "Tartar." 6. A motor cycle has a heavy fly-wheel, in the form of a disc of radius 4 ins. and weight 32 lbs., which is making 1,600 revolutions a minute. The machine and rider weigh 320 lbs. If the rider rounds a curve of 25 ft. radius when travelling at 20 mi./hr., find approximately the horizontal distance that he must move the centre of gravity of himself and his machine owing to the rotation of the fiy -wheel. Note. If the fly-wheel were not spinning he would have to lean over a certain amount in order to turn the corner ; but when the fly-wheel is spinning, he must increase the arm of the gravity coujdIc by a small amount, ^', in order that the fly-wheel may precess. It may be assumed that the gyroscopic couple is the same as if the fly-wheel were in a vertical plane. 7. In the previous question, just before the rider rounds the corner he leans over to the right with an angular velocity of a radian a second, (a) How is the torque supplied which causes the fly-wheel to have this angular velocity ? (6) About what axis was it, and in what direction ? (c) What is the effect on the back wheel of the reaction to this torque ? id) Find its magnitude.

CHAPTER III. DISCUSSION OF THE PHENOMENA DESCRIBED IN THE INTRODUCTORY CHAPTER. 46. If we turn to the Introductory Chapter we shall now see the reasons for the paradoxical behaviour of the tops there described. To take the first instance mentioned, if a top is placed on its toe while spinning, the torque, which would cause it to fall over if it were not spinning, fails in this case to turn over the axle but rotates it, and so the top precesses. Also, if we consider the rule given in Art. 38 for the direction of precession, we see why an ordinary top precesses in the direction of its spin, when viewed, say, from above (Fig. i.), while that in Fig. iii. (a) precesses in the reverse direction : for in the latter case the centre of gravity is below the point of support and the torque acts in the reverse direction. 47. Let us now consider what causes a top that has been spun at an inclination to the vertical to rise to a more vertical (though not necessarily quite vertical) position before settling down to a steady motion. Suppose we take an ordinary top and spin it on a rough board. When it has settled down to a motion in which its axle is practically at a constant angle to the vertical, let us drag the board round in the same direction as the precession. It will be found that the centre of gravity of the top rises, and some of us may remember that the method of resuscitating a dying top is to get it on to one's hand and drag the hand round in the necessary direction. Similarly, if \YQ draw the hand round in the opposite direction the top falls ; so that we can sum up our experiment in the two following statements : (i) hurry the precession, the top rises ; (ii) retard the precession, the top falls. Hence, when a top is spun at an inclination to the vertical and rises, there must be at first something that is hurrying the precession, and eventually (since the top afterwards falls) something that retards it.

48 THE PHENOMENA Now the only forces acting on the top (besides the resistance of the air) are : (i) its weight; (ii) the reaction at the point of contact of the toe with the table. The weight acting vertically downwards cannot hurry or retard the precession. I f^In Fig. 24 let JT and A be respectively the centre and radius of curvature of P's path. The reaction at the toe can be resolved into two components, one S, perpendicular to the plane on which the top is spinning, due to the stiffness of the plane : Fig. 24. and another, R, in the plane due to its roughness. The component of R in the direction PK provides the necessary normal acceleration, while the component along the tangent to the path is either hurrying or retarding the precession. If it is hurrying the precession we see by the rule given in Art. 38, that the axle of the top rises "^ to a more vertical position, and vice versa. Let us consider the cases in detail. Let o) be the velocity of spin, and Q the angular velocity of P about K, and let PN=a. The direction of the friction at P depends on the direction of motion of P, for it will act contrary to the direction of the latter. At the start there must be skidding if the top is dropped on to the plane. Now the actual velocity of P is that relative to A^+that of N \ or, the actual velocity of P (measured into the paper) = a(jd — AQ^. * The rise or fall of a top is more fully discussed in Chapter IV. from considerations of conservation of angular momentum and conservation of energy.

OF THE INTRODUCTORY CHAPTER 49 But at the start w is very big, though it gradually diminishes under the action of the friction, and also the resistance of the air. Hence, at the start we have aco'^ AQy (i) i.e. the top is skidding and P is moving into the paper ; friction accordingly acts in a direction which hurries the precession, and the top therefore rises. In the meantime w is diminishing (and also a as the top gets more vertical), and after a time we have ao) = AQ, (ii) i.e. P has no motion relative to the table, and steady rolling motion becomes possible. And, later, we have ao)

50 THE PHENOMENA [20.] Should we get precession? If so, how would the axle move when the motion is steady ? [21.] What is the condition that a top should /a^^ to its steady position? What conduces to this condition ? [22.] Discuss the rising of that form of top where the "body" to which the spin is given revolves freely on a spindle carrying the toe ; namely, where the toe and spinning body are not rigidly connected. Does it arrive at steady motion more quickly or more slowly than the other kind ? 48. The discussion given in Art. 47 explains the behaviour of the whip-top and loaded sphere described in the Introductory Chapter, as also that of the acorns and hard-boiled eggs : for the effect of friction at the point of contact is to hurry the precession, and so raise the centre of gravity, as in the case of the top. The phenomenon exhibited by the top of Fig. vi., and by the egg of Fig. ix., is caused by a tendency of the liquid to spin about its shortest axis, which overcomes the effect of friction tending to raise the centre of gravity and so bring the top on to its longest axis. To understand this thoroughly it will be necessary to consider under what conditions a body will spin about its shortest axis in preference to any other. 49. Tendency to spin about the least axis. If a rigid body, with three perpendicular axes of symmetry, is free to turn in any direction about its centre of gravity G, and rotation be continuously given to the body about a fixed direction through G, for example the vertical, but no other forces act on it, the body will set itself so that its least axis through G becomes vertical, and it will spin stably in this position. For every particle composing the body tends (owing to what is frequently called centrifugal force) to separate itself as far as possible from the vertical axis, and so increase the moment of inertia about that axis. Hence the body w^ill only spin stably when the moment of inertia is as great as possible ; namely when the least axis of the body through (r is in a vertical position. This can be easily illustrated by taking a stone or any other rigid body attached to a string and, starting with the string vertical, whirling the body round and round oneself as a vertical axis. The body will rise higher and higher until the string is horizontal ; but having arrived at this position it will continue to maintain it, whatever additional spin may be communicated about the vertical axis. Similarly, it will be seen that a liquid or viscous body which is being made to rotate about a vertical axis will tend to change its shape, in such a way that its vertical axis becomes smaller ; and if there are no external forces tending to rotate the body, but rotation has already been communicated about some axis, this axis will tend to become smaller and smaller, as is the case with the Polar axis of the Earth.

OF THE INTRODUCTORY CHAPTER 51 50. We shall now be able to see the reason for the behaviour of the eggs and the tops which were full of liquid. We know (Art. 47) that the shell, even though originally spun about its shortest axis, would, if empty, rise and spin about its longest axis, owing to the friction at the point of contact with the table. But between the liquid and the shell there is comparatively little friction, so that the liquid, once spun about its shortest axis, continues to spin about that axis; and being of much greater mass than the shell it overcomes the latter's tendency to rise on to its longest axis, with the result that the whole body, shell and liquid, spin about the shortest axis. It is clear that the final behaviour of the egg, or top, must depend, amongst other things, on the relative masses of the liquid and shell, and this accounts for the top of Fig. VIII. spinning about its longest axis although full of water. 51. The gyroscopic top (Figs. xv. a, xv. h) is another, and perhaps more remarkable, instance of the precession of the axle. The forces acting on the top are (Fig. 25) : (i) its weight; (ii) the normal reaction S at the point of contact of the top with the spiral coil ; (iii) the tangential reaction F at that point ; (iv) the reaction at the point of support 0. In most models of this top, the centre of gravity is made to coincide with the point of support, though in some it can be adjusted so as to be either above or below as required. For the sake of simplicity we will consider that it is coincident with the point of support — though the following explanation would only require a little modification if this were not so. Let us regard the axle of the top as in the plane of the paper, and let the spindle be rolling (left-handed as viewed from 0, the point of support) along the inside of the coil, into the paper, and be approaching the end of the coil. The motion will be most easily explained by considering first the efiect of the tangential reaction F at the point of contact P, and then the additional efiect of the normal reaction S. (1) Tangential reaction. It is evident that the friction acts into the plane of the paper and perpendicularly to it. This Fig. 25.

52 THE PHENOMENA friction, together with the equal reaction at 0, forms a couple about the horizontal OX, which lies in the vertical plane through the spindle, and consequently the axle of the top tends to set itself towards OX, i.e. towards the coil, downwards, but is prevented from doing so by the normal reaction S of the coil.* The spindle, in consequence, pressing hard against the coil, and thus increasing the friction, rolls or skids along until it reaches the end, when, no longer meeting with any resistance from the coil, it rushes rapidly round the corner. After this it presses up against the coil, since friction now acts out of the paper towards the reader, and the motion is continued on the same principle as before. (2) Normal reaction. The effect of the reaction S is to accelerate the motion of the spindle along the coil. For, if we again take the position of Fig. 25, it is evident that S creates a torque about an axis drawn from 0 into the paper, towards which axis the spindle sets itself by the laws of precession. It should be noticed that if the centre of gravity of the top is at the point of support, the top, when at rest, will balance with its axle at any inclination to the vertical. In this case, even though it is touching the coil, there is no reaction at the point of contact; but the moment the top is spun, any contact with the coil involves friction, and therefore a precessional tendency against the coil. 52. The motion of the toy gyroscope in Figs, xvi.-xxi. is due to the action of the external torque which causes the axle to precess, but there still remain one or two points worthy of remark in connection with the scientific gyroscope of Fig. xxii. It was mentioned that if, while the gyroscope is spinning with the weight attached at X, the screw at Z be tightened, then the gyroscope at once turns about Y'Y. It is interesting to note that the direct cause of this turning is not the onoment of the weight about Y'Y. The moment of this weight tends to set up precession about ZZ', but since the screw at Z is tight, and consequently the frame containing Y'Y is fixed, the pivots at Y and Y' meet with resistance forming a left-handed couple about Z'Z, and, as in the case of the gyroscopic top, it is this resisting couple about Z'Z which turns the gyroscope about Y' Y in accordance with the law of precession. Similarly, referring to Fig. XXIII., it is this resisting couple about a vertical axis which causes the gyroscope to set itself with its axis vertical, and this couple reversed which brings about the somersault. 53. An explanation of the oscillations mentioned in the Introductory Chapter on page 10 will be found in Chapter IV. * This normal reaction is of very considerable magnitude, for it is found that the coil, unless strongly constructed, becomes bent out of shape.

OF THE INTRODUCTORY CHAPTER 53 54. We will now discuss why the "drifting" in still air of a rifle bullet is due to gyroscopic action, while the swerving of a " sliced " golf ball is not. The essential diflerence is that in the case of the rifle bullet the axis of spin is initially a tangent to the path described, while in the case of the golf ball it is a normal. Let us take the latter case first. ^^ Suppose that the ball is travel- "^ ling at velocity v, with a spin o) due to " slicing " in the direction indicated in the figure, which represents a horizontal section of the ball. Let its radius be a. Consider the respective velocities of two points P^, P^ at opposite ends of a diameter which is at right angles to the path of the centre of the ball. The actual velocity of P-^ = v — aoci. „ „ Pg = f -|- aw. But the pressure of the air on the surface of a moving body increases with the velocity. Hence the pressure of the air on Pg is greater than on P^, and consequently the ball will swerve to the right, looking in the direction of motion."^ In the case of the rifle bullet, however, the swerving is in the first instance due to gyroscopic action. The direction of the bullet's spin (due to the rifling of the barrel) is left-handed, viewed from the gun (Fig. 27). The excess of wind pressure ■^ Fig. 27. on the under side of the nose-end of the bullet tends to tilt the point upwards, but results instead, owing to gyroscopic action, in the bullet working over slightly to the left and becoming slightly " broadside on " to the direction of its path. This is not shown in Fig. 27. The consequent increased air *It is for this reason that the face of a "driver" is slightly "laid back"; for, since the ball is "teed up" before being driven, and the club strikes it horizontally, the inclination of the face to the vertical should give a slight under-spin to the ball, which we see from the preceding, causes it to rise against gravity. Otherwise, with the same initial velocity, it would be impossible for the ball to travel as far as it does. It is said that the late Professor Tait, having been given initial data, declared it to be impossible to drive a golf ball beyond a certain distance, which he specified. When his son, the late F. G. Tait, exceeded this distance by a considerable number of yards, the Professor examined the problem more thoroughly and arrived at the above explanation.

54 CELTS pressure on the right-hand side causes the bullet to "drift" to the left. In the case of a boomerang the varying air pressure is due to both of the above causes and to others of a more complicated nature. 55. Celts. Referring to the Celts described in the Introductory Chapter, the explanation of their behaviour is to be found in the direction of the normal reaction at the point of contact with the smooth horizontal plane. In Fig. XI. (Plate i.)^ the surface near the point of contact is approximately spherical, so that when the stone is tilted in any direction the normal reaction passes through the line GG and so has no moment about GG. Hence the stone will have no tendency to turn about GG. But in Figs, xii., xiii., this is not so, and phenomena {a) and (h) are explained in the following way : Phenomena (a). Each base is in these two cases roughly ellipsoidal near the point of contact, but the ellipsoidal surface is " slewed round " slightly, relatively to the main body of the stone. Accordingly, when the stone is slightly displaced from its position of equilibrium, the vertical reaction, which is normal to the ellipsoid at the point of contact, does not pass through GG. Hence it has a moment about GG tending to turn the stone about GG. In Fig. XII. the vertical taps at A and at B result in moments of opposite sign about GG ; in Fig. xiii. they result in moments of the same sign, i.e. from A to B. Phenomena (6). When the stone is spun about the vertical GG, since the conditions are not absolutely perfect, small oscillations will be set up, and the normal reactions already described will be called into play. These may or may not tend to stop the rotation, and so they may either increase or diminish the oscillations. The particular effect of the normal reaction will of course depend on the special shape of each stone. In any of the above cases, the motion, whether reversed or not, will only cease when the original energy of spin has been destroyed by friction or communicated by oscillation to the horizontal surface on which the spinning takes place. A complete analytical investigation of the above phenomena is given by Mr. G. T. Walker in The Quarterly Journal, No. .110, 1896. *For Plates i.-iii. see end of book.

CHAPTER IV. THE STARTING OF PRECESSION. OSCILLATIONS OF THE GYROSCOPE. 56. Conservation of angular momentum. We know that when two particles collide, the total momentum of these two particles, measured in any direction, is the same after impact as before ; for the impulsive reaction between the two produces as much momentum in one particle in one direction as it does in the other particle in the opposite direction. Hence, the moment of the momentum (i.e. the angular momentum) about any fixed line must be the same after impact as before. It follows therefore that, in any system of particles whatever, whether rigidly connected together or not, if internal forces only, and no external forces, act on the system, then, however the configuration of the particles may change, the total angular momentum of the system (i.e. Znir^w) about any fixed line in space remains constant. ILLUSTRATIONS. (1) If a mouse is put in a cylindrical cage capable of revolving freely about a horizontal axle, then, in whatever way the mouse moves, the total angular momentum, about the axle, of the mouse and cage together, remains constant. It is clear to an observer that if the mouse climbs directly up the curved surface of the cage, the cage begins to revolve in the opposite direction, though not necessarily with the same angular velocity as the mouse; for though I-^co-^, the angular momentum produced in the mouse, is equal to /2CO2, the angular momentum produced in the cage, I^ is not necessarily;- equal to /g. (2) If a horizontal disc were constructed which' could turn perfectly freely about a vertical axle, a juggler spinning on the disc could diminish or increase his rate of rotation by extending or dropping his arms, and thereby increasing or diminishing his moment of inertia. A similar action enables a cat to right itself while falling in mid air, legs uppermost, and land on its feet. (See instance 5.)

56 CONSERVATION OF ANGULAR MOMENTUM (3) If a skater is describing a circle about the orange as centre, he can increase his angular velocity about it by leaning over directly towards it, although there is no tangential force tending to increase this velocity. For his moment of inertia about the orange becomes smaller, but his angular momentum about the vertical through the orange remains unchanged."^ If he describes the circle with angular velocity, oo say, he has also an angular velocity co about the vertical axis through the point of contact of his skate with the ice ; for he makes a revolution about this axis in the same time that his centre of gravity makes a revolution about the orange. The pressure exerted by the ice has no moment about the vertical axis through the point of contact. If, while he is describing the circle, he extends his arms, he increases his moment of inertia and therefore diminishes his angular velocity w about the axis in question ; and this diminution of w is shown on the ice by a " flattening " of the curve described — a necessary result, since co is also the angular velocity of his centre of gravity round the orange, which, if diminished, involves a larger radius for the curve.! (4) In the case of the earth and the moon we have another instance of the conservation of angular momentum. The action of the tides on the earth tends to diminish the kinetic energy of the latter, and therefore its rate of spin ; but the angular ononientuvi of the whole system, earth and moon, remains unchanged. Hence, since the angular momentum of the earth itself is diminished by the tides, it follows that the moon must recede from the earth so as to keep the angular momentum of the whole system constant. (5) Cat righting itself in mid-air. Suppose the cat to start falling vertically with no rotation, back downwards and legs extended at full length perpendicular to the body as depicted in Plate ii. Then during its fall the animal turns itself through 180° about some axis, roughly a line running lengthways through the centre of gravity of its body ; but since it starts with no rotation, it can at no time during its fall have any angular momentum about this line, which we will call its axis. (An examination of the photographs given will exclude the idea, sometimes suggested, that the cat gives itself rotation by using the hands for a fulcrum just as it is let go. The first few images show no tendency to turn either to one side or the other.) * In the same way, when water is rotated and then let out of a hole at the bottom of a basin, as the distance r of any particle from the hole diminishes, since the angular momentum remains constant, w increases and the water goes round with increasing speed. The same is true of a stone tied to one end of a string and swung round horizontally while the other end of the string winds up round a fine stick or peg. t It should be remarked that the above reasoning, although in agreement with experiment, is nevertheless not strictly accurate, since we are considering angular momentum about a moving line.

CONSERVATION OF ANGULAR MOMENTUM 57 Let us regard the cat as made up of a fore part and a hind part, whose moments of inertia /^, /^ ^^^ equal when the legs are fully extended at right angles to the body. The photographs"^ given in Plate ii. show that it first contracts its fore legs (thereby making I^ less than I^ and then turns its fore part round. This latter action necessitates the hind part being turned in the opposite direction (since the total angular momentum about the axis is zero) but to a less extent, since I2 is ]> I^. The animal then contracts its hind legs, extends its fore legs, and gives its hind part a turn. This necessitates the fore part being turned in the reverse direction but, again, to a less extent, since I^ is now ^ I^. It will thus be seen that by continued action of this kind the cat can turn itself through any required angle, though at no time has it any angular momentum about its " axis." [23.] If the cat were allowed to fall on to a cushion, resting on a smooth floor, would the fall of the cat move the cushion ? • 57. The foregoing remarks may be summed up in the following two principles connected with the motion of a gyroscope such as we have been considering : 1. Since there is no torque about a vertical axis, the angular tnoTnentwrn about the vertical through the point of support must throughout the motion re7nain constant, i.e. equal to the value, tvhatever it was, at the beginning of the iinotion. 2. The torque, which is about a horizontcd axis, must produce angular momentuTYi about that axis, and it cannot produce angular m^omentum about any other axis. These are of the utmost importance in considering the motion of a gyroscope. An explanation of the phenomenon of precession by the principle of conservation of angular momentum will be found in the Appendix. 58, Why precession increases as a top dies. It will be remembered that if a gyroscope is spun with one end of its axis in the socket of the vertical stand. Fig. xvi., then towards the end of the motion it precesses faster and faster, as it descends, until it finally flies off the stand. Let us apply the principle of conservation of angular momentum to the consideration of this phenomenon. We will suppose the wheel to be spinning initially with angular momentum Jw, its axis being inclined at an acute angle ^^ to the vertical. If we neglect all friction,! then the * These photographs are reproduced from Nature (Nov. 22nd, 1894), by the kind permission of the proprietors. t In reality the friction cannot be entirely neglected. The friction at the socket resists precession, and that at the bearings of the wheel causes a gradual diminution of its spin, but these retardations are not enough to affect the statement that the precession increases as the gyroscope descends. A further discussion, taking some of the friction into account, is given in Art. 66.

58 INERTIA angular momentum about the vertical through the point of support must remain constant throughout the motion, i.e. equal to the initial value IcocosOq. Precession will take place in the saine direction as w viewed, say, from above. As the gyroscope descends, its inclination 0 to the vertical increases, and therefore Jcocos 6 diminishes. [Or the projection of the disc as seen from above diminishes : see Art. 31.] Hence the contribution of angular momentum about the vertical (in the same direction), due to the precession of the frame and wheel (regarded as not spinning), must increase, in order to maintain the constancy of the whole. When the axle becomes horizontal it is clear that the whole angular momentum about the vertical through the point of support is due to the precession of the wheel and frame, and if Iz is the moment of inertia about the vertical for the frame and wheel, and Q the (increasing) azimuthal velocity, we have Iz^ = I(jo cos Oq, Iw cos On or Q I After this the contribution of angular momentum about the vertical from the spinning of the wheel is in the opposite direction to precession (Ico cos 0 has become negative), and since the constancy of the whole is maintained, the angular momentum about the vertical due to precession must continue to increase. This last position of the gyroscope corresponds to the top represented in Fig. ill. (a) of the Introductory Chapter. The same considerations hold good in the case of a top spinning on a fine fixed point, except that in this case the top touches the table before 0 reaches 90". 59. Inertia. We have already alluded in Art. 8, to the resistance due to the inertia of a body, i.e. the resistance which a solid body oflfers, by reason of its mass, to any force applied to it. If the body had no mass it would offer no resistance, provided it were free to move in the direction of the applied force. The following conception of a solid body is frequently useful in illustrating the property of its inertia. Suppose that we imagine a perfectly rigid massless skeleton framework, free to move in every direction, and composed of a great number of small hollow cells with smooth thin walls. Since it is perfectly massless it is incapable of offering resistance to force. Next suppose that each cell is filled with a loosely fitting shot, as in Fig. 28. The whole body has now considerable inertia, due to the mass of the shot which are in the cells : and we may consider a rigid body as composed of an infinite number of small

INERTIA 59 heavy particles (corresponding to the shot in the cells) kept in proximity to one another by a massless framework similar to the one above described. Y X Fig. 28. Fig. 29. We know that a rigid body resists force in every direction ; or, to follow out our analogy, in whatever direction our framework is pushed, a resistance will be experienced, since the moment the framework moves, each shot begins to press against a side of its cell. Let us suppose however that we could conceive an imaginary body which possessed inertia in two directions, say OX and OZ, but not in the direction OY, such a conception might be approximately illustrated by considering the skeleton framework to be composed of long tubular cells, running parallel to OY: for in this case an attempt to push the framework in the direction OY would meet with no resistance. Again, if the tubes were circular in shape as in Fig. 29, and endless, their centres lying on the line OZ, then the body would offer no resistance to being turned about OZ, 60. Start of precession. In Art. 34 we explained that the equation K = Iwil applies only to the maintenance of existing precession and not to the starting of precession. We have yet to consider what happens when precession is first set up. Let a weight be attached to the gyroscope as in Figs. 30 and 31. This will clearly introduce an external couple which acts on the gyroscope, but it should be noticed that there is one already, due to the weight of the wheel and frame. The additional weight will, however, give a clearer illustration. Suppose that the wheel is spun in the direction indicated in Figs. 30 and 31 while its axle XO is held in a horizontal position, and let it be then released so that the attached weight and the weight of the frame and wheel begin to act. It will be observed that the whole system " wobbles " and oscillates considerably, and then finally settles down to a steady precession about the vertical OZ.

60 THE STARTING OF PRECESSION Let us consider the reason for these phenomena, assuming the spin of the wheel as constant throughout the motion. When the gyroscope is released, the axle tends to precess about OZ under the action of the external couple ; but, before it can do so, both the wheel and frame and also the attached weight have to acquire angular momentum about OZ, about which there is no external torque. What we may call the inertia "about OZ'' acts as a resistance to the precession, and causes the wheel to dip downwards (Art. 47) through a small angle. Resistance due to inertia is clearly seen in the case of the attached weight which will "lag behind." Fig. 30. Fig. 31. Fios. 30 and 31. — Gyroscope when released, inclined at a small angle to the horizontal. The angle in the figure is somewhat exaggerated for the sake of clearness. Now when OX dips down out of the horizontal, an observer with his eye vertically above 0 (Art. 31) sees a projection of the disc of the wheel ; so that it is clear that as soon as the axle dips, the spin of the wheel contributes angular momentum about ZO (not about OZ). But since there is no torque about the vertical ZO, the angular momentum of the whole system about this vertical must always be zero, since it was zero originally. Hence, as the gyroscope dips, the frame, weight and wheel must acquire angular momentum about OZ (not about ZO) by the motion of the centre of gravity round OZ. This will cancel that of the wheel about ZO, due to its spin about its axle, and maintain the constant total value zero. In other words the wheel, frame, and weight will precess about OZ.

OSCILLATIONS 61 61. The above phenomenon of the dipping of the gyroscope illustrates also the second principle laid down in Art. 57, i.e. the torque can only produce angular momentum about its own axis. For it is clear that when the system is precessing its kinetic energy is greater than before it was released ; since, as far as the spin of the wheel is concerned, it is the same, and the wheel, frame, and weight are now precessing. The fact is, this excess of kinetic energy is provided by the work done by the torque in turning the system about its own axis, and causing it to dip. Hence the dip is necessary for conservation of energy as well as for conservation of angular momentum. 62. Oscillations. This precession will not at first be steady : for we have seen that whether precession is taking place steadily or not, the angular momentum about the vertical of the frame, weight, and wheel (regarded as not spinning) must be cancelled by the angular momentum about the vertical due to the spin of the wheel — which necessitates the dipping of the axle.^ Hence the precessional velocity depends on the dip, and alters if the dip is altering. There will however be a steady value of Q, consistent with Ico and K, and also a steady angle at which the axle dips to the horizontal : and this steady or mean value of the " dip " and of the precessional velocity Q, is arrived at in the following manner by a series of impulsive jerks and oscillations which are eventually destroyed by friction. The particles which at first, by their inertia, retard the precession, as soon as " dip " takes place acquire, by impulsive action, a velocity of precession greater than the mean velocity Q of the axle, and now, owing to their momentum, have the effect of hurrying the precession. The next instant the precessional velocity of the axle will be greater than that of the particles, and their inertia will again retard precession ; and so on.t But every time the precession of the axle is hurried or retarded the centre of gravity of the system rises or falls (Art. 47). The result is that, in addition to the backward and forward '' wobbling " the gyroscope oscillates up and down about the mean position of " dip," until finally, the oscillations having ■^ By hurrying or retarding the precession the angle made with the horizontal by the axle of the gyroscope can be diminished or increased : but if the system is left to itself it must precess with a small "dip." Although the axle is precessing at an inclination to the horizontal we may still employ the equation K = Iw^ since the angle in question is small. In Chapter VI. this equation is discussed for all inclinations. t The successive retarding and hurrying effect of the inertia can here again be most clearly realised by observing the suspended weight, which swings backwards and forwards in jerks about a mean position. An analogy to this starting of precession may be found in the motion of two railway trucks coupled together at rest, when one receives a sudden impulse. Each in turn assists and hinders the other's motion, until the same steady motion is arrived at for both.

62 OSCILLATIONS been destroyed by friction, an angular velocity Q is reached consistent with the values of K and Iw, i.e., when lot) Steady motion will then follow ; but any attempt to alter this existing state of steady precession will be followed by oscillations similar to those observed at the starting of precession. It should be noticed that during the motion above described, the gyroscopic resistance IwQ (Art. 45) is alternately greater and less than the applied couple. It is clear, then, that the " dip " and consequent oscillations are entirely due to what we may call the inertia "about OZ" of the frame, attached weight, and wheel ; but if they possessed the imaginary property of having no inertia "about OZ" as suggested in Art. 59, there would be no dip and no oscillations. In fact the gyroscope would precess exactly as it does in the case of steady motion where there is no inertia to be overcome. The reactions at the bearings of the wheel contribute slightly to the oscillations, in the same way that the inertia does, but are not the primary cause. The primary cause of the oscillations is the inertia of the system. 63. The reason will now be seen for the phenomenon mentioned in the Introductory Chapter, that if a downward pressure is applied at X (Fig. xxii.), when the gyroscope is spinning slowly, the axle XX' dips appreciably before precession takes place, but a sudden removal of the pressure will leave the system oscillating violently : while if the spin is very fast, the dip is hardly perceptible, and the oscillations are scarcely more than a " shiver." For, in order that steady motion may be arrived at, Q (which equals y-j must be large when w is small, and consequently the impulsive actions and reactions due to the inertia will be large before the required value of Q is reached ; whereas, when to is large, Q, is small, and in this case the dip and oscillations will be small also."'^ A similar explanation to the above accounts for the oscillations or nutations of a top. These are especially conspicuous just before its fall, when it goes through a sort of reeling motion. The spin has by this time been considerably diminished, and *The reason for this may also be partly seen from the parallelogram of velocities. For, in the accompanying figure, if w' represents the velocity created b}' the applied couple about its own axis, it is clear that when w is large the axis of resultant angular velocity is only slightly displaced ; and vice versa.

OSCILLATIONS 63 consequently the friction which in the earlier part of the motion tended to '' damp " the oscillations. 64. The above investigations may now be summed up as follows : Whenever we apply to the axle of a spinning gyrostat a torque which does Tnore than maintain existing precession, this torque produces three results : (a) Oscillations are set up and continue until destroyed by friction. They may be so c[uickiy destroyed, or so slight, that they are hardly appreciable to the eye. (/3) The torque produces, about its own axis, an effect which is not always appreciable to the eye, and corresponds to the dipping of the gyroscope in the preceding articles. Thus we see that work is done on the gyrostat. We shall in future pages allude to this effect of a torque about its own axis as " dip." (y) It produces about an axis perpendicular to itself and to the axle, an effect which is much more appreciable to the eye, and corresponds to precession. Thus we realise the increased kinetic energy due to the work done on the gyrostat. This effect of the torque we shall allude to as precession. Lastly, it must be remembered that there can be no change of angular momentum about an axis round which no torque has acted. 65. Explanation by the principle of energy of " Hurry the precession, the top rises." We shall now see more clearly (compare Art. 47) why a spinning top rises if we hurry its precession. For we have seen that the " hurrying " force produces only a small hurrying effect on the precession (corresponding* to "dip"), but the turning effect (corresponding to precession) is much more appreciable, and raises the top. We are, of course, doing work on the top by hurrying the precession, and since the kinetic energy of precession is not appreciably increased, the potential energy of the top must be increased, i.e. the centre of gravity of the top rises. In the case when friction at the toe alone hurries the precession, the work which raises the top is done at the expense of the kinetic energy of spin of the top. When precession is retarded, work is done by the top, which, since it does not lose much kinetic energy of precession, loses potential energy, i.e. the centre of gravity descends. 66. Explanation by the principle of energy of the increase of precession in a dying top. As in Art. 58, let us consider that a gyroscope is placed in the socket of its stand with its axis at an angle Oq to the vertical, when the angular momentum ■^ As the spin of the top diminishes the hurrying effect is greater. See Art. 63.

64 OSCILLATIONS about its axis is loo. Then Jo) cos ^^ is the angular momentum about the vertical through the centre of the socket, and this angular momentum would remain constant throughout the motion, but for the diminution owing to the frictional couple at the socket on the frame. We can neglect the effect of friction on the spin of the wheel, for it will be small. Now we have seen that the work done by the couple due to gravity = K.E. of precession + K.E. of descent or ascent {i.e. "dip"), and since the latter of these two contributions is not large, it is clear that most of the work done shows itself in K.E. of precession, and for this reason the gyroscope precesses faster as it falls. Again, the work done by the frictional couple at the socket = K.E. of descent (corresponding to precession) + K.E. taken from existing precession (corresponding to " dip "). Of these two contributions the former is the. greater, and therefore the gyroscope also descends more quickly as it falls. The same principles govern the motion of an ordinary spinning top which is gradually falling from a vertical position, or one which has never risen to the vertical, but is descending from its position of steady motion. The gravity couple, by a large number of excessively small " dips," gradually causes the top to fall, and, simultaneously with this, produces the more appreciable increase of precession. The friction couple at the toe (besides destroying the spin of the top) resists the increasing precession — the result, about its own axis, corresponding to " dip " — but contributes, more appreciably, to the falling of the top — the result corresponding to precession. A further discussion of this motion is to be found in Chapters VII. and IX. If the toe of the top is free to move instead of being constrained as in the above instance, the same principles apply ; for we can consider the centre of gravity of the top as a fixed point for purposes of rotation, and discuss its translational motion independently. 67. To prove that when the axle of the gyrostat has been released from its horizontal position, as much energy is lost as is communicated to the system. Let the axle (Fig. 30) on being released descend, by oscillations, through an angle SO before steady motion is reached. Let the angular momentum about the axle be /w, supposed constant throughout : we shall however take into account the friction at 0, which assists in destroying the oscillations. Let Iz be the moment of inertia about OZ of the frame and wheel, and fi the final velocity of precession.

ENERGY LOST IN HEAT. ELECTRICAL ANALOGY 65 Now the angular momentum about the vertical due to the dipping of the wheel is Im sin 80, or loo . SO, since SO is small. The angular momentum about the vertical due to precession is Izi} ; and we have seen that these two are equal in value but opposite in direction. Hence, I(i).SO = IzQ (i) Again, the kinetic energy communicated = 1/ 02 — 2-'- z^^ = ^^IcoQSO, by(i). But the whole work done = KSO = IwQSO; therefore half the work done, i.e. half of the energy communicated, is lost in frictional heat. It should be noticed that the couple K is not quite constant throughout the period under discussion, since it varies between K and K cos 0, where 0 is the inclination of the axle to the horizontal at any particular instant : but since 0 never differs much from SO, the variations in K are negligible. Analogous theorems are known in Electricity and Magnetism. Lord Kelvin first pointed out that if the currents of a system are kept constant by a battery during a displacement in which the electro -kinetic energy is increased by W, an equal quantity W of energy is dissipated from the wires as heat, the battery being thus drawn upon for a double quantity of energy 1W. See Maxwell's Electricity and Magnetism, 3rd Edition, Vol. II., p. 225. See also Vol. I., p. 120, for a similar theory in Electrostatics. E

CHAPTER Y. PEACTICAL APPLICATIONS. I. THE STEEKING OF A TORPEDO. 68. An allusion has already been made on p. 29 to the gyroscopic mechanism employed for automatically steering a torpedo. The patent was originally protected by M. Obry, an Austrian engineer, and sold to the authorities of the Whitehead Torpedo Works at Fiume, by whom it was improved and finally patented in its present form in 1898. The following is a brief outline of the way in which the steering is effected. As the torpedo passes through the impulse tube, a trigger projecting from its upper surface catches against a bolt in the tube and releases a spring by which the gyroscope is spun (see p. 29). Thus, before the torpedo enters the water, the axle of the gyroscope is pointing in the required direction, from which it never deviates (p. 11). Any deviation sideways on the part of the torpedo only alters the position of the gyroscope relatively to the torpedo. The gyroscope is fitted to the torpedo in such a way that this relative change of position opens one of two valves (the other being temporarily closed) connected with the compressed air chamber from which the screws of the torpedo are driven. The air rushing through either valve drives (in one case forwards and in the other backwards) a piston rod connected with a vertical rudder at the stern of the torpedo, and the valves are so arranged that when the torpedo swerves to starboard the rudder steers to port, and vice versa. The middle position when both valves are closed is scarcely ever maintained for an appreciable time, with the result that though the torpedo maintains the direction intended it is by means of a zig-zag path, roughly 2 ft. broad. The mechanism is in addition so arranged that if the position of the object fired at prevents the torpedo being aimed directly at it, it is possible to set the gyroscope (apart from the torpedo) in such a direction that the latter will eventually strike the object, although in the first instance not aimed directly at it. Full particulars and diagrams of the invention can be procured at the Patent Office, Southampton Buildings, London.

SCHLICK'S METHOD OF STEADYING VESSELS AT SEA 67 II. SCHLICK'S METHOD OF STEADYING VESSELS AT SEA. 69. Herr Otto Schlick has recently carried out a series of valuable experiments in the " See-bar," formerly a first-class torpedo-boat of the German Navy, with a view to applying the principle of the gyroscope to counteract the rolling of a vessel at sea. We have seen (p. 42, question 11), that when a paddle steamer heels over to starboard, either by the action of a wave or from some other cause, her bows, owing to gyroscopic action, turn towards the starboard side, and vice versa. Consequently, a paddle steamer rolls less than an ordinary screw steamer, though the gain in this respect is somewhat at the expense of the directness of her course. But it will be noticed that when she heels to starboard, the starboard wheel, by dipping further into the water, exerts more power than the port wheel, and so tends to turn the bows to the port side. Thus the gyroscopic effect of the paddle-wheels tending to steer the vessel to starboard meets with a counteracting tendency from the starboard wheel to steer the vessel to port. Otherwise it would be still more difficult than it now is to keep a straight course with a paddle steamer in stormy weather. If a large fly-wheel were mounted in the middle of a screw steamer on a horizontal axle at right angles to the length of the ship, and were made to revolve rapidly, it is clear that the steamer would become much steadier, but only in so far as she was allowed to make deflections from her course. To obviate this difficulty Herr Schlick places the fly-wheel with its axis vertical, and mounted in a frame (see Plate in.) which can itself turn about a horizontal axle directly athwart ship. The rolling force of the waves is thus counteracted (in part at any rate) by the gyroscopic resistance of the rotating wheel, while the wheel itself turns about the horizontal axle of the frame, but in no way interferes with the direction of the ship's course. Two eflects are thus brought about : (i) the rolling force of the waves is resisted ; (ii) the period of the oscillations is consequently lengthened, so that it no longer synchronizes with the motion of the waves. The latter will thus tend to damp the vessel's oscillations, for it should be remembered that violent rolling is the accumulated eflect of a series of waves each adding to the existing rolling. The oscillations of the fly-wheel itself are discussed in the next article. When the above appliance was first suggested many seamen expressed the opinion that waves would more easily break over a vessel steadied in this manner than if it were free to take up the motion of the sea, but experience has proved that this is not the case. A vertical heaving motion takes place, but the

68 PRACTICAL APPLICATIONS tendency of waves to break over the vessel is less. During experiments conducted on July 17, 1906, the "See-bar" when broadside on to the sea, tvith gyroscope fixed, reached a maximum inclination of 25° to leeward and 15° to windward. Her rolling motion passed through '* phases " gradually increasing and then decreasing. Not long before the gyroscope was set free the rolling attained 15° on each side of the vertical — an arc of rolling of 30°; but immediately the gyroscope came into operation the arc of rolling (out to out) was reduced to 1°. 70. The following detailed account of experiments leading to the more technical applications of the above results is reprinted by kind permission of the Institution of Naval Architects, being an extract from a paper read before the society by Herr Schlick on March 24, 1904. " The phenomena which present themselves in connection with an arrangement of this kind may best be studied by the help of a model, such as that illustrated in Fig. 32. This shows a pendulum which is able to swing to and fro on an axis at n. Above, the pendulum takes the form of a semicircular fork-piece B, A ring is hung on the point of the screw-pins pp in such a manner that it can turn about a horizontal axis through the centre of the fork-piece. In the ring itself, which is so weighted that, when inclined by any means, it will always return to the upright position, a vertical spindle, carrying the fly-wheel F, is set in bearings. If the pendulum be set swinging without the fly-wheel being made to rotate, it will be found to move to and fro with a certain definite period. The frame in which the fly-wheel is borne may first be allowed to move as easily as possible, i.e., with a minimum of friction. If the fly-wheel be then set spinning and the pendulum receive a push, the latter will, in the first place, show a considerably increased period of swing. " The fly-wheel oscillates with its frame during the swings of the pendulum with a so-called phase difference of 90°, that is to say, the swings of the axis of the fly-wheel keep lagging behind those of the pendulum by a quarter of a swing. While the pendulum is passing its central (vertical) position, the axis of the fly-wheel will show its greatest inclination, and when the pendulum is in the outermost position of its swing, the axis of the fly-wheel stands exactly in its middle position. The amplitude, or extent of the swing, of the pendulum will, as theory Fig. 32.

SCHLICK'S METHOD OF STEADYING VESSELS AT SEA 69 tells us, not be influenced thereby, but will remain exactly as great as before. That this is the case will also be readily apparent, since no consumption of energy takes place in the apparatus, the period only being influenced by the increase which takes place in the swinging mass, " Were it possible to fit a fly-wheel of this kind, able to swing in its frame without experiencing friction, into a vessel, this would be advantageous in so far that, to begin with, the rolling motions would become slower, and therefore less unpleasant, and then, on account of the great difference thus produced between their period and that of the waves, they would cease to be of amy consequence. The rolling motions of the vessel would then become considerably less in extent. If the frame which bears the fly-wheel be screwed tight on the model, so that it can no longer turn, the effect hitherto produced by it will cease, and the pendulum will swing with the same period as it would if the fly-wheel were not rotating. " It will readily be seen that the effect produced upon the swings of the pendulum by the rotating fly-wheel can be of greater extent only so long as the plane of the frame bearing the fly-wheel remains approximately vertical. " If the axis of the fly-wheel be inclined at an angle a to the vertical, the moment thus produced, acting against the motion of the pendulum, will be proportional to the value of cos a. Should the axis of the fly-wheel momentarily become horizontal, a position which with a pendulum in violent motion it may almost reach, that is to say, should a = 90° and cosa = 0, the influence of the fly-wheel will disappear altogether. " Since, as already stated, there is a phase difference of 90° between the swings of the pendulum and those of the axis of the fly-wheel, the gyroscopic influence on the pendulum must be least in amount when it is passing the middle position, i.e., at the very position at which it has its greatest angular velocity, because at the same moment the inclination of the axis of the fly-wheel is at its greatest, while the velocity with which it is changing its inclination has become very small or vanished altogether. When, on the other hand, the pendulum has reached its outermost position, and is changing the direction of its motion, thus for an instant reaching a state of rest, the axis of the flywheel then proceeds with its greatest angular velocity through the middle position, the fly-wheel thereby exerting its greatest influence. It will thus be evident that the conditions for the exertion of the greatest possible influence of the gyroscopic action on the pendulum are not present here. In order that the motion of the pendulum may be effectively influenced, the oscillation of the frame with the axis of the fly-wheel will have to be reduced in a suitable degree. In the model illustrated in Fig. 32 this may be most simply effected by tightening the screws pp to a suitable extent, so that they act as a brake on the

70 PRACTICAL APPLICATIONS motion of the frame of the fly-wheel. The swings of the flywheel frame are thus reduced in extent, and the phase difference between the two swinging movements here described now becomes less than 90". " If the experiment be made of setting the pendulum swinging with the brake thus applied to the fly-wheel frame, a very diflerent phenomenon will be observed. The pendulum will indeed still swing with a very considerable period, but the maximum angle attained becomes considerably reduced with each successive swing, so that a state of rest is reached even after about two complete swings. In scientific language, the oscillations of the pendulum experience a damping, in that the energy stored up in it is converted into heat by the friction applied to the fly-wheel frame." The complete paper from which the above is an extract can be procured at 5 Adelphi Terrace, London, W.C. 71. The flrst gyroscopic apparatus for steadying ships which was constructed in England, was made at Newcastle, at the Neptune Works of Swan, Hunter and Wigham Richardson, and was fitted in October, 1908, to the R.M.S. "Lochiel," owned by Messrs. David MacBrayne of Glasgow. It can be thrown in and out of action at will. When it is out of action, the vessel has been observed to roll, out and out, through an arc of 32 degrees, which was reduced to a (total) angle of from 2 to 4 degrees by the action of the gyroscope. The machinery, which occupies very little space in the steamer, is driven electrically and requires very little attention. III. BRENNAN'S MONORAIL. Model exhibited before the Royal Society, May 8, 1907. 72. The gyroscope has recently been employed by Mr. Louis Brennan with striking ingenuity and success to ensure the stability of a heavy car travelling on a single line of rail with its centre of gravity above the level of the rail, as is seen in the accompanying illustrations. The following pages will serve to illustrate the development and growth of his idea from the first elementary principles; but it should be remembered that the application of the gyroscope to methods of locomotion is still in its infancy, and

BRENNAN'S MONORAIL 71 consequently the details of construction in Mr. Brennan's Monorail in its final form will probably differ largely from those which are here explained. The invention,"*^ in its simplest form, consists in affixing to the car a heavy-rimmed fly-wheel, or gyrostat, AB, revolving in the same plane as the road wheels and in the same direction Fig. 33. (Fig. 33), on an axle OX perpendicular to the plane of the paper. If it were drawn, X would be towards the reader. This axle is mounted in a frame or carrier EF, which is pivoted at E and F so that it can turn about an axis OZ coinciding with EF and perpendicular to the axle of the flywheel. This turning can be effected either by muscular power applied to the handle H, or by some automatic arrangement. Suppose that when the car is travelling in a straight line, some force, such as wind pressure, causes it to lean out of the vertical, then, a tilting couple having been applied to the axle of the wheel, the wheel begins to precess about the vertical OZ. But, as in the case of the top, if we " hurry the precession, the top rises," so, if the precession about OZ is hurried, either by applying a force at the handle H or by the automatic control, the fly-wheel will immediately rise to a more vertical position carrying the car with it, and so will restore the vertical position of the car. The control of the precession might be effected by the automatic action of an inverted pendulum. *The provisional and complete speeilications of the Patent, from which this description is taken, can be procured from Messrs, Marks & Clerk, 18 Southampton Buildings, London, W.C. Price 8d.

72 PRACTICAL APPLICATIONS Now let us consider what happens as the vehicle rounds a corner to the right, say. As the vehicle turns, the gyrostat maintains its plane of rotation unaltered in direction, and is thus, relative to the car, displaced out of its central position in the plane FCD. To obviate this displacement, Mr. Brennan makes use of a second gyrostat exactly similar to AB, mounted (preferably) in the same plane on a parallel axle, but rotating in the o2')posite direction ; and the carriers of the two gyrostats are connected by means of gearing, so that the rotation of one carrier in one direction ensures a corresponding rotation of the other carrier in the opposite direction. The movements of the carriers are then controlled as before by a lever or other suitable means. The gyrostat to which the device for accelerating precession is fixed is called the actuating gyrostat. Both are made to rotate in vacuo ; otherwise the resistance of the air at the necessary liigh speed of rotation would be extremely great. It will be noticed that the above principles apply whether the car be moving backwards or forwards : also that the centre of gravity of the vehicle can be made to move laterally in relation to the axis of support without bringing gyroscopic action into play, and thus the car is enabled to move round a curve while maintaining a vertical position. 73. In order to illustrate the stability of Mr. Brennan's car, Messrs. Newton of Fleet Street, London, make a gyroscope Fig. 34. mounted as in Fig. 34, which will be seen to be virtually the same as a model of the invention in its simplest form. Fig. 33. In Fig. 34 an addition has been made to what (at the time of writing) is actually sold by Messrs. Newton, suggested by

BRENNAN'S MONORAIL 73 Professor H. A. Wilson and exhibited by him at a meeting of the Physical Societ}^ of London in November 1907, as a method of automatically hurrying the precession. It consists of a spiral spring 8H, one end of which, 8, is fixed to the rectangular frame A BCD, while the other end, H, is attached to the small crank ZH, which turns about OZ as the wheel precesses, and is rigidly fixed to the carrier EXF. It will be seen that so long as the wheel, and consequently the crank, are in the plane of the rectangular frame BB, the tension in the spring has no moment about OZ', but immediately the wheel precesses and the crank moves out of the plane of the frame, the tension of the spring has a moment which tends to hurry the precession, and consequently causes the wheel and frame to return to their vertical position. But the momentum of the wheel and frame will cause them to overshoot the vertical position,* so that the wheel will again precess out of the plane of the frame, the precession will again be hurried and the previous process will be repeated. Thus oscillations are set up of increasing amplitude, with the result that the frame and gyroscope eventually fall over. Professor Wilson thus illustrates the necessity for " damping " the oscillations set up by any automatic hurrying of the precession. We understand that Mr. Brennan has several means which he employs in order to damp these oscillations, one of which is clearly explained in the following article from the Tmies Engineering Supplement for June 5th, 1907, which is here reproduced by the kind permission of the author, Professor Worthington, late Headmaster of Devonport Naval Engineering College, and that of the Times authorities. Apart from its immediate bearing on the monorail, the article serves as an excellent recapitulation of much that has alread}^ preceded it in previous chapters. "THE BRENNAN MONO-RAIL CAR." " The interest excited by Mr. Brennan's mono-rail car, both among the general public and among engineers, is due partly to the obvious importance of the possible revolution in methods of transport which his car suggests and partly to surprise at the results he has obtained from what we believe is an entirely new application of gyrostats in combination. " The majority even of mechanical engineers, to say nothing of the general public, have too little experience of the properties of the gyrostat to have been able to realize what it is that Mr. Brennan has done, or to appraise his inventions. We say inventions, rather than invention, for the stability of his car depends, as his patent specifications show, on at least three * It will be found that owing to this tendency on the part of the gyroscope to overshoot the central position, we can, by a succession of taps on one side of it» cause it to fall over on that side.

74 PRACTICAL APPLICATIONS distinct inventions— (1) the automatic calling into play of a force tending to accelerate precession, by the rubbing of the axle of spin as it rolls along a guide, an action which may be said to be borrowed from a spinning peg-top ; (2) the regulation of the precession so as to leave the gyrostats, after any disturbance, always with their planes of spin parallel to the rail; (3) the combination of two linked gyrostats spinning in opposite directions, for meeting the exigencies of a curved track. "To understand these points it is necessary to recall the behaviour of Foucault's gyrostat. Let the diagram (Fig. 35) represent a heavy-rimmed disc with its spindle OA horizontal. Fig. 35. Fig. 36. This spindle turns in a ring or frame BAC, pivoted about the horizontal axis BC, at right angles to OA, on the frame BDGE, which frame again is free to turn about the vertical axis DE. If the disc be not spinning, a downward pressure at A will cause A to descend with acceleration. If the disc be spinning slowly in the direction BBC, shown by the arrow, then A will not merely descend, but will move towards B, with rotation about ED, and a sudden removal of the pressure will leave the whole system violently oscillating. " If, however, the disc be spinning very fast, then we find that a downward pressure at A, maintained constant, produces no sensible depression of A, but creates what is called a precessional rotation of the spindle OA, at a constant rate about the axle DE, A moving towards B along the arc CAB, at a constant rate so long as the rate of spin is unaltered. When the pressure ceases, there is only a slight and perhaps imperceptible tremor of the spindle OA, and we are left with the spindle displaced horizontally through an angle which is a measure of the timeintegral of the couple that has been applied about the axis BC as it moved.

BRENNAN'S MONORAIL 75 " If, on the other hand, with the disc spinning very fast, the pressure maintained at A is not vertical but horizontal, in the direction of the precessional rotation just described, then there will be no apparent revolution about ED, but there will be a rotation of the frame BAC about BC, and A will rise, the angle through which OA rises being in this case a measure of the time-integral of the moment of the pressure about ED. " If now, while we maintain a constant downward pressure at A, we also apply a constant horizontal pressure as if to accelerate the horizontal precession of the spindle, then the precession will not indeed be permanently appreciably accelerated, but A will rise at a rate proportional to the horizontal force, and work will be done by the horizontal and against the vertical force. " These are the chief relevant physical facts which lay at Mr. Brennan's disposal and of which he has availed himself with such remarkable skill and success. We will endeavour to explain his arrangement by pointing out its relation to the Foucault gyrostat just described. In the first place the frame EBCD is pivoted on the body of the car at E and D, so that when the car is erect DE is vertical. Mr. Brennan then makes the spindle of his disc into the armature of an electro-motor, whose field-magnets are carried by the pivoted and still balanced frame BAC. When everything is in equilibrium and the car is running erect, the spindle OA is horizontal and at right angles to the rail. We will suppose the car to be running in the direction BC and will refer to A as the right-hand end of the spindle. " Now let a wind-pressure be applied to the left side of the car. The car. begins to turn over relatively to the gyrostat, and thus at once brings down a guide plate G^ (see Fig. 36) fixed to the car, and bent into a circular arc as shown, so as to press on a small roller R^, turning loosely about the end F of the spindle OAF, which now projects beyond the frame BAC. The pressure on this roller causes the spindle to precess from A towards B with an angular velocity proportional to the pressure, and the turning-over of the car is arrested, but at the same moment the friction between the rotating spindle and the roller makes the latter roll along the under side of the guideplate and thus evokes a horizontal f rictional force on the spindle tending to accelerate precession. This causes the end F of the spindle to rise and push back the car against the wind-pressure. Thus the car turns over to meet the wind and is carried over beyond the vertical (perhaps considerably beyond) to a new inclined neutral position at which the moment due to gravity just balances the moment of the wind-pressure. This tilted position is reached, however, with a certain momentum which carries the car beyond it. Up to the instant of reaching this position the moment of the wind-pressure has been in excess of

76 PRACTICAL APPLICATIONS the opposite moment of the gravitational pull, and the impulse that has acted is measured by the angular displacement of the spindle along the guide. From the instant that the car swings past the new neutral position, the moment of the gravitative pull-over to the left exceeds that of the wind-pressure, and the car turning over to the left lifts the guide -plate G-^^ off the roller R-^, and the gyrostat is left to itself, till a very slight further tilting over of the car brings a second curved guide-plate G^ (fixed to the car) to bear on the lower side of a second roller R.^ fitting loosely on a non-rotating sleeve which is part of the frame BAG. This arrests the further turnino^ over of the car and the gyrostat begins to precess back, but this time there is no force accelerating precession, so that the car remains tilted a little beyond the neutral position, till the roller R.2, in passing beyond the middle position, is carried clear of the end of the guide-plate G2 ; the gyrostat is now left again to itself, but the car being released from the pressure of R^ at once falls over a very little more to the left and thus brings to bear on the bottom of the rotating roller R^^ the third guide-plate G^. The spindle now begins to roll over G^ ; its precession is accelerated by friction, and the roller R^ pushes back the guide-plate and so the car up to and beyond the new neutral position of equilibrium, when the turning moment on the car again begins to be reversed, and the car, being now pushed over by the wind to the right of this position, brings its fourth guide-plate G^ to bear on the roller R^, so that further turning is arrested, and the spindle OF precesses back to the middle position. In this way, the oscillations about the new neutral tilted position quickly diminish, and by thus arranging that the car shall overshoot the mark and then return in oscillations of diminishing amplitude, the time-integral of the upsetting couple is reduced to zero, which is the condition that the spindle shall be left at rest in the middle or " ready " position, after any adjustment. Each time the car has to move towards a new neutral position of equilibrium, work is done at the expense of the energy of spin of the disc ; this, and the energy lost in f rictional heat, is made up by the battery which maintains the rotation of the disc. " It is evident that, instead of an upsetting couple due to windpressure, we may have one arising from a lateral shift of load on the car. The same process of self -adjustment will be gone through, the car inclining itself to the left if the shift of load is to the right ; just as a man carrying a load on his right shoulder leans over to the left till the centre of gravity of his system is brought vertically over the supporting base formed by his feet. " We have spoken so far of only a single gyrostat, but it is evident that if we endeavour to travel round a curve the spinning disc will maintain its plane of rotation unaltered in direction, and the car carrying the guides will sweep them

BRENNAN'S MONORAIL 77 round past the end of the spindle F, and thus the middle position will be lost. To remedy this Mr. Brennan employs a second similar gyrostat with an equal disc, spinning at an equal rate in the opposite direction, about a spindle which, when all is in equilibrium, is parallel to or even in a line with the spindle of the first disc. The pivoted frame BAC of the first is so linked to the corresponding frame of the second that any lateral tilt of the first is communicated to the second, but at the same time each of the discs is free to precess. The precession of the second disc is equal to that of the first, but in the opposite direction, and any deviation from this equality and opposition is prevented by toothed gearing which connects the axle ED of the first with the corresponding parallel axle of the second. Such a system oiFers no resistance to turning with the car on account of a curve in the track, while to any upsetting moment it behaves like a single gyrostat of double mass, and enables the car to meet the upsetting moment of the so-called centrifugal forces by leaning over towards the inner side of the curve, exactly as it leant over to meet a wind-pressure. " It should, however, be observed that this adjustment does not get rid of the force tending to displace the rail laterally, and that this can only be completely met by sloping the track on which the rail is laid with exactly the same super-elevation as is required in an ordinary railroad curve (a slope which depends on the velocity prescribed). Mr. Brennan gets rid of the danger of upsetting, but not of the need of providing against displacement of the rail. " It remains to examine what will happen when we pass from a model to a car of larger dimensions. Fortunately, the result works out very favourably, since we find that if we make the linear dimensions of everything ?i-times greater, we can afibrd to spin the gyrostats 7i-times slower and yet secure the same righting eflfect, with the same angular excursion and return of the spindle along the guides. " This result is of great importance, for it means also that the centrifugal stresses in the real gyrostats need not be greater than in the model, and that the rate of spin may be reduced from 7000 per minute in the model to 875 per minute in a car of eight times the size. A greater rate in a smaller gyrostat is however a preferable option. " In this explanatory outline we have been guided by the patent specifications already published ; but we understand that Mr. Brennan has already made important improvements which will not be published till further protection has been obtained." 74. Note on the change of dimensions in a larger car compared with those of a model one. The efficiency of the erecting power of the gyrostat on the car, may be conveniently defined as the ratio of the impulse of the upsetting couple (due,

78 PRACTICAL APPLICATIONS say, to a gust of wind or a shift of load) to the resulting angular precessional displacement, relative to the car, of the axis of the gyrostat; for the larger the impulse required to produce a given displacement, the larger the resistance which the car offers to being upset. Now the arrangement of the car is such, that after some time t the car has no angular momentum about the line of rail ; hence the impulse of applied couple = impulse of erecting couple = Icoe, Q being the precessional velocity and 0 the angle of precession. Hence, the efficiency = /ci). If we now increase the linear dimensions 'Ji-times, the upsetting couple produced by a corresponding lateral displacement of load, say, will be '?^*-times as great : for the displacement is 7^-times greater and the load is '?^^-times greater. Hence, for equal efficiency we must have Im ^^^-times greater. Now / is ?i^-times greater; for I=Mk^ where M is '^i^-times and k is 7i-times as great. Consequently, if a> is ii-times less, we shall have equal efficiency. For further information concerning the Mono -railway, the reader is referred to Professor Perry's article in Nature, March 12, 1908.

CHAPTER VI. STEADY MOTION OF A TOP. 75. The student is reminded at the commencement of this chapter that, when a solid body is under consideration, (i) Angular velocity about any line means total angular velocity — not relative to some moving plane, unless this is expressly stated. (ii) Angular velocity about a line which is moving means (total) angular velocity about the line fixed in space, with which the moving line happens to be coinciding at the instant in question. 76. In the preceding chapter the bodies whose rotation we have discussed have been symmetrical bodies, as, for example, a fly-wheel ; and all the rotations have been about an axis of symmetry, i.e. the axle. If the axis were not an axis of symmetry, an angular velocity about this axis would in general involve angular inoinentum (about this axis, and also) about the two axes perpendicular to it, as is shown in the next article. In this chapter we propose to discuss the equations of motion of an ordinary spinning top, in which case it is clear that only the axle of the top is an axis of symmetry, and any other axis is not. But we shall see in Art. 79 that, since the top is a solid of revolution, any axis perpendicular to the axle of the top is the same for our purpose as if it were an axis of symmetry, and angular velocity about such an axis involves (of course angular momentum about that axis, but) no angular monientuni about perpendicular axes. 11. Rotation about one axis involves in general angular momenta about other axes at right angles to the first. In general, if a body has at any instant an angular velocity about a given axis, this velocity involves an angular monientum about each of two lines perpendicular to the original axis and to each other. For let OZ be the original axis of rotation about which the body has an angular velocity in the positive direction. Let OX, OY, be two straight lines perpendicular to OZ and to each other.

80 PRODUCTS OF INERTIA Let X, y, z be the co-ordinates of any particle P of the body, referred to these axes. Then the angular momentum of the particle P is : About OZ, About OX, About OY, tnccr . r or 7}ir\o. — mcor cos 0 . z oc = — onoor . - . 0 = — '/nooz . x. r — mwr sin 0 . z y = — inoyr .- ,z= — inooz . y. Hence it is seen that, owing to the rotation of the body about OZ, there is a contribution of angular momentum about OX equal to — H^ncoz . x, and about 0 Y equal to — ^mooz . y, in addition to Emr^co, or I^co, about OZ. The expressions Xmzx, Honzy are called products of inertia, but we shall be able to neglect these in dealing with solids of revolution, as will be seen in Art. 79. 78. It should be noticed that although the rotation oo of the body about OZ involves in general angular inonienturn about both OX and OY, yet it involves no component angular velocity of the body about OX and 0Y\ for the resolute of w about OX or OF is cocos90°, which is zero. This can be further illustrated by considering two points A, B moving in parallel lines with the same velocity so that AB i^ perpendicular always to the lines of motion. If the velocity at any instant is v, though the angular velocity of B about A is zero, the angular momentum of B about A is onvc, where c is the distance AB. 79. Particular case of a solid of revolution. If the body is a solid of revolution whose axle is the axis of rotation OZ, as, for instance, an ordinary top spinning vertically on its toe, it is clear (Fig. 38) that for every particle P revolving about OZ in direction {XY) and possessing a definite linear velocity

PRINCIPAL AXES 81 in a definite direction in space, there is a corresponding particle, on the opposite side of the axle OZ to P and equidistant from it, which is moving in the opposite direction in space with the same velocity. As far as rotation about OZ is concerned, these two moving particles cause angular momentum about OZ. But when we consider the angular momentum about OX, UY, we see that the angular momenta of these particles cancel each other. So for all the particles of this solid of revolution. Hence, if the body rotates about its axle OZ with velocity o), it has no other angular momentum due to w besides /^co, and we have already seen that it has no angular velocity due to this rotation besides w. Fig. 38. Fig. 39. Now suppose the body has only a rotation about OX, but OZ is the axle of the solid of revolution (Fig. 39). At a given instant any two corresponding particles P, P' are moving with equal and parallel velocities in planes parallel to YOZ. Resolving these equal velocities in directions OY, OZ, we see that the angular momenta of the particles cancel about both OY and OZ. Hence, in this case, the whole angular momentum of the solid due to the rotation w is represented by Ir^w. An axis such that rotation about it introduces no angular momentum about any perpendicular axis is called a principal axis. It follows that in a top, the axle of the top is a principal axis, as is also any other axis perpendicular to it. 80. Moving axle. It is clear that if the axle of a rotating body moves in any manner it carries the body with it. In fact, the axle is equivalent to an indefinitely fine wire passing through the body. F

82 STEADY MOTION 81. Moving axis. It is sometimes convenient to resolve the angular velocity of a rotating body about three axes mutually at right angles, and these three axes form a frame of reference, which may itself perhaps be moving. If we know the motion of the body relative to this frame, and the motion of the frame, we know the motion of the whole body. If the frame of reference is moving, the axes are called Tnoving axes, and the frame does not necessarily carry the body with it, for all the axes are not necessarily fixed in the body, but may be passing through it. 82. The velocities and momenta of a spinning top in steady motion. By the steady ^motion of a top is meant the motion in which the angular velocity w about the axle of the top remains constant, while the axle has a constant inclination a to the vertical, and is carried round the vertical with constant angular velocity Q, (Fig. 40). This velocity Q, is called the azimuthal velocity, the vertical plane AGZ through the axle of the top being called the azimuthal plane. During steady motion, the centre of gravity G describes a horizontal circle. The lines GC, GA, of which GA is always moving through the material of the top but remaining with GC in the azimuthal plane, are taken as moving axes of reference. Fig. 40. Fig. 41. The velocities. The angular velocity Q of the azimuthal plane about ZG (or -Q about GZ, Art. 20)", can be resolved (Fig. 41) (a) Qsina about GA, (b) Q cos a about GC.

ANGULAR MOMENTA 83 If coa is the angular velocity of the top about GG relative to the aziTTiuthal plane, then the total angular velocity of the top about GG is given by w^Ma+Qcosa, and the total angular velocity about GA is Q, sin a. Thus the angular velocity of the top is the resultant of (i) the spin oo about the axle GG; (ii) the spin Q sin a about the axis GA. About an axis perpendicular to GG and GA there is no angular velocity, since a is constant.* The TrioTYienta. Let the moments of inertia about GG, GA be G, A, respectively. Since GG, GA are principal axes, the angular velocity o) about GG gives an angular momentum component Geo about that axis (and 2 none about any other axis) ; so the angular velocity Qsina about GA gives a component All sin a about GA. Thus the angular momentum of the top is the resultant of Gm and A Q sin a about GG and GA respectively. / / /.^ v^A 83. Velocity axis. The axis of component angular velocity lying in the azimuthal plane we shall call the velocity axis. It is clear (Fig. 42) that it makes an angle /3 with GG given by tan^ = . The Fig. 42. O) axis of total resultant angular velocity is called the instantaneous axis of rotation. 84. Momentum axis. The axis of component angular momentum lying in the azimuthal plane we shall call the momentwin * The motion will perhaps be more easily realised if we consider a small roller such as is sometimes used for mounting photographs. In the accompanying figure, Z'GH, which corresponds to the azimuthal plane, is turning about Z'G with velocity Q, which has about GG a component velocity Q, cos a ; the roller corresponds to the top and turns about its axle GG with velocity w^ relative to the frame, but with total velocity w, where w = w"^ + J^ cos a. When we watch a top spinning, we do not think of its spin relative to the azimuthal plane, but of its total spin about its axle.

84 MOMENTUM AXIS axis. It is clear (Fig. 43) that it makes an angle y with GO , , ^Qsina given by tan y = a ca It will be seen that precession of the momentum axis necessitates a corresponding precession of the axle of the top, since both lie in the azimuthal plane. Hence we may discuss the motion of the top by considering the rotation of the momentum axis (see Art. 85). In steady motion the momentum axis coincides with the axis of resultant angular momentum. 85. Torque required to rotate the momentum axis. Referring to Fig. 20, p. 37, let OX, OY, OZ be any three mutually perpendicular lines, and let OX be the momentum axis at any instant, and take OA to represent the magnitude (if) of the momentum. Let Q be the angular velocity of OX round OZ (see Art. 40), so that after time St, the angular momentum is represented by 0A\ where the angle AOA' = i'lSt. Then if A'B' be drawn parallel to AO, OB' represents the Fig. 43. z B ^ L f/ if/ v/ T I' Y /O Fig. 44. change of angular momentum in time St, since OA and OB' are equivalent to 0A\

ROTATION OF ANGULAR MOMENTUM 85 And, since OB' = AA' = MilSt, the change of angular momentum in time St is MQSt about 0 Y. .'. rate of change of angular momentum is MQ, about OY. This requires for its production a torque MQ, about OY. Hence (Fig. 44) if OL represents the momentum axis of a top whose toe is the fixed point 0, the steady motion may be discussed by considering that OL is rotated about OB, the perpendicular axis in the azimuthal plane, by the torque about OY perpendicular to the azimuthal plane. It will be noticed that the gyroscopic resistance of the top to being turned about OY is MQ (see Art. 45). Co7\ If the axes of resultant angular momentum and angular velocity be any two lines OP, OQ, the angular momentum and velocity can each be resolved into components about OX, OY, OZ and the torques producing the rotations of the component momenta about the three axes can be separately found as above, and equated to the corresponding components of the external acting torques. This method of determining the motion of a top is the one most generally employed, since it is capable of easy extension to the case where the motion is not steady. 86. The steady motion of a top spinning on a fine fixed point. Consider the steady motion of a top spinning with its axle inclined at an angle a to the vertical on a fine point 0 which is F< .^-^ Fig. 45. fixed. An instance of such a motion will be that of a top spinning on a table which is rough enough to prevent slipping, while the toe is considered too fine for the friction to have a

86 FINE PEG moment about the axle, and therefore diminish the spin (see Art. 128). Let the directions of rotation be those marked in Fig. 45. The external forces acting on the top are : (i) its weight Mg at Q\ (ii) a vertical reaction i^ at 0; (iii) a horizontal reaction F at 0. This last must act in the direction marked, since G describes a horizontal circle about OZ, and F, the force causing it to do so, must be parallel to the inward radius. From Art. 82 we see that the momenta are Cw about GO, and J.Qsina about AO, where A is the moment of inertia about the axis through 0. These give component momenta AQ> sin a cos a — Cw sin a about OX, and a component about OZ, the latter of which need not be considered, as it is not rotated and no torque therefore is required on account of it. The former is rotated with angular velocity — Q about OZ, and this requires a torque {AQ, sin a cos a — Cw sin a)( — Q) about OY (Art. 37). But this torque is illf(9'asina, where OG = a. Hence, CmQ sin a — AQ^ sin a cos a = Mga sin a, whence either sin a = 0, or CwQ — AQ^ cos a = Mga. The former gives a = 0 or tt, both of which are possible angles for steady motion. The latter gives a quadratic for Q, i.e. Acosa.Q^-Cw.Q-{-Mag = 0, showing that there are, in general, two possible angular velocities of precession, i.e. ^ _ Cco±s/G^w^ - 4 4>A cos a . Mag, 2 \/AMag .cos a or ft)> C If to) has a smaller value than this, the top cannot spin steadily (compare Art. 102). The larger or the smaller value of Q will be maintained according to the particular initial conditions of motion. Thus if

GYROSCOPIC RESISTANCE 87 the top is started at an angle a with angular velocity «, and either of the above values of Q, it will continue to spin steadily. From the above quadratic equation we see that during steady ^^^^^^ _Ci.^-Mag showing that if G is increased a diminishes, and \i A „ a increases ; hence it appears that a top with a longer leg spins at a greater angle to the vertical. 87. When the majority of rotations in a problem under consideration are right-handed it will be found convenient* to work throughout with right-handed axes. Whichever rotation is employed, the precessional velocity must be considered negative if it does not turn the angular momentum rotated, towards the torque axis both being drawn in the same sense. 88. Equation deduced from gyroscopic resistance. The equation of the preceding article may be obtained by considering that the torque Mga sin a is tending to turn about its axis the two (right-handed) components of angular momentum Ceo, and ^Qsina; but these components, instead of being turned about OY, are precessing about OA and OG with velocities f2sina and Q. cos a respectively. Hence, the gyroscopic resistances to being turned are respectively GwQmia and J.f2^sinacosa (Art. 45). If the torque axis is drawn right-handed we see that GG sets itself towards the torque, while GA sets itself away from the torque. Thus, GwQ^ sin a must be considered a positive resistance, AQ^ sin a cos a must be considered a negative resistance. Since there is no change in angular momentum about the torque axis we have Mga sin a — GooQ^ sin a-\-AiT sin a cos a = 0, or Mga = G(£iQ>^AQ!^ 0.0^ a, as before. The general equations of which the above are a particular case are given in Art. 125. 89. The equation of Art. 86 might also be obtained by rotating the momentum axis instead of the horizontal component of angular momentum. In Fig. 46, let AG be at any instant the axis of precession * *It should be noticed that the term " axis of precession," which is suggested by astronomy, is not strictly in accordance with astronomical usage, where the axis of the earth is said to precess about a line perpendicular to the plane of its orbit, and not to its own axis. In the same way a top spinning at an inclination to the vertical can be said to precess about the vertical ; but we are taking the axis of precession to be the axis perpendicular to the rotating torque and the angular momentum rotated.

88 GYROSCOPIC RESISTANCE round which the angular momentum Cwsecy is rotated with velocity 11 sin (a — y). Fig. 46. We have, employing right-handed rotations, Coo sec y . 1} sin (a — y) = Mga sin a, Cft)Q(sin a — cos a . tan y) = Mga sin a, n c>,( ' ^Qsina\ li/r or GwSi ( sm a — cos a — ^^^ ) = Mga sm a, i.e. CwQ — AQ^ cos a = Mga, as before. It will be noticed that the first of the above equations expresses the fact that 2. I the gyroscopic resistance is equal to the external torque. 90. We see (Fig. 46) that the velocity axis is in this case the instantaneous axis, and makes a constant angle {a — /3) with the vertical OZ : it therefore describes a cone fixed in space having OZ as axis and vertical angle 2(a-^). It is also inclined to OG^ at a constant angle /3, and thus describes a cone fixed in the body having 0(t as axis and vertical angle 2/3. Fig. 47.

THE HODOGRAPH 89 Since points of the bod}^ on the velocity axis are instantaneously at rest, the motion can be represented by the rolling of the latter cone on the former (Fig. 47). [24.] Does all this hold as regards the momentum axis ? [25.] With what angular velocity does the moving cone roll round the fixed one ? 91. Analogy of the hodograph. If a particle P (Fig. 48) is moving along a curve P^P^P^ . . . (not necessarily in one plane), and if from a point 0 fixed in space straight lines OA^, OA^, OA^ are drawn to represent the velocities at the points P^PJP^ . . . , then the velocity of the point A along the curve -4 1^-2^ 3 ... represents the acceleration of the particle P as it moves along the curve P^PgPg, ..., since it represents the rate of change of the velocity. The path A^A^A^...\^ called the hodograph of P. Similarly, if OA^, OA^, represent the raomentum of the particle, then the velocity of the point A along its path represents the force acting on the particle P since it represents rate of change of momentum. Thus if OA^, OA^,..., be successive positions of the axis of resultant angular momentum, and OA^, ... represent the angular momentum of a body at successive instants, then the velocity of A in its path represents the torque acting on the body, since it represents the rate of change of angular momentum. 92. We will now apply this analogy of the hodograph to obtain the equation of Art. 86. In Fig. 46 let OL, which coincides with the momentum axis (Art. 84), represent at any instant the resultant angular momentum of the top. Draw Xilf and LN perpendicular to OZ and OG respectively. Then OM, ML are the vertical and horizontal components of the total angular momentum, and ON, NL are the components along and perpendicular to OH. The linear velocity of L is LM.Q, and represents (Art. 91) the torque causing precession, i.e. Mga sin a.

90 BLUNT PEG Now ifZ = sum of horizontal projections of ON and NL — ON sin a — NL cos a = C Fig. 49. Here, since there is steady motion, co is constant, and therefore there is no component of F perpendicular to the azimuthal plane, the reaction P being along the normal PE. Since G describes a horizontal circle uniformly, if r is its radius, we have F—MQ!^r Also, since G does not move vertically, R = Mg. Considering the angular momentum about GX the horizontal through G, as being made to rotate with velocity Vl about the vertical through G, we have (rotations right-handed) : angular momentum rotated = Co) sin a — AQs sin a cos a about GX, velocity of rotation = Q about GZ, torque producing rotation = Re sin a — F{c cos a-\-y cosec a) about YG.

BLUNT PEG 91 The other components of angular momentum are not rotated. Hence, paying due consideration to sign, (Ceo sin a — AQ sin a cos a) Q = Re sin a — F(c cos a-^y cosec a), or CcoQ sin a ^AQ^sin a cos a = Mgc sin a — ifil^r (c cos a-{-y cosec a). We have, in addition, since P is for the instant at rest, the geometrical condition, that the velocity of P relative to Z + that of L relative to G + that of G relative to the centre of G's circle = 0, or yo) — Q sin a(c-\-y .cot a) — Qr = 0. 94. Steady motion on a surface of revolution. It should be noticed that steady motion can be maintained on any rough surface of revolution in exactly the same way as on a rough horizontal plane. For (Fig. 50) when the steady motion has once started, friction, as before, will act only in the azimuthal plane, the forces and couples acting during successive positions of the top will be the same, and the precessional motion will remain unaltered. Fig. 50. [26.] If the surface of revolution be smooth, under what conditions (if any) will steady motion be possible : (i) if the surface is convex to the top ; (ii) concave ? What direction does the precession take in the latter case ? 95. The motion of a system of two or more gyrostats. Four equal gyrostats have for axes the sides of a light rhoTnhus OBEF, formed of rods jointed together, which hang from 0, and all four are set spinning with equal angular velocities co, and in such a way that all would be spinning in the same direction

92 SYSTEMS OF GYROSTATS if the angles at 0 and E were zero. The mass of each gyrostat is M, and a mass M' is suspended from E. Prove that if the angles at 0 and E are each 2a the whole can move with a steady precession Q provided that (^ +ilfa2)Q2 cos a + (7a)Q = (2if + if )«^. 2(X being the length of a side and G and A the principal moments of inertia of each gyrostat. Let us consider the gyrostats spinning in the direction indicated in Fig. 51. Then (Fig. 51 a) taking the angular momentum rotated and the rotating torque right-handed about OX and YO respectively, it follows that (l is about OZ. Ana'' It V X x' Fig. 51. Fig. 51 (a). Let X, Z be the horizontal and vertical reactions at D, on the gyrostat OD. The angular momentum about OX is : Ceo sin a + (A-^ Ma^) Q sin a cos a. The torque about YO is: Mga sin a -f Z2a sin a — X2a cos a ; .". [(7wsina + (A + l/a2)l'2sinacosa]12 = Mga sin a + Z2a sin a — X2a cos a (i) Now consider the lower gyrostat EB and the torque turning it about E. Referring to Fig. 51 {a) for signs and using left-handed rotations, Q is negative. The angular momentum about EX = Cw sin a-\-{A-\- Ma^) Q sin a cos a.

ROLLING CONE 93 Moment of torque about EY = Mga sin a — Z2a sin a — X2a cos a ', .' . [Co) sin a + (^ + Ala^) Q sin a cos a]( — Q) = Mga sin a — Z2a sin a — X2a cos a (ii) Subtracting (ii) from (i), 2 [Oo) sin a + ( J. + Ma^) Q sin a cos a] Q = 4a^ sin a. But the two lower gyrostats and M'g are supported by the two vertical reactions at F and D. .-. 2Z=2Mg + M'g. Hence the condition for steady motion becomes CwQ + {A -\-Ma^)Q^ cos a = {2M-^M')ag, since a is not zero. [27.] What are the reactions on the axes of the gyrostats at right angles to the azimuthal plane ? [28.] What would be the effect of the two right-hand ones spinning one way and the other two the opposite way ? 96. Cone rolling on a rough horizontal plane. A homogeneous right circular cone rolls on a perfectly rough horizontal plane. Prove that if the vertex remain in contact with the plane T-2 "-^ 1 67r^/t;- cos a ag (4 — 3 cos'^a)' where T is the time of a complete revolution of the axis, a the radius of the base, a the semi-vertical angle of the cone, and k the radius of gyration about a generating line. Consider the steady motion when the vertex 0 is just not touching the plane (Fig. 52). Fig. 52. If CO is the spin about 00 and Q the azimuthal velocity, the angular momenta are Ow and —-4^2 cos a about 00 and OA, where 0, A, are the principal moments of inertia at 0.

The text on this page is estimated to be only 29.78% accurate

94 ROLLING CONE The component momenta are therefore Cw cos a-\-AQ cos a sin a and Gw sin a-{- Ail cos^a about OX, OZ respectively. The latter is not rotated, but the former is rotated with angular velocity Q about OZ and this (Art. 37) requires a torque il [Goo cos a-^AQ cos a sin a] about 0 Y. But since 0 is just not in contact with the plane, the pressure of the plane acts through B, and the moment of the acting torque is i- _, MgDN = Mg\ -^ OG^cosa ^ ^ Lsm a J = Mq -r- -a cot a cos a '^ Lsm a 4 J (4-3cos2a); _ %^/A_Q.^c,2 4 sin a .*. QfCwcosa + ^Qcos asinal = T^-;^ — (4 — Scos^a) (i) '- -■ 4 sm a ^ ^ ^ ' Now consider the motion of the point B of the cone. Since the cone has rotations w about OG and — Qcosa about OA, the parts of the velocity of D due to each, parallel to OY, are respectively — aco and +(acot a)(Qcosa). Hence, since Z) is momentarily at rest, — aw + aQ cot a cos a = 0, 1 , Ocos^a whence co=H ^ . sin a .'. substituting in (i), Q^ -7^ — , G cos a + -4 sin a cos a = -^—^ — (4 — 3 cos^a), Lsm a J 4sma^ ^ and since A sin^a + G cos^a = moment of inertia about OD Q2A;2cosa=^(4-3cos2a), and the periodic time T is 27r/r2, rn9 A 9 ^'*^ COS O. ^a(4 — 3 cos^a) 167r^A:^COsa ^a(4 — 3cos^a)' and when the cone is on the plane, Q being smaller, T^ is greater than this value. [29.] When the cone touches the plane what torque causes it to precess?

EXAMPLES 95 EXAMPLES. Note. In the following examples w and fi are taken so that they would be in the same direction if their axes coincided. 1. Employ the preceding methods to show that the condition for steady motion of a top with a blunt peg (or any solid of revolution) spinning with velocity w on a smooth horizontal plane is (7(ol2 - -4 12^ cos a = illfg'c, where G and A are moments of inertia about the axes in the azimuthal plane through the centre of gravity 6^, c is the distance between G and the point of intersection of the axle and the normal reaction, and a is the angle the axle makes with the vertical. Explain the similarity to the equation in Art. 86. Note. Since the plane is smooth the top "skids" round during the whole motion ; there are no frictional forces at the point of contact, and therefore the centre of gravity G remains at rest — the axle of the top describing a cone, vertex G. 2. A thin circular disc, radius c, spins with velocity w about an axis through its centre perpendicular to its plane, while the rim is in contact with a smooth horizontal plane. If Q, is the azimuthal velocity, show by the above methods that the condition for steady motion is cQ? sin a — 2cwl2 = 4^ tan a, where a is the angle the plane of the disc makes with the vertical. 3. A circular wire ring of radius a rolls on a rough horizontal plane, so that its plane maintains a constant inclination a to the vertical ; if w be the angular velocity of the ring, and Q. the azimuthal motion of its centre, ■^ 4acol2 cosa — aO^sin acosa = 2^sina. 4. Show that the condition for steady motion of a disc spinning as in example 2, but on a rough horizontal plane, is c^w^ cot a (6r + c sin a) = 4^?'^, where r is the radius of the circle described by the centre of the disc. 5. Obtain the equation for steady motion, as in Art. 95, for the case of six gyrostats. Would it be possible with steady motion to spin (a) the two vertical gyrostats in opposite directions ? (6) the two upper ones in the opposite direction to the two lower ones ? 6. One gyrostat, axis 2a, mass J/j, spins on a fine fixed point, while another, axis 26, mass M^., spins on the head of the former. Supposing a steady azimuthal motion 12 to be maintained when each makes an angle a with the vertical, determine equations for Wj, w^, the spins of the two gyrostats, (7^, A^, etc. being the principal moments of inertia through the centres of gravity. Discuss the following points : (i) If the upper one spun the reverse way to the lower one, would steady motion be possible ? (ii) Would steady motion be possible if the upper one made an angle (B with the vertical ? (iii) If the two gyrostats were alike and both inclined at an angle a to the vertical, would a>i = a>2? 7. A gyrostat has a universal joint at a point in its axis and the joint is attached by a string of length ^ to a fixed point ; an angular velocity w is given to the body and 12 is the angular velocity of the centre of gravity about the vertical, 0 the angle which the string,

96 EXAMPLES gravity and the point of suspension, C being the moment of inertia about the axis of figure. Show that when motion is steady, 12 sin )((7w — yli2 cos ^) cos B = Mga sin ( r being the radius of the circle described by the centre of gravity of the cone. Additional problems on steady motion will be found among the Miscellaneous Examples at the end of Chapter IX.

CHAPTER VII. GENERAL MOTION OF A TOP. 97. General motion of a top spinning on a fine fixed point. Hitherto we have only discussed the motion of a top when its axle makes a constant angle a with the vertical. We will now consider the case where this angle may be a variable quantity 0 throughout the motion. It should be particularly remembered throughout this chapter that all frictions are neglected except where otherwise stated. Let 6 be the angle which the axle makes with the vertical through 0 (Fig. 53), w the spin of the top, and i/i~ the azimuthal Y/ Fig. 53. velocity, \/r being the angle the azimuthal plane makes with some plane fixed in space and passing through OZ. Since there are no forces acting about the axle, with regard to which the top is symmetrical, co will remain constant.* * A more rigorous proof of this is given in Art. 128, where it is shown that the condition of symmetry is essential. The reader is reminded that, as in Art. 82, co is eqvial to (0^^ + ^ cos 6, where a>„ is the (now variable) velocity relative to the azimuthal plane. G

98 OSCILLATIONS IN THE AZIMUTHAL PLANE The component velocities of the top are : (JO about OH; yp-sinO about OA; 6 about OY; where OF is perpendicular to the azimuthal plane. The component angular momenta are : Co) about OH', A\p- sin 0 about OA ; AS about OY. Since there are only two unknown quantities (0, \^), only two equations are necessary. Hence the whole motion of the axle of the top may be determined by the two following considerations : (a) The angular momentum about OZ remains constant. (^) The sum of the kinetic and potential energies of the top is constant. 98. Motion of the top when the centre of gravity G has no initial velocity. In this case we see from (a) that Ciio cos 0 + Ayp^ sin^O = Goo cos 0^, (i) if Oq is the inclination to the vertical at which the top is first spun, the axle of the top being initially at rest. From (^) we have i(7a)2 + J^^A^sin2e + i^eH%acose = i(7co2 + i%acoseo, or ^\/r2sin2a4-^^2 = 2i%a(cosOo-cos^) (ii) Equation (i) determines the azimuthal velocity in terms of 0, C(jo(cosO(. — co8 0) ,.... ^ A sin'^6 and the above equations determine the complete motion of the top. 99. The top will oscillate between two limiting values of 0. The elimination of yfr between (i) and (ii) yields (7 V(cos 0^ - cos Of + AW sin^ 0 = 2MAga(Gos 0^ - cos 0) sin^ 0. The stationary values of 0 are given by ^ = 0, i.e. by {(iose-coseQ)[2MAgasm^ O-CW (cos 6^-008 6)] = 0, i.e. 0 = 0o or sin^ 6 = 2\{cos0o — cos0), where 2/\ is written for C'W/2MAga. The latter is cos2 0-2Xcos e + 2\coseo-l =0, whence cosO = X + Vl — 2Xcos ^^^H-A^.

CENTRE OF GRAVITY INITIALLY AT REST 99 Now, X + v/l-SAcosOo + X'^ > X + x/i-2X-\-\^ since cos Oq < 1, i-e- >A + 1-X, i.e. >1. Hence the former root, though giving a real value of cos 0, gives an imaginary value of 0. The latter root gives a real value 0^ for 0 where cos e^ = X- >s/l-2\cos^o + A''^. Hence, ^ = 0 (i.e. the axle momentarily ceases to rise or fall) when 0 = 0q (upper limit) or 0 = 0-^^ (lower limit) : that is, the axle of the top oscillates in the azimuthal plane between two cones (Fig. 54) whose semi- vertical angles are Oq and 0■^^, while the azimuthal plane itself rotates with varying velocity \^. The lower limit is above, at, or below the horizontal position according as 2\cos0q^, =, or

100 CONDITIONS OF STABILITY Relation (a) gives the upper limit of oscillation, i.e. the original position at which the top was spun (see Art. 98). Relation (/3) gives the value of the azimuthal velocity -^ at the lower limit of oscillation. 101. To determine the least value of w which will enable a top to spin, if set down at an inclination 0^ to the vertical. We have seen that the top will first fall to 'a position 0^ and then rise to its original position ; but there must be some angle O2 at which the top fails to spin, either through the peg slipping, namely the constraint at 0 giving way, or through the side of the top touching the ground. Hence the limiting value 0-^ must be reached before this value Og* ^^^^ ^^e necessary condition is O2 ;> 0^, and therefore (1/^)2 >- (\fr\ ; for writing • _ (7a)(l -cos e -/>;)_ (7a)/ 1 k \ ^~ ^(l-cos^e) ~.4U + cosO l-cos^Oj' we see that since k is a positive constant \/r increases with 0. But W,^^"('^°;^.o7r'^^^ from (iii) Art. 98, and (^^)l = ^^ ^^^^ (i^) ^^^' 1^^' . Cw(cos Oq — cos O2) ^^ 2Mga or 0)2 ]> '2 2MgaA sin^O^ C\cos Oq — cos ^2) ' Example. Suppose the radii of gyration of the top are respectively f in. and 2 ins. about the axle and perpendicular axis through the point of support. Let the distance a between the centre of gravity and the toe be 2 ins.; let 0^ = 30° and ^2 = 60^ If 60 is the least angular velocity required ft)2> 2MgaA sin202 (72(cos^Q — cosOg) 252\cos30°-cos60 since (/ = 386 inch/sec'^; 4 (34 .-. w > -^ . V386 X >v/2'049 ^ 289 rad./sec. approx. .*. number of revolutions per sec. must be >47. Thus if the top is thrown from a string coiled round it in circles of average radius 1 in., the hand must be withdrawn from the top at a relative velocity of about 16 mi./hr.

CENTRE OF GRAVITY INITIALLY AT REST 101 102. To determine the least velocity at which a top will spin in a vertical position. In this case we have from equation (iii)^^^-^^' , CHI-cos^) ^ A sin2^ a (JO ^(1+COS0) If now we suppose Q so small that S^ may be neglected, we see that . rj \/r = ^r--r, a constant. Again, from equation (ii) Art. 98, the value of \/r, when 0 is stationary, is given by •,__2i%a(l-cos^) ^^ ^sin^e _ 2%a ~Z(T+cosT) Mga A ' neglecting squares of small quantities. Hence, in order that the top may return to the vertical position after'a small displacement, it is necessary that 0 V Mga 4^2 > ^ ' or 0V>4M^a^. When 0)2 is less than this value, the disturbed motion does not, as the disturbance is indefinitely diminished, tend to return to the vertical motion ; on the other hand, it does not necessarily depart very far from it. Both limits of oscillation in agreement with 0)2 might be near the vertical. The above conditions show that it is very difficult to spin a top for which C is small in comparison with A ; for instance it is almost impossible to spin a pencil on its point. ExaTYiple. Taking the top of the previous article we must ^^^^ _ 4^MgaA > 4 . 386 . 2 . 4 . 642 252 > 284*5 rad./sec. .'. the number of revolutions a second must be >► 45 approx., which is practically the same number as in the preceding case.

102 CHANGE OF SIGN OF AZIMUTHAL VELOCITY Fig. 65. 103. Change of sign of yjr. Let the axle OH (Fig. 55) be taken to represent the angular momentum Goo about itself, and HL the component due at any instant to the resolute of \jr. Then OL represents the total angular momentum about axes in the azimuthal plane ; and if LM is drawn perpendicular to OZ it follows that OM represents the constant angular momentum about the vertical, since 0 has no resolute ^V^sin^ about OZ. Hence L always lies on the fixed horizontal plane through M. This is sometimes called the invariable 'plane. It is clear that if OH is less than OM, HL can never vanish. Consequently \/r does not vanish. If OH is equal to OM, \j/ vanishes when H reaches M, i.e. when the top becomes vertical. If OH is greater than OM, \jr vanishes when H reaches the invariable plane, and changes sign as H rises higher, since HL is then drawn in the opposite direction. 104. Gyroscopic resistance of the momentum axis. The existence of the limiting values of 0 (Art. 99) can be seen from the following considerations. We have seen (Art. 85), that if M is the angular momentum at any instant about OL, of a body to which is being applied a torque about a perpendicular axis, then the resistance the momentum-axis offers to being turned by the torque is MQ, where Q is the precessional velocity already existing about the third axis perpendicular to both. Now (Fig. 55), as the momentum-axis OL is tilted, M increases and also (Art. 85) the precession Q ; consequently the resisting torque increases. Suppose the top is spun initially with its axle at an angle Oq to the vertical : the axle of the top is initially the momentum-axis. As the gravity torque tilts this, the resisting torque is increased. At first it is less than the gravity torque ; then a position is reached at which it is equal to the gravity torque, and at which steady motion would be set up if 6 were zero; and finally, as it increases and exceeds the gravity torque, a position, 0^, is reached where 6 has been reduced to zero. The axle then starts to retrace its path in the azimuthal plane. In the same way, as it retraces its path, a position is reached where the gravity torque is equal to the resisting torque, which is tending to restore the more vertical position, and steady motion would ensue but

GYROSCOPIC RESISTANCE 103 for 0 (now a negative quantity); and finally the position 0 — Oq is reached where 0 is again reduced to zero. Since the angle for steady motion lies between 0^ and 0^, and the tendency is always to return towards it, the position of steady motion must be stable. If there were a slight frictional couple at the toe to diminish w, the limiting positions both of rise and fall would gradually descend until some position 0^ for the lower limit were reached, at which the top either skidded at the point of support or touched the table with its side (see Art. 101). In practice the frictional forces which act on the top generally damp the oscillations so that they are hardly perceptible, and the top appears to descend steadily. See Chapters IV. and IX. 105. Value of the resisting torque at any instant. If x be the angle which OL (Fig. 55) makes at any given instant with the axle of the top, . Ayir sin 0 _ cos Oq — cos 0 sinO from equation (i). Art. 98. The resisting torque MQ = C(jo sec x^ sin (0 — x) = Co)\fr (sin 0 — cos 0 tan x) = Ccoyjr (sin 0 — cot 0 cos 0^ — cos 0) = -^^ (1 — cos Or. COS 6), sm 6^ u / or in terms of 0 only _ (7 V(cos Op - COS e)(l - cos Op cos 0) ~ A sin^O from equation (iii), Art. 98. From the value of tan ^ we obtain cos(0 — x) = cos OqCos xNow since 0 and xfr sin 0 increase together (Fig. 55), it follows that 0 and x increase together; consequently (0 — x) increases with 0. Hence both Ccosecx and ^/r sin (0 — x) have their maximum values at the position 0^, and their minimum values at Oq, showing that the resisting torque is a maximum at the lower limit of oscillation and a minimum at the upper limit. It should be noticed that if C is small compared with A the resisting torque is very small. Compare Art. 102. 106. Equation of motion in the azimuthal plane. It follows that the oscillations in the azimuthal plane are determined by the equation Ae = Mga sin 6 - ^""'^^"^ ^" " "T ^f^ "''' ^" '"' ^- ■ • jA. sm u

104 VALUE OF GYROSCOPIC RESISTANCE Multiplying both sides by 26 and integrating we get the first equation of Art. 99, while ^ ^ Choose,- cos e)_ ^ A sm- 6 Thus the motion of the axle in the azimuthal plane being known, and that of the azimuthal plane, the complete motion of the top is known in terms of 0. If Oq is zero, and 0 small, the above equation reduces to AO = MqasmO ; . o /, — -, Ae = Mgae-^0, neglecting 0'^ and higher powers, while V 4^2 ') >/. = g=co,say. (Art. 102.) The above equations show that the axle oscillates about the vertical with a simple harmonic motion of period — where ^2-'V 4J_2 and the condition for a real oscillation is the condition for stability obtained in Art. 102. Since H the head of the top (Fig. 56) describes a simple harmonic motion in ZR^ the azimuthal plane, while ZR^ rotates uniformly with angular velocity cd^, the velocity of H at any moment is known, and its path in space completely determined. Considering the component velocities of H in the directions HZ, HP, we see that, when viewed from a point in the axis of Z, H will appear to describe a series of loops as in Fig. 63 in periodic .. 27r time — . It will be noticed that the point N oscillates in the fixed plane 27r ZA in periodic time • (See Art. 141.) 6t)j + 0)2 107. Motion of the top when G has an initial velocity. The principles established in Art. 104 show that, whatever be the initial velocity of G, the top still oscillates between two limiting positions. These limits can be found analytically as in Art. 99, by eliminating x/r between the two equations for conservation of angular momentum and conservation of energy.

CENTRE OF GRAVITY INITIALLY IN MOTION 105 The condition that 6 should be zero gives a cubic in cos 0, having one real root between cosO=— 1 and cos0 = cosOo, and another real root between cos 0 = cos 0^ and cos (? = 1. The third root is inadmissible, being greater than unity. If D is the initial anoular momentum about the vertical, which remains constant throughout the motion, we have in this case ^V^sin2e + (7iocos^ = A or, if we write D = (7a)6 cos 0^ where h and 0^ are constants, Cwih cos Oq — cos 0) ^ A sin-0 Also tan ^ _Ayp-^mO _ h cos ^0 - cos Q ~ sin 0 Hence the resisting torque becomes CV(b cos Oq — cos 0)(1 — 6 cos Oq cos 0) ^sin^O and the oscillations in the azimuthal plane are given by ^ A sm-^O the integral of which equation gives the equation of energy. 108. Path in space of H, the head of the top. We can now discuss the general appearance of the path of H in space when viewed from the point Z vertically above the origin. The paths will vary according to the initial conditions of motion ; but in all cases, where oo is sufficiently large to prevent the top from falling altogether, the head will oscillate between the two positions R^, R^, in the azimuthal plane, while the azimuthal plane itself rotates about the fixed vertical. The actual velocity of H in space is that resulting from these two motions. Since \/r is a function of d only, it is the same for all angular positions of H equidistant from Z. It is also clear that H passes and returns through any one position in the azimuthal plane with the same velocity. We will first consider the case in which G has no initial velocity. Here the starting point 8 coincides with R^, the upper limit of H. In Fig. 58 0^ is an acute angle, in Fig. 59 it is obtuse. Since G (and therefore H) has no initial velocity, H must

106 PATH OF THE HEAD OF THE TOP in each case start in the invariable plane (Art. 103). In Fig. 60 Oq, the upper limit, is zero. Other cases can be represented in a similar manner. In Fig. 61, where Oq is acute, an initial velocity v has been given at S to the head of the top in the aziimithal "plane. At this f~^ S'.Ri ! FiC4. 58. Fig. 59. Fig. 60. position \lr is zero, and therefore it will change sign as H returns with velocity v through the same position in the azimuthal plane. The path consequently has a loop as shown. ■tRz I Fig. 62. Fig. 63. In Fig. 62 Oq is acute, and an initial velocity has been given in the direction of precession, causing the top to rise, though not necessarily, to a vertical position. In Fig. 63 Oq is zero, but again an initial velocity v has been given to H in any direction. H therefore returns through its starting point with the same velocity, thus describing a series of loops. 109. Motion of any solid of revolution spinning on a smooth horizontal plane. Considering the case where G has no initial motion, we see that since the plane is smooth, the spin w of the body about its axle remains constant, and G only moves in a vertical line. Let B be the inclination to the vertical at any time of the axle of the body, GN = z, GL = x,PL = y (Fig. 64). We can consider the whole motion of the body as comprised independently of the translational motion of G and the rotational motion about axes through G.

MOTION OF SOLID OF REVOLUTION 107 As before the angular momentum about GC—Gco, „ GA=A\ir sin 0. If the body is put down initially spinning with its axle at an angle Oq to the vertical, and G has no initial velocity, from conservation of angular momentum about the vertical we get A \j/ sin^ O + Goo cos 0 = Gco cos Oq , G(io(cosOq — cosO) or vA sin^O as before (i) Fio. 64. In considering the conservation of energy we must now include the energy of translation. Let Zq be the height of the centre of gravity above the plane originally : we have then Ae^-\-Ayjr^sm^e + Mz^ = 2Mg{zQ-z) (ii) These two equations determine the whole motion, when combined with the geometrical relation z = xcos 0 + y sin 0, X and y being functions of 0, depending on the nature of the meridian curve. By reasoning exactly similar to that of Art. 104, it can be shown that GG will oscillate between two positions Oq, 0-^^, while G consequently oscillates between two positions Gq, G^. At the limits of oscillation i = 0, 0 = 0, and therefore from (ii), Ayjr'^sin^e = 2Mg{zQ-z), whence from (i), G(jo\j^ {cos Oq — cos 0) = 2Mg(zQ — z).

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108 MOTION OF SOLID OF REVOLUTION 110. By the method of gyroscopic resistance. The gyroscopic resistance of the momentum-axis will in this case be the same as in Art. 105. Writing this expression as K, the oscillations in the azimuthal plane are given by where PN=p, and R is given by the equation Mz = R-Mg. As before we have • _ Co)(coii Oo — cos 0) ^~ A sin2 0 •

CHAPTER VIII. MOVING AXES. 111. We will now discuss more fully the motion of a body referred to moving axes, that is, to a "frame of reference" (Art. 81) which is moving in any given manner. The motion of the body relative to the frame when combined with the motion of the frame itself will entirely determine the motion of the body. 112. We will first consider the motion of a point in a plane referred to two moving axes. If P be a point x, y referred to two fixed rectangular axes OX, OY, then the entire motion of P is determined if we know X, y, and consequently dx dy , d^x d^y dt' dt dt^' dt^' or in the fluxional notation, d;, i) and x, y. Now suppose that, 0 remaining fixed, the axes turn about 0 but still remain rectangular (Fig. 65). The motion of the point P is now entirely determined when we know x, y (and therefore X, y and x, y) relative to the moving frame of reference, or moving axes, and the motion of the axes themselves. This latter motion is of course entirely rotational since 0 remains fixed ; and although OX, 0 Y are revolving it must be remembered that at any instant they coincide with two definite, fixed, directions in space. We shall proceed to determine expressions for the total velocities and accelerations of the point P in these two definite directions fixed in space. 113. Linear velocities in two dimensions. Method I. Let OX and 0 Y represent the positions of the axes at the time t, the instant in question, and let OX' OY' represent their positions at the time t + St. Let x(==ON) and y{ = PN) mark the position relative to OX, OY oi the moving point P at the time t, and x-\-Sx{ = ON'),

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110 MOVING AXES y-{-Sy{ — P'N') the position relative to the axes at the time t-\-St when they have arrived at the positions OX', 0Y\ Then if u and v are the component velocities at the time t in the fixed directions in space OX, 0 Y we have ^j^^OL^ON St Fig. 65. But 0Z = projection on OX of the broken line ON'P' = (a; + Sx) cos Se-{y-\- Sy) sin SO = x-\-Sx — ySO, neglecting infinitesimals of the second order. Also ON=^x. Sx-ySe Hence, ic = St dx dO . ,, ,. ., or x — yB. ^. .. , T^ projection on OY of ON'F-PN Similarly, v = hi - — — 5^ = 0 ot ^Ay^-Sy)(io^Se^-{x + Sx)^mSe-y =^^ St ~" _dy do dt df or yi-xO. [30.] Test the dimensions of the above expressions. 114. Method II. These expressions may also be obtained in the following way : '?^ = velocity of P in direction OX = velocity of P relative to iV in this direction + that of JSf — —yO+x.

VELOCITIES IN TWO DIMENSIONS 111 And similarly, V = velocity of P in direction 0 Y = velocity of P relative to N in this direction + that of N It will thus be seen that the total velocities of P in the directions OX, 0 Y are not merely -r. and -j^, but depend also on 6. Cor, If y = 0, X and 0 become the polar coordinates of the point P. Hence, with the more usual notation, we see that the radial velocity is r, and the transversal velocity is rO. 115. Accelerations. We have above found u and v, the rates of change, in definite directions, of x and y, subject to certain conditions. To find the total accelerations in the same directions we must find the rates of change of u and v in these directions, subject to the same conditions. Method I. The component of velocity in the direction OX after time St is (u + Su) cos SO — (v-\- Sv) sin SO. Hence the acceleration in direction OX _ (u-{- Su) cos SO — (v-\- Sv) sin SO — u 8t=o St = u — vO. Similarly, acceleration in direction OY = v-^iiO. 116. Method II. These expressions may also be obtained as follows : The velocities of the moving point P at any instant in the directions OX, OY, considered fixed in space, can be represented by u = ON md v = QN, Fig. 66. but it must be remembered that Q does not represent the position of the moving point.

112 MOVING AXES The required accelerations will be the rates of change of u and V as the axes revolve, and as in Method II. of the previous article we found the rates of change of x and y to be x — yO and y-{-xO, so now the rates of change of u and v will be it — vO and v-{-uO. Cor. It follows as in Art. 114 that, when polar coordinates are employed, the radial acceleration is r — rO'^, ■ 1 d ■ and the transversal acceleration is rO + 2r0 or - • -y- (r'^0). r at ^ ^ These results can also be obtained from first principles by a method similar to that employed in Art. 113. 117. It should be particularly noticed that in the previous articles the velocities and accelerations found are total velocities and accelerations — not relative to the moving frame of reference. Similar expressions can be obtained for a point referred to axes moving in three dimensional space, and for this purpose Method 11. will be found most convenient. 118. Let X, y, z be the coordinates of the point P referred to three rectangular axes OX, 0 Y, OZ (Fig. 67) which form a " frame of reference," while this frame turns with component angular velocities 0^, ^2' ^3 ^'oout the lines OX, OY, OZ — or Fig 67. rather about the lines fixed in space with which OX, OY, OZ happen at the instant under consideration to be coinciding. We will consider that the point 0 remains fixed since we know that if we are discussing the motion of a rigid body and take our origin at the centre of gravity G of the body, then, as

ACCELERATIONS 113 far as rotation about G is concerned, G is equivalent to a fixed point. We will also adopt the following convention for the algebraic signs of rotation about OX, OY, OZ: Rotation about OX is positive in the direction Y to Z, „ OY „ „ „ Z to X, OZ „ „ ., X to Y, namely, the positive direction is that of a left-handed screw as seen from 0 when looking along the axis ; or, is determined by taking the cyclic change of the letters X, Y, Z. 119. Linear velocities in three dimensions. If u, v, %v are the (total) component velocities in the directions of the axes, we have (Fig. 67) '?(, = component velocity of P relative to iT+that of K relative to ^+that of iV relative to 0, and .•. u=^:c—if^^-\-zQ^, and, similarly, by considering relative volocities, nj=zz — x0.2 + y0i. It should be noticed that the dimensions of the expressions are correct. 120. Accelerations. Again, if we take OR, RL, LP (Fig. 67) to represent these velocities u, v, iv, the rates of change of ti, V, w, i.e. the total accelerations, along OX, OY, OZ are v — wO-^-huO^, w — tiO^ + vO^. 121. Angular velocities. Since angular velocity and angular momentum are vector quantities, the foregoing results are also applicable to them. Let (0^, 0^2, 0)3 be the (total) angular velocities of a rigid body about the axes OX, OY, OZ; that is, about the lines fixed in space with which OX, OY, OZ happen to be coinciding at the instant under consideration : it follows that the rates of change of the angular velocities about the moving axes are COi — (O.2O3 -f W3O2 , 0)3 — ^1^2 ~^~ ^2^1 • 122. Angular momentum. If 0 is a fixed point in a body referred to the three moving axes OX, OY, OZ, and h-^, \, h^ are at any instant the components of the angular momentum H

lU EULER'S EQUATIONS of the body about these axes, then the rates of change of the angular momenta are given by 123. General equations of motion of a body having one point 0 fixed. Since the rate of change of angular momentum about an axis is equal to the moment of the external couple about that axis, it follows that if K^, K^, K^ be the external couples acting about OX, OY, OZ, we get the following equations for determining the motion of a body : ^^2 - Ml + Ms = ^^2' 7^3 - 74^2 + Ml = ^3The equations of motion in the form thus obtained were first given by Mr. R. B. Hayward, F.R.S., of St. John's College, Cambridge. They are contained in a paper, published in 1856, in Part I. Vol. X. of the Cambridge Philosophical Transactions. 124. Euler's dynamical equations. In the case of the motion of a single rigid body about a fixed point, or about its centre of gravity, if we take for our moving axes three straight lines fixed in the body, and passing through the fixed point, or through the centre of gravity, we shall have Ql = ^\> ^2 = ^2^ ^3 = 00 3If, further, we take these three lines to be tl:ke principal axes at the fixed point, h^ = ^ c-j-j^ , h^ = Buo^ , 7^3 = Cojg , and the above equations reduce to Awi — (B — C)(jo^o)^ — K^ , B(h.2'-{G—A)w^(t)^ — K2, (7«?3 — {A—B) (jo^oo^ = K^ , which are Euler's equations. 125. General equations of motion of a body deduced from gyroscopic resistance. We have seen in Art. 37 that if angular momentum /oj about OX is precessing about OZ with velocity Q under the action of a torque K about OY, then the gyroscopic resistance offered by the body to the torque K is measured by lijoQ,. Further, it will be remembered (Art. 38) that the direction of precession is determined by the angular momentum setting itself toivards the torque-axis when both are drawn in the same sense.

GYROSCOPIC RESISTANCE 115 In Fig. 68 let us suppose the system to be in motion under the action of external couples K^, K^, K^ about OX, OY, OZ respectively. It is clear that the angular momentum h^ is precessing about OY with velocity 0^, and OZ 0 Similarly for the angular mo3menta h^ and h^. It follows that when the external couple K^ acts about OY the resistance offered is 0. y

116 STEADY MOTION Referring to the equations of Art. 123, we see that the first vanishes, and the second one becomes i.e. GwQ sin a — ^ 1^2 sin aQ cos a = Mga sin a or GmQ — A Q^ cos a = Mga, the equation for steady motion already obtained. The third equation also vanishes. We can in the same way obtain all the equations of steady motion established in Chapter VI. Fig. 69. Fig. 70. 128. General motion of a top on a fine fixed point. If the motion is not steady, taking axes as in Fig. 70, we have 6^= —yfr sin 0; w^= —xfr sin 0 ; ^2 = 0 5 002 = 6 ', 0^ = \lr COS 6; Wg = (0. The equations of motion therefore are Ao)i — Aw^y^r cos 0 H- Gwjd = 0, J. ft)2 + Cwgx/r sin d-\-A w^yjr cos 0 = Mga sin 6, Gw^ — Aco^O — Aoo^yjr sin 0 = 0. Substituting the values of cdj and Wg in the third equation we see (7w3 = 0, i.e. Wg is constant, a result which it will be seen depends on the moments of inertia of the top being the same about axes (1) and (2), i.e. on the symmetry of the top (see Art. 97, footnote).

GENERAL MOTION 117 The other two equations become -A/r sin 0, and (ii) by 20, subtracting (i) from (ii) and integrating A\lrHin^e-\-Ae'^ = E-2Mgacos 0, which is the equation of energy. It will be noticed that of the three equations of motion that about axis (1) gives the condition for constancy of angular momentum about the vertical, that about axis (2) gives the oscillations in the azimuthal plane, and that about axis (3) the constancy of the spin of the top about its axle. In a similar way the equations may be obtained for the general motion of a solid of revolution spinning on a smooth horizontal plane ; but the shortest method is that of writing down the conditions for conservation of angular momentum and conservation of energy. 129. The following is typical of the more advanced problems to be found among the Miscellaneous Examples at the end of Chapter IX. A rough horizontal plane is made to rotate about a fixed vertical axis with constant angular velocity, and the centre of a sphere lying at rest at the point where the axis meets the plane is set in rnotion with a given horizontal velocity. Show that the path of the centre in space is a circle, described with uniform velocity. Let P be the point of contact at any instant of the sphere with the plane, 0 the point where the vertical axis OZ meets the plane. Taking OPX as axis (1), OZ as axis (3), and OY perpendicular to these as axis (2), we have Oi = 0, ^2 = ^' ^^^ ^3 = ^' where 6 is the angle OP makes with some horizontal line fixed in space. Let F-^, F^ be the frictional forces at the point P in directions (1) and (2), /i,/2 the accelerations of the centre of gravity G in these directions, co^, 0)2, Wg the angular velocities of the sphere, m and a the mass and radius respectively. Taking moments about axes through G parallel to (1) and (2), we have %ma^a)i — fmd^o).20 = F^.a = maf^, (i) and fma^(i)2 + 1 ma^co^O = —F^a= — maf^ (ii)

118 ROLLING SPHERE Also if u and v are the velocities of Q in the directions (1) and (2). by Art. 115 f^ = u — v6; f^=v^u6. And, since there is no slipping, the point P of the sphere has the same velocity as the point P of the plane, whence a(jt)2 — u = 0, i.e. ao)2 = u, and aot)-^^-\-v = xQ, i.e. a(ii^ = xil — v, where Q is the constant angular velocity of the plane. Substituting in equations (i) and (ii) we obtain and ^u — ^v6= —^xQO, showing that /2 = ficr2, and f-^^= —fxQO. Taking r, 0 as the polar coordinates of the point P, these equations may be written ^|(r20) = K2r, (iii) and r-re^= -fQrO (iv) Integrating (iii) we obtain r^O = Y r^ + a constant. Since 0 = 0 when r = 0 the constant is zero and Q2 Substituting in (iv) f _|- /^ = Q. Since -t.=1j^ ^^^ above may be written whence the path of P in space is given by r = ccos(0 + a), where c and a are constants, showing that it is a circle. Taking the sum of the squares of r and rO derived from this equation, we see that the velocity is constant and equal to that of projection.

CHAPTER IX. STABILITY OF ROTATION. PERIODS OF OSCILLATION. 130. We shall now consider some cases where a spinning body receives a slight displacement, discussing the conditions under which it will revert to the original motion when the small disturbing force is withdrawn, and the periodic time in which it will oscillate about its original position. 131. Proposition. // OA, OB, 00, are the three principal axes (through a fixed point 0) of a body ivhose rnoments of inertia about these axes are A, B, 0 {either in ascending or descending order of magnitude), then, for small displacements, rotation either about OA or 00 is stable, but about OB unstable. For, considering that the disturbing force is instantly withdrawn, we have, by Euler's equations, since no forces act on the body, ^«,^-(5-6>,

120 STABILITY OF ROTATION which is a real quantity, and the motion is stable. Similarly if the spin Q is about the axis of Wg ; but if it is about the axis of 0)2 , the period becomes 27r / GA ym^ u yi{B-A)(B-cy which is imaginary. Hence the motion is stable for rotation about OA or 00 but unstable about OB. 132. Diabolo which will not spin. The reader will remember that when he is beginning to spin a Diabolo spool by means of the string, unless the two portions of the string are absolutely in the savie i^lane and that at right angles to the axle of the sjpool, then, besides the spin about the axle which he means to communicate, there is also a residuum " wobble " about a transverse axis ; but as he continues his spinning this wobble begins to disappear and the spool settles down to a steady rotation. The reason for this is that the axle of the spool is in general an axis of minimum moment of inertia (though if it is one of maximum moment the same argument applies) and the instantaneous axis tends to revert to the position of the axle of the spool. If we construct a spool which is dynamically equivalent to a sphere we shall find that it is impossible to spin it ; for, since the moments of inertia about all axes are equal, the spool has no tendency to revert to its axle as instantaneous axis of rotation. The result is that the axle wanders about indefinitely in space, though with considerable rapidity. If this Diabolo were already spinning about its axle, it would possess a certain degree of stability ; for, the spin about the axle being large compared to that generated by any disturbingforce about another axis, the instantaneous axis would deviate only to a very slight degree from the axle of the spool. "'^ (See Art. 63, footnote.) A heavy conical sheet projecting equally on either side of the vertex whose semi-vertical angle is equal to tan~^/s/2 (Fig. 71),, has the dynamical properties of a sphere ; but such an ideal construction is impracticable. Since, however, the addition of matter inside the angles AOR, A' OB would increase the moment of inertia about the axis of the cone, while the addition of matter inside the angles AOB, A' OB' would increase the moment of inertia about a transverse axis, we can on such a skeleton cone form a material double hollow cone, in the form of a Diabolo spool which is dynamically equivalent to a sphere ; and this Diabolo we shall find almost impossible to spin. * It is for this reason that in the game of Cup and Ball, when it is required to jerk the ball so that the axial hole catches on the wooden peg, it is necessar} to first give the ball a spin about the axis containing the hole and the point where the string is fastened to the ball ; for then a lateral jerk of the string does not appreciably displace the axis of rotation.

DIABOLO 121 At a meeting of the Physical Society of London, in November, 1907, Mr. C. V. Boys exhibited some interesting experiments with a wooden spool of this description. His spool was pierced with an axial hole, in which condition the moment of inertia was greater about the axle than about a transverse axis. The spool was found easy to spin, and displayed considerable stability. Fig. 71. Fig. 72. The lecturer then inserted through the hole (Fig. 72), a small stick of such a length as to make the two moments of inertia exactly equal, and by attempting to spin the spool illustrated very clearly its marked instability of rotation about any axis whatever, and the impossibility of spinning it. On the same stick being used but pushed slightly from its central position so as to project rather more at one end than the other, the spool again was spun without difficulty, and displayed distinct stability, but at once began to precess owing to the gravity couple called into play by the displacement of the centre of mass. A similar stability of rotation was observed when a longer stick was inserted, causing the moment of inertia to be greater about a transverse axis than about the axis of the cone. It is interesting to notice that in equalising the moments of inertia the adjustment must be very exact before spinning becomes impossible. If a torsion wire is applied as a test it is found that when the difference in periods of oscillation is as much as 1 in 35, spinning is quite easy. The periods should not differ more than one per cent. For several of the above details the author is indebted to the Proceedings of the above-mentioned meeting, published by the Society. 133. A problem of stability relating to Schlick's method of steadying vessels. A fly-ivheel is synfimetrically mounted in a spherical case to which is fixed an axle jperioendicular to the axis of the flywheel. The ends of this axle are mounted inside a holloiv

122 A SIMPLIFIED CASE OF circular cylinder so that the direction of the axle intersects and is 'perpendicular to the axis of the cylinder. The cylinder lies on a rough table and the fly-wheel is set spinning with its axis vertical. Shoiv that if a mass m he attached to the highest point of the cylinder, the system is stable, whatever be the mass of the cylinder, provided the square of the angular velocity of the fly-wheel exceeds mgaAjC'^, ivhere a is the outer radius of the cylinder, A the w^oment of inertia of the case and fiy -wheel about the axle, and G that of the fly-ivheel about its axis. Camb. Math. Tripos, 1908. (2nd Problem Paper.) Let ZOX (Fig. 73) represent the vertical plane through the centre of the fly-wheel and the particle P, where the angle ZOP = e. After any time t let Oc be the projection on this plane of the axle OC of the flywheel, which is rotating in the direction marked, and has precessed through an angle cp towards the reader. If OB is perpendicular to 00 and OA, then OB, originally coincident with the axis of the cylinder, now makes an angle 0 with it. Taking OA, OB, OG as moving axes, and considering the motion of the fly-wheel and case, we have co^= —

SCHLICK'S PROBLEM OF STABILITY 123 or, neglecting squares of small quantities, and combining with (i) Be + ^e-K=0 (ii) Now, considering the motion of the cylinder, and taking moments about the generating line in contact with the table (moment of inertia / for cylinder and particle) J0 = TYiga cos 0 . sin 6 — K, whence from (ii) we have, to the first order, so that the condition for stability is mgaA and the period of a small oscillation is (B+I) 134. Oscillations of a spinning top. Case 1. To find the period of a small oscillation of a spinning top about the position of steady motion where the axle is inclined at an angle a to the vertical. With the preceding notation we have, from conservation of angular momentum, (7ft)(cos a — cos 0) = Ayp- sin^O — AQ sin^a (i) Writing down the equations of motion about the torque axis we have, including gyroscopic resistances, AG = sin e(Mga-G(joxfr-{-A\ir^ cos 0), (ii) while Mga-'GMQ + A^'^cosa = 0 (iii) If 0 becomes a + e and consequently yp- becomes Q + x where € and X ^^© small, we have sin 0 = sin(a + e) = sin a + cos a . e, cos 0 = cos(a + e) = cos a — sin a . e, neglecting squares of small quantities. Hence, from (i), Cco(cos a — cos 0) = Co) sin a . e, and Ayfr sin^O — ^Q sin^a = A (Q + x)(sin^a + 2 sin a cos a.e)- AQ sin^a = 2AQ sinacosa . e + ^x'^^^^^ > .'. J.xsina = ((7a) — 2J.r2cosa)e Mqa — AQ'^ cos a ^ ,..., = —^ ^ . e, irom (iii).

124 OSCILLATIONS Making the same substitution in (ii) we have A'€ = [sin a + cos a . e] [Mga — Cft)(ll-f x) + A (Q2 + 2Qx)(cos a - sin a . e)] = —X sin a(Oa) — 2AQ cos a) — AQ^ sin% . €, employing (iii), HMga-AQ^ cos af , ,.,, • 9 1 r^ ^^2^ - 2MgaA Q^ cos a + MY(^^ _r. ^.6. e 4.*. the period is — , where p = -j^ V^^i^* - ^MgaAiV- cos a + M^a^. Since the quantity under the radical can be written (AQ^-Mga cos af-\-(Mga sin a)^, it is always positive, and we see that the position of steady motion is stable, a result already established in Art. 104. It should be noticed that the node P (Fig. 74) travels uniformly round the circle of steady motion with angular \ J velocity Q ; the periodic time is -77- = -^ .*. the length of PR 2irAQ^ 27r V 2irQ, P Fig. 74. JA^Q^ - 2MgaAQ^ cos a + Mya^' These oscillations have already been mentioned in Chapter IV.,. where they were accounted for by the inertia of the top alternately resisting and hurrying the precession. Where the disturbing force is being continually increased, as for instance when a top is gradually sinking, these oscillations are not regular but instantaneous only about the instantaneous position of steady motion. 135. Case 2. The sleeping top. Definition. A top is said to be asleep when the whole motion of its axle is in the neighbourhood of the vertical, but not necessarily passing through it. If a small displacement be given to a sleeping top, two independent vibrations are set up, namely the oscillations of 0 and \fr about their mean values ; but since i/r is a function of 0 only, it follows that the periods of these two oscillations are

THE SLEEPING TOP 125 the same and the mean values occur simultaneously. The remaining Articles will be devoted to the discussion of this period and the values of 0, \Ja and yfr at any time during the motion. 136. To find the value of 6 at any instant. The equations of conservation of angular momentum and energy give (Art. 107), A-yjr sin^ 0 + Coo cos 0 = Cwb cos 0^, (i) and A-ylr^ sin^ e-\- AO^ -{-2Mga cos 6 = E. (ii) Putting w~A~\' ^^^ 6 = 1 H — ^ . Oq^ where ^ is a constant, the former becomes A\jrO'^ = -^ {&^-^pO^) neglecting cubes and higher powers of 0, i.e. ^ = \^(l+^y (iii) and the initial value oi \[r = \(l+^). Hence we see that /3 is the ratio of the initial excess of \jr over \ to Xj. It should be noticed that if Oq is zero, \^ = Xi, and is therefore constant, as was shown in Art. 102. Combining (ii) with (iii) we obtain reducing to 0^ + X^^O'^ + ^^^^ = /w, a constant, where X2^ = A^^ j-, and the condition for X^ being real is the condition for stability obtained in Art. 106. If 0^ = x, this becomes ^-{-\2^x^ + l3%%'-'^x = 0, (iv) whence 2\d-^2S=^ ^^ or e^ = x = K'^ h' cos (2X2^^ + 2^), or (^jj = K+kcos{2\t^2S\ ^ ^""^ where K, k, and S are constants. (Williamson, Integral Calculus, p. 12.) It follows that the period of oscillation of 0 and yjr about their mean values is — . Xo

126 THE SLEEPING TOP 137. To determine the constants K, k, and S in terms of Oq and 6q. From equation (v) we have i' = - 2X2^^0^ sin (2X2^ + 2(5) : ,. J + X^2(^_^^^2)2^X22/.W. Comparing this with equation (iv) with which it must be identical we get, by equating the constant terms, {K^-lc')\i = ^''\^\ (vi) showing that K is always numerically >• k. B^\ 2 WritinsT ^ } = y^ X2^ we have K^-k^ = y\ (vii) and from (v) iT+Z? cos 2^=1, (viii) A while /c sin 2^ = — ^^ = a say (ix) C^oA.2 These three equations determine K, k, and 2S. From (viii) and (ix) k^ = {l-Kf-\-a'^; .'. using (vii) K^^(l-Kf-a'" = y' 2 .. ^ 2 Also tan 2^ = l-a2_y2 _^l-^2_y2x2 and A;^ = (^^ ""^ ^ j +«^. From the last two equations we see that k may be either positive or negative, and also that S can have two values. If k is taken as positive, we see from equation (ix) that S must have that value which makes sin 26 of the same sign as a. Hence there is no ambiguity. From equation (v) we see that the greatest and least values of (^j are K+k and K—k, and from (vi) the latter limit is always positive ; hence 0 never changes sign but varies between a^ and ctg' where a^, a^ are O^jK — k and O^jK + k respectively. 138. Particular case when K' = k\ It should be remarked that this condition does not necessarily involve K = k, since Oq might be zero, which proves to be the case.

THE SLEEPING TOP 127 If K' = Jc, and only in this case, equation (v) becomes and the motion is simple harmonic. In this case, since K''^ — k'^ = y'^Oq^, we must have either ^q = 0 or y = 0, which gives /3 = 0. ri The latter condition gives, from (iii), '^ = \ = ^-7Hence if K' = Jc the top is either initially vertical or is so displaced that the azimuthal velocity communicated to it is \. The first of these two cases has already been considered in Art. 106 ; the second is clearly of the type discussed in Art. 134, a in this case being small. The path of the head H is Sb simple wave of period ^--. In Art. 140 it is shown that the path can in general be produced by the combined effect of two waves whose periods are ^; — r^— , but it can be drawn without reference to A;^ ± A.2 these considerations b}^ the method employed in the following Article. 139. The path in space of the head of the top as seen from above. We have seen (Art. 137) that 0 never becomes zero, but oscillates between two limits a^ and a^. If a graph be drawn such that the ordinates represent 0^ and the abscissae the time, then equation (v) shows that the re• TT suiting curve is a simple wave of period — . If now we draw a second graph such that each ordinate is the square root of the corresponding one in the first curve, the new graph will give the values of 0 for all time. This second curve is clearly flatter than the first, but repeats after the same period. Now the value of xf/-, if described with the mean value of i//-, is proportional to the time ; hence we may also regard this curve as representing 0 in terms of the angular distance described with the mean value yjr, and we are thus enabled to realise the general appearance of the path in space of the head H of the top. The dotted curve sketched in Fig. 75 shows the path described on the assumption that xlr has its mean value throughout, while the continuous curve gives the actual path. The following considerations will show how they are connected.

128 THE SLEEPING TOP on 2\ Since '\fr = \-^^ ^^ ^, we see that when 0 has its mean value, -v/r also has its mean value ; but as 6 increases i/r diminishes, and vice versa. Let us suppose that the head H is at M when both 6 and \fr have their mean values. This will again occur at M^, M^, etc., namely after successive periods of time ^— • / M f \ Ti • • 1 4 •1 I : ) \ • % • \ \ \ -.. .,-v^ Fig. 75. Now as Q increases beyond its mean value \j^ diminishes ; hence as H moves from M to a^ , the outer limit of 0, the angular distance described is less than that given by the dotted curve for corresponding values of 0. From a/r; .-. ^ = 0 cos \j^ — Oxir sin yj/-. rj = sin 0 sin yp- = 0 sin \/r ; >; = 0 sin xfy + Oxfr cos \fr.

THE SLEEPING TOP 129 With the usual notation, let the axes (1), (2), (3) be drawn so that OZ and (3) are nearly coincident, and the angle between OX and (1) is \jy, that between OY and (1) being f -^ — ^j. The angular velocities about (1), (2), (3) are — i^sin 0, Oy o) respectively. The angular momenta about (1), (2), (3) are — A\frs\nS, A6, C(o respectively, or -A\f^e,AO,Gco. The angular momentum about OX = (-Axj^O) cos \fr-\-Aecos(^-i-\l^)-\- Ceo. i \2 = -A^-\-Gw^. Similarly about OY = A^+Cwri. The torques about these axes are — Magt], Mag^ respectively. Hence, since the rate of change ol* momentum is equal to the torque, and OX, OF are fixed in space, — A ij 4- Co)^ = — Magt], Ai'-\-Cwr] = Mag^; whence -^^2X,;j-(\^-\^)i=0,\ and rj-2\^-{\^-X^^)f] = 0.j The equations can both be satisfied by putting ^=rcos(A^H-e) and >; = rsin(X#-fe), provided that X^ _ 2\^\ + x^2 _ ;^^2 ^ q, i.e. (X-\f = \,\ or \ = \±\2> which shows that the general solution of ^ and rj is ^ = r^ cos { X^ + X2 • i^ + ei } + rg {cos Ai — Xg . ^ + 62} , >; = riSin{Xi + X2. ^H-eJ +r2{sinXi — Xg. ^ + €2}Hence the motion is the combination of two oscillations of 27r periods \±\ 141. Geometrical interpretation of the two oscillations. The coordinates of the projection of H on the plane XY are proportional to ^ and r]. Referring now to Fig. 56, p. 104, and substituting X^, X2 for Wj and C02 respectively, we see that P describes a circle uniformly with velocity Xi + X2, while the image of P in the line ZR^ 1

130 THE SLEEPING TOP describes a circle with uniform velocity X^ — Xg. Moreover ZN is ^Pcos(Xi + X2)^ and PN is -^P sin (Xj + Xg)?^, according as we consider the point P or its image. Hence if we take two circles of radius 2i\, 2r^, described with uniform velocities (Xj + Xg) and (X^ — X2) respectively, by particles P^, P^, whose phases differ by e^ — 63, the coordinates of H, the middle point of P^P 2y referred to fixed axes suitably chosen, are the values found above for ^ and r] respectively. Hence the locus of H gives the projection on the plane XF of the path of the head of the top in space. 142. To determine the value of ^/^ at any instant. Combining equations (iii) and (iv) of Art. 135 and writing {2\t-\-2S)==X' we have ^ ^ K-\-k COS X .-. ^H-X = Xi^ + /3xJ "^^ K + k cos X ~^^^"^2xJa^+/:c. 12 J il -f- HI cos X -Xt + ^- ^ tan- r \ ^ + Z = Xi^ + tan-M^cos(X2^ + (^)[. The constant L can be easily determined from the initial conditions. MISCELLANEOUS EXAMPLES. 1. A fly-wheel is mounted on an axle so that its plane makes an angle a with the axle : show that there is a couple on the bearings of amount (C- A)^^sm aco^a where 12 is the angular velocity about the axle. 2. A conical shell of angle a is made to rotate with uniform angular velocity 12 about a generator, which is vertical. Show that a perfectly rough sphere, of radius a and radius of gyration about a diameter k, can remain in contact with the inner surface of the shell, always at the same point, in a state of steady motion, if its angular velocity about the common normal be / o\ ,, ft , ^ \ o • alt f^ . ^ aa I 1 + T2 ) " sm a - ^2" • 12 sin a tan a + j^ , where R is the distance of the point of contact from the vertex. (Coll. Exam.) 3. A rough inclined plane is made to rotate about a normal axis with constant angular velocity, and a sphere is given any spin and placed upon it. Show that, whatever be the inclination of the plane, the angular velocity of the sphere about the normal to the plane through the point of contact remains constant.

MISCELLANEOUS EXAMPLES 131 4. A heterogeneous sphere has its centre of gravity 6r at a distance c from the centre 0 of the sphere, and the radius of gyration about any line through G is R. The sphere is placed on a fixed smooth horizontal plane, spinning with angular velocity 12 about a radius inclined at an angle a to the vertical ; and at the instant of release the axis OG is in the vertical plane containing the axis of rotation and makes an angle /? with that radius. Show that G describes a horizontal straight line if B^Q^ sin a sin /? —gc . sin^ a + /?. 5. A shell in the form of a prolate spheroid whose centre of gravity is at its centre contains a symmetrical gyrostat, which rotates with angular velocity w about its axis, and whose centre and axis coincide with those of the spheroid. Show that in the steady motion of the spheroid on a perfectly rough horizontal plane when its centre describes a circle of radius c with angular velocity 12, the inclination a of the axis to the vertical is given by {Mhc{a cot a + b) — Ah cos a + C{a sin a + c)] 12^ + C'bioQ — Mgh {a — h cot a) = 0, where M is the mass of the shell and the gyrostat, A the moment of inertia of the shell and the gyrostat together about a line through their centre perpendicular to their axis ; (7, C those of the shell and gyrostat respectively about the axis, a the distance measured parallel to the axis of the point of contact of the shell and plane from the centre, and h its distance from the axis. (Camb. Math. Tripos.) 6. A sphere whose diameter is equal to the difference of the radii of two spherical shells is placed between them, while each shell is made to rotate with uniform angular velocity about some fixed axis through their common centre, though the angular velocity and axis need not be the same in each case. Show that the centre of the sphere will describe a circle uniformly, provided there is no slipping. 7. A rough inclined plane is driven round a vertical axis with constant angular velocity, and a sphere is placed on it. Show that, if the plane is inclined to the vertical at an angle greater than about 18°, steady motion is possible with the centre of the sphere describing a circle in space. 8. A rough horizontal disc can turn about an axis perpendicular to its plane, and a right circular cone, vertical angle a, rests on the disc with its vertex at the axis. If the disc be made to rotate with angular velocity 12, show that the cone takes up an amount of kinetic energy equal to 1 ^2 //cos^a sin^a 2 / V .4 C C and A being respectively the moments of inertia of the cone about its axis and a line perpendicular to the axis through the vertex. 9. A uniform solid right circular cone of vertical angle 2^ is placed on a rough inclined plane of slope a, so that the generator in contact is horizontal. Prove that the cone will always be in contact with the plane, and will oscillate through two right angles, if l+3sin2/5 tana< 9 sin f^ cos (S ' 10. A homogeneous right circular cone, of which the c.g. is at the distance h from the vertex and the semi-vertical angle is a, rolls on a rough inclined plane, starting from rest when it touches the plane along a horizontal line. Prove that, when the generator in contact makes an angle yjr with the horizontal lines in the plane, I^'^ — 2mgh sin^ a sec a sin y sin i//-, I2

132 MISCELLANEOUS EXAMPLES and that the point of action of the resultant normal reaction between the plane and the cone is at a distance A{cos a + 3 sin a tan y sin i//-} from the vertex ; where / is the moment of inertia about a generator, and y is the inclination of the plane to the horizontal. (Coll. Exam.) 11. A solid uniform prolate spheroid whose axes are 2a, 26, 26, spins steadily on a smooth horizontal table. It has angular velocity n about its axis of figure, that axis has angular velocity w about the vertical, and h is the constant height of the centre above the table. Show that and that, if 7i has its least value, (O 2_ ^-^P • f . (Camb. Math. Tripos.) a^ + b^ h 12. A uniform solid sphere of radius c rolls under gravity in contact with a perfectly rough elliptic wire of semi-axes a and 6, whose plane is horizontal : the centre of the sphere moving in a vertical plane through the major axis of the ellipse. Prove that if w be the angular velocity of the sphere when its centre is at a height z above the major axis, 2_ 2gb^{h-z) "^ b^(bz' + 2c') + 5 (^2 + c2 - 62)(a2 - 62)3' the value of z when a)=0 being h, and c > b. 13. A circular disc has a thin rod pushed through its centre perpendicular to its plane, the length of the rod being equal to the radius of the disc. Prove that the system cannot spin with the rod vertical, unless the velocity of a point on the circumference of the disc is greater than the velocity acquired by a body after falling from rest vertically through a height ten times the radius of the disc. (Coll. Exam.) 14. A wheel with 4:?i spokes arranged symmetrically rolls with its axis horizontal on a perfectly rough horizontal plane. If the wheel and spokes be made of a very fine heavy wire, prove that the condition for stability is -,,„ 3 2?^ + 7^ 4 4?^ + Sir where a is the radius of the wheel and V its velocity. (Coll. Exam.) 15. Two light rods OP, FQ, each of length 2a, are smoothly jointed at F, and are the axes of equal gyrostats whose centres of mass are at the middle points of the rods. The gyrostats spin with equal angular velocities n in such directions that both would spin in the same way if OFQ were a straight line. 0 is fixed and Q slides above 0 on a smooth vertical rod OZ. If M is the mass of each gyrostat, A and C its principal moments of inertia, and a mass m is suspended from Q, show that steady motion is possible with a precession 12, in the same sense as the resolved part of any angular velocity n along OZ, provided that k-l lies between unity and zero, where y,___Cn__^ 2{M+m)ga ^~12(^+J/a2) ^^'^ ^~122(^+J/«2)Show further that the motion is always stable. (Camb. Math. Tripos.) 16. Prove that the least velocity v with which a thin circular disc (radius a) must be started in order to roll steadily on a rough horizontal plane in a straight line, or very nearly in a straight line, is given by ^2 > i^a ; and that the period of a small oscillation is 27r {. A{A + Ma'^) \2 'oi^C+Ma^)-MgaA)

MISCELLANEOUS EXAMPLES 133 17. A solid of revolution has an equatorial plane of symmetry, and is rolling with angular velocity w round its axis in steady motion on a perfectly rough horizontal plane, the equatorial plane of the solid being vertical. This motion being slightly disturbed, prove that the period of vibration is 2^, A{A+Ma^) \^ t .C(i>^{C+Ma?)-Mg{a-p)A\ where p is the radius of curvature of the meridian of the solid at the equator, and a the radius of its equatorial circle. 18. A sphere of mass m and radius a contains a symmetrical gyrostat, mass J/, freely pivoted on a diameter of the sphere. The latter is spinning on a rough horizontal plane with angular velocity 12 about this diameter, which is vertical ; and 12' is the angular velocity of the gyrostat. Show that in a small oscillation the point of contact describes an ellipse in time n n {(M-\- m) a^ + mF + J } = {M+ ma^ + niB) 12 + (712', A and C being the moments of inertia of the gyrostat, and k the radius of gyration for the sphere about a diameter. 19. A top, free to turn about a fixed point on its axis, at which the principal moments of inertia are A, A, Cand whose distance from the centre of gravity is A, is started when its axis makes angle - with the vertical drawn upwards so that the spin about the axis is -^ ( — ~^ ) and the angular velocity of its axis in azimuth is 2 1 -^-^ j , the velocity in the meridian plane being zero. Show that the inclination 6 of its axis to the vertical at any time is given by the equation sec^ = l + sech|(^)\}, so that the axis continually approaches the vertical without ever reaching it. (Coll. Exam.)

APPENDIX. Explanation of the precession of a wheel from conservation of angular momentum. If we have a particle P constrained to revolve about a fixed axis OC, then, by the principle of conservation of angular momentum, the nearer it gets to this axis the faster it must revolve, and vice versa. (We may imagine, for instance, the particle P, Fig. A, to be a heavy bead sliding on a wire OA which is turning about OC.) Q -"«'. Fig. a. This acceleration or retardation of rotation represents the action of a single force ^ in the appropriate direction on the particle P. Thus, if the particle is getting nearer to the axis, its accelerated rotation is caused by the action of a certain force in the direction PQ of its rotation ; while if the particle is moving away from OC this force acts in the direction PQ' opposing its rotation. Now let us apply this. We may consider the rotating wheel to be composed of sets of four particles, of which A, B, C, D (Fig. B) are a type, situated symmetrically as shown in the figure, and rigidly attached to one another and to the axle OX (perpendicular to the plane of the paper) about which they are rotating in a direction shown by the arrow-heads. * There will really be a number of forces acting on the particle P due to the action of the various constraints by which it is attached to the frame. The above force is a single force equivalent to this set of forces.

APPENDIX 135 On this system we shall suppose a couple whose axis is YOY' to act in such a direction that it tends to turn A and B out of the paper towards the observer, and C and D into the paper away from the observer. We shall show that any such rotation set up by this couple results in a set of forces being called into play which tend to make the system precess about the third perpendicular axis OZ. Fig. B, By reason of its rotation about OX the particle A is increasing its distance from the axis OY'. Thus, if it has any rotation about this axis due to the couple, there will be a tendency to diminish the speed of this rotation in consequence of the increasing distance of the particle from this axis. This retarding tendency may be represented by a force acting on the particle A perpendicular to the plane of the paper and away from the observer. Again, the particle B is getting nearer to the axis OF of the couple and so must be increasing its rate of rotation about this axis. This acceleration of rotation may be supposed to be caused by a force through B perpendicular to the plane of the paper and towards the observer. Similarly, the effects on the particles C and D may be supposed caused by forces acting on C, D perpendicular to the plane of the paper and respectively toivards and from the observer, since the rotation about YO Y' is causing G and D to move away from the observer. We see that we thus have forces at A and D away from the observer, and at B and C towards the observer. Clearly these forces will make the system precess about the third axis OZ. Thus, when the body is rotating about OX, the application of a couple about YO makes the body turn about the third axis OZ.

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ANSWERS TO EXAMPLES. CHAPTER I. Page 16. 1. 6i rad./sec.^. 2. 800 radians ; 2400 radians. 3. 124950 revolutions. Page 28. 1. (i) ? . JL . I07r = i^ Ib.-ft.-sec. units. (ii) A . 1 l0O7^^- = 2^^ Ib.-ft.-sec. units. o /-x 7 1 200 275 n ., ., 2- ^^>32-24^-T-'^ = 3456^^-^'-"'^-^"^'" (") ^- 2T^C-T^)' = 1^^^*I lb.-ft.-sec. units. 3. 81 : 2 : 9. 4. 12^^ lb.-ft.-sec. units. a 5. 3591|^ poundals. 6. Angular momentum = yf 4 lb.-ft.-sec. units. Energy = I ft.-poundals. 7. - seconds. TT 8. (i) 480 ft.-poundals. (ii) 128x^^3 rad./sec. 9. 4 times. ^"> 32.36.15 = ^"* "'-*^55 (iii) r^ = -00265 ft.-poundals. 11. (i) ?iP^^ sin 0 cos (9 ft. -lbs. .... 11 F (]i ) sees ^ ^ 9216 nP^^ sin ^ cos ^

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CORRIGENDA. ANSWERS TO EXAMPLES. Page 136. 10. (i) 32 4 36~^^ ft. -lbs. (approx.). .... 1 7 25 6400. 7r2. 5 ,0 7 „ , / 22\ ^"^ 2 • 4 • I6 • 144732 "16 '^^^^^^ ^^•■^^- (^^TJ11. (i) ?iPJ^ sin 6* cos 0 inch-lbs. (ii) ^^^ ^^ 768 wPyl/sin^cos^ Page 137. 15. (i) 75.32^.^21 ft. -lb. -sec. units. (ii) 48384 f t. -poundals. ..... 48^21 ^. (ill) — Y^ — = 20 sees, approx. 16. yt2 = -y-in.2. sees.

ANSWERS TO EXAMPLES 137 12. Because a Unite force cannot be produced instantaneously, and therefore some time must elapse before the maximum pull is attained. At the first pull, a smaller force, which is increasing up to a maximum, takes its share in producing momentum ; but later this inferior force is taken up in destroying the existing momentum in the opposite direction to it, and by the time that it is again producing momentum, the maximum force has been attained, and thus the total work done in producing momentum = the maximum pull X the length of the string. Thus the energy, and therefore the momentum produced are greater in the second case. 13. The ratio of the energies produced — i.e. the work done in the two cases, by pulling a string of length I with a force T, is ( ji 1_ ]l • Tl — qIttil - X ; e^Tty-, 14. (i) 1000 ft.-lb.-sec. (ii) 40400 ft.-poundals. (iii) 16 rad./sec. (iv) 12^ seconds. 15. (i) 218^ ft.-lb.-sec. units. (ii) 2304 ft.-poundals. (iii) 1 j\- seconds. 16. ^' = -Vft.2. ,„ 550.400.121 550.605 ,,.. 17. --^^— = 11^H.P. 550.400.121 2750.44 ^^. .^ .. 40 ^ ^ =31f ft.-lbs. 18. (i) 96i^ ft.-poundals. 9A (ii) (a) — , A being the constant of proportion. The same forces are used in each case, but in inverse order, and hence the result is the same. (ill) 9A. (iv) — p^ A, (o=^^-j-^' CHAPTER II. Page 42. 1. 1561 ft. -lbs. 2. 2 7^50 rad./sec. 3. 10 cm. 4. 468 rad./sec. 5. V2 feet. Page 45. 1. 43*816 ft./sec. 2. The wheels on the inside of the curve. 3. The radius =V-^- 4. -i/a^. ^ ft.-poundals. ' tuiZ Z K 5. ^^^^ = 118if ft.-lbs. 6. -41 inches.

138 ANSWERS TO EXAMPLES 7. (a) By the internal stresses. (6) About a vertical axis in a direction which would turn the back wheel towards the centre of the curve, and the head of the machine outwards. (c) The reaction causes the back wheel to be turned away from the centre of the curve, and the head of the machine inwards, thus increasing the risk of skidding. (d) ' —93 ft. -lbs. approximately. CHAPTER VI. Page 95. (a) Yes ; since their axes are merely translated and not rotated, it does not matter whether they are spinning or not. (6) No ; the figure would become skew. 6. Ci Wii2 - (.4 1 + if i^H i/24a2 -f ^ab) il^ cos a = (3/i + 2 i^/,) «^CgWgfi - (^ 2 + ^4^^ + 2a6) 12^ cos a = M2bg. (i) No. (ii) Yes. (iii) No. 7. The same way.

ANSWERS TO QUESTIONS. CHAPTER I. [1.] (a) [l/][Zp. {(i) [M][LJ[T]-\ {y) [M][LY[T]-\ [2.] 448 Ib.-ft.'. [3.] It is involved in the mass M. [4.] AOn, [b.] Pt = M{v^-v). [io.] K.E^hMvK [7.] Ps = J J/( V - i'^). CHAPTER II. [8.] (i) It upsets it. (ii) It causes it to precess. [9.] (i) The further end dips. (ii) The further end turns to the left, (iii) The near end dips further. [10.] {a) The front of the car is depressed. (6) The back of the car is depressed. [11.] {a) The bows turn to starboard. {h) She heels to port. [12.] The "Centrifugal Force." CHAPTER III. [13.] There is a small frictional couple at the toe, due to the toe not being a mathematical point. [14.] Because if the top begins to lean over, the gravity-couple about the edge of support causes it to precess ; and if the precession is hindered by a rough surface, the top falls down. [15.] There is more friction to "hurry" the precession. [16.] The radius a is greater and thus w is reduced at a greater rate. [17.] The final direction of friction is opposite to its initial direction. [18.] In both cases the result of the air is the same. But if the peg is blunt the effect of friction mentioned in question 13 is more marked than in the case of the fine peg. On the other hand, friction acts at a larger arm in the case of the blunt peg, so that a smaller frictional force is capable of restoring w to the value necessary for steady motion. [19.] It would move in a vertical straight line ; or if the motion were steady it would be at rest.

140 ANSWERS TO QUESTIONS [20.] Yes. The axle would describe a cone with vertex G. [21.] €HM < J.12. The heavier the top, the better it fulfils this condition ; for we have a big torque acting, and consequently a large 12. [22.] The motion becomes steady more quickly. For the large spin of the body is only partially communicated to the loose spindle ; hence the «(o (of the spindle) is never ver}^ large, and soon = J 12. CHAPTER IV. [23.] No. For since it has no angular momentum about its axis, there will be no horizontal impulse on the cushion. CHAPTER VI. [24.] No. For points of the body on the momentum-axis are not instantaneously at rest. [25.] 0) cos ^ + 12 sin a sin ^. [26.] (i) If convex, the direction of R will not allow G to move in a horizontal circle, so that steady motion is impossible. (ii) If concave, steady motion will be possible. The direction of precession will depend on which side of the axle the reaction R lies, i.e. on the angle at which the top spins and the direction of the common normal at the point of contact. [27.] Zero, during steady motion. [28.] Either 12 would be zero ; or the figure would fold up about OE^ according to the nature of the joints at 0 and E. [29.] The torque of Mg and R. about D ; for the vertical reaction of the plane on the cone, although equal to Mg., does not act through the line of action of the latter. GLASGOW ; PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD.

Models of the Spinning Tops mentioned in this Chapter can be obtained from Messrs. Newton & Co., Scientific Instrument Makers, 3 Fleet Street, London, E.G.

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Plate I. Fig. XL Fig. XII. c 1 L^,..i ^ y A V Fig. XIII. [see pp. 7 and 54

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Plate III. riioto : Llnclerwood and Underwood. Schlick's Apparatus for Steadying Ships. Photo : Underwood and UnderwooJ. Model exhibited before the Royal Society' in 1907. [see p. 67.

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