ft^y DYNAMICS OF ROTATION BV THE SAME A UTHOR A Study of Splashes With 197 Illustrations from Instantaneous Photographs Medium 8vo. LONGMANS, GREEN AND CO. London, New York, Bombay, Calcutta, and Madras DYNAMICS OF ROTATION AN ELEMENTARY INTRODUCTION TO RIGID DYNAMICS BY A. M. WORTHINGTON, C.B., MA., F.R.S. FORMERLY HEADMASTER AND PROFESSOR OF PHYSICS AT THE ROYAL NAVAL ENGINEERING COLLEGE, DEVONPORT NEW IMPRESSIOJSr LONGMANS, GREEN, AND CO. 39 PATERNOSTER ROW. LONDON FOURTH AVENUE & 30TH STREET, NEW YORK BOMBAY, CALCUTTA, AND MADRAS 1920 r^t ^^ Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/dynamicsofrotatiOOwortrich PREFACE TO THE FIRST EDITION Many students of Physics or Engineering, who from want either of mathematical aptitude, or of sufficient training in the methods of analytical solid geometry, are unable to follow the works of mathematical writers on Kigid Dynamics, must have felt disappointed, after master- ing so much of the Dynamics of a Particle as is given in the excellent and widely-used text-books of Loney, or Garnett, or Lock, to find that they have been obliged, after all, to stop short of the point at which their know- ledge could be of appreciable practical use to them, and that the explanation of any of the phenomena exhibited by rotating or oscillating rigid bodies, so interesting and obviously important, was still beyond their reach. The aim of this little book is to help such students to make the most of what they have already learnt, and to carry their instruction to the point of practical utility. As a matter of fact, any one who is interested and observant in mechanical matters, and who has mastered the relations between force, mass, and acceleration of velocity of translation, will find no difficulty in appre- hending the corresponding relations between couples, moments of inertia, and angular accelerations, in a rigid 462658 vi Preface. body rotating about a fixed axle, or in understanding the principle of the Conservation of Angular Momentum. Instead of following the usual course of first developing the laws of the subject as mathematical consequences of D'Alembert's Principle, or the extended interpretation of Newton's Second and Third Laws of Motion, and then appealing to the experimental phenomena for verification, I have adopted the opposite plan, and have endeavoured, by reference to the simplest experiments that I could think of, to secure that the student shall at each point gain his first ideas of the dynamical relations from the phenomena themselves, rather than from mathematical expressions, being myself convinced, not only that this is the best way of bringing the subject vividly and without vagueness before the learner, but that such a, course may be strongly defended on other grounds. These considerations have determined the arrangement of the chapters and the limitations of the work, which makes no pretence at being a complete or advanced treatise. My best thanks are due to those friends and pupils who have assisted me in the revision of the proof-sheets and in the working of examples, but especially to my colleague, Mr. W. Larden, for very many valuable suggestions and corrections. A. M. W. Devonport, 31«« OdL 1891. PREFACE TO THE SIXTH EDITION The demand for successive editions of this book has afforded opportunities for considerable improvements since its first issue. Errors and omissions kindly pointed out by readers and friendly critics have been rectified, while the continued use of the book as a text-book with my own students has enabled me to detect and alter ambiguous phrases, and in some places to improve the arrangement of the argument. The use of the Inertia-Skeleton, introduced on p. 64, has proved so satisfactory a simplification for non- mathematical students, to whom a momental ellipsoid would be only a stumbling-block, and could be used so readily for further extensions, in the manner indicated on pp. 122 and 123, that I hope I may be pardoned for calling attention to it. Experiments with a gyroscope, made by the students themselves with Chapter XIII. as guide, have proved very satisfactory and interesting, and may usefully include a deduction of the rate of spin from an observation of the rate of precession, after the moment of inertia of the wheel has been determined by means of the oscillating table figured on p. 80. vll viii Preface, In the interests of clear teaching, the convention (which I am glad to see has been adopted in America) has been adhered to throughout, of using the word * pound ' when a force is meant, and ' lb/ when a mass is meant, and I have ventured to give the name of a * slug to the British Engineer's Unit of Mass, i.e. to the mass in which an acceleration of one foot-per-sec.-per-sec. is produced by a force of one pound. A. M. W. Devonport, 11^^ Junt 1906. CONTENTS. CHAPTER I. DEFINITIONS OP TERMS AND PRELIMINARY KINEMATICS. Page 1. Rigid Body. 2. Angular Velocity. 2. Rate of Revolution. 3. Relation between (y) and (w). 3. Angular Acceleration. 3. Uniformly Accelerated Rotation. 5. Examples. 6. Geometrical Representation of Angular Velocities and A> celerations. 7. On the Use of the word Moment 8. Definition of Torque. 8. Definition of Equal Torques. 8. Fundamental Statical Experiment. 8. Measure of Torque. 8. Unit Torque. 9. British Absolute Unit of Torque. 9. Gravitation or Engineer's British Unit of Torqua 9. Distinction between * pound ' and ' lb.' CHAPTER II. ROTATION UNDER THE INFLUENCE OF TORQUE. Page 11. Proposition I. ,, 12. Proposition II. „ 13. Methods of Experimental Verification. Contents, Page 14. Variation of the Experiments. „ 15. Familiar Instances. „ 15. The Analogue of Mass in Rotational Motion. ,, 17. Rotational Inertia. ,, 17. Definition of the Unit of Rotational Inertia. „ 18. Examples for Solution. ,, 18. To Calculate the Rotational Inertia of any Rigid Body. ,, 18. Proposition iii. „ 19. Rotational Inertia of an Ideal Single -particle System. „ 20. Moment of Inertia. ,, 20. Unit Moment of Inertia. ,, 21. Definition of Angular Momentum. „ 22. To find the Kinetic Energy of a Rigid Body rotating about a Fixed Axle. „ 23. Work done by a Couple. ,, 23. Analogy with the Expression for the Work done by a force in Rectilinear Motion. ,, 24. Change of Kinetic Energy due to a Couple. „ 24. Radius of Gyration. „ 25. Numerical Examples. „ 30. Note to Chapter ii. D'Alembert*s Principle. CHAPTER III DEFINITIONS, AXIOMS, AND ELEMENTARY THEOREMS NECESSARY FOR DEALING WITH MOMENTS OF INERTIA — ROUTH'S RULE AND ITS APPLICATION. Page 33. Definition of Moment of Inertia of an Area. „ 33. Definition of Moment of Inertia of a Volume. ,, 34. Axiom. ,, 34. Illustration. „ 34. Axiom. „ 35. Proposition l Contents, xi Page 36. Routh's Rule for Finding the Moment of Inertia about an Axis of Symmetry in certain cases. , 36. Examples of the Application of Dr. Routh's Rule. „ 37. Theorem of Parallel Axes. „ 38. Proposition ii. „ 39. Applications. „ 40. Proposition III. ,, 42. Examples for Solution on Chapters i., il., and ill. CHAPTER IV. MATHEMATICAL PROOFS OF THE DIFFERENT CASES INCLUDED UNDER routh's RULE. Page 46. To Find I for a Uniform Thin Rod about a Perpendicular Axis through one end. „ 47. Corollary. „ 48. Rectangle. „ 48. Circular Disc. „ 50. Thin Rod by Integration. „ 50. Circular Disc by Integration. „ 51. Moment of Inertia of an Ellipse. „ 52. Sphere and Cone. „ 52. Sphere by Integration. ,, 53. Exercises. CHAPTER V. rUKTHER propositions CONCERNING MOMENTS OF INERTIA — PRINCIPAL AXES— GRAPHICAL REPRESENTATION OF INERTIA-CURVilS AND SUB- FACES— EQUIMOMENTAL SYSTEMS— INERTIA SKELETONS. Page 55. Proposition iv. ,, 56. Propositions v. and VL „ 58. Proposition vii. XI 1 Contents. Page 60. Graphical Construction of Inertia-Curves and Surfaces. ,, 62. Diagrams of Inertia Curves. „ 63. Construction of Moment of Inertia Surface, „ 64. Equimomental Systems — Proposition Viii. „ 64. Inertia Skeleton — Proposition ix. CHAPTER VI. SIMPLE HARMONIC MOTION. Page 67. Definition of Simple Harmonic Motion. ,, 68. Definition of Period. ,, 69. Definition of Phase. ,, 69. Expression for the Period or Time of a Complete Oscillation. CHAPTER VII. AN ELEMENTARY ACCOUNT OF THE CIRCUMSTANCES AND LAWS OP ELASTIC OSCILLATIONS. Page 70. Perfect or Simple Elasticity. 70. Hooke's Law. 71. Illustrations of Hooke's Law. 72. Oscillations due to Elasticity. 73. Ratio of Acceleration to Displacement. 73. Expression for the Time of a Complete Oscillation. 74. Applications. 75. Extension to Angular Oscillations. 76. Applications. 76. Equivalent Simple Pendulum. 77. Examples. 79. Oscillating Table for Finding Moments of Inertia. 8L Examples for Solution. Contents. xiii CHAPTER VIII. CONSERVATION OF ANGULAR MOMENTUM. Page 82. Analogue in Rotation to Newton's Third Law of Motion. ,, 83. Application of the Principle in cases of Motion round a fixed Axle. „ 83. First Example. „ 84. Second Example. „ 85. Third Example. ,, 85. Fourth Example. „ 87. Consideration of the Kinetic Energy. „ 87. Other Exemplifications of the Principle of the Conservation of Angular Momentum. „ 88. Graphical representation of Angular Momentum. „ 89. Moment of Momentum. ,, 89. Conservation of Moment of Momentum. „ 91. General Conclusion. „ 91. Caution. ,, 91. Ballistic Pendulum. ,, 93w Examples. CHAPTER IX. ON THE KINEMATICAL AND DYNAMICAL PROPERTIES OF THE CENTRE OF MASS. Page 94. Evidence of the Existence for a Rigid body of a point pos- sessing peculiar Dynamical Relatious. „ 95. Experiments (1), (2), and (3). „ 96. Experiments (4) and (5). ,, 96. A Couple causes Rotation about an Axis through the Centre of Gravity. „ 97. Experiment (6) with a Floating Magnet „ 98. Experiment (7). XIV Contents. Page 99. Definition of Centre of Mass. ,, 100. Proposition I. — (Kinematical.) On the Displacement of the Centre of Mass. ,, 101. Pure Rotation and Translation. „ 101. PropositioQ ii. — (Kinematical. ) On the Velocity of the Centre of Mass. „ 101. Proposition iii, — (Kinematical.) On che Acceleration of the Centre of Mass. „ 102. Summary. „ 102. Corresponding Propositions about Moments. ,, 103. Proposition iv. On the Resultant Angular Momentum ,, 104. Proposition V. Resultant Moment of the Mass-accelerations. „ 104. Proposition vi. On the Motion of the Centre of Mass of a body under External Forces. „ 105. Proposition vii. On the Application of a Couple to a Free Rigid Body at Rest. „ 105. Proposition viii. The Motion of the Centre of Mass- does not afifect Rotation about it. ,1 106. Independent treatment of Rotation and Translation. „ 106. On the Direction of the Axis through the Centre of Mass about which a Couple causes a free Rigid Body to turn. Caution. „ 107. Total Kinetic Energy of a Rigid Body. „ 108. Examples. „ 110. Examples for Solution. CHAPTER X. CENTRIPETAL AND CENTRIFUGAL FORCES. Page 111. Consideration of the Forces on the Axle. „ 111, Proposition. Uniform Motion of a Particle in a Circle. ,, 112. Use of the terms 'Centripetal Force' and 'Centrifugal Force.' ,, 113. Centripetal Forces in a Rotating Rigid Body. ,, 113. Rigid Lamina. „ 115. Extension to Solids of a certain type. 116. Convenient Dynamical Artifice. Contents xv Page 117. Centrifugal Couples. „ 118. Centrifugal Couple in a body of any shape. „ 119. Centrifugal Couples vanish when the Rotation is about a Principal Axis. ,, 121. Importance of Properly Shaping the Parts of Machinery intended to Rotate rapidly. „ 121. Equimomental Bodies similarly rotating have equal and similar Centrifugal Couples. ,, 121. Substitution of the 3-rod Inertia-Skeleton. ,, 123. Transfer of Energy under the action of Centrifugal Couples CHAPTER XI. CENTRE OF PEKCUSSIOM.. Page 125. Thin Uniform Rod. „ 126. Experiment. „ 127. Experiment. ,, 128. Illustrations — Cricket Bat, Door. ,, 128. Centre of Percussion in a Body of any Form. CHAPTER XIL ESTIMATION OF THE TOTAL ANGULAB MOMENTUM. Page 130. Simple Illustrations. „ 132. Additional Property of Principal Axes. ,, 133. Total Angular Momentum. „ 133. The Centripetal Couple. „ 135. Rotation under the influence of no Torque. The Invarlabia Axis. CHAPTER Xm. ON SOME OF THE PHENOMENA FBESENTEP BT SPINNING BODIEa Page 136. Gyroscope. „ 137. Experiments (1), (2), ^nd (3). xvi Contents. Page 138. Experiment (4). 139. Definition of Precessioiu 139. Experiment (5). 140. Experiments (6), (7), and (8). 141. Experiments (9) and (10). 141. Precession in Hoops, Tops, etc. 142. Further Experiment with a Hoop. 143. Bicycle. 143. Explanation of Precession. 145. Analogy between Steady Precession and Uniform Motion in a Circle. 145. Calculation of the Rate of Precession. 148. Observation of the 'Wabble.* 150. Explanation of the Starting of Precession. 152. Gyroscope with Axle of Spin Inclined. 153. Influence of the Centrifugal Couple. 154. Explanationof the eflfects of impeding or hurrying Precession. 154. The Rising of a Spinning Top. 156. Calculation of the 'Eflfort to Precess.' 157. Example (1) Precessional Forces due to the wheels of a railway -engine rounding a curve. 157. Precessional Stresses on the machinery of a pitching, rolling, or turning ship, 158. Example (2) Torpedo-boat turning. 159. Miscellaneous Examples. 1 60. Appendix ( 1 ) On the terms A ngular Velocity and Rotational Velocity. 161. Appendix (2) On the Composition of Rotational Velocities. 161. Appendix (3) The Parallelogram of Rotational Velocities. 164. Appendix (4) Evaluation of the steady precessional velocity of a gyroscope or top with the axis of spin inclined. 166. Appendix (5) Note on Example (4) p. 86. 166. Appendix (6) On the connection between the Centripetal Couple and the residual Angular IVTomentun^^ DYNAMICS OF EOTATION. CHAPTER L DEFINITIONS OF TERMS AND PKELIMINARY KINEMATICS. Rigid Body. — A body in Dynamics is said to be rigid (i.e. stiff) so long as the forces acting upon it do not change the relative positions of its parts. We shall deal, at first, chiefly with such familiar rigid bodies as a fly-wheel turning on its axle ; a cylindrical shaft ; a grind- stone ; a door turning on its hinges ; a pendulum ; a magnetic compass-needle ; the needle of a galvanometer with its attached mirror. It should be observed that such a body as, for example, a wheelbarrow being wheeled along a road is not, taken as a whole, a rigid body, for any point on the circumference of the wheel changes its position with respect to the rest of the barrow. The wheelbarrow consists, in fact, of two practically rigid bodies, the wheel and the barrow. On the other hand, a sailing-boat may be regarded as a rigid body so long as its sails are taut under the influence of the wind, even though they be made of a material that is far from rigid when otherwise handled. So also a stone whirled by an inextensible string consti- tutes, with the string, a single body which may be legarded as rigid so long as the string is straight. A Dynnmus of Rotation, Angular Velocity. — When a rigid body turns about a fixed axis, every particle of the body describes a circle about this axis in the same time. If we conceive a radius to be drawn from the centre of any such circular path to the particle describing it, then, if the rotation be uniform, the number of unit angles swept out in unit time by such a radius is the measure of what is called the angular velocity of the body, or its rotational velocity. The unit of time invariably chosen is the second, and the unit angle is the 'radian,' i.e. the angle of which the arc is equal to the radius. Hence, in brief, we may write Angular velocity (when uniform) = Number of radians described per second. The usual symbol for the rotational or angular velocity of a body is w (the Greek omega). "When the rotational velocity is not uniform, but varies, then its value at any instant is the number of radians that would be swept out per second if the rate of turning at that instant remained uniform for a second.* Rate of Revolution. — Since in one revolution the radius describes 27r radians, it follows that the number of revolutions made per second when the angular velocity is w, is — , and that when a body makes one revolution per second, it describes 2t unit angles per sec, and has therefore an angular velocity = w = 27r. Thus a body which makes 20 turns a minute has an angular velocity ?^g'<^??=?;. Tangential 5pced. — The linear velocity {v) of a particle * See Appendix (1). Defiriitions of Terms, describing a circle of radius r about a fixed axis is at any instant in the direction of the tangent to the circular path, and is conveniently referred to as the tangential speed. Relation between v and w. — Since a rotational velocity radians per sec. corresponds to a travel of the particle over an arc of length rw each second, it follows that t? = ro> or CD = — . r Very frequent use will be made of this relation. Examples, — (1) A rotating drum 4 feet iu diameter is driven by a strap which travels 600 feet a minute and without slipping on the drum. To find the angular velocity — 600 o) = — = 60 =5 radians per sec, (2) A wheel 3 feet in diameter has an angular velocity of 10. Find the speed of a point on its circumference. = 1*5 X 10 feet per sec. = 15 feet per sec. Angular Acceleration. — When the rate of rotation of a rigid body about a fixed axle varies, then the rate of change of the angular or rotational velocity is called the angular or rotational acceleration, just as rate of change of linear velocity is called linear acceleration. The usual symbol for angular acceleration is w. Thus w is at any instant the number of radians per second that are being added per second at the instant under consideration. We shall deal at first with uniform angular accelerations, for which we shall use the less general symbol A. Uniformly accelerated Rotation.— If a rigid body Dynamics of Rotation, start rotating from rest with a uniform angular acceleration A, then after t seconds the angular velocity w is given by o> = A^. If the body, instead of being at rest, had initially an angular velocity w^, then at the end of the interval of / seconds the angular velocity would be ^+JAO^ = a)„if+iAj!^ (ii) By substituting in (ii) the value of t given in (i) we obtain the equation (o2=w^2_i.2A(9 (iii), which connects the angular velocity w with initial velocity w^ and the angle d swept through. The student will observe that these equations are precisely similar to and are derived in precisely the same way as the three fundamental kinematic equations that he has learned to ^ It is not considered necessary to reproduce here the geometrical or other reasoning by which this is established. See Garnett's Elementary Dynamics, and Lock's Dynamics for Beginners. Definitions of Terms. flse in dealing with uniformly accelerated rectilinear motion of a particle, viz.: — v=^U'\-at ..,.»... (i) s=M^-|-^a/' . ...... (ii) v'^—u'-\-1as (iii) Example 1. — A wheel is set gradually rotating from rest with a uniform angular acceleration of 30 units of angular velocity per sec. In what time will it acquire a rate of rotation of 300 revolutions per minute ? Solution. — 300 revolutions per minute is an angular velocity of 300x27r ,. , . , .„ , .. • A • 300x27r — — — radians per sec, which will be attained in — - — — - sec. = !!:sec.= ?i^ sec. = 1-0472 sec. 3 3 Example 2. — A wheel revolves 30 times per sec. : with what uni- form angular acceleration will it come to rest in 12 sec, and how many turns will it make in coming to rest ? Solution. — Initial angular velocity = a)„ = 30 x 2?? = GOtt. This is destroyed in 12 sec, .*. angular acceleration = --r^ = -57r = — 15 "708 radians per sec, each second. The — sign means that the direction of the acceleration is opposite to that of the initial velocity w^, which we have tacitly assumed to be + in writing it equal to GOtt. The angle described in coming to rest is obtained at once from the 3rd of the fundamental equations now that we know the value of A. Thus :— a)2=a,,2 + 2A^ O2=(607r)2-107r5 .-. 107r^=(607r)2 .*. ^ = 3607r =3607r revolutions. -=180 ro volution a. Dynamics of Rotation. Example 3. — A wheel rotating 3000 times a minute has a uniform angular retardation of tt radians per sec. each second. Find when it will be brought to rest, and when it will be rotating at the same rate in the opposite direction. 3000 revolutions per min. = 3000 x Stt 60 = IOOtt radians per sec, and will therefore be destroyed by the opposing acceleration tt in 100 sec. The wheel will then be at rest, and in 100 sec. more the same angular velocity will have been generated in the opposite direction. (Compare this example with that of a stone thrown vertically up and then returning.) Geometrical Representation of Rotational Veloci- ties and Accelerations. — At any particular instant the motion of a rigid body, with one point fixed, must be one of rotation with some definite angular velocity about some axis fixed in space and passing through the point. Thus the rotational velocity is, at any instant, completely represented by drawing a straight line, of length proportional to the rotational velocity, in the direction of the axis in question, and it is usual to agree that the direction of drawing and naming shall be that in which a person looking along the axis would find the rotation about it to be right- ^^^ J handed (or clockwise). Thus the line OA would correspond to the direction of rotation indicated in the fig. If we choose to conceive a body as affected by simultaneous component rotations about three rectangular intersecting axes, we shall obtain the actual axis and rotational velo- city, from the lines representing these components by the parallelogram law. (For illustration and proof see Appendix (2) and (3). Definitions of Terms. In the same way rotational acceleration about any axis fixed in space may be represented by drawing a line in its direction (with the same convention), and simultaneous rotational accelerations may be combined according to the parallelogram law. On the Use of the word Moment. — The word moment was first used in Mechanics in its now rather old-fashioned sense of * importance ' or ' consequence/ and the moment of a force about an axis meant the importance of the force with respect to its power to generate in matter rotation about the axis ; and again, the moment of inertia of a body with respect to an axis is a phrase invented to express the importance of the inertia of the body when we endeavour to turn it about the axis. When we say that the moment of a force about an axis varies as the force, and as the distance of its line of action from the axis, we are not so much defining the phrase * moment of a force,' as expressing the result of experiments made with a view to ascertaining the circumstances under which forces are equivalent to each other as regards their turning power. It is important that the student should bear in mind this original meaning of the word, so that such phrases as ' moment of a force ' and ' moment of inertia ' may at once call up an idea instead of merely a quantity. But the word ' moment ' has also come to be used by analogy in a purely technical sense, in such expressions as the * mo- ment of a mass about an axis,' or ' the moment of an area with respect to a plane,' which require definition in each case. In these instances there is not always any corresponding physical idea, and such phrases stand, both historically and scientifi- cally, on a different footing. 8 Dynamics of Rotation. Unfortunately the words ' moment of a force ' are regarded by some writers as the name rather of the product * force X distance from axis ' than of the property of which this product is found by experiment to be a suitable measure. But happily for the learner the difficulty thus created has been met by the invention of the modern word torque to express * turning power.' Definition of Torque. — A force or system of forces which has the property of turning a body about any axis is said to be or to have a torque about that axis (from the Latin torgueo, I twist). Definition of Equal Torques. — Two torques are said to be equal when each may be statically balanced by the same torque. Fundamental Statical Experiment. — Torques are found to be equal when the products of the force and the distance of its line of action from the axis are equal. Experi- ments in proof of this may be made with extreme accuracy. The result may also be deduced from Newton's Laws of Motion. Measure of Torque. — The value of a torque is the value of this product. This again is a matter of definition. Unit Torque. — Thus the unit force acting at unit distance is said to be or to have unit torque, and a couple has unit torque about any point in its plane when the product of its arm and one of the equal forces is unity. Definitions of the Terms, 9 British Absolute Unit of Torque.— Since in the British Absolute system, in which the lb. is chosen as the unit of mass, the foot as unit of length, and the second as unit of time, the unit of force is the poundal, it is reasonable and is agreed that the British absolute unit of torque shall be that of a poundal acting at a distance of 1 foot, or (what is the same thing, as regards turning) a couple of which the force is one poundal and the arm one foot. This we shall call a poundal-foot, thereby distinguishing it from the foot-poundal, which is the British absolute unit of work. Gravitation or Engineer's British Unit of Torque. — In the Gravitation or Engineer's system in this country, which starts with the foot and second as units of length and time, and the pound pull {i.e. the earth's pull on the standard lb.) as unit of force, the unit of torque is that of a couple of which each force is 1 pound and the arm 1 foot. This may be called the * pound-foot.' * Distinction between * pound ' and * lb.* — The student should always bear'in mind that the word pound is used in two senses, sometimes as a force, sometimes as a mass. He will find that it will contribute greatly to clearness to follow the practice adopted in this book, and to write the word ' pound ' whenever a force is meant, and to use the symbol * lb.' when a mass is meant. Axis and Axle. — An axis whose position is fixed rela- tively to the particles of a body may be conveniently referred to as an axle. * On this system the unit mass is that to which a force of 1 pound would give an acceleration of 1 foot-per-second per second and is a mass of about 32*2 lbs. It is convenient to give a name to this practical unit of inertia, or sluggishness. We shall call it a ' slug.' CHAPTEE II. ROTATION UNDER THE INFLUENCE OF TORQUE. The student will have learnt in that part of Dynamics which deals with the rectilinear motion of matter under the influ- ence of force, and with which he is assumed to he familiar, that the fundamental laws of the subject are expressed in the three statements known as Newton's Laws of Motion. These propositions are the expression of experimental facts. Thus, nothing but observation or experience could tell us that the acceleration which a certain force produces in a given mass would be independent of the velocity with which the mass was already moving, or that it was not more difficult to set matter in motion in one direction in space than in another. We shall now point out that in the study of the rotational motion of a rigid body we have exactly analogous laws and properties to deal with : only that instead of dealing with forces we have torques ; instead of rectilinear velocities and accelerations we have angular velocities and accelerations ; and instead of the simple inertia of the body we have to con- sider the importance or moment of that inertia about the axis, which importance or moment we shall learn how to measure. It will contribute to clearness to enunciate these corre- sponding laws with reference first to a rigid body pivoted Rotation under the Infiuence of Torque. 1 1 about a fixed axle, i.e. an axis which remains fixed in the body, and in its position in space ; and although it is possible to deduce each of the propositions that will be enunciated as con- sequences of Newton's Laws of Motion, without any further appeal to experiment, yet we shall reserve such deduction till later, and present the facts as capable, in this limited case at any rate, of fairly exact, direct experimental verification. Proposition I. — Tlie rate of rotation of a rigid body revolving about an axis fixed in the body and in space cannot be changed except by the application of an external force having a moment about the axis, i.e. by an external torque. Thus, a wheel capable of rotating about a fixed axle cannot begin rotating of itself, but if once set rotating would con- tinue to rotate for ever with the same angular velocity, unless acted on by some external torque (due, e.g. to friction) hav- ing a moment about the axis. Any force whose line of action passes through the axis will, since this is fixed, be balanced by the equal and opposite pressure which fixes the axis. It is true that pressure of a rotating wheel against the material axle or shaft about which it revolves does tend to diminish the rate of rotation, but only indirectly by evoking friction which has a moment about the axis. It is impossible in practice to avoid loss of rotation through the action of friction both with the bearings on which the body is pivoted and with the air; but since the rotation is always the more prolonged and uniform the more this friction is diminished, it is impossible to avoid the inference that the motion would continue unaltered for an indefinite period could the friction be entirely removed. The student will perceive the analogy between thia first ""12 Dynamics of Rotation, Proposition and that known as Newton's First Law of Motion. Proposition II. — The angular acceleration or rate of change of angular velocity produced in any given rigid mass rotating about an axis fixed in the body and in space is proportional to the moment about the axis of the external forces applied, i.e. to the value of the external torque. To fix the ideas, let the student think first of a wheel rotating about a fixed shaft passing through its centre, and to this wheel let us apply a constant torque by pulling with constant force the cord AB wrapped round the circumference. [It may be well to point out here that if the wheel be accu- rately symmetrical, so that its centre of gravity lies in the axis of the shaft, then, as will be shown in the chapter on the Centre of Mass, since the centre of gravity or centre of mass of the wheel does not move, there must be some other equal and opposite external force acting on the body. This other force is the pressure of the axle, so that we are really apply- ing a couple as in Fig. 2 ; but this latter force has no moment about the axis, and does not directly affect the rotation.] Our Proposition asserts that (1) So long as the torque has the same value, i.e. so long as the cord is pulled with the same force, the Rotation U7ider the Ltfluence of Torque, 1 3 acceleration of the angular velocity of the wheel is uniform, so that the effect on the wheel of any torque, in adding or subtracting angular velocity, is independent of the rate at which the wheel may happen to be rotating when the torque is applied. (2) That a torque of double or treble the value would pro- duce double or treble the acceleration, and so on. (3) If several torques be applied simultaneously, the effect of each on the rotation is precisely the same as if it acted alone. Also it follows (4) That different torques may be compared, not only statically but also dynamically, by allowing them to act in turn on the same pivoted rigid body in a plane perpendicular to the axis, and observing the angular velocity that each generates or destroys in the same time. Methods of Experimental Verification. — Let an arrangement equivalent to that of the figure be made. AB is an accurately centred wheel turning with as little friction as possible on a horizontal axis, e.g. a bicycle wheel on ball bearings. Round its circumference is wrapped a fine cord, from one end of which hangs a mass C of known weight (W), which descends in front of a graduated scale. FIO. 4. 14 Dynamics of Rotation, It will be observed that C descends with uniform accelera- tion. This proves that the tension (T) of the cord BC on the weight is uniform, and from observation of the value (a) of the acceleration, that of the tension is easily found, being given by the relation W-T _a W 'g (where ^ is the acceleration that would be produced in the mass by the force W alone), and T multiplied by the radius of the wheel is the measure of the torque exerted. Thus the arrange- ment enables us to apply a known and constant torque. But since the linear acceleration of C is uniform, it follows that the angular acceleration of the wheel is uniform. By varying the weight W, the torque may be varied, and other torques may be applied simultaneously by means of weights hung over the axle, or over a drum attached thereto, and thus the proportionality of angular acceleration to total resultant torque tested under various conditions. It will be observed that in the experiments described we assume the truth of Newton's Second Law of Motion in order to determine the value of the tension (T) of the cord ; but it is possible to determine this directly by inserting between and B a light spring, whose elongation during the descent tells us the tension applied without any such assumption. Variation of the Experiments. — Instead of using our known torque to generate angular velocity from rest, we may employ it to destroy angular velocity already existing in the following manner : — Let a massive fly-wheel or disc be set rotating about an axis with a given angular velocity, and be brought to rest by Rotation under the Influence of Torque. 1 5 a friction brake which may be easily controlled so as to maintain a constant measurable retarding torque. It will be found that, however fast or slowly the wheel be rotating, the same amount of angular velocity is destroyed in the same time by the same retarding torque ; that a torque r times as great destroys the same amount of angular velocity in — of the time; while if a second brake be applied simultaneously the effect of its retarding couple is simply superadded to that of the first. It may be remarked that the direct experimental verifica- tions here quoted can be performed with probably greater accuracy than any equally direct experiment on that part of Newton's Second Law of Motion to which our 2nd Proposition corresponds^ viz. that 'the linear acceleration of a given body is proportional to the impressed force, and takes place in the direction of the force.' Thus, our second Proposition for rotational motion is really less far removed than is Newton's Second Law of Motion from fundamental experiment. Familiar Instances. — Most people are quite familiar with immediate consequences of these principles. For example, in order to close a door every one takes care to apply pressure near the outer and not near the hinged side, so as to secure a greater moment for the force. A workman checking the rotation of any small wheel by friction of the hand applies his hand near the circumference, not near the axis. The Analogue of Mass in Rotational Motion.— In the study of rectilinear motion it is found that if after making experiments op some giv^n t>ody we pass to another, the" 1 6 Dynamics of Rotation. same forces applied to the second body do not, in general, produce in it the same accelerations. The second body is found to be less easy or more easy to accelerate than the first. We express this fact by saying that the 'inertia' or 'mass' of the second body is greater or less than that of the first. Exactly the same thing occurs in the case of rotational motion, for experiment shows that the same torque applied to different rigid bodies for the same time produces, in general, different changes of angular velocity. Thus, the pull of a cord wrapped round the axle of a massive fly-wheel will, in say 10 seconds, produce only a very slow rotation, while the same torque applied to a smaller and lighter wheel will, in the same time, communicate a much greater angular velocity. It is found, however, that the time required for a given torque to produce a given angular velocity does not depend simply on the mass of the rigid body. For, if the wheel be provided as in the figure with heavy bosses, and these be moved further from the axis, then, although the mass or inertia of the wheel, as re- gards bodily motion of the whole in a straight line, is unaltered, yet it is now found to be more difficult to accelerate rotationally than before. The experiment may be easily made with our bicycle wheel of Fig. 4, by removing alternate tensional spokes and fitting it with others to which sliding masses can be conveniently attached. With two wheels, however, or other rigid bodies, precisely similar in all respects except that one is wc^de of ^ lighter Rotation under the Infltience of Torque. 1 7 material than the other, so that the masses are different, it is found that the one of less mass is proportionately more easy to accelerate rotationally. Hence we perceive that in studying rotational motion we have to deal not only with the quantity of matter in the body, but also with the arrangement of this matter about the axis ; not solely with the mass or inertia of the body, but with the importance or moment of this inertia with respect to the axis in question. We shall speak of this for the present as the Rotational Inertia of the body, meaning that property of the body which determines the time required for a given torque to create or destroy in the body a given amount of rotational velocity about the axis in question. Definition of the Unit of Rotational Inertia. — Just as in the Dynamics of rectilinear motion we may agree that a body shall be said to have unit mass when unit force acting on it produces unit acceleration, so in dealing with the rotation of a rigid body it is agreed to say that the body has unit rota- tional inertia about the axis in question when unit torque gives it unit angular acceleration, i.e, adds or destroys in it, in one second, an angular velocity of one radian per sec. If unit torque acting on the body takes, not one second, but two, to generate the unit angular velocity, then we say that the rotational inertia of the body is two units, and, speaking generally, the relation between the torque which acts, the rotational inertia of the body acted on, and the angular acceleration produced, is given by the equation Angular acceleration ==- ; — %r—, —. Rotational inertia Just as in rectilinear motion, the impressed force, the mass B 1 8 Dynamics of Rotatio7i. acted on, and the linear acceleration produced, are connected by the relation A 1 ,• Force Acceleration = . mass Examples for Solution. — (1) A friction brake which exerts a con- stant friction of 200 pounds at a distance of 9 inches from the axis of a fly-wheel rotating 90 times a minute brings it to rest in 30 seconds. Compare the rotational inertia of this wheel with one whose rate of rotation is reduced from 100 to 70 turns per minute by a friction couple of 80 pound-foot units in 18 seconds. Ans. 25 : 24. (2) A cord is wrapped round the axle, 8 inches in diameter, of a massive wheel, whose rotational inertia is 200 units, and is pulled with a constant force of 20 units for 15 seconds, when it comes off. What will then be the rate of revolution of the wheel in turns per minute? The unit of length being 1 foot, and of time 1 second. Ans, 4*774 turns per minute. To calculate the Rotational Inertia of any rigid body. — We shall now show how the rotational inertia of any rigid body may be calculated when the arrangement of its particles is known. We premise first the following : — Proposition III. — The ^rotational inertia* of any rigid hody is the sum of the * rotational inertias ' of its constituent parts. That this is true may be accurately ascertained by trials with the experimental wheel of Figs. 4 and 5. Let the wheel, unloaded by any sliding pieces, have its rotational inertia determined by experiment with a known torque in the manner already indicated, and call its value I„. Then let sliding pieces be attached in certain noticed positions, and let the new value of the rotational inertia be Ii. Then, according to our proposition, Ii— I, is the rotational inertia of the sliders. If this be the case, then the increase of rotational inertia Rotation under the Influence of Torque. 1 9 produced by the sliders in this position should be the same, whether the wheel be previously loaded or not. If trial be now made with the wheel loaded in all sorts of ways, it will be found that this is the case. The addition of the sliders in the noticed positions always contributes the same increase to the rotational inertia. Rotational Inertia of an ideal Single-particle System. — We now proceed to consider theoretically, in the light of our knowledge of the dynamics of a particle, what must be the rotational inertia of an ideal rigid system consisting of a single particle of mass m connected by a rigid bar, whose mass may be neglected, to an axis at dis- tance (r). Let be the axis, M the particle, so that 0M=7*, and let the system be acted on by a torque of L units. v^ This we may suppose to be 2 — — ) ^ *^ . , - _. , , FIG. 0. due to a force P acting on the particle itself, and always at right angles to the rod OM, and of such value that the moment of P is equal to the torque, i.«. Pr = L or P= -. r The force P acting on the mass m generates in it a linear P . P . acceleration a = — in its own direction. — is therefore the m w amount of linear speed generated per unit time by the force in its own direction, and whatever be the variations in this linear speed (r), — is always equal to the rotational velocity co, and therefore the amount of rotational velocity generated per 20 Dynamics of Rotation. unit time, or the rotational acceleration, A, is -th of the r linear speed generated in the same time, . . P Pr t.e. A=— =— ». rm mr^ L Torque mr'* But A= ^q"" . ; (See p. 17.) rotational mertia .*. The rotational inertia of a single particle of mass m at a distance r from the axis=m7''. Any rigid body may be regarded as made up of such ideal single-particle systems, and since the rotational inertia of the whole is the sum of the rotational inertias of the parts, we see that if wzi, Wj, m„ ... be the masses of the respective particles, r,, rj, ^-g, . . . their distances from the axis, then The rotational inertia of the body = 2(7727''). This quantity 1{mr*) is generally called the Moment of Inertia of the body. The student will now understand at once why such a name should be given to it, and the name should always remind him of the experimental properties to which it refers. We shall from this point onward drop the term ' rotational inertia,' and use instead the more usual term 'moment of inertia,' for which the customary symbol is the letter I. Unit Moment of Inertia. — We now see that a particle 'Rotation under the Influeiice of Torque, 2 1 of unit mass at unit distance from the axis has unit moment of inertia. It is evident also that a thin circular hoop of unit radius and of unit mass rotatinsr o about a central axis perpen- dicular to the plane of the circle, has also unit moment - ... - ... FIG. 7. no. 8. of inertia; for every particle may with close approximation be regarded as at unit dis tance from the centre. In fact, I=2(7nr») =2(m) = 1. The same is true for any segment of a thin hoop (Fig. 8) of unit radius and unit mass, and it is also true for any thin hollow cylinder of unit radius and unit mass, rotating about its own axis. Thus the student will find it an easy matter to prepare accurate standards of unit moment of inertia. A thin cylinder or hoop, of one foot radius and weighing 1 lb., will have the unit moment of inertia on the British absolute system. We shall call this the Ib.-foot^ unit. The engineer's unit is that of one slug (or 32-2 lbs.) at the distance of 1 foot, i.e. a slug-foot^. Definition of Angular Momentum.— Just as the pro- duct mass X velocity, or (mv), in t^ranslational motion is called momentum, so by analogy when a rigid body rotates about a fixed axle, the product (moment of inertia) x (angular or rota- 22 Dynamics of Rotation, tional velocity), or (Iw), is called angular or rotational mo- mentum.* And just as a force is measured by the change of momentum it produces in unit time, so a torque about any axis is measured by the change of angular momentum it produces in unit time in a rigid body pivoted about that axis, for since ■A.=^ L=IA. To find the Kinetic Energy of a rigid body rotat- ing at)0ut a fixed axle. — At any given instant every particle is moving in the direction of the tangent to its cir- cular path with a speed v, and its kinetic energy is therefore equivalent to Jmz;* units of work, and since this is true for all the particles the kinetic energy may be written 2( j. But for any particle the tangential speed v—rm where r is the distance of the particle from the axis and w is the angular velocity ; .2 -.a .*. kinetic energy = S — - — units of work, and in a rigid body a> is the same for every particle ; .*. the kinetic energy =a)'|2(mr*) units of work, = JIw' units of work, t The student will observe that this expression is exactly * When the body is not moving with simple rotation about a given fixed axis, w is not generally the same for all the particles, and the angular momentum about that axis is then defined as the sura of the angular momenta of the particles, viz. 2(mr-u>). t It will be remembered that the unit of work referred to will depend on the unit chosen for I. If the unit moment of inertia be that of 1 lb. at distance of one foot, then the unit of work referred to will be the foot-poundal (British Absolute System). If the unit moment of inertia be that of a 'slug' at distance of one foot, then the unit of vtox-k referred to will be the foot-pound. Rotation under the hifluence of Torque, 23 analogous to the corresponding expression \mv'*^ for the kinetic energy of translation. Work done by a Couple.- -When a couple in a plane at right angles to the fixed axis about which a rigid body is pivoted, turns the body through an angle ^, the moment of the couple retaining the same value (L) during the rotation, then the work done by the couple is L^. For the couple is equivalent in its effect on the rotation to a single force of magnitude L acting at unit distance from the axis, and always at ^'Z^IZ"^^ right angles to the same radius during the //^ ^^\ j\ rotation. f < ► y\ \ * ^^^ } I In describing the unit angle, or 1 radian, yV / / this force advances its point of application \^!!^22^^1^^ through unit distance along the arc of the fio. 10. circle, and therefore does L units of work, and in describing an angle 6 does ltd units of work. Analogy with the expression for the work done by a force, in rectilinear motion. — It will be observed that this expression for the measure of the work done by a couple is exactly analogous to that for the work done by a force in rectilinear motion, for this is measured by the pro- duct of the force and the distance through which it acts measured in the direction of the force. If the couple be L poundal-foot units, then the work done in turning through an angle 6 is LO foot-poundals. If the couple be L pound-foot units, then the work done will be L^ foot-pounds. / 24 Dymamics of Rotation, Change of Kinetic Energy due to a Couple.— When the body on which the couple acts is perfectly free to turn about a fixed axis perpendicular to the plane of the couple, it is easy to see that the work done by the couple is equal to the change in the kinetic energy of rotation. For if A be the angular acceleration, w^ the initial, and w the final value of the angular velocity, then (see equation iii. p. 4) 2A ' L T' and A=y- ... L^=^Ia>'-JI(u/ = Final kinetic energy— Initial kinetic energy. Radius of Gyration. — It is evident that if we could condense the whole of the matter in a body into a single particle there would always be some distance Iz from the axis at which if the particle were placed it would have the same moment of inertia as the body has. This distance is called the radius of gyration of the body with respect to the axis in question. It is defined by the relation M being the mass of the body and equal to the sum of the masses of its constituent particles. [We may, if we please, regard any body as built up of a very great number (n) of eoual particles, each of the same mass, Rotation under the Influence of Torque, 25 which are more closely packed together Avhere the matter is dense, less closely where it is rare. Then M=wm and 2(mr'')=m2r', so thatA;*=m — = — , nm n i.e. k* is the value obtained by adding up the squares of the distances from the axis of the several equal particles and dividing by the number of terms thus added together. Tliat is, we may regard ¥ as the average value of the square of the distance from the axis to the several constituent equal par- ticles of the rigid body.] In a few cases, such as those of the thin hoops or thin hol- low cylinder figured on p. 21, the value of the radius of gyration is obvious from simple inspection, being equal to the radius of the hoop or cylinder. This is approximately true also for a fly-wheel of which the mass of the spokes may be neglected in comparison with that of the rim, and in which the width of the rim in the direction of a radius is small compared to the radius itself. Numerical Examples. — We now give a number of numerical examples, with solutions, in illustration of the prin- ciples established in this chapter. After reading these the student should work for himself examples 1, 3, 6, 9, 10, 14, and 15, at the close of Chapter III. Example 1. — A wheel weighing 81 Z6s., and whose radius of gyration is 8 inches, is acted on by a couple whose moment is 5 vound-foot units for half a minute ; find the rate of rotation produced, \st Method of <5>o/wf ion.— Taking 1 lb. as uuit mass. The unit force is the pouiidal ; ,-. I( = MP) = 81xr^y = 81xilb.-ft.2units = 36unit8. 26 Dynamics of Rotation. Moment of force or torque = 5X5r poundal-ft. units = 5x 32 = 160 units (nearly) ; angular acceleration = A = torque ^160^40 moment of inertia 36 9 radians per sec. each second ; the angular velocity generated in half a minute =a) = Ai= — X 30 radians per sec. 9 ^ 400 ,. =-^ radians per sec. o 400 1 , = -^ X — - turns per sec, 3 ZTT = -^ X '1589 turns per sec. = 127r2 turns per minute. u 2rwZ Method of Solution. — Taking the unit of force as 1 pound, then the unit of mass is 1 slug = 32 lbs. (nearly), 81 the mass of the body is ^ slugs. Torque = 6 pound-foot units ; 1 1 ,■ A torque ^ 9 40 .*. angular acceleration = A = t—V^ — i^ = 5 -^ o = Tr moment ol mertia 8 9 radians per sec. each second ; .'., as before, the rate of rotation produced in one half-min. = 1271 2 turns per minute. Example 2. — Find the torque which in one minute will stop the rotation of a wheel whose mass is 160 lbs. and radius of gyration 1 ft. 6 in. and which is rotating at a rate of 10 turns per second. Find also the number of turns the wheel will make in stopping. 1st Solution. — Using British absolute units. The unit of mass ia I lb., the unit of force 1 poundal. I = Mfc2 = 160 X (- ) units = 360 units. Angular velocity to be destroyed = a) = 10 x 27r radians per sec. = 207r ; Rotation tinder the Influence of Torq2te, 2 7 /. this is to be destroj'ed in 60 sec. ; .*. angular acceleration required SOtT TT ,. , , x=—-=- radians per sec. each second. The torque required to give this to the body in question = moment of inertia x angular acceleration = 360 X 5- ■= 1207r poundal-foot units 1207r 15 , ., . = — — =-r TT pound-ft. units. The average angular velocity during the stoppage is half the initial velocity, or 5 turns per second, therefore the number of turns made in the 60 seconds required for stopping the wheel = 60 x 5 = 300. 2nd Solution. — Using Engineer's or gravitation units. The unit force is 1 pound. The unit mass is 1 slug = 32 lbs. nearly. T TIT72 160 /3\2 .^ 45 . I = M.k^=-:r^xi — ) units =— units. The angular velocity to be destroyed = 10 x 27r radians per sec The time in which it is to be destroyed is 60 sec; .*. angular acceleration = A= — ^ =- radians per sec. each sec. The torque required to give this to the body in question 45 IT 15 = lxA=— X— -=— TT pound-ft. units as before. Example 8. — A cord, 8 feet long, is tm-apped round the axle, 4 inches in diameter, of a heavy wheel, and is pulled with a constant force of 60 pounds till it is all unwound and comes off. The wheel is then found to he rotating 90 times a minute ; find its moment of inertia. Solution. — Using British absolute units. The unit of mass is 1 lb. and of force 1 poundal. The force of 60 pounds = 60 x 32 poundals. This is exerted through a distance of 8 feet ; ,*. the work done by the force = 8 x 60 x 32 ft. -poundals. The K.E. of rotation generated = ^ Ioj- = ^ I x I' — -— — j . 28 Dyna77zics of Rotation, Equating the two we have ilx 9772=8x60x32; .•/l = 2_x8x60x32^^_f,,^^.^3^ 97r2 It will be observed that this result is independent of the diameter of the axle round which the cord is wound, which is not involved in the solution. The torque exerted would indeed be greater if the axle were of greater diameter, but the cord would be unwound .propor- tionately sooner, so that the angular velocity generated would remain the same. Using Engimer's or gravitation units, the solution is as follows: — The unit of force is 1 pound and of mass 1 slug. The work done by the 60 pound force in advancing through 8 feet =8 X 60 = 480 ft. pounds. The K E. of rotation generated = JTa)- = H x (^^ ^ ^- Vfoot-pounds of work. Equating the two we have ilx97r2=480j 2x480. 2x480x32, i = ^ .^ (slug-ft."* umts) 97r2 lb.-ft.2 units as before Example 4. — A heavy wheel rotating 180 times a minute is brought to rest in 40 sec. by a uniform friction of 12 pounds applied at a dis- tance of lb inches from the axis. How long would it take to be brought to rest by the same friction if two small masses each weighing 1 lb. were attached at opposite sides of the axis, and at a distance of two feet from it. Solution. — 1st. Using Engineer's or gravitation units. The unit of force is 1 pound and of mass 1 slug. In order to find the effect of in- creasing the moment of inertia we must first find the moment of inei tia 1, of the unloaded wheel. This is directly as the toroue reouired to 60 15 X 40 100 Rotation under the Injtuence of Torqtie. 29 stop it, directly as the time taken to stop it, and inversely as the angular velocity destroyed in that time. Thus 12x^x40 I — ^^ ^ 180 X 2iT slug-foot^ units. The moment of inertia in the second case is l2=Ii + 2mr' = I + — X22 9 100^8 . ,, «= 1- ^r^ approximately. Thus the moment of inertia is increased in the ratio 100 8^ !■> ^ "^32 IT and the time required for the same retarding torque to destroy the same angular velocity is therefore greater in this same ratio, and is now 40 sec. + ^ x , ka ^ 40 sec. = 40'31416 sec oZ lUU Or, using absolute units, thus The unit of mass is 1 lb., the unit force 1 poundal — The moment of inertia Ii of the unloaded wheel is directly as the torque required to stop its rotation, directly as the time required, and inversely as the angular velocity destroyed in that time, and is equal 15 12x32XpX40 ^ 180X2. lb-^t.^""^ts, 60 ^ 32x15x40x60 .^ , . , , » or 1,= X — ^ units (approximately) = Ib.-ft.' units. IT 30 Dynainics of Rotation. The moment of inertia in the second case = T2=Ii + 2mr2 = ?25? + 8; TT /. the moment of inertia is increased in the ratio of 3200 _ 3200 l-o : ; 77 IT and therefore the time required for the same retarding torque to destroy the same angular velocity is increased in the same proportion, and is now 8x_rr 3200' Note to Chapter II. In order to bring the substance of this chapter with greater vivid- ness and reality before the mind of the student, we have preferred to take it as a matter of observation and experiment that the power of a force to produce angular acceleration in a rigid body pivoted about a fixed axle is proportional to the product of the force and its distance from the axis, i.e. to its moment in the technical sense. But this result, together with the fact that what we termed the ' rotational inertia ' of a body is given by 2(mr2), might have been obtained as a direct deduction from Newton's Laws of Motion. We now give this deduction, premising first a statement of d'Alembert's Principle, which may be enunciated as follows : ' In considering the resultant mass-acceleration produced in any direction in the particles of any material system, it is only necessary to consider the values of the external forces acting on the system.* For every force is to be measured by the mass-acceleration it pro- duces in its own direction (Newton's Second Law of Motion), and also every force acts between two portions of matter and is accom- panied by equal and opposite reaction, producing an equal and opposite mass-acceleration (Newton's Third Law). The action and reaction constitute what we call a stress. When the two portions of matter, between Avhich a stress acts, are themselves parts of the system, it follows that the resultant mass-acceleration thereby pro- duced in the system is zero. The stress is in this case called an internal stress, and the two forces internal forces. But though the forces are internal to the system, yet they are external, or, as Newton Rotation under the Influence of Torqtie. called them, ' impressed ' forces on the two particles respectively. Hence, considering Newton's Second Law of Motion to be the record solely of observations oji particles of matter, we may count up the forces acting in any direction on any material system and write them equal to the sum of the mass-accelerations in the same direction, but in doing so we ought, in the first instance at any rate, to include these internal forces, thus -, / external forces \ , -^ / internal forces ^^^^ /'mass- accelerations^ I in any direction J ^ \} in same direction J ^ I in same direction J We now see that 2(internal forces) = 0. Hence we obtain as a deduction external forces'^ _^ /'mass-accelerations'\ ^in any direction/ \ in same direction /' or 2E = 2(ma). This justifies the extension of Newton's law from particles to bodies or systems of particles. If any forces whatever act on a free rigid body, then whether the body is thereby caused to rotate or not, the sum of the mass-accelerations in any direction is equal to the sum of the resoliites of the applied forces in the same direction. Now, since the line of action of a force on a particle is the same as the line of the mass-acceleration, we may multiply both the force and the mass-acceleration by the distance r of this line from the axis, and thus write the moment about any axis of "j ( moment of the mass-accelera- the force, on any particle, [• = S tion, along that line, of the along any line, J ( same particle, and, therefore, summing up the results for all the particles of any system, we have {moments about any axis of) C moments about the same all the forces acting on the > = 2 < axis of the mass-accele- particles of the system ) ( rations of the particles, „ /moments of the external^ ,^ /moments of the intemalX °'^i, forces J+^V forces ) = 2 /moments of the mass-X \ accelerations. / Now, not only are the two forces of an internal stress between two 32 Dynamics of Rotation. particles equal and opposite, but they are aXoti^ the same straight line* and hence have equal and opposite moments about any axis what- ever, hence the second term on the left side of the above equation is always zero, and we are left with „ /moments of the external\ ^^ /moments of the massA \ forces / ~ \ accelerations. / Now, we may resolve the acceleration of any particle into three rectangular components, one along the radius drawn from the particle perpendicular to the axis, one parallel to the axis, and one perpen- dicular to these two. It is only this latter component (which we will call ap) that has any moment about the axis in question, and its moment is rap, where r is the length of the radius. Thus the moment of the mass-acceleration of any particle of mass m may be written mrap. Now, in the case of a particle which always retains the same dis- tance (r) from the axis, ap is the rate of increase of the tangential speed 17, and if o) be the angular velocity about the axis, v=r(o. So that ap = rate of increase of rw. Also, r being constant, the rate of increase of rw is r times the rate of increase of o). Hence, in this case, ap=rd), and if, further, the whole system consists of particles so moving, and with the same angular velocity, i.e. if it is a rigid body rotating about a fixed axle, then for such a body so moving 2 (moments of the mass-accelerations) = 2mr-ra>. »=o)2mr*. Hence, in this case 2 (moments of the external forces) = angular acc° x 2{mr^ ., 1 1 *• External torque or the angular acceleration = ^. — -i — • 2 mr^; * This is, perhaps, not explicitly stated by Newton, but if it were not true, then the action and reaction between two particles of a rigid body would constitute a couple giving a perpetually increasing rotation to the rigid body to which they belonged, and affording an indefinite supply of energy. No such instance has been observed in Nature. CHAPTEE III. DEFINITIONS, AXIOMS, AND ELEMENTARY THEOREMS NECES- SARY FOR DEALING WITH MOMENTS OF INERTIA. ROUTH'S rule and its APPLICATION. Constant use will be made of the following Definitions and Propositions. Definition. — By a slight extension of language we speak of the moment of inertia of a given area with respect to any axis, meaning the moment of inertia which the figure would have if cut out of an indefinitely thin, perfectly uniform rigid material of unit mass per unit area, so that the mass of the figure is numerically equal to its area. This dynamical defini- tion becomes purely geometrical, if we say that the moment of inertia, with respect to any axis, of an area A, and of which the indefinitely small parts a^ ttj, a,, . . . are at distance r„ fj, . . . from the axis, is equal to =^ar'). It will be observed that the area may be either plane or curved. Definition. — In the same way the moment of inertia about any axis of any solid figure or volume V, of which Vi r, f, . . . are the indefinitely small constituent parts, may be defined as 34 Dynamics of Rotation, Axiom. — The moment of inertia of a body with respect to any axis is the sum of the moments of inertia of any con- stituent parts into which we may conceive it divided, and similarly the moment of inertia with respect to any axis of any given surface or volume is equal to the sum of the moments of inertia of any constituent parts into which we may con- ccive the surface or volume divided. This follows from the definitions just given. Illustration. — Thus the moment of inertia of a peg-top, shaped as in the figure, about its axis of re- volution, is equal to the moment of inertia of the hemispherical dome of wood ABC + that of the conical frustum ABDE+that of the conical point of steel DE. Axiom. — It is evident that the radius of gyration of any right prism of uniform density about any axis perpendicular to its base is the same as that of the base. For we may conceive the solid divided by an in- definite number of parallel planes into thin slices, each of the same shape as the base. Thus, if k be radius of gyration of the basal figure, and M the mass oJ the prism, the moment of inertia is MA;' units, and this holds whether the axis cuts the figure as O2O',, or does not cut it as OiO'i. Thus the problem of finding the moment of inertia of an ordinary lozenge-shaped compass needle, such as that figured, reduces to that o, o; via. 12. Moments of Inertia — Elementary Theorems, 35 of finding the radius of gyration about 00' of the horizontal cross-section ABGD. Proposition I. — The moment of inertia of a lamina about any axis Oz pei-pendicular to its plane, is equal to the sum of its moments of inertia about any two rectangular axes Ox and Oy in its plane, and intersecting at the point where the axis Oz meets the plane of the lamina. Or^ in an obvious notation. Proof. — From the figure we have at once I,=S(7wr') = l,m{x'+f) =^mx^-\-^my* Fia, 14. Example. — We have al- ready seen that a thin hoop of radius r and mass m has a moment of inertia Mr^ about a central axis perpendicular to its plane. Let I be its moment of inertia about a diameter. Then I is also its moment of inertia abont a second diameter perpendicular to the former; .*. by this pro- position 2I = Mr«i .. l = Mr2 2 ' i.e., the moment of inertia of a hoop about a diameter is only half that about a central ^3ti9 perpendicular to the plane of the hoop. 36 Dynamics of Rotation. Routh's Rule for finding the Moment of Inertia about an Axis of Symmetry in certain cases.— When the axis about which the moment of inertia is required passes through the centre of figure of the body and is also an axis of symmetry, then the value of the moment of inertia in a large number of simple cases is given by the following rule of Dr. Routh :— Moment of inertia about an axis of symmetry — AT V ^"""^ ^^ ^^ squares of the perpendicular semi-axes 3, 4, or 5, Tj _ sum of the squares of the perpendicular semi-axes or *; 371-375 The denominator is to be 3, 4, or 5, according as the body is a rectangle, ellipse (including circle), or ellipsoid (including sphere). This rule is simply a convenient summary of the results obtained by calculation. The calculation of the quantity 2(mr') is, in any particular case, most readily performed by the process of integration, but the result may also be obtained, in some cases, by simple geometry. We give in Chapter IV. examples of the calculation in separate cases, and it will be seen that they are all rightly summarised by the rule as given. Examples of the Application of Dr. Eouth's Rule. — To find the radius of gyration in the following cases : — (1) 0/ a rectangle of sides (2a) and (2b) about a central axis perpendicular to its plane. Here the semi-axes, perpendicular to each other and to the axis in question, are a and b ; therefore, apply- ing the rule, we have (2) Of the same rectangle about a central axis in its plane per- pendicular to one side (b). Here the semi-axes, perpendicular to (i? w Mo7nents of Inertia — Elementary Theorems. 37 each other and to the axis in question, are 6 anciO (see fig. 17), (since the figure has no dimensions perpendicular to its own plane) ; " 3 ~ 3 * (3) Of a circular area of radius r about a central axis perpendicular to its plane. Here the semi-axes, perpen- dicular to each other and to the axis of symmetry in question, are r and r ; / applying Routh's rule I.2_!_L-L=1_. rio.r. *= - 4 2 (4) Of a circular area about a central axis in the plane of the circle. The semi-axes, perpendicular to each other and to the axis in question, are r and o ; .*, applying Routh's rule 4 4 (5) Of uniform sphere about any central axis 5 5 (6) The moment of inertia of a uniform thin rod about a central axis perpendicular to its length. I = Massx =Massx— . 3 3 Theorem of Parallel Axes.— When the moment of inertia of any body about an axis through the centre of mass (coincident with the centre of gravity *) is known, its moment of * The centre of gravity of a body or system of heavy particles is de- fined in statics as the centre of the parallel forces constituting the weights of the respective particles, and its distance x from any plane is shown to be given by the relation _ WiXi + w^^ + w^x^+ . . . +tg,a?« ^~ Wi + w^+ +m;, _ :^{wx) 38 Dynamics of Rotation, inertia about any parallel axis can be found by applying the following proposition : — Proposition IL — The moment of inertia of any body about any axis is equal to its moment of inertia about a parallel axis through its centre of mass, plus the moment of inertia which the body would have about the given axis if all collected at its centre of mass. Thus, if I be the moment of inertia about the given axis, T^ that about the parallel axis through the centre of mass, and R the distance of the centre of gravity from the given axis, and M the mass of the body. I = I,+MR». Proof. — Let the axis of rota- tion cut the plane of the dia- gram in 0, and let a parallel axis no. 18. ^^ through the centre of mass (or centre of gravity) of the body cut the same plane in G, and let P be the projection on this where w^, W2 . . . . are the weights of the respective particles, and Xi, X2 . . . . their distances from the plane in question. Now, since the weight (w) of any piece of matter is found by ex- periment to be proportional to its mass or inertia (w), we may substi- tute (m) for (w) in the above equation, and we thus obtain _ _ 2(rwa;) *~ ^m ' For this reason the point in question is also called the centre of mass, or centre of inertia. If the weight of {i.e. the earth -pull on) each particle were not pro- portional to its mass, then the distance of the centre of gravity from ^(ivx) any plane would still be — — ' : but the distance of the centre of mass "Silnix] from the same plane would be - — : and tha two points would not then coincide. Mome7its of Inertia — Elementary Theorems, 30 plane of any particle of the body. Let m be the mass of the particle. OP and GP are projections of the radii from the two axes respectively. Let PN be perpendicular to OG. Then, since 0P'=dG* + GP»-20G.GN ; .-. 2(mOP-=)=2(mOG»)+2(mGP')-26G.2(77?GN) = MOG''+2(777GP'')-0, for, since G is the projection of the centre of mass, the posi- tive terms in the summation 2(mGN) must cancel the negative. (The body in fact would balance aboub any line through G.) Thus, I = MIIHV Applications. — (1) To find the moment of inertia of a door about its hinges. Regarding the door as a uniform thin lamina of breadth a and mass M, we see that its moment of inertia, about a parallel axis through Its centre of gravity, is L=M «)'+"•= M 12' I=MS+M(|y = M|-'. no. 19. (2) To find the moment of inertia of a uniform circular disc about a tangent in its plane. ^^^''+^ I,=M (by Routh's rule), and I=I,+Mr« =Mg+r«) = M|r«. (3) To find the moment of- inertia of a uniform 40 Dynamics of Rotation, bar or other prism about a central axis perpendicular to its lengthy where the bar is not thin. (For example of a bar-magnet of circular cross-section suspended by a fine thread as in the fig.) For the sake of being able to deal with a case like this, which is of very common occurrence, we shall prove the following : — FIG. 21, Proposition 111.— The moment of .inertia of any uniform right prism, of anrj cross section whatever about a central axis perpendicular to the line joining the centres of gravity of the ends, is equal to the moment of inertia of the same prism considered as a thin bar, plus the moment of inertia that the prism would have if condensed by endwise contraction into a single thin slice at the axis. Proof — Let g, g^, be the centres of gravity of the ends of the prism. g r?; ^ FIQ. 22. Imagine the prism divided into an indefinite number of elementary thin slices by planes parallel to the ends. The Moments of tnerha — Elementary Theorems, 41 line ^, ^1, contains the centre of gravity of each slice and of the whole prism. Let r be the distance of any one of these slices from the centre of gravity (G) of the whole prism, and m the mass of the slice. Then the moment of inertia i of this slice about the given axis 00' is, by the theorem of parallel axes, given by z=z,+mr^, where \ is the moment of inertia of the slice about a parallel axis through its centre of gravity ; .'. the whole moment of inertia I required is I=2(i,+mr2) and 2t, is the same as the moment of inertia I, of all the slices condensed into a single slice ; thus the proposition is proved. This theorem is of use in questions involving the oscillationg of a cylindrical bar magnet under the influence of the hori- zontal component of the earth's magnetic force. 42 Dynamics of Rotation. Examples for Solution. (Jn <^e.s«, as in all other Examples in the book, the anstvers given are approximate only. Unless otherwise stated, the value of g is taken as 32 feet per second each second, instead 0/ 32 19.) (1) A heavy wheel has a cord 10 feet long coiled round the axle. This cord is pulled wiih a constant force of 25 pounds till it is all unwound and comes ofiF. The wheel is then found to be rotating 5 times a second. Find its moment of inertia. Also find how long a force of 5 pounds applied at a distance of 3 inches from the axis would take to bring the wheel to rest. Ans. (1) 16-2 lb.-ft.2 units. (2) 12-72 sec. (2) A uniform door 8 feet high and 4 feet wide, weighing 100 lbs., swings on its hinges, the outer edge moving at the rate of 8 feet per second. Find (1) the angular velocity of the door, (2) its moment of inertia with respect to the hinges, (3) its kinetic energy in foot-pounds, (4) the pressure in pounds which when applied at the edge, at right angles to the plane of the door, would bring it to rest in 1 second. Ans. (1) 2 radians per sec. (2) 533-3 lb.-ft.2 units. (3) 33-3 (nearly). (4) 8-3 pounds (nearly). (3) A drum whose diameter is 6 feet, and whose moment of inertia is equal to that of 40 lbs. at a distance of 10 feet from the axis, is employed to wind up a load of 500 lbs. from a vertical shaft, and is rotating 120 times a minute when the steam is cut off. How far below the shaft-mouth should the load then be that the kinetic energy of wheel and load may just suffice to carry the latter to the surface ? Ans. 41 -9 feet (nearly). (4) Find the moment of inertia of a grindstone 3 feet in diameter and 8 inches thick ; the specific gravity of the stone being 2-14. Ans. 709-3 lb.-ft.2 units. Examples on Chapters /., //., and III. 43 (5) Find the kinetic energy of the same stone when rotating 6 times in 6 seconds. Ans. 303*7 ft. -pounds. (6) Find the kinetic energy of the rim of a fly-wheel whose exter- nal diameter is 18 feet, and internal diameter 17 feet, and thickness 1 foot, and which is made of cast-iron of specific gravity 7*2, when rotating 12 times per minute. {N.B. — Take the mean radius of the rim, viz. 8f feet, as the radius of gyration.) Ans. 23360 ft. -pounds (nearly). (7) A door 7^ feet high and 3 feet wide, weighing 80 lbs., swings on its hinges so that the outward edge moves at the rate of 8 feet per sec. How much work must be expended in stopping it ? Ans. 853*3 foot-poundals or 26*67 foot-pounds (very nearly). (8) In an Atwood's machine a mass (M) descending, pulls up a mass (m) by means of a fine and practically weightless string passing over a pulley whose moment of inertia is I, and which may be regarded as turning without friction on its axis. Show that the ac- celeration a of either weight and the tensions T and t of the cord at the two sides of the pulley are given by the equations . . . (i) . . . (ii) . . . (iii) where r= radius of pulley. What will equation (iii) become if there is a constant friction of moment Q) about the axis ? Ans. a r^CT-t-f) I (9) A wheel, whose moment of inertia is 60 Ib.-ft.* units, has a horizontal axle 4 inches in diameter round which a cord is wrapped, to which a 10 lb. weight is hung. Find how long the weight will take to descend 12 feet. Ans. 11*66 sec. (nearly). Z)»V^ \ o ^ ^ V A ^ / /< / and of mass (w), and co-ordinates x, y, z. Let OP = r, and let the distances AP, BP, CP, of P from the axes of X, y and z respectively, be called r„ r^, and r^ Then the moment of inertia of the particle P about X is mrl=m{y' •\-z^)^ „ y is mrlz=m{z^ -\-x^)^ „ zismrl=m{x^-\'y*\ Fia.2SA. therefore, for the whole body, the moment of inertia about the axis of a;, or Ij., = 2my'+2w2' „ „ „ „ y, or I^, = 2m2''+277ia;' „ „ „ „ z, or I, =2ma;' + 2m3/' Therefore I,+Iy+I,=2(2mxH27wy'+2m0'). Now this is a constant quantity, for x^-\-y''-\-z^=r^ Therefore ?7w;'+my'+m^*=mr' for every particle. Therefore ^rnx'' + 2my' + ^mz" = 2mr' = Constant. Therefore Ijj+ 1^4-1,= Constant, and this is true whatever the position of the rectangular axes through the fixed point. Proposition V. — In any plane through a given point fixed in the body, the axes of greatest and least moment of inertia^ for that planej are at right angles to each other. For let us fix, say, the axis of z ; this fixes the value of I„ and therefore Tj.+Iy= Constant. Hence, when I^ is a maximum I^ is a minimum for the plane xy, and vice versd. Proposition VI. — If about any axis (Ox) through a fixed point of a body, the moment of inertia has its greatest value, then Principal Axes, 57 ohoui some axis (Oz), at right angles to Oa;, it will have its least value ; and about the remaining rectangular axis (Oy) tJie moment 0/ inertia will be a maximum for the plane yZj and a minimum for the plane xy. For, let ug suppose that we have experimented on a body and found, for the point 0, an axis of maximum moment of inertia, Ox. Then an axis of least moment of inertia must lie somewhere in the plane through O, perpendicular to this, for if in some other plane through there were an axis of still smaller inertia, then in the plane containing this latter axis, and the axis of x we could find an axis of still greater inertia than Ox, which is contrary to the hypothesis that Ox is a maximum axis. Next, let us take this minimum axis as the axis of z. The moment of inertia about the remaining axis, that of y, must now be a maximum for the plane yz. For 1^^ being fixed, If+I»=coiistant, and therefore I^ is a maximum since I, is a minimum. Again, I, being fixed, IjB+Iy= constant, and therefore !„ is a minimum for the plane xy, since I, is a maximum. Definitions. — Such rectangular axes of maximum, minimum, ind intermediate moment of inertia are called principal axes for the point of the body from which they are drawn, and the moments of inertia about them are called principal moments of inertia for the point ; and a plane containing two of the principal axes through a point is called a principal plane for that point. When the point of the body through which the rectangular axes are drawn is the Centre of Mass, then the principal axes are called, par excellence, the principal axes of the body, and the moments of inertia about them the principal moments of inertia of the body. 58 Dynamics of Rotation. It is evident that for such a body as a rigid rod, the moment of inertia is a maximum about any axis through the centre of mass that is at right angles to the rod, and so far as we have gone, there is nothing yet to show that a body may not have several maximum axes in the same plane, with minimum axes between them. We shall see later, however, that this is not the case. Proposition VII.— To show that the moment of inertia (Iqp) ahmt any axis OP making angles a, yS, y, with the principal axes through any point 0, for which the principal moments of inertia are A, B, and C respectively y is AcosV -|-Bcos'/?-|- Ccos'y. It will conduce to clear- ness to give the proof first, for the simple case of a plane lamina with respect to axes in its plane, Let ahc be the plane lamina, Ox and Oy any rectangular axes in its plane at the point O, and about these axes let the moments of inertia be (A') and {B') respectively, and let it be required to find the moment of inertia about the axis OP, making an angle 6 with the axis of x. Let M be any particle of the lamina, of mass (w), and co- ordinates X and y. Diaw MN perpendicular to OP to meet it in N. Then the moment of inertia of the particle M about OP is mMN'. Draw the ordinate MQ, and from Q draw QS no. 24a. Principal Axes. 59 meeting OP at right angles in S. Then MN^ = OM»-ON» =a;»+y'-"(OS+SN)' and OS is the projection of OQ on OP, and therefore equal to a;cos6^ and SN is the projection of QM on OP, and therefore equal to y sin^ .-. MN'=a;»+2/»-(a;cos^-f2/sin^)' =a;'(l-cos»6')+i/Xl-sin'^)-2sin^cos% =a;'sin' Q 4-y'cos'^ ^— 2sin0cos^a;y ,*, Iop= SmMN^ = coB^ dlmy^ + sm^62mx^ — 2&mdcos62mxy =A'cos'^+B'sin2^-2sin(9cos^2ma:y. We shall now prove that when the axes chosen coincide with the principal axes so that A' becomes A and B' B, then the fac- tor ^mxi/j and therefore the last term, cannot have a finite value. For since the value A of the moment of inertia about 0, is now a maximum, Iqp cannot be greater than A, so that A— lot cannot be a —ve quantity whatever be the position of OP. i.e. A —Acos^d— Bsm^0-{-2sm9coB6^mxy cannot be— 2;e, i.e. AsmW^Bsm^9-\-2sindcos6^2mxy cannot he—ve^ now, when OP is taken very near to Ox, so that is infinitesi- mally small, then also sin^ is infinitesimally small, while cos0 is equal to 1, and so that if "Zmxy has a finite value, the two first terms of this expression, which contain the square of the small quantity sin^ may be neglected in comparison with the last term, and according as this last term is -^-ve or — i;e, 80 will the whole expression be -{-ve or —ve. Now, whether the small angle ^ is -{-ve or — w, cos^ is always -\-ve, and ^{rnxy) is always constant ; neither of these factors then changes signs with 6 ; but sin^ does change sign with 6; so that, the last term, and therefore the whole ex- pression is — ve when ^ is --ve and very small. Hence it is impossible that 2mxy can have a finite value, 6o Dynamics of Rotation, But Irnmj is constant whatever be the value of ^, and there- fore is zero or infinitesimally small even when Q is finite; therefore, finally, Iop=^cos2^+^sin2^ [If we prefer to describe the axis OP as making angles a and y8 with the rectangular axes of x and y respectively. Then in the above proof we have everywhere cosa for cos^, and cos/5 for sin^, and Iop=^cos 2a+5 cos2/3.] The proof of the general case for the moment of inertia lop of a solid body of three dimensions about any axis OP, making angles a, ^ and 7, with maximum, minimum, and intermediate rectangular axis, Oa;, Oy, Oz is exactly analogous to the above, only we have 0M»=a;'+2/'+s', instead of OM»=a;»+y* and ON=a:cosa-l-ycos/3+2;cosy, instead of ON=a:cosa+ycos^, and co3*a4-cos*^-|-cos'y=l, instead of cos''a+cos'j8=l, whence it at once follows that instead of the relation Iop=^'cos'a-f ^'cos'/?— 2cosacos/i^2wia:y, we obtain Iop=^'cosV+^cos'/34-(7cos'7— 2cosacos^2ma:y — 2cos/?cosy 2m?/2!— 2cosycosa27n2!a:. And, as before, when A'=Ay and B'=Bi or C=C, each of the last three terms can be shown to be, separately, vanishingly small, and therefore finally Iop=^cos'a+^cos'/3+ C'cos^'y. Graphical Construction of Inertia-Curves and Surfaces. — Definition. — By an ' inertia-curve ' we mean a plane curve described about a centre, and such that every radius is proportional to the moment of inertia about the axis through the centre of mass whose position it represents. Similarly, a moment of inertia surface is one having the same property for space of three dimensions. Principal Axes, 6i It is evident that we can now construct such curves or surfaces when we know the principal moments of inertia of the body. (I.) Construction of the inertia curve of any plane lamina for axes in its plane. Draw OA and OB at right angles, and of such lengths that they represent the maximum and minimum moment of in- ertia on a con- venient scale, and draw radii between them at intervals of, say, every 10°. Then mark off on these in succession the corresponding values of the expression OAcos'^-t-OBsin'^, (which may be done graphically by a process that the student will easily discover), and then draw a smooth curve through the points thus arrived at. In this way we obtain the figure OA of the diagram (Fig. 25a) in which the ratio ^^ was taken Ux> 2 equal to -. Complete inertia curves must evidently be sym- metrical about both axes, so that the form for one quadrant gives the shape of the whole. If OA were equal to OB the curve would be a circle, for if maximum and minimum values of the radius are equal, all values are equal no. 25a. 62 Dynamics of Rotation. Figure 26a shows in a single diagram the shape of the ,(2) Fio. 26a. curves when -^-j. has the vahies — , — , -, and — respectively. Ui> o 2 1 Principal Axes. 63 (II.) Condrudion of Moment of Inertia Surface. — Let any section through the centre of mass be taken, containing one of the principal axes of the body (say the minimum axis O^), and let the plane zOO of this section make angles AOC= 6 and BOG =(90° — ^) or <^, with the axes of x and y respec- rio. 27a. tively. Then, from what has been said, the intersection OC of this plane with that of xy will be a maximum axis for the section ZOO, and the value loo o^ the moment of inertia about it will be Iqc=^cos'^+-Scos'<^. Let the length of OD represent this value. The length of any radius OP of the inertia curve for the section is ^cos'a-f Scos'^+Ccos'y. Let the angle COP, or 90° — 7, which OP makes with the plane of xy be called 5. Then cos'a=cos'AOP OP OP OD* OP* OD>^OP' =cos'^cos'8 OB* OB" OD* and cos'/?=cos"BOP=====x^=cos'<^cos'a 64 Dynamics of Rotation, Therefore lop =^cos^^cos'8+5cos''<^cos^S+Ccos'y = IqcCOs'^S + (7cos*y. Therefore the inertia curve for the section zOC may he drawn in precisely the same way as for a plane lamina, and this result holds equally well for all sections containing either a maxi- mum or minimum or intermediate axis. Inspection of the inertia curves thus traced (Fig. 26a) shows that there is, in general, for any solid (except in the special case when the curve is a circle), only one maximum axis through the centre of mass, and one minimum axis, with a corresponding intermedij^te axis. Equimomental Systems.— Proposition Ylll.—Any two rigid bodies of equal mass, and for which the three principal mo- ments of inertia are respectively equal, have equal moments of inertia about all corresponding axes. Such bodies are termed equimomental. That such bodies must be equimomental about all corre sponding axes through their centres of mass follows directly from the previous proposition ; and since any other axis must be parallel to an axis through the centre of mass, it follows from the theorem of parallel axes (Chapter iii. p. 37) that in the case of bodies of equal mass, the proposition is true for all axes whatever. Any body is, for the purposes of Dynamics, completely represented by any equimomental system of equal mass. Inertia Skeleton.— Proposition IX. — For any rigid body there can be constructed an equimomental system of thres uniform rigid rods bisecting each other at right angles at its centre of masSy and coinciding in direction with its principal axes. Principal Axes, 65 For let aa\ hh\ cd (Fig. 27a) be three such rods, coiuciding respectively with the principal axes, Ox, Oy, Oz^ and let the moment of inertia of aa' about a perpendicular axis through Obe A' while that of hV is B' and that of cd is C Then, for the system of rods, If, therefore, the body in question has corresponding principal moments A, ^, C equimomental therewith when B-\-a=A a'hA'=B A'+B'=0 ''X the system of rods becomes (i) (ii) (iii) These three equations enable us to determine the values of A', By and (7, to be assigned to the rods. By addition we have, or A'\B'^(y=::\{A^B^C) whence subtracting B'+C=A we have A'=i(B+C^A) and similar expressions for B' and C Such a system of rods we may call an inertia skeleton. Such a skeleton, composed of rods of the same material and thick- ness, and differing only in length, presents to the eye an easily recognised picture of the dynamical qualities of the body. The moment of inertia will be a maximum about the £ 66 Dynamics of Rotation. direction of the shortest rod, and a minimum about the direction of the longest. [It may be mentioned that, for convenience of mathematical treatment of the more difficult problems of dynamics, advan- tage is taken of the fact that any solid can be shown to be equimomental with a certain homogeneous ellipsoid whose principal axes coincide with those of the solid. Also that if we had chosen to trace inertia curves by making the radius everywhere inversely proportional to the radius of gyration^ i.e. to the square root of the moment of inertia, then the curve for any plane would have been an ellipse, and the inertia- surface an ellipsoid.] CHAPTER VI. SIMPLE HARMONIC IMOTION. The definition of Simple Harmonic Motion may be given as follows : — Let a particle P travel with uniform speed round the cir- cumference of a fixed circle, and let N be the foot of a per- pendicular drawn from P to any- fixed line. As P travels round the circle N oscillates to and fro, and is said to have a simple harmonic motion. It is obvious that N oscillates between fixed limiting positions N, Ni which are the projections on the fixed line of the extremities A and B of the diameter parallel to it, and that at any instant the velocity of N is that part of P's velocity which is parallel to the fixed line, or, in other words, the velocity of N is the velocity of P resolved in the direction of the fixed line. Also the acceleration of N is the accelera- tion of P resolved along the fixed line. Now the acceleration of P is constant in magnitude, and always directed towards the centre C of the circle, and is equal to — =7-(u' (PC)a>' ; consequently the acceleration of 68 Dynamics of Rotation. N = (u'xtlie resolved part of PC in the direction of the fixed line=w' X (NO), being the projection of C on the fixed line. Thus we see that a particle with a simple harmonic motion has an acceleration which is at any instant directed to the middle point about which it oscillates, which is proportional to the displacement from that mean position, and equal to this displacement multiplied by the square of the angular velocity of the point of reference P in the circle. We shall see, very shortly, that the extremity of a tuning- fork or other sonorous rod, while emitting its musical note of uniform pitch performs precisely such an oscillation. Hence the name ' Simple Harmonic' The point in the figure corresponds to the centre of swing of the extremity of the rod or fork, and the points N„ Ni to the limits of its swing. The time T taken by the point N to pass from one ex- tremity of its path to the other, and back again, is the time 27r taken by P to describe its circular path, viz., — . This is defined as the * Period,* or * Time of a comjplde oscillation of N. It is evident that if at any instant N have a position such as that shown in the figure, and be moving (say) to the left, then 27r . after an interval ~ it will a'^rain be o> ° in the same position and moving in the same direction. Hence the time of a complete swing is sometimes defined as the interval between two consecutive passages of the point through the same position in the same direction p Fio. 33. Simple Ha^nnonic Motion. 69 The fraction of a period that has elapsed since the point N last passed through its middle position in the positive direc- tion is called the phase of the motion. Since the acceleration of N at any instant =NOxo)2 = displacement x w^ 2 acceleration at any instant "corresponding displacement or, abbreviating somewhat, 0)= / acceleration V displacement* Consequently Since 1 = — 0) 'p_.2^X /displacement V acceleration * acceleration The object of pointing out that the time of oscillation has this value will be apparent presently. It must be carefully noticed that to take a particle and to move it in any arbitrary manner backwards and forwards along a fixed line, is not the same thing as giving it a simple harmonic motion. For this the particle must be so moved as to keep pace exactly with the foot of the perpendicular drawn as described. This it will only do if it is acted on by a force which produces an acceleration always directed towards the middle point of its path and always proportional to its dis- tance from that middle point. We shall now show that a force of the kind requisite to produce a simple harmonic motion occurs very frequently in elastic bodies, and under other circumstances in nature. CHAPTEE VII. AN ELEMENTARY ACCOUNT OF THE CIRCUMSTANCES AND LAWS OF ELASTIC OSCILLATIONS. I. For all kinds of distortion, e.g. — stretching, compress- ing, or twisting, the strain or deformation produced by any- given force is proportional to the force, so long as the strain or deformation is but small. Up to the limit of de- formation for which this is true, the elasticity is called 'perfect' or * simple*: 'perfect,' because if the stress be removed the body is observed immediately and completely to recover itself; and 'simple,' because of the simplicity of the relation between the stress and the strain it produces. In brief — For small deformations the ratio -- — :- is strain constant. This is known in Physics as Hooke's Law. It was expressed by him in the phrase ' ut tensio sic vis.' Illustrations of Hooke's Law. m\ ¥ FIG. 84. (1) If, to the free end A of a long thin horizontal lath, fixed at the other end, a force w be applied which depresses the end through a small distance d, then a force ^w will depress it through a distance 2d, Sw through a distance Zd, and so on. 70 Elastic Oscillations. 71 (2) If the lath be already loaded so as to be already much bent, as in the fig., it is, nevertheless, true if the breaking-strain be not too nearly approached, that the application of a small additional force at A will produce a further deflection proportional to the force applied. But it must not be expected that the original force w will now produce the original depression dj for w is now applied to a different object, viz., a much bent lath, whereas it was origin- ally applied to a straight lath. Thus w will now produce a further depression d' and 2w „ „ „ 2(Z' 3m; „ „ „ 3d' where d' differs from d. (3) A horizontal cross-bar is rigidly fixed to the lower end of a long thin vertical wire; a couple is applied to the bar in a horizontal plane, and is found to twist it through an angle 6 : then double the couple will twist it through an angle 20, and so on. This holds in the case of long thin wires of Bteel or brass for twists of the bai through several complete revolutions. (4) A long spiral spring is stretched by hanging a weight W on to it (Fig. 37). If a small extra weight w produces a small extra elongation «, Then „ 2w „ „ 2<5, and „ 3w „ „ 3«, and so on. Similarly, if a weight w be subtracted from W the shortening will be e, and „ 2uj „ „ „ 2«, and so on. This we might expect, for the spring when stretched by the weight Gj- -OTn ■5^ no. 36. ^l Dynai7iics of Rotation. Ww is so slightly altered from the condition in which it was when stretched by W, that the addition of w must produce the same elongation e as before ; therefore the shortening due to the removal of w must be e. From these examples it will be seen that the law enunciated applies to bodies already much dis- torted as well as to undistorted bodies, but that the value of the constant ratio stress Fio. 87. corresponding small strain is not generally the same for the undistorted as for the distorted body. 2. If a mass of matter be attached to an elastic body, as, for instance, is the weight at A in Fig. 35, the cross-bar AB in Fig. 36, or the weight W in Fig. 37, and then slightly displaced and let go, it performs a series of oscillations in coming to rest, under the influence of the force exerted on it by the elastic body. And at any instant the displacement of the mass from its position of rest is the measure of the distortion of the elastic body, and is therefore proportional to the stress between that body and the attached mass. Hence we see that the small oscillations of such a mass are performed under the influence of a force which is propor- tional to the displacement from the position of rest. 3. We shall consider, first, linear oscillations, such as those of the mass W in Fig. 37, and shall use for this constant ratio — — ^^^H£ the symbol R, the force being expressed in displacement absolute units. It will be observed that E measures the resisting power of the body to the kind of deformation in question. For if the displacement be unity, then R=the Elastic Oscillations. ^3 corresponding force : thus, E is the measwe of resistance tJie body offers when subjected to unit defoi'maiion.^ We shall consider only cases in which the mass of the elastic body itself may be neglected in comparison with the mass M of the attached body whose oscillations we study. 4. If the force be expressed in a suitable unit, the accelera- tion of this mass at any instant is , and is directed towards the position of rest. Since the mass M is a constant quantity, and since the ratio -=-. — = is constant and displacement equal to K ; therefore, also the ratio ^^^£^I^£^^ is constant displacement and = |. 5. Now it is, as we have seen, the characteristic of Simple Harmonic Motion that the acceleration is proportional to the displacement from the mean position. Consequently we see that when a mass attached to an elastic body, or otherwise influenced by an * elastic ' force, is slightly displaced and then let go, it performs a simple harmonic oscillation of which the corresponding Time of a complete oscillation = 27r^ /displ acement V acceleration 6. Hence (from § 4) we have for the time of the complete linear oscillation of a mass M under an elastic force, T = 2;r /I VR whatever may be the 'amplitude' of the oscillation, so long as the law of 'simple elasticity' holds. * This is sometimes called the modulus of elasticity of the body for the kind of deformation in question, as distinguished from the modulus of eiasticitv of the tmJmtance. 74 Dynamics of Rotation. 7. Applications. — (l) A 10 Ih. mass hangs from a long thin light spiral spring. On adding 1 oz. the spring is found to be stretched 1 inch; on adding 2 ozs., 2 inches. Find the time of a complete small oscillation of the 10 lb. weight. Here we see that the distorting force is proportional to tlie dis- placement, and therefore that the oscillations will be of the kind examined. We will express masses in lbs., and therefore forces in poundals. Since a distorting force of ^^ pounds ( = 3| = 2 poundals) produces a displacement of ^^ ft. .-. the ratio ,. f^^^^ =E = -|- = 24. displacement jV = 4'05 sec. (approximately). (2) A mass of 20 lbs. rests on a smooth horizontal plane midway between two upright pegs, to which it is attached by light stretched elastic cords. (See fig.) SO lbs FIO. 38. It is found that a displacement of ^ an inch towards either peg calls out an elastic resistance of 3 ozs., which is doubled when the displacement is doubled. Find th( time of a complete small oscilla- tion of the mass about its posUion of rest. force 3 X ^ X 32 abs. units. Here K = ^j^pig^^g^g^^^- ^ ^144 .-. T = 27r. /M -Stt /20; V R V 144 sec. r= 2*34 sec. (approximately). 8. The student will now perceive the significance of the limitation of the argument to cases in which the mass of the elastic body itself may be neglected. If, for example, the Elastic Oscillations. 75 spring of Fig. 37 were a very massive one, the mass of the lower portion would, together with W, constitute the total mass acted on by the upper portion ; but as the lower portion oscillated its form would alter so that the acceleration of each part of it would not be the same. Thus the considerations become much more complicated. Hence, also, it is a much simpler matter to calculate, from an observation of the ratio R, the time of oscillation of a heavy tvQ. 39. mass W placed on a light lath as in the figure, than it is to calculate the time of oscillation of the lath by itself. 9. Extension to Angular Oscillations.— Since any conclusion with respect to the linear motion of matter is true also of its angular motion about a fixed axle, provided we sub- stitute moment of inertia for mass ; couple for force ; angular distance for linear distance ; it follows that when a body performs angular oscillations under the influence of a restoring couple whose moment is proportional to the angular displacement, then the time of a complete oscillation is ^ Vi '''■ where I is the moment of inertia with respect to the axis of oscillation and R is the ratio = — ,. P. : the angular displacement couple being measured in absolute units. 76 Dynamics of Rotation. M) FIO. 40. Applications. — (l) Take the case of a simple pendulum of length 1 and mass m. Wlien the displacement is 6, the moment of the restoring force is m<7 X OQ (see fig.) = mgl sin 6 = mgl 6 ii 6 is small. • Tt— "^o^T^e"^ of couple _''^^^^^nr corresponding displac*" 6 Also 1—ml^ V ri,lg as also may be shown by a special inves- tigation, such as is given in Garnett's Dynamics, Chap. V. (2) Next take the case of a body of any shape in which the centre of gravity G is at a distance I from the axis of suspen- sion 0. As before, when the body is displaced through an angle 6, the moment of the restoring couple is mgl sin 6=mgl 6 it 6 is but small, and p_moment of couple _mg'Zd_ , angular displac*" " * 6 T=27r /X no. 41. 10. Equivalent Simple Pendu- lum. — If K be the radius of gyra- tion of the body about the axis Elastic Oscillations. 71 of oscillation, then I = ?'Jv-, and Let L be the length of a simple pendulum which would have the same period of oscillation as this body. The time of a com- plete oscillation of this simple pendulum is 27r / -. For this to be the same as that of the body we must have or L = K-* Hxamples. — (l) A thin circular hoop of radius r hung over a peg swings under the action of gravity in its own plane. Find the lejujth of the equivale7it simj)le pendulum. Here the radius of gyration K is given byK2 = r2 + /2. And the distance I from centre of gravity to point of suspension is equal to r. .'. length of equivalent simple pen- dulum, which is equal to " + 7-2 ^ riQ. 42. is, in this case, — The student should verify this by the experiment of hanging, together with a hoop, a small bullet by a thin string whose length is the diameter of the hoop. The two will oscillate together. (2) A Korizontal bar magnet^ of moment inertia I, makes n complete oscillations per sec. Deduce from this the value of the product Mil where M is the magnetic moment of the magnet, and H the strength of the eartWs horizontal field. Let ns be the magnet. {See Fig. 43.) Imagine it displaced through an angle 6. Then since the magnetic moment is, by definition, the value of the couple exerted on the magnet when placed in a uniforn\ field of unit strength at right angles to the lines of force, it follows 78 Dynamics of Rotation, that when placed in a field of strength H at an angle 6 to the lines of force the restoring couple =MH sin B. = MH^ when 6 is small. . -p ^ restoring couple _MH^ angular displac'- ~ B = MH. And T = 27r V R N.B. MH orMH=^\ -The student of physics will remember that by using the same magnet placed mag- netic E. and W., to deflect a small needle situated in the line of its axis, we can M H* by combining the result of an oscilla- tion-observation of MH with that of a find the value of the ratio Thus M deflection-observation of — , we obtain Kio. 43. the value of H at the place of observa- tion. (8) A bar magnet oscillates about a central vertical axis under the injiuence of the earth's horizontal Jield^ and performs 12 complete small oscillations in one minute. Two small masses of lead, each weigh- ing one oz., are placed on it at a distance of 3 inches on either side of the axis, and the rate of oscillation is now reduced to 1 oscillation in 6 seconds. Find the moment of inertia of the magnet. Let the moment of inertia of the magnet be I oz.-inch^ units. Then the moment of inertia of the magnet with the attached masses is 1 + 2x1 x3-=(I + 18) oz.-inch2 units. The time of a complete oscillation of magnet alone is 5 sec. Thus 27r and 2fr \/e~' /l + l V R 6. Elastic Oscillations. 79 /l + 18_6 V —1 s"' 1 + 18 36 0' -T"^25* .-. 1 = 40.909 oz.-mch2 units. II. Oscillating Table for finding Moments of Inertia. — A very useful and convenient apparatus for find- ing the moment of inertia of small objects such as magnets, galvanometric coils, or the models of portions of machinery too large to be directly experimented upon, consists of a flat light circular table 8 or 10 inches in diameter, pivoted on a vertical spindle and attached thereby to a flat spiral spring of many convolutions, after the manner of the balance-wheel of a watch, under the influence of which it performs oscilla- tions that are accurately isochronous. See, Fig. 4:3a. The first thing to be done is to determine once for all the moment of inertia of the table, which is done by observing, first, the time T^ of an oscillation with the table unloaded, and then the time Ti of an oscillation with a load of known moment of inertia Ix—e.g. the disc may bfe loaded with two small metal cylinders of known weight and dimensions placed at the extremities of a diameter. Then, since and Ti = 27r^ /L±l» V R Tj'-T/ I^ having thus been determined, the value of I for any object laid on the disc, with its centre of gravity directly over the 8o Dynamics of Rotation. axis, is found from the corresponding time of oscillation T by the relations T=27r^/ir V R audT=27r /l±L V li whence 1=1 q^2 rp Fio. 43a, Examples on Chapters VI and VII 8 1 Examples for Solution. (1) A thin heavy bar, 90 centimetres long, hangs in a horizont:il position by a light string attached to its ends, and passed over a peg vertically above the middle of the bar at a distance of 10 centi- metres. Find the time of a complete small oscillation in a vertical plane containing the bar, under the action of gravity. Ans. 1'766 , . . . seconds. (2) A uniform circular disc, of 1 foot radius, weighing 20 lbs., is pivoted on a central horizontal axis. A small weight is attached to the rim, and the disc is observed to oscillate, under the influence of gravity, once in 3 seconds. Find the value of the small weight. Ans. 1*688 lbs. (3) A bar magnet 10 centimetres long, and of square section 1 centimetre in the side, weighs 78 grams. "When hung horizontally by a fine fibre it is observed to make three complete oscillations in 80 seconds at a place where the earth's horizontal force is '18 dynes. Find the magnetic moment of the magnet. Ans. 202*48 . . . dyne-centimetre units. (4) A solid cylinder of 2 centimetre radius, weighing 200 grams, is rigidly attached with its axis vertical to the lower end of a fine wire. If, under the influence of torsion, the cylinder make 0*5 complete oscillations per second, find the couple required to twist it through four complete turns. Ans. 3200 Xtt^ dyne-centimetre units. (5) A pendulum consists of a heavy thin bar 4 ft. long, pivoted about an axle through the upper end. Find (1) the time of swing ; (2) the length of the equivalent simple pendulum. Ans. (1) 1*81 seconds approximately; (2) 2*6 feet. (6) Out of a uniform rectangular sheet of card, 24 inches x 16 inches, is cut a central circle 8 inches in diameter. The remainder is then supported on a horizontal knife-edge at the nearest point of the circle to a shortest side. Find the time of a complete small oscilla- tion under the influence of gravity (a) in the plane of the card ; (6) in a plane perpendicular thereto. Ans. (a) 1*555 seconds ; (6) 1*322 seconds. (7) A long light spiral spring is elongated 1 inch by a force of 2 pounds, 2 inches by a force of 4 pounds. Find how many complete small oscillations it will make per minute with a 3 lb. weight attached. Ans. 1527 CHAPTEE VIII. CONSERVATION OF ANGULAR MOMENTUM. Analogue in Rotation to Newton's Third Law of Motion. — Newton's Third Law of Motion is the statement that to every action there is an equal and opposite reaction. This law is otherwise expressed in the Principle of the Conservation of Momentum, which is the statement that when two portions of matter act upon each other, whatever amount of momentum is generated in any direction in the one, an equal amount is generated in the opposite direction in the other. So that the total amount of momentum in any direction is unaltered by the action. In the study of rotational motion we deal not with forces but with torques, not with linear momenta but with angular momenta, and the analogous statement to Newton's Third Law is that 'no torque, with respect to any axis, can be exerted on any portion of matter without the exertion on some other portion of matter of an equal and opposite torque about the same axis. ' To deduce this as an extension of Newton's Third Law, it is sufficient to point out that the reaction to any force being not only equal and opposite, but also in the same straight line as the force, must have an equal and opposite moment about any axis. The corresponding principle of the conservation of angular momentum is that by no action of one portion of matter 82 Conser'vation of Angular Momentum. ^2> on another can the total amount of angular momentum, about any fixed axis in space, be altered. Application of the Principle in cases of Motion round a fixed Axle. — We have seen (p. 21) that the ' angular or rotational momentum ' of a rigid body rotating about a fixed axle is the name given, by analogy with linear momentum (mv), to the product Io>, and that just as a force may be measured by the momentum it generates in a given time, so the moment of a force may be measured by the angular momentum it generates in a given time. 1st Example of the Principle.— Suppose a rigid body A, say a disc whose moment of inertia is Ij, to be rotating with angular velocity (o^ about a fixed axle; and that on the same shaft is a second disc B of moment of inertia Ij, and which we will at first suppose to be at rest. Now, imagine the disc B to be slid along the shaft till some projecting point of it begins to rub against A. This will set up a force of friction be- tween the two, the moment of 'which will at every instant be the same for each, consequently as much angular momentum as is destroyed in A will be imparted to B, so that the total quantity of angular momentum will remain unaltered. Ultimately the two will rotate together with the same angular velocity 12 which is given by the equation no. 44. m Q: 84 Dynamics of Rotation. If the second disc had initially an angular velocity Wj, then the equation of conservation of angular momentum gives us Ii+Ia ' which, it will be observed, corresponds exactly to the equation of conservation of linear momentum in the direct impact of inelastic bodies, viz. : — (Wli + wig) V— mxVx + 7712^2. 2nd Example. — A horizontal disc whose moment of inertia is I„ rotates about a fixed vertical axis with ^ I angular velocity Wi. Imagine a particle of any mass to be detached from the rest, and no. 45. connected with the axis by an independent rigid bar whose mass may be neglected. At first let the particle be rotating with the rest of the system with the same angular velocity w,. Now, let a horizontal pressure, always at right angles to the rod and parallel to the disc, be applied between them so that the rotation of the particle is checked, and that of the remainder of the system accelerated {e.g. by a man standing on the disc and pushing against the radius rod as one would push against the arm of a lock-gate on a canal), until finally the particle is brought to rest. By what has just been said, as much angular momentum as is destroyed in the particle will be communicated to the remainder of the disc, so that the total angular momentum will remain unaltered. We may now imagine the stationary non-rotating particle transferred to the axis, and there again attached to the remainder of the system, without affecting Conservation of Angular Afoinenlum. 85 the motion of the latter. If 1 2 is now the reduced moment of inertia of the system, and Wj its angular velocity, we have, by what has been said, l2«a=Ii<»i II Or, we may imagine the particle, after having been brought to rest, placed at some other position on its radius, and allowed to come into frictional contact with the disc again, till the two rotate together again as one rigid body. If I3 be now the moment of inertia of the system, we shall have Ist03 = l2a)2 = Ii ^M o will remain constant, there being no externa] force with a moment about the axis to in- \ crease its amount. But it is not so apparent ^^^ in this case how the increase of angular ^^g ^y velocity that accompanies the diminution of moment of inertia has been brought about. For simplicity, consider instead of a finite mass M a par- ticle of mass m at distance r from the axis when rotating with angular velocity w. The moment of inertia I of the particle is then mr^ and the angular momentum =Iw but ria=.'o the tangential speed ; .'. the angular momentum =mrz;, thus for the angular momentum to remain constant v must increase exactly in proportion as r diminishes, and vice versa. In the 6ase in question the necessary increase in v is effected by the resolved part of the central pull in the direction of the motion of the particle. For the instant this pull exceeds the value 1^^ j of the centripetal force necessary to keep the particle moving in its circular path, the particle begins to be drawn out of that path, and no longer moves at right angles to the force, but partly in its direction, and with increasing velocity, along a spiral path. Conservation of Angular Momentum. 87 This increase in velocity involves an increase in the kinetic energy of the particle equivalent to the work done by the force. Consideration of the Kinetic Energy. — It should be observed, in general, that if by means of forces having no moment about the axis we alter the moment of inertia of a system, then the kinetic energy of rotation about that axis ia altered in inverse proportion. For, let the initial moment of inertia I, become Ij under the action of such forces, then the new angular velocity by the principle of the con- servation of angular momentum is toj = Wj X =i and the new value of the rotational energy is JliwJ = Jl2a>?X \ = (original energy) x — i. la The student will see that in Example 2, p. 84, the stoppage of the particle with its radius rod in the way described involves the communication of additional rotational energy to the disc, and that, in Example 3, the pulling in of the cord attached to the sliding masses communicated energy to the system, though not angular momentum. Other Exemplifications of the Principle of the Conservation of Angular Momentum.— (i) A juggler standing on a spinning disc (like a music-stool) can cause his rate of rotation to decrease or increase by simply extending or drawing in his arms. The same thing can be done by a skater spinning round a vertical axis with his feet close together on well-rounded skates. ss Dynamics of Rotation. (2) When water is let out of a basin by a hole in the bottom, as the outward parts approach the centre, any rota- tion, however slight and imperceptible it may have been at first, generally becomes very rapid and obvious.* (3) Thus, also, we see that any rotating mass of hot matter which shrinks as it cools, and so brings its particles nearer to the axis of rotation, will increase its rate of rotation as it cools. The sun and the earth itself, and the other planets, are pro- bably all of them cooling and shrinking, and their respective rates of rotation, therefore, on this account increasing. If the sun has been condensed from a very extended nebu- lous mass, as has been supposed, a very slow rate of revolu- tion, in its original form, would suffice to account for the present comparatively rapid rotation of the sun (one revolu- tion in about 25 days). Graphical representation of Angular Momentum. — The angular momentum about any line of any moving body or system may be completely represented by marking off on that line a length proportional to the angular momen- tum in question. The direc- tion of the corresponding rotation is conveniently in- dicated by the convention that the length shall be named in the direction in which a right-handed screw would advance through its nut if turning with the same rotation. Thus OA and OB in Figs. 48 and 49 would represent angular momenta, as ^ It can be shown that other causes besides that mentioned may also produce the effect referred to. no. 48. Conservation of Angular Momentum. 89 shown by the arrows. Since a couple has no moment about any axis in its plane and has the same moment about every axis perpendicular to its plane, and is measured by the angular momentum it generates in unit time about any such axis, it follows that a line drawn parallel to the axis of a couple and of a length proportional to its moment, equally represents both the couple and the angular momentum it would generate in unit time, and hence the angular momenta generated by couples can be combined and resolved exactly as we combine or resolve couples. Thus if a body whose angular momentum has been generated by the action of a couple and is represented by OA, be acted on for a time by a couple about a perpendicular axis, this cannot alter the angular momentum about OA, but will add an angular momentum which we may represent by OB perpendicular to OA. Then the total angular momentum of the body must be represented by the diagonal OC of the parallelogram AB (Fig. 50). And in general the amount of angular momentum existing about any line through is represented by the projection on that line of the line representing the total angular momentum in question. Moment of Momentum. — The phrase * angular or rota- tional momentum' is convenient only so long as we are dealing with a single particle or with a system of particles rigidly connected to the axis, so that each has the same angular velocity ; when, on the other hand, we have to con- sider the motions of a system of disconnected parts, the principle of conservation of angular momentum is more con- 90 Dynamics of Rotation, veniently enunciated as the * conservation of moment of momentum.' By the moment of momentum, at any instant, of a particle about any axis is meant the product (mi^) of the resolved part imv) of the momentum in a plane perpendicular to the axis, and the distance (p) of its direction from the. axis ; or the moment of momentum of a particle may be defined and thought of as that part of the momentum which alone is concerned in giving rotation about the axis, multiplied by the distance of the particle from the axis. Since the action of one particle on another always involves the simultaneous generation of equal and opposite momenta along the line joining them (see Note on Chapter ii.), it follows that the moments about any axis of the momenta generated by such interaction are also equal and opposite. Hence in any system of particles unacted on by matter outside there is conserva- tion of moment of momentum, or, in algebraical language, ^{mvp) — constant. The moment of momentum of a particle as thus defined is easily seen to be the same thing as its angular momentum iw. For, as we have seen — see Appendix (1) — a>= ^ and i (by definition) =mr^ ,\ i(i)=mvj). General Conclusion. — The student will now be prepared to accept the conclusion that if, under any circumstances, we observe that the forces acting on any system cause an altera- tion in the angular momentum of that system about any given fixed line, then we shall find that an equal and opposite altera- tion is simultaneously produced in the angular momentum about the same axis, of matter external to the system. Conservation of Angular Momentum. 91 Caution. — At the same time he is reminded that it is only in the case of a rigid body rotating about a fixed axle that we have learned that the angular momentum about that axle is measured by Io>. He must not conclude either that there is no angular momentum about an axis perpendicular to the actual axis of rotation ; or that Iw will express the angular momentum about an axis when a> is only the component rotational velocity about that axis. Thus if a body, consisting of two small equal masses mm, united by a massless rigid rod be rotating, say right-handedly, about a fixed axis oy, bisecting the rod and making an acute angle with it, then it is evident that, at the instant represented in the diagram, though the rota- tion is about oy and has no com- ponent about ox, yet, on account of the velocity of each mass per- pendicular to the plane of the paper there is actually more angular momentum (left-handed) about ox than there is (right-handed) about oy. This point will be fully discussed in Chapter xii. FIG. 51. Ballistic Pendulum. — In Robins's ballistic pendulum, used for determining the velocity of a bullet, we have an interesting practical application of the principle of conserva- tion of moment of momentum. The pendulum consists of a massive block of wood rigidly attached to a fixed horizontal axle above its centre of gravity abou^ which it can turn 92 Dynamics of Rotation. freely, the whole being symmetrical with respect to a vertical plane through the centre of mass perpendicular to the axle. The bullet is fired horizontally into the wood in this plane of symmetry perpendicular to the axle, and remains embedded in the mass, penetration ceasing before the pendulum has moved appreciably. The amplitude of swing imparted to the pendulum is observed, and from this the velocity of the bullet before impact is easily deduced. Let I be the moment of inertia of the pendulum alone about the axle, M its mass, d the distance of its centre of gravity from the axis, and let Q be the angle through which the pendulum swings to one side. Then, neglecting the relatively small moment of inertia of the bullet itself, the angular velocity o) at its lowest point is found by writing Kinetic energy of^ mork subsequently done pendulum \ — \ against gravity in rising at lowest point, J I through angle ^, JI(u'' = M(7(^(l-cos^), an equation which gives us w. Now, let V be the velocity of the bullet before impact that we require to find, m its mass, and I the shortest distance from the axis to the line of fire. Then writing moment of momentum about^ _ rangular momentum about axis before impact, J "~ \ axis after impact, we have mvl=\ia^ which gives us v. The student should observe that we apply the principle of conservation of energy only to the frictionless swinging of the pendulum, as a convenient way of deducing its velocity at its lowest point. Of the original energy of the bullet the greater part is dissipated as heat inside the wood. Examples on Chapter VIII. 93 In order to avoid a damaging shock to the axle, the bullet would, in practice, be fired along a line passing through the centre of percussion, which, as we shall see (p. 124), lies at a distance from the axis equal to the length of the equivalent simple pendulum. Examples. (1) A horizontal disc, 8 inches in diameter, weighing 8 lbs., spins without appreciable friction at a rate of ten turns per second about a thin vertical axle, over which is dropped a sphere of the same weight and 5 inches in diameter. After a few moments of slipping the two rotate together. Find the common angular velocity of tha two, and also the amount of heat generated in the rubbing together of the two (taking 772 foot-pounds of work as equivalent to one unit of heat). Ans. (i) 7 '6 19 tarns per sec. „ (ii) -008456 units of heat. (2) A uniform sphere, 8 inches in radius, rotates without friction about a vertical axis. A small piece of putty weighing 2 oz. is projected directly on to its surface in latitude 30° on the sphere and there sticks, and the rate of spin is observed to be thereby reduced by iV- Find the moment of inertia of the sphere, and thence its specific gravity. Ans. (i) 7 J oz.-foot2 units. „ (ii) -0332. (3) Prove that the radius vector of a particle describing an orbit under the influence of a central force sweeps out equal areas in equal times. (4) A boy leaps radially from a rapidly revolving round-about on to a neighbouring one at rest, to which he clings. Find the eflfect on the second, supposing it to be unimpeded by friction, and that the boy reaches it along a radius. (6) Find the velocity of a bullet fired into a ballistic pendulum from the following data : — The moment of inertia of the pendulum is 200 lb. -foot^ units, and it weighs 20 lbs. The distance from the axis of its centre of gravity is 3 feet, and of the horizontal line of fire is V feet ; the bullet penetrates as far as the plane containing the axis and centre of mass and weighs 2 oz. The cosine of the observed swing is ^. Ans. 950*39 feet per sec. (taking ^ = 32-2.) CHAPTEE IX. ON THE KINEMATICAL AND DYNAMICAL PROPERTIES OF THE CENTRE OF MASS. Evidence of the existence for a Rigid Body of a point possessing peculiar dynamical relations. — Suppose a single external force to be applied to a rigid body previously at rest and perfectly free to move in any manner. The student will be prepared to admit that, in accordaiice with Newton's Second Law of Motion, the body will experience an acceleration proportional directly to the force and inversely to its mass and that it will begin to advance in the direction of the applied force. But Newton's Law does not tell us explicitly whether the body will behave differently according to the position of the point at which we apply the force, always assuming it to be in the same direction. Now, common experience teaches us that there is a difference. If, for example, the body be of uniform material, and we apply the force near to one edge, as in the second figure, the body begins to turn, while if we FIG. 53. , i n 1 • 1 ^ apply the force at the opposite edge, the body will turn in the opposite direction. It is always possible, however, to find a point through which, if the force be ap- 94 Fio 52. Properties of the Ce^itre of Mass. 95 plied, the body will advance without turning. The student should observe that if, when the force was applied at one edge of the body, as in Fig. 2, the body advanced without turning, precisely as we may suppose it to have done in Fig. 1, this would not involve any deviation from Newton's Law applied to the body as a whole, for the force would still be producing the same mass-acceleration in its own direction. It is evidently important to know under what circum- stances a body will turn, and under what circumstances it will not. The physical nature of the problem will become clearer in the light of a few simple experiments. Experiment 1. — Let any convenient rigid body, such as a walking- stick, a hammer, or say a straight rod conveniently weighted at one end, be held vertically by one hand and then allowed to fall, and while falling let the observer strike it a smart horizontal blow, and observe whether this causes it to turn, and which way round ; it is easy, after a few trials, to find a point at which, if the rod be struck, it will not turn. If struck at any other point it does turn. The ex- periment is a partial realisation of that just alluded to. Experiment 2. — It is instructive to make the experiment in another way. Let a smooth stone of any shape, resting loosely on smooth hard ice, be poked with a stick. It will be found easy to poke the stone either so that it shall turn, or so that it shall not turn, and if the direction of the thrusts which move the stone without rotation be noticed, it will be found that the vertical planes containing these directions intersect in a common line. If, now, the stone be turned on its side and the experiments be repeated, a second such line can be found intersecting the first. The intersection gives a point through which it will be found that any force must pass which will cause motion without turning. Experiment 3. — With a light object, such as a flat piece of paper or card of any shape, the experiment may be made by laying it, with a very fine thread attached, on the surface of a horizontal mirror dusted over with lycopodium powder to diminish friction, and then tugging 96 Dynamics of Rotation. at the thread ; the image of the thread in the mirror aids in the alignment. The thread is then attached at a different place, and a second line on the paper is obtained. If a body, in which the position of the point having these peculiar properties has been determined by any of the methods described, be examined to find the Centre of Gravity, it will be found that within the limits of experimental error the two points coincide. This result may be confirmed by the two following experiments. Experiment 4. — Let a rigid body of any shape whatever be allowed to fall freely from rest. It will be observed that, in whatever position the body may have been held, it falls without turning (so long at any rate as the disturbing effect of air friction can be neglected). In this case we know that the body is, in every position, acted on by a system of forces (the weights of the respective particles) whose resul- tant passes through the centre of gravity. Experiment 5. — When a body hangs at rest by a string, the direc- tion of the string passes through the centre of gravity. If the string be pulled either gradually or with a sudden jerk, the body moves upward with a corresponding acceleration, but again without turning. This is a very accurate proof of the coincidence of the two points. We now pass to another remarkable dynamical property, which may be enunciated as follows : — ' If a couple he applied to a non-rotating rigid body that is perfectly free to move in any manner ^ then the body will begin to rotate about an axis passing through a point not distinguishable from the centre of gravity.' This very important property is one which the student should take every opportunity of bringing home to himself. If a uniform bar, AB, y^ G R free to move in any man- I • I A ' y" ner, be acted on by a couple ,10.54. whose forces are applied Properties of the Centre of Mass. 97 as indicated, each at the same distance from the centre of mass G, then it is easy to believe that the bar will begin to turn about G-. But if one force be applied at A and the other Fio. 55. v\Q. 56. ^ at G itself, as in Fig. 55, or between A and G, as in Fig. 56, then it is by no means so obvious that G will be the turning point. The matter may be brought to the test of experiment in the manner indicated in the following figure. V ^s= no. 67. Experiment 6. — A Magnet NS Ues horizontally on a square-cut block of wood, being suitably counterpoised by weights of brass or lead, 80 that the wood can float as shown in a large vessel of still water. The whole is turned so that the magnet lies magnetic east and west, and then released, when it will be observed that the centre of gravity G remains ^ vertically under a fixed point P as the whole ^ The centre of gravity must, for hydrostatic reasons, be situated in the same vertical line as the centre of figure of the submerged part of the block. G * 98 Dynamics of Rotation, turns about it. It is assumed here that the magnet is affected by a horizontal couple due to the earth's action. We now proceed to show experimentally that when a rigid body at rest and free to move in any manner is acted on by forces having a resultant which does not pass through the centre of Gravity, then the body begins to rotate with an- gular acceleration abofui the centre of Gravity, while at the same time the centre of gravity advances in the direction of the resultant force. Experiment 7. — Let any rigid body hanging Fio. 68w freely at restby a string be struck a smart blow vertically upwards. It will be observed that the centre of gravity rises vertically^ while at the same time the body turns about it, unless the direction of the blow passes exactly through the centre of gravity. [It will be found convenient in making the experiment for the observer to stand so that the string is seen projected along the vertical edge of some door or window frame. The path of the Centre of Gravity will then be observed not to deviate to either side of this line of projection. The blow should be strong enough to lift the centre of mass considerably, and it is well to select an object with considerable moment of inertia about the Centre of Gravity, so that though the blow is eccentric the body is not thereby caused to spin round so quickly as tP strike the string and thus spoil the experiment] Properties of the Centre of Mass, 99 We have now quoted direct experimental evidence of the existence in the case of rigid bodies of a point having peculiar dynamical relations to the body, and have seen that we are unable experimentally to distinguish the position of this point from that of the centre of gravity. But this is no proof that the two points actually coincide. Our experiments have not been such as to enable us to decide that the points are not in every case separated by -^^^ inch, or even by yjg- inch. "We shall now proceed to prove that the point which has the dynamical relations referred to is that known as the Centre of Mass, and defined by the following relation. Let TWi, mj, ma, ... be the masses of the constituent particles of any body or system of particles; and let Xi, a^aj ajg, . . . be their respective distances from any plane, then the distance a of the centre of mass from that plane is given by the - miaji 4-^23^2+ . . . relation x— mi+mj+mj-f . . ^'^ ^= ^' That the centre of mass whose position is thus defined coincides experimentally with the centre of gravity, follows, as was pointed out in the note on p. 38, from the experimental fact, for which no explanation has yet been discovered, that the mass or inertia of diff'erent bodies is proportional to their weight, i.e. to the force with which the earth pulls them. Our method of procedure will be, first formally to enunciate and prove certain very useful but purely kinematical pro- perties of the Centre of Mass, and then to give the theoretical proof that it possesses dynamical properties, of which we have selected special examples for direct experimental demon* stration. lOO Dynamics of Rotation, By the student who has followed the above account of the experimental phenomena, the physical meaning of these pro- positions will be easily perceived and their practical import- ance realised, even though the analytical proofs now to be given may be found a little difficult to follow or recollect. Proposition I. — (Kinematical.) On the displacement of the centre of mass. If tJie particles of a system are displaced from their initial positions in any directions, then the displacement d experienced by the centre of mass of the system in any one chosen direction is con- nected with the resolved displacements di, d^, d^, . , , of the respective particles in the same direction by the relation ^_ m,d,+m^di-\- .... -fmA Wi+m,+ .... +?w„ ^ ^(md) or d= -^ — -. zm Proof. — For, let any plane of reference be chosen, perpen- dicular to the direction of resolution, and let x be the distance of the centre of mass from this plane before the displacements, x' its distance after the displacements, T"v,«« - 2(7wa;) . - ^m(x-\-d) ^(mx) 2{md) ~ 2m ■*■ 2m ••• ^-^=^=^^- Q.E.D. If ^{md)=o, then d=o, i.e. if, on the whole, there is no mass-displacement in any given direction, then there is no displacement of the centre of mass in that direction. Properties of the Centre of Mass. loi Definitions. — If a rigid body turns while its centre of mass remains stationary, we call the motion one of pure rotation. When, on the other hand, the centre of mass moves, then we say that there is a motion of translation. Proposition II. — (Kinematical.) On the velocity of the centre of mass of a system. If v,, Va, v^ ... be the respective velocities in any given direction at any instant of the particles of masses mj, m„ mj, etc., of any system, then the velocity v of their centre of mass in the same direction is given by the relation This follows at once from the fact that the velocities are measured by, and are therefore numerically equal to, the displacements they would produce in unit time. Proposition III. — (Kinematical.) On the acceleration of the centre of mass. If ai, a^, . . . be the accelerations in any given direction, and at the same instant of the respective particles of masses m^, m^ . . . of a system, then the acceleration a of their centre of mass in the same direction at thai instant, is given by the relation zra This follows from Proposition II., for the accelerations are measured by, and are therefore numerically equal to, the velocities they would generate in unit time. 102 Dynamics of Rotation, Summary. — These three propositions may be conveniently summed up in the following enunciation. /- mass-disjplacements\ The sumofihe resolutes in any direction of the < momenta > \ mass- accelerations J of the particles of any system is equal to the total mass of the r displacement \ system multiplied hy the < velocity > , in the same direction, \ acceleration ) of the centre of mass. Corresponding to these three Propositions are three others referring to the sum of the moments about any r mass-displacements \ axis of the-( momenta > of the particles of a system, V mass-accelerations / and which may be enunciated as follows : — *The algebraic sum of the moments about any given fixed r mass-displacements \ axis of the < momenta > of the particles of any system \ mass-accelerations ) is equal to the sum of the moments of the same quantities about a parallel axis through the centre of masSj plus the moment about the given axis C displacement \ of the\ velocity >of the centre of mass, multiplied by the ^ acceleration ^ 77ites5 of the whole system. Since the moment of the mass-displacement of a particle has no special physical significance, we will begin at the second link of the chain and give the proof for the angular momenta. Properties of the Centre of Mass. lo o Proposition IV. — (Kinematical.) The angular momentum of any system of particles about any fixed axis, is equal to the angular momentum about a parallel axis through the centre cf muss + the angular momentum which the system would have about the given axis if all collected at the centre of mass and moving with it. Proof. — Let the plane of the diagram pass through a particle P and be perpendicular to the given fixed axis and let G be the pro- jection on this ])lane of the centre of mass. Join OGr. -^^ S. Let PQ represent the resolute {v) of 1 ~ ~ ^ / the velocity of P in O FIG. 59. the plane of the diagram ; PS the resolute v' of this velocity perpendicular to OG. Draw OM (=p) perpendicular to PQM; GT parallel to PQ and GN (=/) parallel to GM. Then the angular momentum of P about 0=pmv=mvx OM=zmv(TM.-^OT)=mrp'-{-mvOG^=p'mv+mv'OG. Therefore, summing for all the particles of the system, Total angular momentum about = ^(pm.v) = ^(p'mv)-\' '2(0Gmv')=2{p'mv)-\-0G'2(mv') = ^(p'mv)-\-0Gv^mj where v' is the velocity of the centre of mass perpendicular to OG. This proves the proposition. Corollary. — If the centre of mass is at rest v'=0 and "Spmv ■=2p'mv, thus the angular momentum of a spinning body whose centre of mass is at rest is the same about all parallel txes. It is very important that the student should realise 164 Dynamics of Rotation, this. He will easily associate it with the fact that the angular momentum measures the impulse of the couple that has produced it, and that the moment of a couple is the same about all parallel axes. Proposition V. — (Kinematical). In exactly the same way, substituting accelerations for velocities, we can prove that 2(pma) = ^p'ma + 0Ga'2w. Proposition VI. — (Dynamical.) On the motion of the centre of mass of a body under the action of external forces. We shall now show that The acceleration in any given direction of the centre of mass of a material system algebraic sum of the resolutes in that direction of the external forces ~ mass of the whole system For, by Newton's Second Law of Motion (see note on Chapter II), (the algebraic sum of the ex-\ _ /the algebraic swn of the mass-\ ternal forceSf ) \ accelerations^ J 2E=2(?7ia); hit by III. 2(ma)=a2m; - 2E 2.m which is what we had to prove. This result is quite independent of the manner in which the external forces are applied, and shows that when the forces are constant and have a resultant that does not pass through Properties of the Centre of Mass, 105 the centre of mass (see Fig. 53), the centre of mass will, nevertheless, move with uniform acceleration in a straight line, so that, if the body also turns, it must he about an axis through the centre of mass. Proposition VII. — (Dynamical.) The application of a couple to a rigid body at rest and free to move in any manner, can only cause rotation about some axis through the centre of mass. For, by Proposition VI, Acceleration of centre of mass=— — , but in the case of a couple 2E=0 for every direction, so that the centre of mass has no acceleration due to the couple, which, therefore (if the body were moving), could only add 'rotation to the existing motion of translation. Proposition VIII. — (Dynamical.) When any system of forces is applied to a free rigid body, the effect on the rotation about any axis fixed in direction, passing through the Centre of Mass and moving with it, is independent of the motion of the Centre of Mass. For, by the note on Chapter 11. , p. 32, 2 (moments of the mass- \ ^ ,, . r .. ^ , . , Resultant moment of the accelerations about any > = ^ , . . „ - . - I external forces, axis fixed in space) ' or '2(pma) = li but, by Proposition V. (see Fig. 59, p. 103), ^pma)=^p'ma) + 0G l{ma') .\ l(pma)+OGr ^ma')=h. to6 Dynamics of Rotation. If, now, the centre of mass be, at the instant under considera- tion, passing through the fixed axis in question (which is equivalent to the axis passing through the Centre of Mass and moving with it), OG=0 and the second term vanishes and 2(1^ ma) =Ly i.e. the sum of the moments of the mass-accelerations about such a moving axis = resultant moment of the external forces, precisely as if there had been no motion of the Centre of Mass. This proposition justifies the independent treat- ment of rotation and translation under the influence of external forces. On the direction of the Axis through the Centre of Mass, about which a couple causes a free Rigid Body to turn. — Caution. — The reader might be at first disposed to think that rotation must take place about an axis perpendicular to the plane of the applied couple, especially as the experiments quoted do not reveal the contrary ; but it should be observed that the experiment of the floating magnet was not such as would exhibit satisfactorily rotation about any but a vertical axis. It is not difficult to show that rotation will not in general begin about the axis of the couple. To fix the ideas, let us imagine a body composed of three heavy bars cross- ing each other at right- Fio. 60. angles, at the same point 0, which is the centre of mass of the whole system, and let Properties of the Centre of Mass, 167 the bar AB be much longer and heavier than either of the other two CD and EF, and let this massive system be embedded in surrounding matter whose mass may be neglected in comparison. It is evident that the moment of inertia of such a system is much less about AB than about CD or EF, or that it will be easier to rotate the body about AB than about CD or EF. Hence, if a couple be applied, say by means of a force through the centre of mass along EF, and an equal and opposite force at some point P on the bisector of the angle DOB, then this latter force will have equal resolved moments about CD and about AB. But rotation will begin to be generated more rapidly about the direction of AB than about that of CD, and the resulting axis of initial rotation will lie nearer to AB than to CD, and will not be perpendicular to the plane of the couple. In fact, the rods EF and CD will begin to turn about the original direction of AB, considered as fixed in space, while at the same time the rod AB will begin to rotate about the axis CD, considered as fixed, but with a more slowly increasing velocity. We shall return to this point again in Chapter xii. Total Kinetic Energy of a Rigid Body.— When a body rotates with angular velocity (w) about the centre of mass, while this has a velocity (v), we can, by a force through the centre of mass destroy the kinetic energy of translation (JMt;') leaving that of rotation (i^Iw*) unaltered. Thus, the total kinetic energy =JMt;' + |Ia>". In the examples that follow on p. 110, this consideration often gives the readiest mode of solution. io8 Dynamics of Rotation^ Examples. (1) Two Tnasses M and m, of which M is the greater, hang at the ends of a weightless cord over a smooth horizontal peg, and move under the action of gravity ; to find the acceleration of their centre of mass and the upward pressure of the peg. Taking the downward direction as + ve, the acceleration of M is M while that of m is —g M-m Hence substituting in the M + m ----- - ^ M+m' general expression for the acceleration of the centre of mass, ^i,., 5^?^ we have - _ My(M - m) -mgr(M - m) _ (M-m)^ ^ (M + m)2 ~^(M + m)2* The total external force which produces this acceleration is the sura of the weights - the push P of the peg ; .-. (M + m)^-P = (M + m)^P^"'^^' '{M + mf '(M+m)2 i^!^ absolute units of force. M + m (2) A uniform solid sphere rolls without slipping down a plane inclined at an angle 6 with the horizontal ; to find the acceleration of its centre and the tangential force due to the friction of the plane. It is evident that if there were no friction the sphere would slide and not roll, and therefore that the accelera- tion (a) of the cen- tre C, which we wish to find is due to a total force mg sin ^ - P parallel to no. 60a. Properties of the Centre of Mass, 109 the plane, where P is the friction. ^_. ^>^sin — - — ^ y^ijere m = the mass of the sphere, m p = gr sin ^ - - . . . . (i) Now, the moment of the force (P) with reference to a horizontal axis through C is Pr, and, therefore, calling the angular acceleration of the sphere A, and its radius of gyration fc, Pr=AI=AxTOA:2 , (ii) P^Ai2 " m r ,'. substituting in (i) . . k¥- r Now, since the sphere is at any instant turning about the point of contact with the plane, we have © = — and A = - (iii) T T /. substituting in the equation, we get a=gsme-—j In the case of a sphere F=^^— — ^=-— r* 5 5 5 ^ 7 Hence, equating the total force to the mass-acceleration down the plane, mg sin 6 -F^mg sin 6 x - 2 P=— -mgrsiu^. [This question might also have been solved from the principle of the Conservation of Energy.] I lO Dynamics of Rotation, Examples for Solution. (1) Show that when a coin rolls on its edge in one plane, one-third of its whole kinetic energy is rotational. (2) Show that when a hoop rolls in a vertical plane, one-half of its kinetic energy is rotational (3) Show that when a uniform sphere rolls with its centre moving along a straight path, f of its kinetic energy is rotational. (4) Find the time required for a uniform thin spherical shell to roll from rest 12 feet down a plane inclined to the horizontal at a slope of 1 in 50. Ans, 8 seconds (nearly). (5) You are given two spheres externally similar and of equal weights, but one is a shell of heavy material and the other a solid sphere of lighter material. How can you easily distinguish between khem? (6) A uniform circular disc, half an inch thick and 12 inches in radius, has a projecting axle of the same material half an inch in diameter and 4 inches long. The ends of this axle rest upon two parallel strips of wood inclined at a slope of 1 in 40, the lower part of the disc hanging free between the two. The disc is observed to roll through 12 inches in 53*45 seconds. Deduce the value oig correct to 4 significant figures. Ans. gr = 32*19 /.s.s. (7) What mass could be raised through a space of 30 feet in 6 seconds by a weight of 50 lbs., hanging from the end of a cord passing round a fixed and a moveable pulley, each pulley being in the form of a disc and weighing 1 lb, Ans. 84*02 lbs. Instructions. — Let M be the mass required. Its final velocity at the 30 end of the six seconds will be twice the mean velocity, i.e. 2 x V /•*• 6 = 10/. 5. From this we know all the other velocities, both linear and angular — taking the radius of each pulley to be r. Equate the sum of the kinetic energies to the work done by the earth's pull. Remember that the fixed pulley will rotate twice as fast as the moveable one. (8) A uniform cylinder of radius r, spinning with angular velocity a), about its axis, is gently laid, with that axis horizontal, on a hori- zontal table with which its co-eflicient of friction is /*. Prove that it will skid for a time -— ^ and then roll with uniform velocity ^. 3/xgf "^3 CHAPTER X. CENTRIPETAL AND CENTRIFUGAL FORCES. We have, so far, dealt with rotation about a fixed axis, or rather about a fixed material axle, without inquiring what forces are necessary to fix it. We shall now consider the question of the pull on the axle. Proposition. — Any particle moving with uniform angular velocity w round a circle of radius r must have an acceleration rco^ towards the centre, and must therefore be acted on by a force mrm^ towards the centre, where m is the mass of the particle.^ Let us agree to represent the velocity {v) of the particle at A by the length OP measured along the radius OA at right angles to the direc- tion of the velocity. Then the velocity at B is represented by an equal length OQ measured along the radius OB, and the velocity added in the interval is (by the triangle of velocities) represented by the line PQ. If the interval of time considered be very short, B is very near to A and Q to P, and PQ is sensibly perpendicular to the radius * Since w= -, ru}^= _, and it is proved in text-books on the dynamics of a particle, such as Gamett's Elementary Dynamics and Lock's Dynamics, that the acceleration of a point moving uniformly in a circle with speed v is towards the centre, and is — : thus the Student will be already familiar with the propositior^. We give, however, a rather difiereot proof. Ul 1 1 2 Dynamics of Rotation, OA, and therefore the velocity it represents is along this radius and towards the centre. This shows that the addition of velocity, i.e. the acceleration, is towards the centre. Let the very short interval in question be called {di). Then PQ represents the velocity added in time {dt\ i.e. the acceleration X {di). PQ _ acceleration x {di) •*• OP V But ^ = angle POQ=a>((^0 acceleration X (c?0 _ /^a V — \ f acceleration =t;o>=r(i>*. Hence, if the particle have a mass tw, the centripetal or centre-seeking jorce required to keep it moving with uniform speed in a circle of radius r is a force of — or mrtii^ units. The unit force is here, as always, that required to give unit acceleration to unit mass. Thus, if the particle has a mass of m lbs., and moves with speed v feet per second in a circle of radius r feet, the force is — or mra>' poundals ; while if the particle have a mass of m grams and move with velocity of v centimetres per second in a circle of radius r centi- metres, then the centripetal force is m— dynes. Illustrations of the use of the terms ' Centripetal Force' and * Centrifugal Force.'— A small bullet whirled round at the end of a long fine string approximates to the case of a heavy particle moving under the influence of a centripetal force. The string itself is pulled away from the centre by the bullet, which is said to exert on it a centrifugal force. Similarly a marble rolling round the groove at th^ Centripetal and Centrifugal Forces, 1 1 3 rim of a solitaire-board is kept in its circular path by the centripetal pressure exerted by the raised rim. The rim, on the other hand, experiences an equal and opposite centrifugal push exerted on it by the marble. In fact, a particle of matter can only be constrained to move with uniform angular velocity in a circle by a centri- petal force exerted on it by other matter, and the equal and opposite reaction exerted by the body in question is in most cases a centrifugal force. Thus, when two spheres attached to the ends of a fine string rotate round their common centre of gravity on a smooth table, each exerts on the string a centrifugal force. In the case, however, of two heavenly bodies, such as the earth and moon, rotating under the influ- ence of their mutual attraction about their common centre of gravity, the force that each exerts on the other is centripetal. We cannot in this case^ermre anything corresponding to the connecting string or to the external rim. Centripetal Forces in a Rotating Rigid Body.— When we have to deal, not with a single particle, but with a rigid body rotating with angular velocity w, and of which the particles are at different distances, r,, rj, r„ etc., from the axis, it becomes necessary to find the resultant of the forces (mirito*), (TTijrjw'), etc., on the several particles. Rigid Lamina. — We take first the case of a rigid lamina of mass M turning about an axle perpendicular to its plane. Here all the forces lie in one plane, and it is easily shown that the resultant required is a single force, through the centre of mass of the lamina, and equal to MEw', where R is the distance from the axis to the centre of mass ; [and MRw', again, is equal to M— , where V is the speed of the centre \\ of mass in its circular path]. rt4 Dynamics of Rotation, This may be shown at once from the following well known proposition in Statics : * If two forces be represented in Fio. 63. Fio. 64. magnitude and direction by m times OA and n times OB, then their resultant is represented in magnitude and direc- tion by (m+w) times OC, C being a point which divides the line AB, so that the ratio "^^-Z CB m •B (For proof sec Greave's Stalks, p. 18.) For let A and B be any two particles of the lamina, and let their masses be m and w, then the force along OA is mw^OA, and that along OB is fiw'OB; therefore, by the proposition quoted, the resultant force is (m-{-7i)(o'OC, and passes through 0, which, since it divides the distance AB inversely as the masses, is the centre of mass and centre of gravity of the two particles. This resultant may next be combined with the force on a third particle of the rigid system, and so on till all are included. Centripetal mid Centrifugal Forces, 1 1 5 Exlension to Solids of a certain type.— By piling up laminae whose centres of gravity all lie on the same Fio. 7a line parallel to the axis, as indicated in the diagrams (Figs. 66-70), we may build up solids of great variety of shape, and 116 Dynamics of Rotation, by then combining resultants on the several laminae, we see that in order to keep the body rotating with uniform angular velocity, we require only a single force passing through its centre of gravity, and directed towards the axis and equal to MRw*, where M is the mass of the whole body. The requisite force might, in such a case, be obtained by connecting the centre of gravity of the body to the axis by a string. The axis would then experience a pull MRw', which changes in direction as the body rotates. If the axis passes through the centres of mass of all such laminae, then R = 0, and the force disappears, and the axis is unstrained. It is often of high importance that the rapidly rotating parts of any machinery shall be accu- rately centred, so that the strains and consequent wear of the axle may be avoided. Convenient Dynamical Artifice. — It should bo observed that the single force applied at the centre of mass would not supply the requisite centripetal pressure to the individual particles elsewhere if the body were not rigid. If, for example, the cylinder AB rotating as indicated about 00' consisted of loose smooth particles of shot or sand, it would be necessary to enclose these in a rigid case in order that the single force a])plied at G should maintain equilibrium. The particles between G and A would press against each other and against the case, and tend to turn it round one way, while those between G and B would tend, by their centrifugal pressure, to turn it the other way. Now, it is very convenient in dealing with problems involving the con- sideration of c* applied at the centre of mass of the body, and a couple in a plane parallel to the axis; but the axis of this couple will not, except in special cases, be perpendicular to the plane containing the centre of gravity and the axis of rotation. Centripetal and Centrifugal Forces, 1 1 9 This result may be reached by taking, first, any two par- ticles of the body, such as A and B in the diagram, of masses m and n respectively, and showing that the centrifugal forces 'p and ^ exerted by each are equivalent to two forces along CA' and CB' (the direc- tions of the projections of y and 2 on a plane perpendicular to the axis and containing the centre of mass of the two particles), together with the two couples py and qc[. Then the two coplanar forces along CA' and CB' have, as before (see p. 114), a resultant (m+ri)(u'CG, while the two couples combine into a single resultant couple In a plane parallel to or containing the axis of rotation but not parallel to CG. In this way, taking all the particles in turn, we arrive at the single force through the centre of mass of the whole and a single couple. Centrifugal Couples vanish when the rotation is about a Principal Axis, or about an Axis parallel thereto. — It is obvious that in the case of a thin rod (see Fig. 72) there is no centrifugal couple when the rod is either parallel or perpendicular to the axis of rotation, which is then a principal axis (or parallel to a principal axis), and it is easy to show that for a rigid body of any shape the centrifugal couples vanish when the rotation is about a principal axis. I20 Dynamics of Rotation. y s > -.^ X y >< 7xy [ / D X rio. 73a. Froof. — Let us fix our attention on any particle P of a body which rotates with uniform positive angular velocity Wy, about a fixed axis Oy passing through the centre of mass of the body. Let O^ and Oz be any two rectangular axes perpendicular to Oy. The centripetal force on the particle is always equal to mrin^ (see Fig.YSA), and its component parallel to Ox is —mxiHy^ (negative in sign because it tends to decrease a:), and this changes the value of the momentum of the particle perpendicular to the plane yz. The moment about Oz of this component of the centripetal force is —iHymxy and measures the rate at which angular momentum is being generated about Oz. The sum of the moments of such com- ponents for all the particles of the body is — (u/2ma:y, and this with its sign changed, or in^^mxyy is the measure of the centri- fugal couple about Oz. Now l.mxy vanishes when either x or y is a principal axis of the body (see pp. 59 and 60). Heuc3 there is no centrifugal couple when the body rotates about a principal axis. It follows that a rigid body rotating about a principal axis, and unacted on by any external torque, will rotate in equili- brium without the necessity of being tied to the axis. But in the case of bodies which have the moments of inertia about two of the principal axes equal, the equilibrium,, as we have seen, will not be stable unless the axis of rotation is the axis of greatest moment. Centripetal and Centrifugal Forces, 121 Importance of properly shaping the parts of machinery intended to rotate rapidly. — In coimecLion with this dynamical property of principal axes, the student will now recognise the importance of shaping and balancing the rotating parts of machinery, so that not merely shall the axis of rotation pass through the centre of mass, but it shall also be a priiici])al axis^ since in this way only can injurious stresses on the axle be completely avoided. Equimomental bodies similarly rotating have equal and similar centrifugal couples.— Pvw/.— Let ^u t/u ^i be any three rectangular axes of the one body (1), and iCa, ^2* 2^2 the corresponding axes of the other (2), and let A', B', C be the respective moments of inertia about these axes. Then about any other axis, in the plane xy making any angle a with (x), ^ ( = 90° — a) with (?/), and 7 ( = 90°) with {z)f the moment of inertia of (1) is (as we see by refer- ring to p. 60), A'cos' a-fB'cos'yS— 22wza;iyi, cos a cos )8, while that of (2) about a corresponding axis is A' cos* a-fB'cos'jS— 227wa:2?/aCosacosy8 (for the terms involving cosy as a factor disappear since cos 7 = COS 90° = 0), and, since the bodies are equimomental, these two expressions are equal, therefore 2mxiyi = l.mx^yi. Therefore for equal rates of rotation about either x or y, the centrifugal couples about {z) are equal, and this is true for all corresponding axes. Substitution of the 3-rod inertia-skeleton. — This result justifies us in substituting for any rotating rigid body 122 Dynamics of Rotation, its three-rod inertia-skeleton, the centrifugal couples on which can be calculated in a quite simple way. We will take first a solid of revolution, about the axis of minimum inertia C. For such a body the rod C is the longest, and the two rods A and B are equal, and these two, together with an equal length measured off the cen- tral portion of the third rod (C), combine to form a system dynamically equivalent to a sphere for which all centri- fugal couples vanish about all axes ; there thus remains no. 78a for consideration only the excess at the ends of the rod (see Fig. 73b). The centrifugal couple is in this case obviously about an axis perpendicular to the plane (xy) containing the rod C and the axis of rota- tion (y), and its value, as we have seen, is w'^mxy; now if r be the distance of a particle from the origin 0, a:=r sin Q and !/=r cos ^, .'. iii^^mxy-=iii^ sin 6 cos B ^mr^, and 2mr'= moment of inertia about z of the projecting ends of the rod C = moment of inertia of the whole rod about a perpen- dicular axis— the moment of inertia of rod A about a perpendicular axis, = i(A+B-C)-J(B+C-A) (see p. 65) =A-0 Therefore the centrifugal couple = a>*( A— C) sin ^cos 0. If had been the axis of maximum moment of inertia then the rod would have been the shortest of the three rods instead of the longest, and we should have had a defect instead of an Centripetal and Centrifugal Forces. 123 excess to deal with, and the couple would have been of the opposite sign and equal to w'(C — A) sin ^cos Q. We shall make use of these results later on in connection with a spinning-top and gyroscope. (See Appendix.) If all three moments of inertia are unequal, we could describe a sphere about the shortest rod as diameter, and should then have a second pair of projections to deal with. We could find, in the way just described, the couple due to each pair separately and then combine the two by the parallelo- gram law. We shall, however, not require to find the value of the couple except for solids of revolution. Transfer of Energy under the action of Centri- fugal Couples. — Returning again to our uniform thin rod as a conveniently simple case, let us suppose it attached in the manner indi- cated in either figure (Figs. 74 and 75), so as to turn freely in the framework about the axle CC, while this rotates about the fixed axis 00'. The rod, if liberated in the position shown, while the frame is rotating, will oscillate under the influence of the centrifugal couple, swinging about the mean position ah. It is impos- Bible in practice to avoid friction at the axle CO', and these 124 Dynamics of Rotation. oscillations will gradually die away, energy being dissipated as frictional heat. To the question, Where has this energy come from ? the answer is, From the original energy of rota- tion of the whole system, for as the rod swings from the position AB to the position aJ, its moment of inertia about 00' is being increased, and this by the action of forces having no moment about the axis, consequently, as wo saw in Chapter viii. p. 87, the kinetic energy due to rotation about 00' (estimated after the body has been fixed in a new position) must be diminished in exactly the same pro- portion. Thus, O if the whole system be rotat- ing about 00', and under the influence of no external torque, and with the rod initially in • ° no. 75. the position AB, then as the rod oscillates, the angular velocity about will alternately decrease and increase ; energy of rotation about the axis 00' being exchanged for energy of rotation about the axis CC CHAPTER XL CENTRE OF PERCUSSION. X ^, G Let a thin rod AB of mass m be pivoted at about a fixed axle perpendicular to its length, ^ and let the rod be struck an impulsive blow (P) at some point N, the direc- tion of the blow being perpendicular to the plane containing the fixed axle and the rod, and let G be the centre of mass of the rod (which is not neces- saiily uniform). Suppose that simultaneously with the impulse (P) at N there act at G two opposed impulses each equal and parallel to (P). This will not alter the motion of the rod, and the blow is seen to be equivalent to a parallel impulse (P) acting through the centre of mass G, and an impulsive couple of moment PxGN. On account of the former the body would, if free, immediately after the im- pulse be moving onwards, every part with the velocity v=. _1P}_ iP)_ B Fio. 76. (P) m ' On account of the latter it would be rotating about G .,, , , ., (P)XNG with an angular velocity about an axis Oy making an angle with the minimum axis C. The centri- petal couple is in the plane yx containing the axis C, and its moment about ^!=a>x angular momentum about x. (See Fig. 8lA.) The angular velocity cu may be resolved into two com- ponents about the principal axes, viz., w sin Q about OA and (0 cos Q about OC. The angular momentum about OA is then A(o sin ^, and about OC is Cw cos 0} The sum of the * It is only because OA and OC are each principal axes that we can write the angular momentum about them as equal to the resolved part Qf the angular velocity x the moment of inertia. FIQ. 81a. Total Angular Momentum. 135 resolutes of these about Ox is — Atosin ^cos ^+C(ucos ^sin^=--(A— C)a)cos ^sin 9. This multiplied by w or — ta2(A— C) sin Q cos ^ is therefore the moment of the centripetal couple about z required to maintain the rotation. This result with the sign changed is the value of the centrifugal couple, and agrees with that obtained in a different way on p. 122. Rotation under the influence of no torque. — A rigid body of which one point, say its centre of mass, is fixed can only move by turning about that point, and at any instant it must be turning about some line, which we call the instantaneous axis, passing through that point. Every particle on that line is for the instant stationary, though, in general, it will be gaining velocity (such particles will in fact have acceleration but not velocity). Hence after a short interval of time these same particles will no longer be at rest, and will no longer lie on the instantaneous axis. If, however, the axis of rotation is a principal axis, and no external forces are acting, there will be no tendency to move away from it, for there will be no centrifugal couple. We thus realise that if such a body be set rotating and then left to itself its future motion will depend on the direction and magnitude of the centrifugal couple. After it is once abandoned, however, the axis of total angular momentum must remain fixed in space ; it is therefore often termed the invarialle axis. CHAPTER XIII. ON SOME OF THE PHENOMENA FRESENTED BY SPINNING BODIES. The behaviour of a spinning top, when we attempt in any way to interfere with it, is a matter that at once engages and even fascinates the attention. Between the top spinning and the top not spin- ning there seems the di (Terence almost between living matter and dead. While spinning, it appears to set all our pre- conceived views at defiance. It stands on its point in apparent contempt of the conditions of statical stability, and w^hen we endeavour to turn it over, seems not only to resist but to evade us. The phenomena presented are best studied in the Gyroscope, which may be described as a metal disc AB (see Fig. 82) with a heavy rim, capable of rotating with little friction about an axle CD, held, as shown in the figure, by a frame, so that the wheel can turn either about the axle CD, or (together with the frame CD) about the axle EF, perpendicular to CD, or about the axle m FIG. 82. Phenomena presefited by Spinni7ig Bodies. 137 GH, perpendicular to every possible position of EF, or the wheel may possess each of these three kinds of rotation simultaneously. The axle CD we shall refer to as the axle of spin, or axle (1), the axle EF we shall call axle (2), and the axle GH, which in the ordinary use of the instrument is vertical, we shall call axle (3). Suppose now the apparatus to be placed as shown in the figure, with both the axle of spin and axle (2) horizontal, and let rapid rotation be given to it about the axle of spin CD. Experiment 1. — If, now, keeping GH vertical, we move the whole bodily, say by carrying it round the room, we observe that the axle of rotation preserves its direction unaltered as we go. This is only an Illustration of the conservation of angular momentum. To change the direction of the axle of spin would be to alter the amount of rotation about an axis in a given direction, and would require the action of an external couple, such as, in the absence of all friction, is not present. Experiment 2.— If, while the wheel is still spinning, we lift the frame-work CD out of its bearings at E and F, we find we can move it in any direction by a motion of translation, without observing any- thing to distinguish its behaviour from that of an ordinary non-rotat- ing rigid body : but the moment we endeavour in any sudden manner to change the direction of the axle of spin an unexpected resistance is experienced, accompanied by a curious wriggle of the wheel Experiment 3. — For the closer examination of this resistance and wriggle let us endeavour, by the gradually applied pressure of smooth pointed rods (such as ivory penholders) downwards at D and upwards at C, to tilt the axle of spin — axle (1) — from its initial direction, which we will again suppose horizontal, so as to produce rotation about EF — axle (2). We find that the couple thus applied is resisted, but that the whole framework turns about the vertical axle GH — axle (3) — and continues so to turn as long as the pressures are applied, ceasing to turn when the couple is removed : the direction of the 1 3 8 Dynamics of Rot atioii, rotation about axle (3) is counter-clockwise as viewed from above when the spin has the direction indicated by the arrows. (See Fig. 83.) Experiment 4. — If, on the other hand, we endeavour by means of a gradually applied horizontal couple to impart to the already spinning wheel a rotation about axle (3), we find that instead of such rotation taking place, the wheel and its frame begin to rotate about the axle (2), and continue so to rotate so long as the couple is steadily applied. The direction of this rotation is that given in Fig. 84 below, and Fio. 83. Flo. 84, the effects here mentioned may be summarised by saying that with the disc rotating about axle (1) the attempt to impart rotation about a perpendicular axle is resisted, but causes rotation about a third axle perpendicular to both. In each diagram the applied couple is indicated by straight arrows, the original direction of spin by unbroken curved arrows, and the direction of the rotation produced by the couple by broken curved arrows. It should be noticed that it is only for convenience of reference that we suppose the axis of spin to be initially hori- zontal. Had this axis been tilted, and axle (3) placed per- pendicular to it, the relation of the directions would be the same. Definition. — The rotation of the axle of spin in a plane per- pendicular to that of the couple applied to it is called a pre- Phenome7ia presented by Spinning Bodies. 139 cessional motion — a phrase borrowed from Astronomy — and we shall speak of it by that name. The application of the couple is said to cause the spinning wheel to ' precess.' Rule for the direction of Precession.— In all cases the following Kule, for which the reason will be apparent shortly, will be found to hold. The Precession of the axle of spin tends to convert the existing spin into a spin about the axis of the couple, the spin being in tht direction required by the couple. Experiment 6. — The actions just described may be well exhibited by attaching a weight at or D, as in the accompanying figure no. 85. no. 8A, (Fig. 85), or still more strikingly, by supporting the frame CD on a point P, by means of a projection DK, in whose lower side is a shallow conical hollow, in the manner indicated in the figure (Fig. 86). 1 40 Dynamics of Rotation. If the wheel were not spinning it would at once fall, but instead of falling it begins when released to travel with precessional motion round the vertical axis HP, and even the addition of a weight W to the framework at C will, if the rate of spin be sufficiently rapid, produce no obvious depression of the centre of gravity of the whole, but only an acceleration of the rate of precession round HP. It will, indeed, be observed that the centre of gravity of the whole does in time descend, though very gradually, also that the precession grows more and more rapid. Each of these effects, however, is secondary, and due, in part at any rate, to friction, of which we can never get rid entirely. In confirmation of this statement we may at once make the two following experiments. Experiment 6. — Let the precession be retarded by a light hori- zontal couple applied at and D. The centre of gravity at once descends rapidly. Let the precession be accelerated by a horizontal couple. The centre of gravity of the whole begins to rise. Thus we see that any friction of the axle GH in Fig. 85, or friction at the point P in Fig. 86, will cause the centre of gravity to descend. Experiment 7. — Let Experiment 6 be repeated with a much smaller rate of original spin. The value of the steady precessional velocity will be much greater. Hence we see that friction of the axle of spin might account for the gradual acceleration of the precessional velocity that we observe. Experiment 8. — Let us now vary the experiment by preventing the instrument from turning about the vertical axle (3), which may be done by tightening the screw G (Fig. 82), the base of the instrument being prevented from turning by its friction with the table on which it stands. If we now endeavour as before to tilt the rotating wheel, we find that the resistance previously experienced has disappeared, and that the wheel behaves to all appearance as if not spinning. Phenomena presented by Spinning Bodies, 141 Experiment 9. — But if the stem GH be held in one hand, while with the otlier a pressure is applied at C or D to tilt the wheel, its 'effort to precess ' will be strongly felt. Experiment 10. — Let us now loosen the screw G again, but fix the frames CD, which may be done by pinning it to the frame EF, so as to prevent rotiiiion about the axle EF. It will now be found that if, as in Experiment 4, we aj^ply a horizontal couple, the previously felt resistance has disappeared ; but here, again, the ' effort to precess' will be strongly felt if the framework CD be dismounted and held in the hand, and then given a sudden horizontal twist. Precession in Hoops, Tops, etc. — It needs only the familiarity that most of us obtain as children with hoops, tops, bicycles, etc., to recognise that we have in these also the very same phenomenon of precession to explain Thus, when a hoop rolling away from us is tilted over to the left, it nevertheless does not fall as it would if not rolling. Since the centre of gravity does not descend, the upthrust at the ground must be equal to the weight of the hoop, and must /^' ^ constitute with it ^ couple ^^^--"'^ " tending to turn the hoop over. We observe, however, that instead of turning over, the hoop turns to the left, i.e. it takes on a precessional motion. If we forcibly attempt with the hoop-stick to make it turn more quickly to the left, the hoop at once rears itself upright again (compare Experiment 6). It is true that when the hoop is bowling along a curved path of radius R in an inclined position, as shown in the 142 Dynamics of Rotation, figure, there is a couple acting on it in a vertical plane, due to the centrifugal force ^ , and the lateral friction of the ground. But this will not account for the curvature of the track, nor can it be the sole cause of the hoop not falling over, for if the hoop be thrown from the observer in an inclined position, and spinning so as afterwards to roll back towards him, it will be observed not to fall over even while almost stationary, during the process of 'skidding,* which precedes the rollinGf back. o Further Experiment with a Hoop. — It is an instruc- tive experiment to set a small light hoop spinning in a ver- tical plane, in the air, and then, while it is still in the air, to strike it a blow with the finger at the extremity of a horizon- tal diameter. The hoop will at once im^ over about that diameter. If the experiment be repeated with the hoop not spinning, the hoop will not turn over, but will rotate about a vertical diameter. This experiment will confirm the belief in the validity of the explanation above given of the observed facts. That a spinning top does not fall when its axis of spin is tilted is evidently an instance of the same kind, and we shall show^ (p. 154) that the behavipur of a top in raising itself from an inclined to an upright position is due to an acceleration of the precession caused by the action of the ground against its peg, and falls under the same category as the recovery of posi- tion by the hoop, illustrated in experiments 4 and 6 with the gyroscope. * See also p. 70 of a Lecture on Spinning Tops, by Professor John Perry, F.R. S. Published by the Society for Promoting Christian Know- ledge, Charing Cross, London, W.C. Phenomena presented by Spinnhig Bodies. 143 Bicycle. — In the case of a bicycle the same causes operate, but the relatively great mass of the non-rotating parts (the framework and the rider) causes the effect of their momentum to preponderate in importance. It is true that when the rider finds himself falhng over to his left, he gives to his driving-wheel, by means of the handles, a rotation to his left about a vertical axis, and that this rotation will cause a pre- cessional recovery on the part of the wheel of the erect position. How considerable is this effort to precess may be readily appreciated by any one who will endeavour to change the plane of rotation of a spinning bicycle wheel, having first, for convenience of manipulation, detached it in its bearings from the rest of the machine. But if the turn given to the track be a sharp one, the momentum of the rider, who is seated above the axle of the wheel, will be the more power- ful cause in re-erection of the wheel. It should also be noticed that the reaction to the horizontal couple applied by the rider will be transmitted to the hind wheel, on which it will act in the opposite manner, tending to turn it over still further, and at the same time to decrease the curvature of the Fio. 88, track, and thus the effect of the centrifugal and friction couple already alluded to in reference to the motion of a hoop. Explanation of Precession. — That the grounds of the apparently anomalous behaviour of the gyroscope may be fully apprehended, it is necessary to remember that the principle of the conservation of angular momentum implies 144 Dynamics of Rotation. (i) That the application of any external couple involves tne generation of angular momentum at a definite rate about the axis of the couple ; and (ii) That no angular momentum about any axis in space can be destroyed or generated in a body without the action of a corresponding external couple about that axis. Now, if the spinning wheel were to turn over under the action of a tilting couple as it would if not spinning, and as, without experience, we might have expected it to do, the latter of these conditions would be violated. For, as the wheel, whose axis of spin was, let us suppose, originally hori- zontal, turned over, angular momentum would begin to be generated^ about a vertical axis without there being any corresponding couple to account for it; and if the tilting continued, angular momentum would also gradually disappear about the original direction of the axle of spin, and again without a corresponding couple to account for it. On the other hand, by the wheel not turning over in obedience to the tilting couple, this violation of condition (ii) is avoided, and by its precessing at a suitable rate condition (i) is also fulfilled. For, as the wheel turns about the axis of precession, so fast does angular momentum begin to appear about the axis of the couple as required. 1 When the wheel is simply spinning about axis (1) the amount of angular momentum about any axis in space drawn through its centre, is {see p. 89) proportional to the projection in that direction of the length of the axle of spin. Or again, the amount of angular momentum about any axis is proportional to the projection of the circular area of the disc which is visible to a person looking from a distance at its centre along the axis in question. Thus, if the axis were to begin to be tilted up, a person looking vertically down on the wheel would begin to see some of the flat side of the wheel. The student \iill find this a convenient method of following with the eye and estimating the development of angular momentum about any axis. Phenomena presented by Spinning Bodies, 145 Analogy between steady Precession and uniform Motion in a Circle. — To maintain the uniform motion of a particle along a circular arc requires, as we saw on p. Ill, the application of a force, which, acting always perpendicular to the existing momentum, alters the direction but not the magnitude of that momentum. Similarly, for the mainten- ance of a steady precession, we must have a couple always generating angular momentum in a direction perpendicular to that of the existing angular momentum, and thereby alter- ing the direction but not the magnitude of that angular momentum. We showed (pp. Ill, 112) that to maintain rotation with angular velocity w in a particle whose momentum was mv, required a central force of magnitude mviHy and we shall now find in precisely the same way, using the same figure, the value of the couple (L) required to maintain a given rate of precession about a vertical axis in a gyroscope with its axle of spin horizontal. Calculation of the Rate of Precession. — Let w be the rate of precession of the axle of spin. Let I be the moment of inertia of the wheel about the axle of spin. Let 12 be the angular velocity of spin. Then 112 is the angular momentum of the wheel about an axis coinciding at any instant with the axle of spin.^ It is to be observed, that in the absence of friction at the pivots, the rate of spin about the axle of spin remains imaltered. ^ The student is reminded that, on account of the already existing precession, the angular momentum about the axle of spin would not be 10 if this axle were nut also a principal axis, and ^t right-angles to the axis of precession (see p. 132). K 146 Dynamics of Rotation. Let us agree to represent the angular momentum 112 about the axle of spin when in the position OA by the length OP measured along OA. Then the angular momentum about the axle when in the position OB is represented by an equal length OQ measured along OB, and the angular momentum added in the interval is re- presented by the line PQ. If the interval of time con- sidered be very short, then OB is very near OA, and PQ is perpendicular to the axle OA. This shows that the ansrular momentum added, and therefore the external couple required to maintain the precession, is perpendicular to the axle of spin. > Let the very short interval of time in question be called {di)^ then PQ represents the angular momentum added in time (c//), t.tf. (the external couple) x idi), PQ_ external couple x (txV) •'• op~ m • But ^=angle POQ=a,(rf/); . external couple X {di) , •,.. '• w- — ^-^='»(''"' or external couple = Iflw. The analogy between this result and that obtained for the maintenance of uniform angular velocity of a particle in a circle becomes perhaps most apparent when written in the following form : — ^ Tq rotate the linear fliQwejitum mv with angular velocity Phenomena presented by Spinning Bodies. 147 w requires a force perpendicular to the momentum of magni- tude mv.w. ' While * To rotate the angular momentum Ifi with angular velocity w requires a couple, about an axis perpendicular to the axis of the angular momentum, of magnitude I12(u.' Since then L=Ifl(o _ L^ or the rate of precession is directly proportional to the mag- nitude of the applied couple, and inversely as the existing angular momentum of spin. That the rate of precession (w) increases as the rate of spin 12 diminishes has already been shown (see Experiments 5 and 7). But the result obtained also leads to the conclusion that, when the rate of spin is indefinitely small, then the rate of precession is indefinitely great, which seems quite contrary to experience, and requires further examination. To make this point clear, attention is called to the fact that our investigation, which has just led to the result that w=_^, applies only to the maintenance of an existing precession^ and not to the starting of that precession from rest. Assum- ing no loss of spin by friction, it is evident that there is more kinetic energy in the apparatus when precessing especially with its frame, than when spinning with axle of spin at rest. In fact, if i be the moment of inertia of the whole apparatus about the axle, perpendicular to that of spin, round which precession takes place, the kinetic energy is increased by the amount Jf(o', and this increase can only have been derived from work done by the applied couple at starting. Hence, 148 Dynamics of Rotation, in starting the precession, the wheel must yield somewhat to the tilting couple. Observation of the * Wabble/— This yielding may be easily observed if, when the wheel is spinning, comparatively slowly, about axis (1), we apply and then remove a couple about axis (2) in an impulsive manner, for example by a sharp tap given to the frame at C. The whole instrument will be observed to wriggle or wabble, and if close attention be paid, it will be noticed that the axle of spin dips (at one end), is quickly brought to rest, and then begins to return, swings beyond the original (horizontal) position, comes quickly to rest, and then returns again, thus oscillating about a mean position. Meanwhile, and concomitantly with these motions, the framework CD begins to precess round a vertical axis, comes to rest, and then swings back again. The two motions together constitute a rotation of either extremity of the axle of spin. If the rate of spin be very rapid, these motions will be found to be not only smaller in amplitude, but so fast as not to be easily followed by the eye, which may discern only a slight 'shiver' of the axle. Or, again, a similar effect may be observed to follow a sudden tap given when the whole is precessing steadily under the pressure of an attached extra weight. It will probably at once, and rightly, occur to the reader that the phenomenon is due to the inertia of the wheel and its attached frame, etc., with respect to rotation about the axis of precession. To any particular value of a tilting couple, and for a given angular momentum of spin about axis (1), there must be, as we have seen, so long as the couple is applied, an appropriate corresponding value for the preces- Phenomena presented by Spinning Bodies. 149 sional velocity, but this velocity cannot be at once acquired or altered. The inertia of the particles remote from the axis of precession enables them to exert forces resisting preces- sion, and we have seen as an experimental result (Experi- ments 6 and 8), that when precession is resisted the wheel obeys the tilting couple and turns over, acquiring angular velocity about the axis of the couple. But the parts that resist precessional rotation must, in accordance with the principle that action and reaction are equal and opposite, themselves acquire precessional rotation. Hence, when the impulsive couple, having reached its maximum value, begins to diminish again, this same inertia has the effect of hurrying the precession, and we have also seen in Experiment 6, that to hurry the precession is to produce a (precessional) tilt opposite to the couple inducing the precession, and this action destroys again the angular velocity about the axis of the applied couple which has just been acquired. The wabble once initiated can only disappear under the influence of frictional forces.^ Thus the wabbling motion is seen to be ^ We can now see in a general way in what manner our equation must be modified if it is to represent the connection between the applied couple and the rate of precession during the wabble. The yielding under the applied couple implies that this is generating angular momentum about its own axis by the ordinary process of generating angular acceleration of the whole object about that axis, and thus less is left unbalanced to work the alternative process of rotating the angular momentum of spin. In fact, if our equation is to hold, we must write (in an obvious notation) L-l2W2=wx angular momentum about horizontal axis perpendicular to the axis of the couple. But the motion being now much more complicated than before, the angular momentum about the horizontal axis that is being rotated can no longer be so simply expressed. As we have seen, it is not inde- pendent of COj* 150 Dynamics of Rotation. the result of forces tending first to check and then to accelerate precession, a phenomenon that has been already observed. But to observe one phenomenon, and then to point out that another is of the same kind, cannot explain both, and it is still desirable to obtain further insight into the physical reactions between the parts, which enables a couple about axle 2 to dart precession about axle 3, and vice Explanation of the Starting of Precession.— Suppose that we look along the horizontal axis of spin at the broad-side of the disc spinning as indicated by the arrow (Fig. 90), and that there is applied to it a couple about axle (2) tending, say, to make the upper half of the disc advance towards us out of the plane of the diagram, and the lower half to recede. We shall show that simultaneously with the rotation that such a couple produces about axle (2), forces are called into play which start precession about (3). All particles in quadrant (1) are increasing their distance from the axis (2), and therefore (see pp. 85 and 86) checking the rotation about (2), producing, in fact (on the massless rigid structure within the cells of which we may imagine them lying as loose cores), by reason of their inertia, the effect of a force away from the observer applied at some point A in the quadrant. Similarly, all particles in quad- rant (2) are approaching the axis (2), and therefore by their momentum perpendicular to the plane of the diagram are accelerating the rotation about (2), producing on the rigid structure of the wheel the effect of a pressure towards the observer at some point B. In like manner, in quadrant (3), in which the particles are receding from axis (2), they exert Phenomena presented by Spinning bodies. 1 5 1 on the rigid structure a resultant force tending to check the rotation about (2), equal and opposite to O) that exerted at A, and passing through a point C similarly- situated to A. Again, in quadrant (4) the force is away from the observer, is equal to that at B, and passes through the similarly situated point D. These four forces con- stitute a couple which does not affect the rotation about (2), but does generate pre- cession about (3). On the other hand, when precession is actually taking place about axis (3), we see, by dealing in precisely the same way with the several quadrants, and considering the approach or recession of their particles to or from axis (3), that the spin produces a couple about axis (2) which is opposed to and equilibrates the external couple that is already acting about axis (2), but which does not affect the rotation about axis (3). If, when precession about (3) is proceeding steadily, the external couple about (2) be suddenly withdrawn, then this opposing couple is no longer balanced, and the momentum of the particles initiates a wabble by causing rotation about (2).i ' Some readers may tind it easier to follow this explanation by 152 Dynamics of Rotation. Gyroscope with Axle of Spin Inclined.— It will be observed that we have limited our study of the motion of the spinning gyroscope under the action of a tilting couple fo the simplest case of all, viz., that in which the axle of spin is perpendicular to the ver- tical axle, which there- fore coincides with the axis of precession. If we had experimented with the axle of spin inclined as in Fig. 91, then the axis of precession, which, as we have seen, must always be perpendicular to the axis of spin, would have been itself inclined, and pure rotation about it would have been impossible owing to the manner in which the frame CD is attached to the vertical axle. The former precessional rotation could be resolved into two components, one about the vertical axis which can still take place, and one about a horizontal axis which is prevented. Now, we have seen that the effect of impeding the preces- sional rotation is to cause the instrument to yield to the imagining the disc as a hollow massless shell or case, inside which each massive particle whirls round the axis at the end of a fine string, and to think of the way in which the particles would strike the flat sides of the case if tins were given the sudden turn about axle 2. Pkcnojnena presented by Spinning Bodies. 153 tilting couple. Hence we may expect to find that the sudden hanging on of a weight, as in the figure, will cause a more marked wabble of the axle of spin than would be produced by an equal torque suddenly applied when the axle of spin was horizontal. This may be abundantly verified by experi- ment. It will be found that if the instrument be turned from the position of Fig. 91 to that of Fig. 85, and the same tap be given in each case, the yield is far less noticeable in the horizontal position, although (since the force now acts on a longer arm) the moment of the tap is greater ; and if other tests be applied, it will be observed that the quasi-rigidity of the instrument, even when spinning fast, is notably dimin- ished when the axle of spin is nearly vertical, i.e. when nearly the whole of the precession is impeded. Pivot-friction is liable to be greater with the axle of spin inclined, and this produces a more noticeable reduction of the rate of spin, with a corresponding increase of tilt and acceleration of the precession, which (as we show in the Appendix) would otherwise have a definite steady value. The precession also is now evidently a rotation about an axis which is not a principal axis of the disc, and on this account a centrifugal couple is called into play, tending, in the case of an oblate body like the gyroscope disc, to render the axle more vertical, i.e. to help the applied couple, if the weight is hung at the lower end of the axle, as in the figure, but to diminish the couple if the weight is hung from the upper end. It must be remembered, however, that the disc of a gyro- scope can only precess in company with its frame, CD, and the dimensions and mass of this can be so adjusted that the disc and frame together are dynamically equivalent to a sphere, 154 Dynamics of Rotation, every axis being then a principal axis as regards a common rotation of disc and frame. In this manner disturbance by the centrifugal couple may be avoided. In deah'ng with a peg-top moving in an inclined position with proces- sional gyration about a vertical axis (see Fig. 93), such centrifugal forces will obviously need taking into ac- count. With a prolate top, such as t hat figured, the effect of the centri- fugal couple will be to increase the applied couple and therefore the rate of precession ; with a flattened or oblate top like a teetotum, to diminish it. The exact evaluation of the steady precessional velocity of gyroscope or top with the axis of spin inclined will be found in the Appendix. FI03. 92 AND 93. Explanation of the Effects of Impeding or Hurry- ing Precession. — Though we have throughout referred to these effects as purely experimental phenomena, the ex- planation is very simple. The turning over of the gyroscope, when the steady precession is impeded, is itself simply a precessional motion induced by the impeding torque. Kefer- ence to the rule for the direction of precession (p. 139) will show that the effect either of impeding or hurrying is at once accounted for in this way. The Rising of a Spinning Too. — ^Ve have already Phenomena presented by Spinning Bodies. 155 (p. 142)" seen that this phenomenon would follow from the action of a torque hurrying the precession, and have intimated that it is by the friction of the peg with the ground or table on which the top spins that the requisite torque is provided. We shall now explain how this friction al force comes into play. The top is supposed to be already spinning and precess- ing with its axis in- clined as indicated in Fig. 93. The relation between the directions of tilt, spin, and pre- cession is obtained by the rule of page 139, and is shown by the arrows of Fig. 94, repre- senting the peg of the top somewhat enlarged. The extremity of the peg is always somewhat rounded, and the blunter it is, the farther from the axis of spin will be the part that at any in- stant is in contact with the table. On account of the preces- sional motion by which the peg is swept bodily round the horizontal circle on the table, this portion of the peg in contact with the table is moving forwards, while, on the other hand, on account of the spin, the same part is being carried backwards over the table. So long as there is relative motion of the parts in contact, the direction of the friction exerted by the table on the peg will depend on which of these two opposed velocities is the greater. If the forward, precessional velocity is the greater, then the friction will oppose precession and increase the tilt ; while if the backward linear velocity due ■^%On -:^. no. ©4. 1^6 Dynamics of Rotation. to the spin is the greater, then the peg will skid as it sweeps round and the friction will be an external force aiding pre- cession, and the top will rise to a more vertical position. When the two opposed velocities are exactly equal, then the motion of the peg is one of pure rolling round the horizontal circle : there is then no relative motion of the parts in con- tact, parallel to the table, and the friction may be in either direction, and may be zero. With a very sharp peg, of which the part in contact with the table is very near the axis of spin, the backward linear velocity will be very small, even with a rapid rate of spin ; so that such a top will less readily recover its erect position than one with a blunter peg. Also on a very smooth surface the recovery is necessarily slower than on a rough one, as may easily be seen by causing a top which is spinning and gyrating and slowly erecting itself on a smooth tray, to move on to an artificially roughened part. The explanation here given, though somewhat more de- tailed, is essentially the same as that of Professor Perry in his charming little book on Spiniiing Tops already referred to, and is attributed by him to Sir William Thomson. We will conclude by recommending the student to spin, on surfaces of different roughness, such bodies as an egg (hard- boiled), a sphere eccentrically loaded within, and to observe the circumstances under which tlie centre of gravity rises or does not rise. Bearing in mind the explanation just given, he should now be able to accownt to himself for what he will observe, and to foresee what will happen under altered con- ditions. Calculation of the * Effort to Precess.'— We saw, Phenovtena presented by Spinning Bodies. 157 in Experiments 9 and 10, that when precession is prevented an ' effort to precess ' is exerted by the spinning body against that which prevents it. Thus, in the experiments referred to, pressures equivalent to a couple were exerted by the axle of the spinning wheel on its bearings. If 0) be the rate at which the axle of spin is being forcibly turned into a new direction, the;i wlfi is the rate at which angular momentum is being generated about the axis per- pendicular to the axis of = — = rr— = — radians per second. r 528 12 *^ /. Moment of couple required = 2lQa) absolute units. = 1200 pound-foot units (very nearlyj. Applying the rule for the direction of precession, we see that this couple will tend to lift the engine off the inner rail of the curve. [We have left out of consideration the inclination which, in prac- tice, would be given to the wheels in rounding such a curve, since this will but slightly affect the numerical result.] Similar stresses are produced at the bearings of the rotating parts of a ship's machinery by the rolling, pitching, and turn- ing of the ship. In screw-ships the axis of the larger parts of such machinery are in general parallel to the ship's keel, and will therefore be altered in direction by the pitching and 1 5 8 Dynamics of Rotation . turning, but not by the rolling. There appear to be no trustworthy data from which the maximum value of w likely to be reached in pitching can be calculated. As regards the effect of turning, the following example, for which the data employed were taken from actual measure- ments, shows that the stresses produced are not likely in any actual case to be large enough to be important. Example (2). — A torpedo-boat with propeller making 270 revolu- tions per minute, made a complete turn in 84 seconds. The moment of inertia of the propeller was found, by dismounting it and observ- ing the time of a small oscillation, under gravity, about a horizontal and eccentric axis, to be almost exactly 1 ton-foot^. Required the processional torque on the propeller shaft. Solution — 270 X 27r Q = — -— — = 28"3 radians per second. 60 ^ O— 1 1 to = — = _— radians per second. 84 147 1 = 2240 lb. -foot2 units. .•, torque required = iQto absolute units. = 2240 X 28-3 x il poundal-foot units, 147 = 148*4 pound-foot units (very nearly). This torque will tend to tilt up or depress the stern according to the direction of turning of the boat, and of rotation of the propeller. Miscellaneous Examples. 159 MISCELLANEOUS EXAMPLES. L Find (a) the total angular momentum, (6) the position of the axis of total angular momentum, (c) the centrifugal couple in the two following cases : — (i) A uniform thin circular disc of mass M and radius r, rotating \vith angular velocity to about an axis making an angle B with the plane of the disc. (ii) A uniform paraUelipiped of mass M and sides 2a, 26, and 2c, rotating with angular velocity (o)) about a diagonal. 2. A wheel of radius (r) and principal moments of inertia A an(? B, inclined at a constant angle {&) to the horizon rolls over a horizontal plane, describing on it a circle of radius R, in T sec. Find (1) the position at any instant of the actual axis of rotation and the angular velocity about it ; (2) the angular momentum about this axis ; (3) the total angular momentum ; (4) the position of the axis of total angular momentum ; (5) the magnitude of the external couple necessary to maintain equilibrium. 3. Referring to Fig. 85, p. 139, if the moment of inertia of the spinning gyroscope about CD is 3000 gram-cm.^ units, and ii CD = 10 cm. and the value of the weight hung at D = 50 grams, and the rate of precession is observed to be 1 turn in 25 seconds, find the rate of spin of the gyroscope. 4. What would be the answer to the last question if the axis of spin had been inclined at an angle of 45°, as in Fig. 91, p. 152, the moment of inertia of the wheel about EF being 1800 gram-cm.^ units, and the principal moments of inertia of the frame CDEF being 2000 and HOG units respectively ? APPENDIX (1) ON THE TERMS ANGULAR VELOCITY AND ROTATIONAL VELOCITY. We can only speak of a hody as having a definite angular velocity with respect to an axis, when every particle of the body has the same angular velocity about that axis, i.e. where the body, at the instant under consideration, is actually rotating about the axis in question. Thus for a hody angular velocity means always rotational velocity, and either term may be used indifferently. But a particle may have a definite angular velocity with respect to an axis about which it is not rotating. Thus let P be a particle in the plane of the paper, moving with some velocity V, which may be inclined to the plane of the paper, but which has a resolute v in the plane ^/ of the paper in the direc- y;:^ tion APB (say). Let Qyx^ be any axis perpendicular g ^r^ to the plane of the paper. r^^- ^^' In any infinitesimal interval of time (dt) let the particle be carried from P to a point whose projection on the plane of the paper is F; then W = vdt. In the interval {dt) the projection of the radius vector has swept out the angle POP' ( = c?^), and ^ is called the angular velocity of the CLt particle about the axis 0, at the instant in question. , . , , dd Z.POF The measure of this angular velocity (<^)= ^= — ^7~ ^ dt " rdt ■" rdt " dt r r* Appendix, i6i Thus the angular velocity (w) of any particle with respect to any axis at distance r, is obtained by finding the resolute {v) of its velocity, in a plane perpendicular to the axis, and draw- ing from the axis a perpendicular (p) on the direction of this resolute, then w=^^ (2) ON THE COMPOSITION OF ROTATIONAL VELOCITIES. Definition, — If a rigid body is rotating about some axle A, fixed to a frame, while the frame rotates about some axle B, fixed to a second frame, which in its turn rotates about a third axle C, fixed (say) to the earth, then the motion of the body relative to the earth's surface at the place where is fixed, is said to be compounded at any instant of the three simultaneous rotations in question about A, B, and C, con- sidered as fixed in the positions they occupy at that instant. A similar definition applies to any number of simultaneous rotations. (3) THE PARALLELOGRAM OF ROTATIONAL VELOCITIES. Enunciation. — If the motion of a rigid body of which one point is fixed may at any instant he described by saying that it is rotating about the intersecting axes OA and OB toith two simultaneous rotational velocities represented by the lengths OA and OB then, at the instant in i question, the actual motion of y^ the body is a rotation about and n/ , ^ represented by OD, the diagonal / ^ "- ^ ^ ^ of the parallelogram AB. b/ p^S^ ^ Proof — Let yAN'. .-. (DAP=(OyAN' = K.OB.AN' = K X area AB. = Kx2x area of AOAD. = K.ODxAP .-. a,=K.OD i.e.. OD represents the resultant rotational velocity on the scale already chosen. The resultant OD may now be combined with a third component rotational velocity OC in any other direction and so on to any number of components. Conversely^ any rotational velocity may be resolved ac- cording to the parallelogram law into three independent rectangular components, as intimated in the text (p. 6). The Parallelogram of rotational accelerations follows at once as a corollary, and thus rotational velocity, and rotational acceleration are each shown to be a vector quantity. It is important, however, that the student should realise that rotational displacements, if of finite magnitude, are not vector quantities, for the resultant of two simultaneous or successive finite rotational displacements is not given by the Appendix, 163 parallelogram law, and the resultant of two such successive finite displacements is not even independent of the order in which they are effected. To convince himself of this, let the reader place a closed book on its edge on the table before him, and keeping one corner fixed let him give it a right-handed rotation of 90°, first about a vertical axis through this corner, and then about a horizontal axis, and let him note the position to which this brings the book. Then let him replace the book in its original position and repeat the process, changing the order of the rotations. He will find the resulting position to be now quite different, and each is different also from the position which would have been reached by rotation about the diagonal axis. Hence we cannot deduce the parallelogram of rotational velocities from that of finite rotational displacements as we can that of linear velocities from that of finite linear displacements. Composition of simultaneous rotational velocities about parallel axes. — The student will easily verify for himself that the resultant of simultaneous rotational velocities w^ and w^ about two parallel axes A and B is a rotational velocity equal to a>a+w^ about a parallel axis D which divides the distance between A and B inversely as (d^, and w^g. If ilia and w^ are equal and opposite (graphically repre- sented by a couple) then the resultant motion of every particle of the rigid body is easily seen to be a translation perpendicular to the plane containing the two axes and equal to the rotational velocity about either multiplied by the distance between them. A farther extension is now also easy, and the student will realise that just as any system of forces reduces to a single force through some arbitrarily chosen point and a couple, 80 any system of simultaneous rotational velocities of a rigid body about any axes whatever, whether intersecting or not, reduces to, or is equivalent to, a rotational velocity about an 1 64 Dynamics of Rotation, axis through some arbitrarily chosen point, together with a motion of translation. (4) PRECESSION OF GYROSCOPE AND SPINNING TOP WITH AXIS INCLINED. The value (w) of the steady precessional velocity of a gyro- scope whose axis is inclined at an angle Q to the vertical, where an external tilting couple of moment L is applied about the axis EF (see Fig. 91) may be found as follows. Referring still to Fig. 91, let the vertical axis of precession be called (y) and the axis EF of the couple, («), and the hori- zontal axis in the same plane as the axle of spin (a;). Let C be the moment of inertia of the disc about the axle of spin, A its moment about a perpendicular axis, and let 12 be the angular velocity of spin relative to the already moving frame. (1) Let the dimensions of the ring have been adjusted in the way mentioned on p. 153 so that the rotation about y in- troduces no centrifugal couple. Then the value of the angular momentum about (x) is simply Ci2 sin ^, and to rotate this about (y) with angular velocity (w) will require a couple (L) about (z) equal to wCft sin 6. Whence a)=--__^. W sm 6 It follows that with a gyroscope so adjusted the rate of steady precession produced by a weight hung on as in Fig. 91 will be the same whether the axis be inclined or horizontal for the length of the arm on which the weight acts, and therefore the couple L, is itself proportional to sin 6. N.B. — The resolute of w about the axis perpendicular to EF and CD is A as before (p. 147). (2) Let the ring and disc not have the adjustment men- tioned, and let the least and greatest moments of inertia of Appendix, 165 the ring be C and A' respectively. If the disc were not spinning in its frame, i.e. if 12 were zero, we should require for equilibrium a centripetal couple (see p. 122) equal to — (A— C)a)'sin ^cos ^— (A'— C')co''sin^cos^. On account of the spin an additional angular momentum C12 sin Q is added about a^ to rotate which requires an additional couple cdCI2 sin Q. Whence the total couple required = L=Cfi(osin^-(A-C-A'-0Vsin6'cos^, which gives us w. In the case of a top precessing in the manner indicated in Fig. 96, the tilting couple is myl sin 0, and the only differ- ence in the solution is ^ that there is no frame, sothat A'=OandC'=0. But it will be observed that our 12 still means the velocity of spin rela- tive to an imaginary frame swinging round with the top. The quad- ratic equation for co thus becomes mgl = C12a> — (A-CKcos^. We might, if we had preferred it, in each case have simply found by (fftgr) no. 97. resolution the total angular momentum about (x) after the manner of page 134, and, multiplying this by o>, have obtained the value of the couple about z. But by looking at the matter in the way suggested the student will better realise the fact that the centripetal couple is that part of the applied couple which is required to rotate the angular momentum contri- buted about X by the precessional rotation itself. 66 Dynafnics of Rotation. (5) NOTE ON EXAMPLE (4) p. 86. A VERY simple and beautiful experimental illustration, which is almost exactly equivalent to that indicated in the text, is the following : — Let a long, fine string be hung from the ceiling, the lower end being at a convenient height to take hold of, and let a bullet or other small heavy object be fastened to the middle of the string. Holding the lower end vertically below the point of suspension let the string be slackened and the bullet caused to rotate in a horizontal circle. On now tightening the string the diameter of this circle will contract and the rate of revolution will increase ; on slackening the string the reverse happens [Conservation of Angular Momentum]. The kinetic energy gained by the body during the tightening is equivalent to the work done by the hand + a very small amount of work done by gravity, since the smaller circle is in a rather lower plane than the larger. (6) ON THE CONNEXION BETWEEN THE CENTRIPETAL COUPLE AND THE RESIDUAL ANGULAR MOMENTUM. It is convenient to think of the centripetal force which acts on any uniformly rotating particle of mass m (see fig. 97) as the force which is required to rotate \^v the momentum {mv) of the particle at the required rate. The force -^r =imriD^=mrn)'X(o=invx about a fixed axle Oy passing through the centre of mass 0. Take Ox in the plane of the paper as the axis of a:, Pi KIO. 09. and the axis Oz perpen- dicular to the plane of the paper. First let the rod be perpendicular to Oy. Oy is then a prin- cipal axis. There is no angular momentum about any line in the plane xz, and no centri- petal couple. Next let the rod be inclined as shown, and let it be passing through the plane xy. Oy is no longer a principal axis, and there is now a centripetal couple (of moment ymx(ii^^y!m'oiiii)^ = (ji^^mxy) and also angular momentum about Ox (the value of which is ymxu)-\-ym'x(o=(oI,mxy). At the instant in question there is no angular momentum about Oz, for each particle is moving parallel to 0^, but after a quarter-turn the amount of angular momentum at present existing about Ox will be found about O^. Thus the total residual angular mo- mentum is rotated by the cen- tripetal couple whose value is equal to the residual angular momentum rotated x the rate of rotation. It should be ob- served that during the quarter-turn from x to z^ the cen- tripetal couple will also gradually destroy the angular momentum previously existing about Ox. The same is true even in the most genepl case of a body Fia. 100. f68 Dynamics of Rotation, of three unequal moments of inertia, rotating about any non-principal axis, Oy, through the centre of mass O. For the residual angular momentum {(Oy^mxy) about Ox, when combined by the parallelogram law with that about Oz (equal to (Dy^mzy) will give a total residual angular momentum about some line OP in the plane xz. The centripetal couple is in the plane yOP, and equal to the angular momentum about OP X the rate of rotation. Explanaiion of the criterion for centre of Percussion : — The reader will now be better able to realise the signi- ficance of the criterion for the existence of a centre of percussion given on p. 129. Let him think of a uni- form, rectangular, thin board ABCD swinging freely about a fixed axle AB along its upper hori- zontal edge, and loaded with a uniform massive diagonal bar BD. We wish to find if, and where, the front of the board can be struck, so as to give no impul- sive shock to the axle, and we have already learnt that the blow must be struck at right angles to the board, and so that the board, if free, would begin to rotate about EOF drawn through the centre of mass O parallel to the axle AB. Further we know that both for board and rod separately, and therefore for the two together, the blow must be delivered at a distance § of BC from the fixed axle. But where the lar rotates about EF, it will have left- handed angular momentum about GH also, and if we struck our blow at P on HG, we could not impart any such angular momentum, which therefore could only be derived from an impulsive pressure of the axle forward at B and backward Appendix. 169 at A. If, however, we shift P to the left, to some point P', keeping it always at the same distance from AB, we can give the angular momentum required about GH. The axle will then experience no strain, and it is easy, when the masses of board and rod are known, to calculate the shift required which fixes the position of the centre of pressure. In this case every particle of the system, when rotation begins, moves perpendicularly to the paper, and there is no angular momentum about an axis through at right angles to the paper. But if the bar, still centred at 0, were inclined to the board at any angle (other than 90°), there would be suddenly acquired angular momentum also about an axis through 0, parallel to the blow, which could not be imparted by the blow, but only by impulsive pressures up and down at A and B. Hence in this case there would be no centre of percussion. Thus the criterion is that with rotation about EOF, the axis of total residual angular momentum shall be HG, i.e. shall be in the plarie containing the fixed axle and the centre of mass, and therefore, as we have seen, the centrifugal couple must lie in this plane, and this is the form in which the criterion was given — not because the centrifugal forces come into play, but because it is generally easier from inspection to form a fairly accurate impression of the position of the plane of the centrifugal couple than it is to realise the direction of the residual angular momentum. INDEX Acceleration of centre of mass, 101, 104. linear, 3. of particle moying round a circle, 111. Acceleration (angular or rota- tional), 3, 12. of centre of mass, 104. composition of, 7. geometrical representation of, 6. mass, 30, 102. proportional to torque, 12, 30. ratio of, to displacement in simple harmonic motion, 73. relation of, with torque and rotational inertia, 17. uniform, 3. variation of, with distribu- tion of matter, 17. Angle, unit of, 2. Annulus (plane circular), radius of gyration of, 54 (4). Area, moment of, 7. moment of inertia of, 33. radius of gyration of, 37, 38. Artifice (dynamical), for questions involving centripetal force, 116. Atwood's machine, 43 (8). Axes, of greatest and least moment of inertia, 56. principal, 57, 132. theorem of parallel, 37. Axis, 9. about which a couple causes a free rigid body to rotate, 96, 105. instantaneous, 135. invariable, 135. Axle, 9, 137. Axle, pressure on, 126, 129. of spin, 137. Bar, see Rod. Bat (cricket), centre of percussion of, 128. Bicycle, 143. Boat (sailing), a rigid body, 1. Body (rigid), 1. centre of gravity of, 37. centre of percussion in, 128. centrifugal couple in, 118. centripetal force in, 113. efifect of couple on, 96, 105, 106. equimomental, 64, 121. modiilus of elasticity of, 73. motion of, with one point fixed, 2, 6. point of, having peculiar dynamical relations, 94-98. spinning, 136-158 ; see also Gyroscope and Top. total kinetic energy of, 107. Brake (friction), 15. Carriage, effective inertia of, 45 (16). Centre (of gravity), 37, 96, 97 (footnote). Centre (of inertia), 38 (footnote). Centre (of mass), 38 (footnote), 94-110,99. acceleration of, 101, 104. displacement of, 100. velocity of, 101. Centre(ofpercussion), 125-129, 126, criterion for, 129, 168. Compass needle, moment of inertia of, 34. X71 172 Dynamics of Rotation, Couple, change of kinetic energy due to, 24. effect of, on free rigid body, 96, 105, 106. on spinning body, 139. restoring, 75. unit, 9. work done by, 23. Couple (centrifugal), 117-120. effect of, on peg-top, 154. of equimomental bodies, 121. transfer of energy under action of, 123. Couple (centripetal), 133. connection of, and residual angular momentum, 166. elimination of, in gyroscope. 153. Curve, precessional force due to wheels of railway engine round- ing a, 157. Curves (inertia), graphical con- struction for, 60-64, 66. Cylinder (thin hollow), radius of gyration of, 25. D'Alembbrt's Principle, 30. Deformation, proportional to force, 70. unit of, 70. Disc, moment of inertia of, 39, 50. kinetic energy of rolling, 110(1). Displacement of centre of mass, 100. ratio of acceleration to, in simple harmonic motion, 73. Door, 15, 42 (2) and (7). centre of percussion of, 128. moment of inertia of, 39. Earth, rotation of, 88. Effort to precess, calculation of, 156. Elasticity, modulus of, 73 (foot- note). perfect or simple, 70. Ellipse, moment of inertia of, 51. Ellipsoid, moment of inertia of, 53. Energy, transfer of, under action of centrifugal couple, 12.3. Energy (kinetic), change of, due to couple, 24. due to variation of the moment of inertia, 87, 166. of precessing spinning body, 147. of rolling disc, 110 (1). of rolling hoop, 110 (2). of rolling sphere, 110 (3/ total, of rigid body, 107. Engine (railway), precessional force due to wheels of, on a curve, 157. Examples, on angular oscillations, 76-78. on angular velocity, 3. on conservation of angular momentum, 83-86, 87. on effort to precess, 157. on equivalent simple pen- dulum, 77, 78. on properties of centre of mass, 108, 109. ■ on radius of gyration, 53. on rotational inertia, 18, 25-30. on simple harmonic motion, 74. for solution, 42-45, 53, 54, 81, 93, 110, 159. on turning of ship, 158. on uniform angular accelera- tion, 6, 6. Experiments, on behaviour of spinning bodies, 137, 138, 142. on centre of percussion, 126. on equality of torque, 8. on existence of rotational inertia, 16. on floating magnet, 97. on Hooke's Law, 70, 71. on point of a body having peculiar dynamical relations, 94-98. Index, -^Th Experiments on precession, 139- 141. on proportionality of torque and angular acceleration, 13, 14. on value of rotational inertia, I 18. Figure (solid), moment of inertia of, 33, 60. Flywheel (with light spokes and thin rim), radius of gyration of, 25. Foot-pound, 22 (footnote). Foot-poundal, 9, 22 (footnote). Force, centrifugal, 111-124. centripetel, 111-124, 112, 113. connection between centri- fugal and centripetal, 112. elastic, 73. moment of, 7, 8. precessional, due to wheels of railway engine rounding a curve, 157. turning power of, 7. Fork (tuning), motion of, 68. Friction, 11. brake, to check rotation, 15. entirely removed, 11. moment of, 11. pivot, effect on gyroscope, 153. Gravity (centre of), %te Centre (of gravity). Gyration (radius of), aec Radius (of gyration). Gyroscope, 136. with axle of spin inclined, 152, 164. Hogke's Law, 70. Hoop, equivalent simple pendulum of, 77. experiments with, 141, 142. kinetic energy of rolling, 110 (2). moment of inertia of thin, 35. precession of, 141. Hoop, radius of gyration of thin, 25. Ideal single particle system, 19. Inertia, 16. the cause of wabble of spin- ning body, 148. curves, graphical construc- tion for, 60-64, 66. effective, of a carriage, 45 (16). skeleton, 64, 121. surfaces, 60-64, 66. Inertia (rotational), 17, 19,' 30. calculation of, of rigid body, 18. relation of, with torque and angular acceleration, 17. unit of, 17. Inertia (moment of), 7, 20, 34. about any axis, 38, 58. of area, 33, 48. axes of greatest and least, 56. calculation of, 46. of compass needle, 35. of disc, 39, 48, 50. of door, 39. effect of change of, on kinetic energy, 87, 166. of ellipse, 51. of ellipsoid, 53. general case for, of solid, 60. of hoop, 35. of lamina, 35, 54, 58. maximum and minimum, 56. of a model compared to that of real object, 45 (15). of a peg-top, 34. principal, 57. of prism, 34, 40. by oscillating table, 79. of rod, 37, 40, 46, 50. Routh's rule for, 36. skeleton, 64, 121. of solid figure, 33. of sphere, 37, 52. sum of, of rigid body about three rectangular axes, 55 surface, 63. 174 Dynamics of Rotation. Inertia (moment of), unit, 20. of wheel and axle, 44 (10). JnoGLEB (spinning), 87. Lamina, centrifugal force in rigid, 113. inertia curve for, 61. moment of inertia of, 35, 54, 58. Lath, bending of, 70, 71. time of oscillation of, 75. Law (Hooke's), 70. Laws of Motion (Newton's), 10. analogues in rotation to, 11, 12, 18, 82. Lb., distinction of, with pound, 9. Lb. -foot 2, 21. Machinery, importance of proper shape for rapidly revolving, 121. pitching and rolling effect on, of ship, 157. Magnet, oscillating, 41, 77, 78. floating, 96. Mass, acceleration, 102. analogue of, in rotational motion, 25. centre of, see Centre (of mass). displacement, 102. moment of, 7. proportional to weight, 38 (footnote). unit of, 9 (footnote). Model, moment of inertia of, com- pared to real object, 45 (15). Moment of area, 7. of friction, 11. of force, 7, 8. of inertia, sec Inertia (mo- ment of). of mass, 7. of mass displacement of par- ticles, 102. of momentum, 89. Momentum (angular), 21. about principal axes, 134 (footnote). Momentum,connection of residual, with centripetal couple, 166. conservation of, 82. graphical representation of, 88. moment of, 89. of system of particles, 103. total, 130-135, 133. Motion, Laws of, see Laws of Motion (Newton's). precessional, 138. round a fixed axle, 83. simple harmonic, 67-69, 73. of tuning fork, 68. Needle (compass), moment of inertia of, 34. Newton's Laws of Motion, %et Laws of Motion (Newton's). Oscillation, angular, 75. of cylindrical bar magnet, 41. elastic, 70-81. of heavy spiral spring, 75. Parallelogram of rotational velocities, 6, 161. of rotational accelerations, 7, 162. Peg-top, 8te Top. Pendulum, ballistic, 91, 93 (5). equivalent simple, 76. simple, 76. Percussion (centre of), see Centre (of percussion). Period of simple harmonic motion, 68. Phase of simple harmonic motion, 69. Plane (principal), 57. Planets, rotation of, 88. Pound, distinction of, with lb., 9. foot, 9. two senses of word, 9. Poundal, 9. foot, 9. Power (turning), see Torque. Precession, 139, 164. Index, 175 Precessiou, aualogy of, aud uniform motion in a circle, 145. calculation of rate of, 145. direction of, 139. eflFect of hurrying or imped- ing, 154. effect of, on wheels of railway engine rounding a curve, 157. explanation of, 143. in hoops, tops, etc., 141. starting of, 150. Pressure, impulse, on axle, 126, 129. Prism, moment of inertia of, 34, 40. Radian, 2. Radius (of gyration), 24. of annulus, 54 (4). of area, 36, 37. of flywheel with light spokes and thin rim, 25. of square about diagonal, 53(1). of thin hollow cylinder, 25. of thin hoop, 25. of triangle, 53 (2), 54 (3). of uniform sphere, 37. of uniform spherical shell, 54 (5). of various solid figures, sec Moment of inertia. Rectangle, moment of inertia of, 36, 48. Resistance of a body submitted to unit deformation, 73. Revolution (rate of), «€e also Rotation. effect of torque on, 11. Rod, moment of inertia of, 40, 46, 48, 50. motion of sonorous, 68. rotating loaded massless, 91. Rotation, composition of, 6. effect of torque on rate of, 11. ' , about principal axis on centripetal couple, 119. Rotation, effect of, of free rigid body independent of motion of centre of mass, 105. pure, 101. rate of, 2. under the influence ot torque, 10-32. under the influence of no torque, 135. uniformly accelerated, 3. of water escaping by- hole in basin, 88. Routh's rule for moments of in- ertia, 36. examples of, 36, 37. Second, 2. Shell, radius of gyration of spheri- cal, 54 (5). Ship, effect of rolling and pitching on machinery of, 157. Shrinking, effect of, of cooling bodies on rotation, 88. Skater (spinning), 87. Skeleton (inertia), 64, 121. Skidding of hoop, 142. Slug, 9 (footnote). Slug-foot 2, 21. Speed (tangential), 2. Sphere, moment of inertia of, 52. rolling along inclined plane, 108, 110. Spring (spiral), stretching of, 71. oscillation of heavy, 74. Square, radius of gyration of, 63 (1). Stone whirled by string, a rigid body, 1. Strain proportional to force, 70. Stress, relation to strain, 70. precessional, on machinery of pitching, rolling, or turning ship, 157. Substance, modulus of elasticity of, 73. Sun, rotation of, 88. Surface (inertia), graphical con- struction for, 60-64, 66. UNIVEl Dynamics of Rotation, Table (oscillating), for moments of inertia, 79. Tin^e, of complete oscillation, 68, 73,75. unit of, 2, 9. Top (peg), centrifugal couple in, 154. moment of inertia of, 34. behaviour of spinning, 136, 142. precession of spinning, 141. rising of spinning, 155. Torque, 8. action of, on spinning top, 154. application of known and constant, 14. British absolute unit of, 9. comparison of different, 13. effect of, on rate of rotation, 11. effect of several simultane- ous, 13. engineer's unit of, 9. equal, 8. equality of, 8. gravitational unit of, 9, measure of, 8. • proportionality of, with an- gular acceleration, 12, 30. relation of, with acceleration and rotational inertia, 17* retarding, 15. ■ unit, 8. Translation, 101, Triangle, radius of gyration of, 53 (2), 54 (3). Unit of angle, 2. of force, 9. of length, 9. of mass, 9 (footnote). moment of inertia, 20, 21. of rotational inertia, 17. of time, 2, 9. of torque, 9. of work, 9, 22 (footnote). Velocity (angular or rotational), 2, 160. composition of, 6, 161, 163. destruction and generation of, by torque, 14. geometrical representation of, 6. parallelogram of, 6, 161 of centre of mass, 101. Wabble of spinning top, 148. Water rotation in escaping by hole in basin, 88. Weight and mass, 38 (footnote). Wheel and axle, moment of in- ertia of, 44 (10). Wheelbarrow not a rigid body, 1 Work done by a couple, 23. unit of, 22 (footnote). Printed by T. and A. Constable, Printers to His Majesty at the Edinburgh University Press, Scotland THIS BOOK IS DUE ON THE LAST DATE STAMPED BEIiOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. 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