CRZTICAL ORPINIONS of the FIRST EDITION.'This seems a very useful little volume, both as an Elementary School Text-Book and as suitable for the use of candidates preparing for matriculation, preliminary scientific, and other examinations. It is well put together, not too advanced for the end in view, and the exercises and illustrations are of a clear and pertinent character.' LANCET.'This- little work is, as it claims to be, well suited for beginners, a very small knowledge of simple arithmetical rules and a smattering of algebra only being required to work out tolerably difficult problems. The treatment of the subject is novel, motion being considered before rest or equilibrium. Numerous examples are given throughout the book to be worked out by the student, and questions given at the Preliminary Scientific Examinations of the University of London for the last few years render the book very useful to the student.' MIDLAND COUNTIES HERALD.'In the study of Mechanics, as with Arithmetic, the principles are too frequently left unmastered, and a carefully set problem proves a stumbling-block, not generally from ignorance of formulke, but from inability to apply the formula to the case in question. Preparation for examination in Elementary Mechanics requires a thorough grounding in the principles, and at least as much exercise in their application as is found necessary in Arithmetic and Algebra. Mr. MAGNUS'S book seems admirably adapted to furnish these two requirements...... The order is scientifically logical, and appears to possess advantages in teaching. The principles are lucidly explained and illustrated; examples are worked out, and each lesson is followed by a series of graduated questions, requiring a thorough grasp of principles for their solution. An appendix gives a number of examination papers set at various public institutions.' SCHOOLMASTER.'Mr. MAGNUS'S little work on Elemennlay 1lfechlanics is a great improvement upon the few similar ones already in use, and will help many teachers and private students out of great difficulty. The importance of the subject is so generally recognised that almost every educated man is expected to know something of its leading principles; and there is scarcely a public examination held in which it is not either a necessary or an optional subject. Yet, strange to say, it has hitherto been no easy matter to find a text-book at once simple enough to meet the requirements of the beginner, and at the same time full and complete enough to serve as a basis for more advanced reading. Henceforth, however, it will only be necessary to name the one before us, and the difficulty will be at an end. Following the example recently set by the highest authorities on the subject, the Author has departed consider O)pinious of the Pre-s. ably from the usual arrangement of the topics discussed. Kinematics, or the consideration of some of the simplest principles of motion, without regard to the quantity or quality of the matter moved, comes first. This is followed by the laws of Dynamics, under which head is considered the matter that is set in motion, and the cause or force producing it. The third and last part embraces all the problems connected with bodies at rest, which are discussed under the head of Statics, or the science of equilibrium. In addition to these subjects there is a brief but clear exposition of the doctrine of Energy, which affords a capital view of the connexion between the science of mechanics and other branches of physics. All the important propositions are accompanied and illustrated by numerical examples worked out in the text; and to each section are added progressively arranged exercises, which the student will find excellent tests of his knowledge of the principles laid down. To render the work more especially serviceable to candidates for the matriculation and preliminary scientific examinations of the University of London, all questions set during the last few years have been classified and appended to the several chapters to which they correspond.' LEEDS MERCURY.' The style is lucid, the solved exercises carefully chosen, the work compact. An intelligent boy ought in a few months to be able to make himself master of the greater portion of this small book which Mr. MAGNUS has aimed at making sufficiently elementary to be placed in the hands of a beginner. What we consider to be higher praise is that we believe it to contain nothing that the student will have to unlearn in a subsequent portion of his career. We can recommend it as a trustworthy introduction to more advanced text-books.' NATURE. Mr. MAGNUS'S book is admirably written from first to last, and shows remarkable powers of organising knowledge so as to adapt it to the gradually developing mind of the learner, and this shews the Author in his true colours as a thoroughly practical teacher. It is evident that the book is not made up of mere proposals or suggestions of what might be done, but it is a record of what has been done. It is plain that all these lessons have been given, that they are the results of experience; and in this consists much of their value.... Mr. MAGNUS realises the idea that a definition should not only explain the real meaning, but should be so worded that it could not possibly be taken to mean anything else, and this will save much confusion and misunderstanding on the part of the pupil. The definitions are in fact brief essays; and thus the pupil, instead of learning the mere words by heart, will, after reading them, become imbued with their spirit, will be interested in their application; and when this point is reached, half the difficulty of the study will have been removed.' JEWISH CHRONICLE. L ESSONS IN ELEMENTARY MECHANICS. INTRODUCTORY to the STUDY of PHYSICAL SCIENCE, DESIGNED FOR THE USE OF SCHOOLS, ACADEMIES, AND SCIENTIFIC INSTITUTIONS. _wRTIT: I-T. /ET RO'S 2T: EXET=C ISSES. BY PHILIP MAGNUS, B.Sc., B.A., LIFE-GOVERNOR OF UNIVERSITY COLLEGE, LONDON. IT'rITr EI'ENDATIOATS AND INTRODUCTTON BY PROF. DEVOLSON WOOD, OF STEVENS INSTITUTE OF TECHNOLOGY. S:ECO-lD E DIT1TJS, TETvIS:ED. NEW YORK: JOHN WILEY & SONS, 15 ASTOR PLACE. 1876. Entered according to Act of Congress, in the year 1876, by DEVOLSON WOOD, In the Office of the Librarian of Congress at Washington, D. C JOHN F. TRow & SON, PRINTERS AND STEREOTYPERS, 205-213 z.ast 12th St., NEW YORE. PREFACE. ELEMENTMXlY Nworks Oll sc ientific subjects are very desirable. It is h1owever, too often the case that ill attelmptillg to produce thelm-n they are really made more cliflilcnlt thlan the liore advanced wvorlks, ald ill too many imnstanlces spectal cases are treated as if they were,Jemmerai, alnd elrroleons iml)pressin]s are mlae 111)(,1 thle miind of the learnler. Ietter nIot teacll a subnject tllan tea(cl errors. It costs molre to illleit ai rl wat OlIe sil)l)osed to be true thllan to learill co(lTet('ly tle i most I)bstruse princip;ll es. Mecllallies 1has lhad its share of errloleous illstlil'tio() ill b7neReflvt(',1/ Woort/S,; but -we 1elieve thlat tlis one is trulyx elementary, looicall3, c(orrect, and well adapted to meet the wants of the yonno (on1 this sul)ject. The article o01n " /)M,?TsiT'e iFhces" has been re-wnrTittenl for tlhe America-n editioin, and (so changed1 from1 thie original as to con11formn with the wr1iter's 5views of the s111nbje(t, and, as we tlik, Inlak(de it to Con1fo'rm more -learly with tile v iewns,f tlle Author as expresse(l ill ()tlc( par ts of tile wol(,,-. PREFACE. My system of teaching agrees with that of the Author ill regard to presenting some of the principles of Dynamics very early in the course, and I have adopted the plan in my larger work on Mechanics, which is now in press. A student gains a clearer and more general idea of the action of forces by considering their effect upon moving bodies than by considering them ill Equilibrium. The Author has carried this part of the subject, considerably beyond what is usually attempted in elementary works. It is unlecessary to dec(ide which is the more useful, Statics or Dynamics. They are equally indispensable. Both have,lumleronus applications in the Arts and Sciences, and should receive their appropriate share of study. The Author is direct and explicit in his statemlent of principles. The numerous and well-selected "Exercises " and problems are a grand feature of the work. The student who masters them cannot fail to have a good knowledge of the subject, and be able to apply its principles whenever required in practical life. DTi VOLSON WOOD. HOBOKEN, July,? 1876. PRE ACE. IN THESE LESSONS, which are intended for the use of those who have had no previous instruction in the subject, I have endeavoured to bring into prominence the leading principles of Mechanics, and to exemplify them by simple illustrations; and with the view of showing the connection between this science and other branches of Physics, some few pages have been set apart to a brief exposition of the Doctrine of Energy. In arranging the contents of this volume I have deviated considerably from the plan usually adopted, and have been guided by the general principle that the idea of Motion is more elementary than that of Force, and that two Forces, at least, must combine -to produce Equilibrium. In accordance with this view the subject of Statics has been made to depend on the laws of Dynamics, and these are preceded by VI PREFACE. a discussion of some of the simplest principles of Motion. I cannot help thinking that the theory of Equilibrium occupies too prominent a position in many of our Text-books, and that the student obtains, in the problems of Statics, a very inadequate idea of Force and of its modes of expression. In the present arrangement I have followed that order which appears to me to be most logical, and which experience in teaching has shown to be practically aidvantageous. The book contains that amount of matter which a pupil may be expected to acquire in a first year's course of instruction in the subject. It is divided into a number of Sections, which may serve as separate lessons, and should be studied in the order in which they occur. All important propositions are illustrated by numerical examples, worked out in the text, and the lessons are furnished with exercises progressively arranged. To render the work more especially serviceable to candidates for the Matriculation and Preliminary Scientific Exami. nations of the University of London, all questions set during the last few years have been classified, and appended to the several chapters to which they correspond. Answers to the Exercises and Examination Questions are given at the end of the volume, PREFACE. Vii In writing these lessons, I have had in view the want, which is very generally felt, of a School Textbook, sufficiently elementary to be placed in the hands of a beginner, and yet affording a trustworthy basis for the subsequent work of the student. To what extent I have succeeded in meeting this want, I must leave others to determine. I take this opportunity to express my thanks to my friend Mr. BENJAMIN KIISCH, Barrister-at-Law, for the assistance he has afforded nme in revising the proofs, and for many valuable suggestions, whilst the work was passing through the press. P.M. January 1875. PREFACE TO THE SECOND EDITION. TIE DEMAND for a Second Edition within a few months after the appearance of the First, has enabled me to make a few corrections in the Answers to the Examples, and some slight additions to, and alterations in the text, which will, I trust, increase the usefulness of this little work. P. M. Yovember 1875. CONTENTS. INTR OD UCTIONT. ~1. Motion-~ 2. Rest-~ 3. Motion in Molecules-~ 4. Varieties of Motion-~ 5. Physics defined-~ 6. Mechanics defined... Page 1 JIINEMIA'Y CS-310i TION. CHAPTER I. MIEASUREMIEN' OF 0MOTION. LESSON 1. UIIFORM AID ACCELERATED-SPACE DESCRIBED. ~ 7. Velocity — 8. Uniform Velocity-~ 9. Variable Velocity-~ 10. Examples-~ 11. Space described~ 12. Examples-~ 13. Space described in a particular Second-~ 14. V2 = 2fs-~ 15. Relation between Final and Initial Velocity and Space described-Exercises.. 10 IL COMPOSITION OF VELOCITIES. ~ 16. Resultant Velocity-~ 17. Formulae of Motion-~ 18. Composition of Velocities not in the same Straight Line-~ 19. Parallelogram of Velocities -~ 20. Polygon of Velocities —~ 21, Composition of :X CONTENTS. LESSON Uniform and Uniformly Increasing Velocity~ 22. Law of Composition of Velocities- ~ 23. Resolution of Velocity-~ 24. Examples-Exercises Page 20 III. GEOMETRICAL REPRESENTATION OF MOTION. ~ 25. Representation of Time and Velocity-~ 26. Of Space described-~ 27. Of Uniform Motion~ 28. Of Uniformly Accelerated Motion —Examination...... 30 CHAPTER II. FALLING BODIES. IV. 1ODIES FALLING FREELY. ~ 29. Application of General Principles of Motion-~ 30. Time of Rising-~ 31. Whole Time of Flight-~ 32. The height to vhich a body rises~ 33. Examples..... 40 V. MOTION ON AN INCLINED PLANE. ~ 34. Acceleration up or down an Inclined Plane -~ 35. Determination of Motion —~ 36. Examples -~ 37. Time of falling down a Chord of Vertical Circle-~ 38. Line of quickest descent from a point to a straight line-Exercises-Examination. 46 D iYNAMIICS-FORCE. CHAPTER III. M1EASUREMENT OF FORCE. LI. MASS-MOMENTUM-UN,'IT OF FORCE-WVEIGHT.1. ~ 39. Matter, Mass-~ 40. Momentum-~ 41 CONTENTS. X] LESSON Density-~~ 42-3. Force-~ 44. Unit of Force~ 45. Gravitation-~ 46. Weight and Mass-~ 47. Centre of Gravity-~ 48. Pressure. Page 58 V7II. DYNAMICAL FORMULA - ATWOOD1S MACHINE - PROBLEMS. ~ 49. Fundamental Equation of Dynamics-~ 50. Relation between P, W, and f-~ 51. Atwood's Machine-~ 52. Problems-Exercises. 66 VIII. IMPULSrVE FORCES. ~ 53. iMeasure of an Impulsive Force-~ 54 Examples-Exercises —Examination. 80 CHAPTER IV. NEWTON S LAWS OF MOTION. IX. THE FIRsT AND SECOND LAWS. ~ 5'5. Laws of Motion-~ 56. Law I.-~ 57. LaW II. 90 X.THE- THE-IRD LAW OF MOTION. ~ 58. Statement of the Law-~ 59. Statical Reaction-~ 60. Tension-~ 61. Impact-~ 62. Examples-~ 63. Recoil of a Gun-Examination. 94 CHAPTER V. ENERGY. XI. WORK-FRICTION. ~ 64. Definition of Work-~ 65. Unit bf Work -~ 66. Measure of Work done-~ 67. Friction~~ 68-9. Measures of Friction-~ 70. Laws of~ 71. Co-efficient of-~ 72. Examples.. 102 Xll CONTENTS. LESSON XII. VARIETIES OF ENERGY —CONSERVATION OF ENERGY. ~ 73. Definition of Energy-~ 74. Kinetic and Potential Energy-~~ 75-6. Conversion of Heat into Mechanical Energy, and of Mechanical Energy into Heat-~ 76 (a). Dissipation of Energy-~ 77. Relations between Heat and Work —~ 78. Law of Conservation of Energy-~ 79. Totality of Physical Energy-~ 80. Transmutation of Energy-~ 81. Indestructibility of Energy-~ 81. (a). Relation between Force, Momentum, and Energy-ExercisesExamination..... Page 110 CHAPTERI VI. MACHINES. XIII. GENERAL REMARKS - APPLICATION OF LAW OF ENERGY. ~ 82. Definition of a Machine-~ 83. Relation between Power, Weight, and Friction-~ 84. Mechanical Advantage and Disadvantage-~ 85. Simple Mechanical Powers.. 127 XIV. LEVEuR-WHEEL-AND-AXLE. ~ 86. Principle of the Lever-~ 87. Three kinds of Levers-~ 88. Examples of Levers-~ 89. Weight of Lever Considered-~ 90. Examples — Exercises-~ 91. Wheel-and-Axle; its Mechanical Advantage-~ 92. Windlass-Capstan —~ 93. Toothed Wheels-Exercises... 131 XV. THE PULLEY. ~ 94. Pulley described-~ 965. Fixed Pulley~ 96. Single Moveable Pulley-~ 97. Combinations — First System-~ 98. Weights of Pulleys Considered-~~ 99-100. Second and Third SystemsExercises... 143 XVI. THE INCLINED PLANE-WEDGE-ScREW. ~~ 101-2. Ratio of Wto P, when VY moves uni CONTENTS. Xii formly, and P acts parallel to the Plane-~ 103. Ratio of W to P when P acts parallel to Base of Plane-Case of Weight supported' by two Forces -~ 104. Examples-~ 105. The Wedge-~ 106. Wedge Moved by Series of Impulsive Forces~ 107. The Screw described-Ratio of W to P, when P just resists WY-Exercises-Examination...... Page 154 STA1TICS-R EST. CHIAPTEPR VII. THEORY OF EQUILIBRIUM. LESSON XVII. GENERAL CONSIDERATIONS-FORiCES IN THE SAME STRAIGHT LINE. ~ 108. Problem of Statics-~ 109. Method of Estimating and Representing Statical Forces~ 110. Forces in the same Straight Line~~ 111-13. Forces in Equilibrium acting at a Point-Exercises.167 XVIII. COMPOSITION OF FORCES ACTING AT A POINT, BUT NOT IN THE SAME STRAIGHT LINE. ~ 114. Parallelogram of Forces-~ 115. Experimental Verification-~ 116. Resultant of several Forces acting at the same Point-~ 117. Formula for Resultant of two Forces-~ 118. Special Cases-~ 119. Examples-Pulley with Cords Inclined-~ 120. Resultant of three Forces acting at a Point, but not in the same Plane~ 121-3. Geometrical Properties of the Resultant. a. 171 XlV CONTENTS. LESSON XIX. GEOMETRICAL CONDITION OF EQUILIBRIUM WHEN TWO OR 3MORE FORCES ACT AT A POINT. ~ 124. Triangle of Forces-~ 125. Extension of this Proposition-~ 126. Examples-Equilibrium on Inclined Plane-~ 127. Polygon of Forces-Exercises,.. Page 181 XX. RESOLUTION OF FORCES -ANALYTICAL CONDITIONS OF EQUILIBRIUM WHEN ANY NUMBER OF FORCES ACT AT A POINT. ~~ 128-9. Components of a Force-~ 130. Projection of a Force along a Line-~ 131. Composition of resolved parts of Forces-~ 132. Method of Finding the Resultant by resolving the Forces-~ 133. Examples-~ 134. Conditions of Equilibrium-Exercises,... 191 XXI. FORCES NOT MEETING AT A POINT-PARALLEL FO RCES. ~ 135. Like and Unlike Parallel Forces-~ 139. Resultant of two Parallel Forces-~ 137. Determination of its Position-~ 138. Experimental Verification-~ 139. Examples-~ 140. Resultant of several Parallel Forces —~ 141. Centre of Parallel Forces-~ 142. Condition of Equilibrium -~ 143. Examples-~ 144. Resolution of a Force into Parallel Components-~ 145. Example -~ 146. Couples-Exercises.. 200 XXII. FORCES PRODUCING ROTATION —M.OMENTS. ~ 147. Rotation-~ 148. The two elements on which the Rotatory effect of a Force depends~ 149. Moment of a Force about a Point-~ 150. Geometrical Representation-~ 151. No Moment about a Point in Line of Action of Force — ~ 152. Positive and Negative Moments-~ 153. >Moments of Forces about a Point in their Re CONTENTS.:XV LESSON suitant-~ 154. Equilibrium of Moments-~ 155. Application to Lever-~ 156. Examples-~ 157. Balances-~ 158. Common Balance-~ 159. False Balance-~ 160. Roman Steelyard-~ 161. Danish Steelyard-~ 162. Examples of other Balances-~~' 163-5. General Properties of Moments-Exercises.. Page 217 XXIII. GENERAL CONDITIONS OF EQUILIBRIUM OF FORCES IN ONE PLANE-RECAPITULATION. ~ 166. Conditions of Equilibrium for any number of Forces in one Plane-~ 167. Recapitulation- General Results-~ 168. Examples -Examination...... 233 CHAPTER VIII. CENTRE OF GRAVITY.,XXIV. PROPERTIES OF CENTRE OF GRAVITY-EQUILIBRIUM OF A'ODY ON A HARDn SURFACE. ~ 169. Definition of Centre of Gravity-~ 170. Equilibrium of a Body Suspended at a Point~ 171. Centre of Gravity of Uniform Laminme~~ 172-3. Equilibrium of a Body resting on a Hard Surface-~ 174. Three kinds of Equilibrium ~~ 175-6. Position of Centre of Gravity in Stable and Unstable Equilibrium-~ 177. Energy of a body in its three States of Equilibrium-~ 178. Limiting Position of a Body on an Inclined Plane 248 XXV. METHODS OF FINDING THE CENTRE OF GRAVITY OF PARTICLES AND BODIES. ~ 179. Centre of Gravity of two Heavy Particles-~~ 180-2. of a number of Heavy Particles-~ 183. Of Symmetrical Bodies-~ 184. Of a Triangle-~ 185. Example-~ 186. Of the Xvi CONTENTS. LESSON Perimeter of a Triangle-~ 187. Of any Rectllinear Figure-~ 188. Examples-Exercises Page 259 XXVI. METHODS OF FINDING THE CENTRE OF GRAVITY OF BODIES JOINED TOGETHER, AND OF PARTS, OF BODIES. ~ 189. Centre of Gravity of two Bodies — ~~190-1. Examples-~ 192. Centre of Gravity of the Frustum of a Body-~ 193. Example~~ 194-5. Centre of Gravity of a number of Particles-~ 197. Of the Surface of a Cone-~ 199, Of a Pyramid-~ 200. Of a Cone —,ExercisesExamination. 273 A PPENDIX. A. EXAMINATION PAPERS SET AT VARIOUS INSTITUTIONS 289 B,. ANSWERS... 305 ELEMENTARY MECHANICS, INTRODUCTION. ~ 1. Notion.-Our earliest observations must have shown us that some things are moving and that others appear to be at rest. We know what motion means when we watch the rising of the sun, the passage of a bird through the air, the waving of the trees in the wind, or the rushing of the waves to the sea-shore. Every variety of matter seems to be endowed with the faculty of movement. The stone falls to the ground, the flower opens its petals to the sun, and living creatures of every size and shape move in endless ways. ~ 2. Rest.-We are equally familiar with bodies in a state of rest. Nothing seems more immovable than the earth on which we stand. The various things we see around us,-the books that lie upon our shelves, the pictures that hang upon our walls, are all apparently at rest, and we expect them B ~2 INTRODUCTION. to remain so unless they happen to be disturbed by some external cause. A little thought will show us that this state of rest is not as simple as it seems. Let us suppose that we are travelling in a railwaycarriage, and that another train is moving in the same direction on adjoining rails. After a time it overtakes us and then the two trains move on side by side with equal speed. In this case all sense of motion will be lost; the train at which we are looking and the carriage in which we sit will equally appear to be at rest. This simple illustration is sufficient.to make us see that objects may be moving when we suppose them to be stationary, and that the evidence of our senses cannot wholly be trusted. Now the earth on which we stand is in the position of the second train. It is moving round the sun with a considerable velocity; but as we are moving with it and at the same rate, it appears to us to be at rest. Let us consider, further, the condition of those bodies; which although absolutely moving with the earth are,: relatively to us, at rest, Take the picture hanging on the wall. The picture is suspended by cords which hang over a nail. If these cords were to break, or the nail were to give way, the picture, we know very well, would at once fall to the ground. It appears, therefore, that the picture, although at rest, is really tending to fall and is only prevented from obeying its natural tendency by the cords and INTRODUCTION. 3 nail that hold it back. What is true of the picture is true of all things that are in any way supported. Each article of furniture in this room would faill through the floor, if the floor were not strong enough to support it. There is a vessel on the table filled with water, and in the side of the vessel is a cork. The water appears to be motionless. Remove the cork and the water immediately begins to flow out. This water then is endowed with a tendency to motion which the sides of the vessel resisted. The air of the room is seldom still. But suppose for a moment that there is no kind of draught. Let a window or fireplace be opened, let the air be freed in some direction from restraint, and it will at once obey its tendency and begin to move. In these examples no reference has been made to the cause or causes that are supposed to produce the several movements indicated. But the causes themselves do not come within the range of our observing faculties. All that observation teaches us is, that bodies tend to move. ~ 3. Notion in Molecules.-If we examine matter more minutely we shall find that it consists of molecules or very small portions, and that among these particles movements are constantly taking place which are separately hidden from the eye, but the total result of which can be discerned. We have all noticed, perhaps, that on a line of rails a certain space is left between the separate pieces at B 2 4 INTRODUCTION. the points where they are joined together. This space is left, because it is found that the length of the rails increases in hot weather and decreases in cold weather; and if the line of rails consisted of one continuous bar of iron fixed at its two extremities, it would become bent and twisted in order to find room for its expansion. If we put a liquid into a glass vessel and place the vessel over the flame of a spirit lamp, we shall very soon observe that the liquid is rising in the vessel, and after a time it will begin to boil, and its particles will be violently agitated. We know how seldom perfect stillness prevails in the atmosphere. The wind is always blowing somewhere. Now the motion of the air is caused directly or indirectly by the sun's heat,-and the sun's heat is ever varying in intensity. These examples serve to show that heat is a6companied by motion: but this motion takes place among the particles themselves of which the body consists. The body, as a whole, does not move from place to place; but with every variation in its temperature there is a corresponding movement among the particles which compose it, We can take another illustration. Most persons know what happens if a stick of sealing-wax be rubbed on flannel and then held over some scraps of paper. The pieces of paper will at once begin to move towards the wax, and may be made to stand on end under its influence. They are electrified, and in that condition they tend to move. Now, we cannot here INTRODUCTION. a say to what extent the particles of every body are thus influenced. There may exist opposing tendencies to motion, and in that case no visible effect will be prod uced. But whenever electricity is developed the particles of the body are agitated, and motion, or the tendency to motion, results. We have hitherto considered inanimate matter. Let us now see what happens in the animal and vegetable world. A plant or animal may be said to kdiffer from a piece of lifeless matter by its growth alnd decay. Now growth implies a continuous change in the particles of which a body consists. A living organism cannot preserve its old particles and at the same time acquire new. It increases by the decay of old and the substitution of new matter. In this respect animate bodies increase much in the same way as a merchant's capital. A capitalist cannot grow rich by hoarding: on the contrary, he must become daily poorer, for how parsimonious soever he may be, he must consume a portion of what he possesses to support life. It is only by spending money, by buying and selling, by constantly exchanging capital and allowing it to be used by labourers as food, that capital can increase. The same is true of living tissue. In growing it undergoes continuous decay, and the decay is continuously repaired. When the process of reparation does not proceed as rapidly as that of decay, the plant begins to fade and the animal to die. In this continuous decay and reproduction 6 INTRODUCTION. we have a further example of motion among the molecules of bodies. We thus see that bodies themselves and their molecules are constantly in motion or tending to move: that absolute rest nowhere exists; and that what we call rest, which is really rest relatively to us, can be analysed into counteracted tendencies to motion. As motion is thus universally present, we. are sensible of what it is, without being able tot define it. It does not admit of explanation; for there is no condition, in which matter exists, that is simpler or more elementary. ~ 4. Varieties of Motion.-There are different kinds of motion. Let us see what they are. If a body moves from one place to another place it is said to undergo translation. It may move in a straight line or in a curve. The run of a billiard-ball is,an example of motion in a straight line. The flight of an arrow and the course of the planets illustrate what. is meant by curvilinear motion. When a body moves about a fixed point or axis, round which the particles describe concentric circles, the body is said to rotate. A door rotates on its hinges, a wheel on its axle. It frequently happens that several motions are combined. A body tending to move in two different directions may be found to move in a straight line between them, or to INTRODUCTION. 7 assume a curvilinear motion as the result of these two tendencies. Thus the path of a cricket-ball is a curved line, which is the joint effect of the tendency of the ball to move in the direction in which it was thrown, and of its tendency to fall to the earth. Further the motion of rotation is frequently combined with that of translation. This occurs when a wheel rolls along the ground. The motion of the earth round the sun is the result of two tendencies to motion in different directions, producing a curve called the orbit, and of rotation about a fixed axis passing through the poles. When a body is under the influence of opposing tendencies to motion, which exactly counterbalance one another, it is said to be in equilibrium. There is another kind of motion, to which the name undulatory has been applied. It exists under a variety of forms, but may roughly be described as the movement of a particle to and from a particular point. This displacement of the particle is often called its excursion, and is, in all cases, very small. When a series of particles undergo successively this kind of to-and-fro motion a wave is said to be produced. The peculiarity of wave-motion is that although the particles never move beyond the limits of an excursion they appear to undergo translation. If we fix our eyes on a piece of wood floating on a sea-wave, we shall observe that whilst the wave approaches ever nearer to the shore, the piece of wood maintains INTRODUCTION. its position, rising and falling continuously. The apparent motion of translation is the result of the up-and-down movement of each'successive particle in its own place. This kind of motion is further illustrated, when the wind sweeps over a field of corn and bends the several ears in succession. In this case it is evident that each stalk of corn cannot move out of its own place, and yet the eye can follow the wave as it passes from one end of the field to the other. In an undulation the particular motion of each particle is well illustrated by the movement of the bob of a pendulum, which vibrates to and from the lowest point in its path. ~ 5. Physics Defined.-The science of Physics embraces the consideration of bodies and molecules under every variety of motion, and is subdivided according to the particular effect the several kinds of motion produce upon the senses. Thus the passage of a bird in the air produces a sense of locomotion, the resistance of a heavy body tending to fall to the earth a feeling of pressure. Certain visible motions produce the sensation of heat, others give rise to those of sound, light, electricity, &c. Physical science is divided into two main branches according as the motion to be considered is the motion of a body as a whole, or of the undulations of the particles of which it consists. It has INTRODUCTION. 9 been suggested to call the one branch of the subject Molar Physics,' as treating of motion in mass, and the other' Molecular Physics,' as treating of the motion of molecules. The flight of a rifle-bullet, the blow of a cricket-bat, the ascent of a balloon are questions of Molar Physics; whilst problems concerning heat, light or sound belong to the other branch of the subject. ~ 6. Mechanics.-The term tMechanics has generally been employed, and is adopted in the present volume, to embrace the science of the motion and equilibrium of bodies. It involves the consideration of matter in its three forms, solid, liquid and gaseous. But the following pages will treat of solid matter only. The subject will be considered under three heads. Under the first will be discussed some elementary principles of motion, apart from the consideration of the quantity or quality of the matter moved. This subject is called Kinematics, or the science of pure motion. Under the second head the matter that is set in motion and the cause, or force producing it, will be considered. To this division of.the subject the name Dynamics, or the Science of Force, has been given. The third part will embrace certain problems connected with bodies at rest, and these will be discussed under the head of Statics, or the Science of Equilibrium. 10 KINEMTIACS — MOTION. KINEMATICS(-MOTION. CHAPTER I. 3MEASUREMIENT OF MOTION. I. Uniformn-Accelerated.Motion —Space described. ~ 7. Velocity. —The first question we have to determine is, how motion may be measured and numerically represented. Now we measure all things by their effects, and the visible effect of motion is change of place. When we consider motion with reference to time we obtain the idea which is embodied in the word Velocity. The introduction of the idea of time distinguishes kinematics from a purely geometrical science. Velocity, or rate of motion, is measured by the amount of change of place, i.e. by the space traversed in a given time. Motion may be uniform or variable. When uniform, equal spaces are described in equal times, and the velocity is constant. In variable motion, MEASUREMENT OF MOTION. 11 the velocity continually changes. In measuring velocity certain units of time and space are adopted. The unit of time is everywhere one second; the unit of length is one foot in England, but is different in different countries. In England, therefore, the velocity of a moving body is measured by the number of feet traversed in one second; and a body is said to be moving with a unit of velocity when it moves through one foot in one second. The unit of velocity may be briefly indicated as a foot-second. When the velocity is variable, it is, at any moment, measured by the space through which the moving body would pass in one second, if it were to continue to move throughout that second with the velocity which it had at that particular moment. ~ 8. Uniform Velocity. —If v be the uniform velocity of a moving body, v equals the number of,feet which a body traverses in one second, and 2 v is the number of feet traversed in 2 seconds. 3 v,,,, 3, tv,,,,,, t If s = space traversed in t seconds, then s t v. This is the fundamental proposition of uniform motion. Suppose a body to be rotating about a fixed 1 2 KINEMATICS-MOTION. point, like the sail of a windmill, and that it sweeps out an angle, the measure of which is 0, in one second, then 0 is said to be the angular velocity of the body, and t 0 will be the angle described in t seconds. If a be the measure of the arc, t a will be the space traversed in t seconds. ~ 9. Variable Velocity.-The velocity of a body may increase or decrease, uniformly or variably. If it increase or decrease uniformly, the motion is said to be. uniformly accelerated or retarded. If the velocity vary, but not uniformly, the motion is said to be un-uniformly accelerated or retarded. As problems connected with variable acceleration are very complicated, requiring for their solution the higher parts of mathematics, we shall consider, in the following pages, uniform acceleration only. Uniform acceleration or retardation is measured by the increase or decrease of the velocity per second. Thus, suppose a body is found to be moving at the beginning of three successive seconds, with the velocity of 10 ft, 15 ft., and 20 ft. respectively, the body is said to be moving with a uniform acceleration of 5 ft. per second. So too, if a body started with a velocity of 60 ft. per second, and at the end of the first second was moving with a velocity of 50 ft. only, and at the end of the next second with a velocity of 40 ft., the body would be MEASUREMENT OF MOTION. 13 said to be moving with a minus acceleration or retardation of 10 ft. It is clear that all propositions with respect to uniformly increasing velocity are equally true with respect to uniformly decreasing velocity. Let f represent the acceleration of a moving body, then The velocity gained in 1 second is f,,,, 2 seconds is 2 f 3f,, t,, 3f ),,,, t,, tf If we call v the velocity gained or lost in t seconds when a body is moving with a uniform acceleration off feet per second, then v = tf. ~ 10. Examples.-(1) A body starting from rest has been moving for 5 minutes, and has acquired a velocity of 30 miles an hour; what is the acceleration of the body in feet per second? 30x 1760 x 3 Here, the vel. 60= ft. per second = 44. 11p andv=tf.. 44=5 x 60 xf. f=ft. persec. (2) If a body move from rest with a unifolin acceleration of 2- ft. per sec., how long must it be moving to acquire a velocity of 40 miles an hour? 14 KINEMATICS -MOTION. 40 x 1760 x 3 1-76 Here. vel. - 40 x 1760 x 3 176 ft. per sec. 60 x 60 3 2 176 2 and f = 2.-. = t x -. t 88 sees. (3) What velocity does a body acquire in 3 minutes, if its motion is accelerated at the rate of 32 ft. per sec.? Here f - 32, t= 3 x 60,.- v =- 32 x 3 x 60 = 5760. ~ 11. Space Described.-To find the space through which a body passes when it moves with a uniform acceleration is a somewhat difficult problem, requiring higher mathematics than we are supposed to have at our command. Later on, we shall show how this problem may be solved by a purely geometrical method, but now we shall content ourselves with explaining how it immediately follows from a simple and-almost self-evident proposition. Suppose a train to pass a certain station at the rate of 20 miles an hour, and to pass another station, one hour afterwards, at the rate of 30 miles an hour, and that the velocity has increased uniformly throughout the interval. Now, it is very evident that those two stations are 25 miles apart; for the mean velocity with which the train has moved is 25 miles per hour, and since the velocity has increased uniformly throughout the interval, we may clearly assume that the space described in the first MEASUREMENT OF MOTION. 15 half-hour was as much less than it would have beer, if the velocity had been uniform and 25 miles per hour, as in the second half-hour it was greater. The train will, therefore, have described the same space in the hour, as it would have done if the velocity had been uniform and equal to 20 + 80 2 i.e. 25 miles. Enunciating this principle generally we may say:-The space described in any given tine by a body moving with a uniform acceleration equals the space that it would have described, if it had been moving throughout the given time with a uniform velocity equal to the mean of its initial and terminal velocities. We are now able to determine the space described in t seconds, when a body moves with an acceleration off feet per second..First. Let the body start from rest. In this case -the initial velocity = o, and the velocity after t seconds, that is the terminal velocity = tf,.* mean 0 + tf - tf vel. f t; and if a body move for t seconds with a velocity equal to f, the space described is t x tf (~ 8). t2 f Secondly. Suppose the velocity at starting to 16 KINEMATICS-MOTION. be -u, then t seconds afterwards the velocity will be lt + tf, and the mean velocity is e q ~ - + t f t f 2 2 space described in t seconds -- s = t iu + Generally, if u be the velocity at the beginning of a time t, and v the velocity at the end of that period, the space described in the time t = t'h + v ~ 12. Examples.-(1) A body commences to, move with a velocity of 30 ft. per sec., and its velocity is each second increased by 10 ft. per sec. Find the space described in 5 Eeconds. Initial vel. = 30; Final vel. = 30 + 50 = 80 30 +- 80: mean vel. 2 —~ 55 and space described = 5 x 55 = 275 feet. (2) Find the acceleration, if a body starting with a velocity of 10 ft. per sec. describes 90 ft. in 4 secs. Let f = acceleration; Initial vel. - 10; Final vel. = 10 + 4 f.. mean vel. = 10 + 10 + 4.f 10 + 2f. 2. space described=4 (10 + 2f)=40 + 8f= 90 ft... - 6 feet per second. MEASUREMENT OF MIOTION. 17 ~ 13. To find the space described in any particular second, when a body moves with a uniform acceleration f. If the body start from rest the space described in the first second is 0 + f -.f 2 2 Space described in 2nd second isf + 2f 3f 2 2 3rd 2.f + 3f_ 5f 2 2 4th 3f + ~4f 7f 2 2 Hence, it will be seen, by looking at the numbers 1, 3, 5, 7, that the space described in the tth second equals the tth odd number multiplied by. or s = (2 t- 1) So also, the space described in the first t seconds of the motion equals the sum of the first t odd numbers multiplied by f or s= t2as before, (~ 11). 2 2 ~ 14. From the two formulae v = tf(~ 9) and s _ t2f (~ 11) we obtain, by eliminating t, a third formula which connects the velocity acquired with the space described. Thus: C 1 8 KINEMATICS —MOTION. v=tfand. t = v; also s 2 t f 2 v2 2 2 f 2 2f v- 2fs. Examples. —(1) Find the velocity of a body, which starting from rest with an acceleration of 10 feet per sec. has described a space of 20 ft. v2 = 2fs = 2 x 10 x 20 = 400; v = 20. (2) What space must a body traverse to acquire a velocity of 50 ft. per sec. if it move with a uniform acceleration of 5 ft.? 50 x 50-2 x 5 x s.. s=250ft. ~ 15. To find the relation between the velocity and space described, when a body starts with an initial velocity u, and moves with an acceleration f LetI h equal the space which the body, starting from rest, would have described under the accelerationf in acquiring the velocity ut. Then u2 = 2f h; and if v be the final velocitr which the body possesses, after describing the space s, v equals the velocity which the body would have acquitred if it had started from rest, and passed through /I + s feet V2 2f(h + s) = 2fh.+ 2fs = tu2 + 2fs =2 u u2 + 2fs. MEASUREMENT OF MOTION. 19 This equation gives the relation between the final velocity, the initial velocity and the space traversed. Example. —Through what space must a body pass under an acceleration of 5 ft. per sec., so that its velocity may increase from 10 ft. to 20 ft. per sec.? Here 202 - 102 + 2 x 5 x s; 400 - 100 400 100 30 ft. 10 EXERCISES. 1. In what time will a body moving' with an acceleration of 25 feet per second acquire a velocity of 1000 feet. per second? 2. What space will a body describe in 6 seconds, moving: with an acceleration of 160 yards per minute? 3. With what velocity must a body start, if its velocity be: retarded 10 feet per second and it come to rest in 12 seconds? 4. In how many seconds will a body describe 1400 feet, moving from rest with acceleration of 7 feet per second?' 5. Through what space will a body move in 4 seconds with an acceleration of 32'2 feet per second? 6. A body moving from rest with a uniform acceleration describes 90 feet in the 5th second of its motion, find the acceleration and velocity after 10 seconds. 7. What is the velocity of a particle which moving with an acceleration of 20 feet per second has traversed 100W feet? In the examples the body is supposed to start from rest, unless otherwise stated. c KINEMATICS-MOTION. 8. A body is observed to move over 45 feet and 55 feet in 2 successive seconds, find the space it would describe in the 20th second. 9. With what velocity is a body moving after 4 -seconds if its acceleration be 10 feet per second? 10. A body moves from rest with an acceleration of 360 yards per minute, and 4 seconds afterwards another body begins to move with an acceleration of 32 feet per second. When will tile latter overtake the former? 11. What velocity must a body have so that, if its velocity be retarded 10 feet per second, it may move over 45 feet? 12. What velocity will be gained by a particle that moves for 5 seconds with an acceleration of 12 feet per second? II. Composition of Velocities. ~ 16. Resultant Velocity.-If a body tend to move with several different velocities, the velocity with which it actually moves is called the resultant FIG. 1. A B C A C B velocity, and those several velocities are called comnponents. If a body tend to move with a velocity i which would take it from A to B in one second, and likewise with a velocity u' which would take it from B to C in IMEASUREMENT OF MOTION. 21. the same straight line in one second, then at the end of the second the body will be found at C, as if it had moved with a velocity ut + u'. So too, if the body have several tendencies to uniform motion in the same straight line, the resultant velocity will be the algebraical sum of the component velocities. Cases of the composition of velocities in the same line occur when a body is moving on something which is itself in motion, as when a boat is descending or ascending a river. Suppose the velocity of the stream to be 3 miles an hour and the vessel to be sailing at the rate of 8 iniles an hour in still water, then the actual velocity of the vessel is 5 or 11 miles an hour according as the vessel is sailing up or down stream. When a man paces the deck of a steamer, which is sailing down a river, the actual velocity of the man is the algebraical sum of the velocity of the steamer and of the stream, and of the rate at which the man is walking. ~ 17. If a body tend to move with a uniform velocity and a uniform acceleration' in the same straight line, or if it be moving with a certain accele-: ration along a line, which is, itself, moving uniformly, the resultant velocity at the end of any given time will be the sum or difference of the uniform velocity. and the velocity acquired during that time. If u be The word acceleratioan is used, here and elsewhere, to express a ueziforimly increasing velocity. 22 KINEMATICS-MOTION. the uniform velocity,f the acceleration, and t the time v=u+tf; and if s1 be the space the body would describe in t seconds, if moving with the uniform velocity ul, and' s2 the space it would describe, if moving with the acceleration fl, then if s be the space actually described, s = si ~ s2 zt U + (j~~8 11) If there be several uniform velocities and several accelerations, t2(f, +.f2 +...) s= t (u ++ u2 +. ) 2 An instance of the composition of a uniform velocity and two different accelerations occurs when a body is projected up or down a rough incline. The three formulae, already determined, v -u tf t2 f s - t + -2 = U2 + 2s (~ 15) are sufficient for the solution of all problems which are concerned with rectilinear motion resulting from the composition of a uniform velocity and uniform acceleration in the same straight line. ~ 18. Composition of Velocities not in the same straight line. —Suppose a body tend to move with MEASUREMENT OF MOTION. 23 a uniform velocity ut which would take it from A'to B in one second, and with a uniform velocity u' which would take it from A to D in one second, then at the end of the second the body will be found at C, where B C is equal and parallel to A D. Moreover, the body will have moved along A C, and A C represents the resultant velocity. That A Cwill be the path of the body may be seen by supposing the body to be moving along A Xwhilst the line A XT moves parallel to itself with its extremity in A Y. Then if A B be divided into any number of equal parts, say four, and A D into FIG. 2. bA B X whle the body moves from a like number of parts, while the body moves from A to 6b, the point A with the line A B will move from A to d,, and the body will be at cl at the end of the first quarter of a second; and for the same reason the body will be at c2, c3 at the end of each subsequent quarter of a second. The points cl, c2, C3 can be proved to be in the same straight line by equality of triangles, and since A cI, C1 C2, c2 C3 are equal, the motion along A C is uniform. 24 KIINEMATICS-MOTION. ~ 19. Parallelogram of Velocities.-The foregoing proposition is known as the.parallelogram of velocities, and may be enunciated thus: —If a body tend to move with two uniform velocities represented by the two sides of a parallelogram, drawn through a fixed poznt, then the resultant velocity will be represented by the diagonal of this parallelogram that passes throutgh the same point. Since the velocities A B and A D are equivalent to A G, it is clear that if a body tend to move with three velocities represented by A B, A 1D and C A, the body will remain at rest; and since B C is equal to A D, the three velocities that neutralize one another can be represented by A B, B C and C A,-the three sides of a triangle takern in order. ~ 20. It follows from the foregoing that if a. body tend to move simultaneously with several velocities, which would take it (fig. 3) from 0 to A, from O to B, from 0 to C, from 0 to D in one second, and if A B' be drawn equal and parallel to 0 B, B' C' equal and parallel to 0 C and C' D' equal and parallel to 0 D, then since 0 B' is the diagonal of the parallelogram formed by 0 A, 0 B, it represents the resultant of these two velocities, and 0 C' represents, for the same reason, the resultant of the velocities O B' and 0 C, i.e. of O A, 0 B, and 0 C; and similarly 0 D' represents the final resultant of the several velocities. WVe see, therefore, that if a body have M1EASUREMENT OF MOTION. 25 these several tendencies to motion, it will be found at the end of a second or of any given time at the same point D', as if it had moved first from 0 to A, then from A to B', thence from B' to C', and finally from C' to -D', i.e. along the sides of a polygon which respectively represent the velocities. And if the point D' had coincided with 0, or the FIG. 3. 0 D \'1 c 7~ —----- ~D' body had had an additional velocity represented in magnitude and direction by D' 0, the body at the end of the second, or of any less period of timne, would have been at 0; in other words it would have remained at rest. If, therefore, the several velocities, with which the body tends to move, can be represented in magnitude and direction by the sides of a closed polygon taken in order, the body will be at rest; but if the velocities are represented by the sides of an open polygon, the body will move, and the resultant velo 26 iKINEMATICS-MOTION. city ivilI be. represented by the straight line that closes the polygon. ~ 21. Composition of uniform Velocity and Acceleration.-Suppose a body tend to move with a unilbrm velocity which would take it from A to B in one second, and likewise with an acceleration that would take it from A to D in one second; then, at FIG. 4. A B fs~~\ ~ the end of the second the body will be found at C where B C is equal and parallel to A D, just as if it had moved from A to B and from B to C in the second; but the body will not have moved along the diagonal A C. For, since the velocity along A D is not uniform, the spaces described in equal intervals of time will not be equal along A D, whilst they are equal along A B, and therefore the points cl, c2, c3 will not lie in a straight line. In this case, therefore, the path is a curve, and the nature of the curve depends on the magnitude of the acceleration. The MEASUREMENT OF MOTION. 27 path-of a shot projected at a certain angle to the horizon, is a curve resulting from the composition of a uniform velocity in one direction and an acceleration in a different direction. So, also, if a body tend to move with two different accelerations in different directions, the diagonal of the parallelogram will represent the resultant acceleration, although the path of the body may be along some other line. ~ 22. All these results may be summed up in one general law: Whien a body tends to move with several different velocities in' different directions, the body will be, at the end of any given time, at the same point, as if it had moved with each velocity separately. This is the fundamental law of the composition of motions, and it shows that all problems which involve simultaneous tendencies to motion' may be treated as if those tendencies were successive. ~ 23. Resolution of Velocity.-As the diagonal of the parallelogram,'the sides of which represent the component velocities, was found to represent the resultant velocity, so any velocity represented by a certain straight line may be resolved into component velocities represented by the sides of the parallelogram of which that line is the diagonal. Suppose a body urged by a velocity that would take it from 0 to A in one second, then if O C A B be 28 KINENIATICS —IOTION. any parallelogram described on 0 A as diagonal, the body would equally be at A at the end of one second, if urged by two velocities, which would separately take it from 0 to C and 0 to B in one second, and therefore 0 B, 0 C represent the components of this velocity. If 0 B and 0 C be at right angles to each other then OA2 -- B2 + 0 C2 and if Xbe the comFIG. 5. 4Y ponent along 0B and Y the component along 0 C, and if R be the original velocity along 0 A, we have X-::b OB: OA orX =OBR 0 A Y: i:: OC: OA orY y OC R OA ~ 24. Examples.-(1) A body tends to move with velocities of 30 feet and 40 feet per sec. along two straight lines at right angles to each other; find the resultant velocity. Let V = resultant velocity, then V2 = 302 + 402 = 2500.. V= 50 feet per sec. MEASUREMENT OF MIOTION. 29 (2) A body is moving in a certain direction at the rate V feet per second, find the components of its velocity along lines inclined to its direction at angles of 30~, 45~, 60~ respectively. If the angle A 0 B is 450 it follows that 0 B = BA and OA2 = 2 0B2. OB A, and if V be the velocity along 0 A, its component along 0 B is V FIG. 6. A A o B 0 B O B If the angle A 0 B is 30~, it follows that 0 A 2 A B,.OA2 =G B2 + or OB 0 A 2 and the component of the velocity is V 2 3 If the angle A 0 B is 60~, 0B = 2 and the component required is - As these results frequently occur, they should be very carefully remembered. 30 KINEMATICS- MOTION. EXERCISES. 1. A body is simultaneously urged to move with velocities of 50 feet, 21 feet, and 25 feet respectively; can the body remain at rest? 2. A body whilst moving vertically downwards with a uni-: form velocity of 10 feet per second is urged horizontally with an acceleration of 5 feet per second;find its distance from starting point after 2 seconds. 3. A body tends to move in a certain direction with an acceleration of 32 feet per second, but is constrained to move in a direction inclined at an angle of 450 to the original direction; find the component of its acceleration in the latter direction. 4. A body moving with a uniform velocity of 30 miles an hour has its velocity accelerated 10 feet per second in the same direction; find the space traversed in a quarter of a minute. 5. A body is moving at the rate of 40 miles an hour when its velocity is retarded at the rate of 6 inches per second; when and where will it stop? 6. A body tends to move with equal velocities of 10 feet per second in two directions inclined at 1200 to each other; find its path and resultant velocity. III. Geometrical Representation of Motion, ~ 25. In this Lesson we shall show how the subjects that have already been considered, viz., time, uniform "and variable velocity, and space described may be geometrically represented. MEASURE3IENT OF MNOTION. 31 Let OX be a line limited towards 0, unlimited towards X, on which units of length correspond to units of time; so that if the points A, B, C be equally distant from 0, and 0 A represent one second, 0 B would represent two seconds, and-so on. Now let the velocity with which a body is moving at any particular time be represented by a vertical FIG. 7. 0 sC 0 A B C i drawn through one of the points 0, A, B, C, &co Thus, if OP represents the velocity at 0, and -A Q at A, then A 0Q is greater than O P, in the same proportion as the velocity at A is greater than the velocity at 0, and these lines 0 P, A Q, B R,.. indicate the number of feet per second with which the body is moving at the points 0, A, B.... In the same way, if all the corresponding verticals be drawn for periods of time between O, A, B... and their extremities be joined, the line P Q R S F is called the curve of velocity. It must not be supposed that the curve of velocity is the same thing as the path of a body. 32 KINEMATICS — MOTION. Motion might take place in a straight line and yet the curve of velocity might be represented by the annexed figure. The curve of velocity is merely a graphic representation of the increase and decrease in the rate of motion at successive intervals of time. If we draw the lines P a, Q b parallel to the line O X, Q a and R b will represent the increase of the velocity during the first two seconds; and if this increase were uniform these lines would represent the acceleration. When the acceleration varies, it is measured at any point by the velocity, which would be added in a unit of time, if the velocity increased uniformly throughout such time. If therefore tangents be drawn to the curve at the points Q and R the acceleration at Q would be represented by r b, and at R by s c. Thus r b measures the rate at which the velocity of the body is increasing per second at the particular moment of time indicated by 0 A, and when the velocity already acquired is A Q. In this way we obtain a graphic representation of the velocity and acceleration of the body at any instant of time. ~ 26. We have now to show how the space described in any given time may be graphically represented. Let 0 Z (fig. 8) represent any interval of time, A 0 the velocity at 0, YZ the velocity at Z, and AP Y the curve of motion as before. Let F G be a very small interval of time MIEASUREMENT OF MOTION. 33 r.. Let F P be the velocity at the beginning of ihe time r, G Q thle velocity at the end. T.hen^ the, space described in the time r must be greater thani the space that would be described if the velocity]' F were uniform throughout the interval, and less than the space that would be described with.the uniform velocity Q G. That is, the true space must lie between PF x F G andQ G x F G (since s=v t); but P F x F G - the rectangle P G, and Q QG FIG. 8. A 0 _ k z X x FG = the rectangle Q F. Therefore the true space described is represented by a figure the magnitude of which lies between the rectangles P G and Q F. Now the whole time 0 Z is made up of the sum of such intervals as F G, and therefore the whole space described in the time 0 Z is somewhere between the number of units of area in the sum' of all the rectangles like P G, and the sum of all the rectangles like Q F. But the sum of each of these sets of rectangles approaches nearer and nearer D 34 KINE1MATICS -MOTION. to the area of the whole figure 0 A YZ, as F G is made smaller and smaller; and can be made to differ from 0 A: Y.Z by as small a quantity as ever we please.'It' thus appears that the space described lies between two quantities; that each of these quantities becomes ultimately equal to 0 A. YZ, as F G diminishes without limit; and, therefore,' that the space described:. equals the number of units of, area in the figure 0 A YZ. We have thus proved that the space described in any given time may be represented by the number of units of area contained by the two verticals of velocity, the included line of time, and the portion of the curve intercepted between these two verticals. The problem of finding the space described in any time resolves itself into that of finding the area of a curve. In all but the simplest cases a knowledge of higher mathematics is necessary. ~ 27. In uniform motion the velocity at different intervals 6of time remains the same. The curve FIG. 9. 4_P io t a iK becomes therefore in this case a straight line,: parallel to. the line of time, and the space described in: t seconds equals the area 0 L: O0 K x KL = t. va MEASUREMENT OF MOTION. 35 ~ 28. —To find the space described in t seconds when a body moves with a uniform acceleration. Inr this case the increments of velocity for successive seconds are constant. First. Let the body start from rest. Then it O X be the line of time, and O A, A B... represent FIG. 10. L O A B C IIK seconds, and if P A represent the velocity at A, Q B at B and R C at C; and if P b, Q c be drawn parallel to 0 X, then P A = Q b = i c =f, the acceleration, and O P Q can be proved to be a straight line..Let LKrepresent the velocity aftert seconds, then 0 K = t and L K = tf, and the space described in t seconds equals the area of the triangle 0 L If = t x tf 2 t2f 2 The same reasoning as was employed to establish the general proposition (~ 26) might have been used to prove this independently..Secondly.;Let the body start with a given velocity. D2 36 IINEMATICS — MOTION. Let u equal this initial velocity, f the acceleration, andt the time. Then 0 P = u and Q a - f, where 0 A represents one second. Let 0: — t. Then the FIG. 11. -~ A,' T K space described is represented by 0 P L K, which is equal to the rectangle 0 H + triangle P L H =PO x OK+- LIH x PH = ut + tf.t t2f' s= tu +t. Let S T bisect 0 1K, and through S draw M S SN parallel to 0 ~K. Then since the triangle P M S is equal to the triangle S L N, the area OPLK:=.area OMN K ='STx OK OP+LK 2-= x OK 2 t. where u' is the terminal velocity. Or, the space' described, in any given time, when a MEASUREMENT OF MOTION. 37 body starts with a certain velocity which is uniformly accelerated, is equal to the space which would have been described, if the body had moved throughout the given time with a uniform velocity equal to the mean of the initial and terminal velocities (~ 11). EXAMINATION. 1. Distinguish between motion of translation, rotation, and vibration. 2. What is meant by uniform velocity? How is it measured? 3. Two bodies start from A to B and from B to A, two points 80 yards apart, at the same time; the one moves uniformly at the rate of 10 ft. per see., the other at the rate of 12 ft. per sec.; where will they meet? 4. If a particle 10 inches from a given point revolve round it 7 times in 22 sees., find the velocity of the particle. 5. Two men A and B start at the same moment in the same direction, from two points 1,500 ft. apart; if A walk 4 miles an hour and B 3l miles an hour, whero will A overtake B? 6. When is a velocity said to be uniformly accelerated? 7. A body begins to move with a velocity of 100 ft. per sec., and at the end of 7 sees. its velocity is 65 ft. How: much is the velocity retarded a second? 8. Show how it is that the space described in any time, when a body moves with a uniform acceleration, isproportional to the square of the time. 9. Enunciate the law of the composition of velocities. 10. A body is simultaneously impressed with three uniformvelocities, one of which would cause it to move 10 ft. North in 2 sees., another 12 ft. in one see. in the same 38 KINEMATICS —IOTION. direction'; and a third 21 ft. South in 3 sees. Where will the body be in 5 sees.? 11. Explain the proposition known as the parallelogram of velocities. 12. A body tends to move horizontally, with a-uniformvelocity of 12 ft. per sec., and also vertically downwards with a uniform velocity of 8 ft. per sec.- determine the position after 3 sees. 13. A body begins to move with an acceleration of 8 a,. per. sec., and its velocity is at the same time retarded 2 ft. in 3 sees.; find the space described in 3 sees. 14. Explain why it is dangerous to jump out.of a railway carriage in motion. 75. A body is projected horizontally from tile top of a cliff with a velocity of 500 ft. per' sec.; it, reaches tlie ground in 3 sees,;. find its distance from the foot of the cliff. 16. If a person is walking in a straight line, in what direction must he throw a ball upwards, that it may return into his hand? 17. If a ball be thrown out of the window of a railway carriage in motion, in vwhat direction will it seem to' fall, and in what direction will it really fall? 18. A body moving uniformly with a velocity of 10'ft. per sec. is suddenly impressed with an acceleration of 32 ft. per sec. in the same. direction; what space will be described in the 3rd second of its accelerated motioii? 109. A body moves with a velocity.of 10 ft. per sec. in a given direction,; find the velocity in a direction inclined at an angle of 30~ to the original direction. 20. What acceleration along a certain line is equivalent to an acceleration of 20 ft. per sec. in a direction that makes an angle of 45~ with that line? 21. A particle moves with a uniformiy increasing velocity. Show that the. whole space described is proportional'to the square of the time from the beginning of the MEASUREMENT OF MOTION. 39 motion. -(Matriculationz xctm. Unliv. Lon., Jan. 1871.) 22. A balloon is carried along by a current of air moving from east to west at the rate of 60 miles an hour, having no motion of its own through the air, and a feather is dropped from the balloon. What sort of a path will it appear to describe, as seen by a man in the balloon? —(Matric., June 1874.) to40 KKINETMATICS-MOTION. CHAPTER II. FALLING BODIES. IV. Bodies fiallin.g freely. ~ 29. Most of the preceding principles of motion are well illustrated by falling bodies. When a body is allowed to fall freely it is found to acquire a veloc ity of about 32-2 feet per second every second of its motion, so that it is said to move with an acceleration of 32-2. This acceleration, which for the sake of convenience is represented in books on Mechanics by the letter'g,' can be shown to vary with the distance of the body from the etlrth's centre. Thus at the summit of a high mountainr g is found to be less than near the surface of the earth; and at the equator, in consequence of the peculiar configuration of' the earth, it is less than in the neighbourhood of tile poles. Thus the velocity which a body (17) N102g -20 (18) 24 lbs. nearly. 1 + /-~; 1+ ~'a'3 310 ANSWERS TO EXERCISES XXI. Exr'ncIsrs. Pp. 215-7. (1) 2: 3. (2) 70-7, 50~-. (3) 20 ins. (4) 67 ins. from one end. (5) 2 up median line from angular point. (6) 21/2 P along'the diagonal. (7) 40 ins. fromn greater force. (8) 12 ins. (9) 47- ins. (11)'- rad. (12) 64 lbs.; 3a lbs. (13) 194 ins. from one end. (14) 29 lbs. (15) 14 lbs. (16).3 Of middle line from one of the sides. (17) 71M ins, from 3 lbs. weight. (18) 40 lbs.; 24 lbs. (19) 16 lbs.; f7 of the length from the 7 lbs. (20) As 4: 3 1'. XXII. EXERCISES. Pp. 232-3. (1) 1: 1/3. (2) W. (3) 1.2 lbs. (4) 1/3 lbs. (5) (8-31/6) x 9'6 ins. from one end. (6) 2/'3: 1. (7) 61/2 ozs. (8) 5 ins. (9) 5 ins. from end where weight acts. (10) 6 lbs. (11) 12 ins.; 27 lbs. EXAMINATION (CrAPTR VII.) - Pp. mAO-7. (2) 40;ins.; 20 ins. (3) 2/831 lbs. (4) 6 lbs. (8) 3': 34: 1-:' 14. (10) 10 lbs. (11) 1'74 ft. (12) 72 (4- /3) ins. from 4 lbs. end. (15) 4\/2-,/ lbs. (17) 81/3; 1613 lbs. (18) 10/3. (24) 1/201; /146; /91. (25) (a>) 50 lbs. 25 ins.; 40 lbs. 7'5 ins. (b) 62 lbs., 31 ins.; 72 lbs., 13'5 ins. (26) ~ 126. (27) 11'25; 27'4. (28) 15 lbs., 131bs. (29) ~ 155. (30) 27'73. (35) ~ 128. (38) A pushing force at 600 with the plane, or vertically upwards; R =6/3 or 0. (39) 6Z7 ft. from one end. (42) 4: 9. (43) 1 in. from one end. (44) 2,7; 1'351/. (46) ~ I54. (47) If D) be the middle point of A, C' the point in the stringr to which the weihllt is attached, and C E the perpendi AND EXAMINATION QUESTIONS. 31 cular on.4 B in its position of equilibrium -E = 2; and D E 7 when AB is horizontal, the pivot is 6 ins. from I). (48) ( + j/3); (5- V3) ozs. (49) /5 lbs., 2 JV lbs. (51) 12 ins. (54) If C be the nail, B the end of stick against wall, C B=, 3 ft.; the pressure on the nail is 8$/3 ozs., and that on the wall 8 (,/9-1) ozs. XXV. EXERCISES. Pp. 271-2. (2) 240 ft. (3) 20 (/2 + 1) = 4828 lbs. (4)'- of length from one end. (5) 450~. (6) V/3: 1. (7) 10 lbs. (8) ~ of median line from centre. (9) 4 diagonal from centre. (10) I up median line fiom base. (11) 4 ft. from the other end. (13) If a = side of square, Centre of Gravity is from centre of square. (14) 8 ins. up 1-8 median line of larger A from common base. (15) -/i of length of rod from middle point. (16) 3x/3: 5. XXVI. ExERcisEs... 285. (1) If A, B, be the middle points of the longer and shorter arms, C the angular point, and if A )D = A B, then the C of G is ~ up D C. (2) 7-9- ins. from heavier ~3 3 3' Y3a weight. (3) 27'-~- feetfrom the base. (4) ( a-3312 from the common side, where a is the side of the square. (5) 32 ins. from base. (6) One-sixth of radius from centre of large circle. (7) ~- of diagonal from centre. EXAMINATION (CHAPTER VIII.) Pp. 285-7. (2) If C and B be the middle points of the two parts and CB be bisected in D, D must be vertically under A, the point of suspension. The position of the wire may be found by drawingr a horizontal line throug,(h A, 312 ANSWERS TO EXERCISES. and marking off a distance A E equal to /4. of a, where a is length of half the wire, and then dropping a vertical BEF=- a. If FA be joined and an equilateral A 17 described on it, the two sides will be the position of the wire. (4) If G be point required and G F be perpendicular to B C, then CF = "7a, and G F 7=a where a = CB. (5) 3'464 ins. (8) Reckoning from corner opposite to that from whicll rectangle is taken X=- 3L ins., Y = 11 ins. (9) If C be the centre, and A the point,to which weight is fixed, mark off A G = I inch along A C, and G Twrill be vertically under point of suspension. (10) 3'26 ins. fiom top. (11) 54 ins. from base of shorter. (12) 3'2 ins. from heavier weight. (16) 8: 7. (18) -9 of line joining 9 ozs. and 3 ozs. from the 9 ozs. weight. THE END.