UC-NRLF 
 
 SbM 707 
 
REESE LIBRARY 
 
 OF T1IK 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Q 
 
 Received ^ yr^^^tf^_ ., 188 <f*~ 
 
 Accessions No. 3&f3</ Shelf No. 
 

 -7 
 
isT^fejj 
 
 OF THE T r 
 
 UNIVERSITY 
 
LESSONS 
 
 IN 
 
 ELEMENTARY DYNAMICS 
 
 ARRANGED BY 
 
 H. G. MADAN, M.A. 
 
 FELLOW OF QUEEN'S COLLEGE, OXFORD; ASSISTANT MASTER 
 IN ETON COLLEGE 
 
 ' Omnia mutantur, nihil interit.' 
 
 OVID. 
 
 W. & R. CHAMBERS 
 
 LONDON AND EDINBURGH 
 
 1886 
 
Edinburgh : 
 Printed by W. & R. Chambers. 
 
PREFACE. 
 
 IT seems to be pretty generally agreed that some branch of 
 Physics should form an early, if not the very earliest part of a 
 scientific education, involving, as it does, an examination of 
 those general and fundamental properties which all kinds of 
 matter possess to a greater or less extent, and by which we 
 recognise things as being forms of matter. Of the various 
 branches into which Physics is divided, Mechanics has an 
 undoubted claim to the first place, from the simplicity and 
 exactness of its laws, the readiness with which they can be 
 demonstrated, and the multitude of practical illustrations of 
 them which meet us on every side in the ordinary movements 
 of life, in games, and in work. The study of Chemistry or of 
 Electricity may be more amusingsand attractive on account of 
 the brilliant experiments associated with it, but the exact 
 explanation of chemical and electrical phenomena is far less 
 easy and far less certain than that of mechanical laws, and is, 
 indeed, hardly within the grasp of minds previously untrained 
 in scientific methods and ideas.* 
 
 An endeavour is made in this book to explain some of the 
 properties of matter, Newton's Laws of Motion, and the modern 
 conceptions of Energy and Work, in such a manner as involves 
 only the most elementary knowledge of mathematics. A boy's 
 mind is eminently practical ; he is not at once struck with the 
 force and beauty of such expressions as S = ^gt 2 or W = mv^/ 2 ; 
 he is apt to get bewildered by, and to fail to take in, abstract 
 mathematical demonstrations such as most elementary books 
 take a too early opportunity of putting before him. Things 
 cannot at first be presented to his mind in too concrete a form : 
 illustrations drawn from the phenomena of every-day life, and 
 the games which he enjoys, add a vigour and interest to his 
 
 * Chemistry and Electricity will probably themselves, at no very distant period, 
 be brought under the head of branches of Dynamics. 
 
2 PREFACE. 
 
 work such as may be wanting when lie is set down to cal- 
 culate wearily the present value in farthings of a million dollars 
 put out at compound interest about the date of the Flood, or to 
 find how many square yards of paper are required to cover 
 Westminster Hall. But he begins to think that there is ' some- 
 thing in it,' when he finds that in all his movements, in walking, 
 cricket, fives, rowing, racing, &c., he is (unconsciously, it may 
 be, but none the less really) obeying a few very simple Laws of 
 Motion, and that even these are little more than an expansion of 
 the great principle of Energy. 
 
 Further, a beginner wants precise definitions of terms, and 
 cannot easily pick out what he wants from such casual mention 
 or diffuse explanation of them as appears sufficient to many 
 writers. An attempt is here made to give shortly and exactly 
 the real scientific meaning of such expressions as ' above/ 
 * below,' * on the same level,' ' out of the perpendicular,' * weigh- 
 ing ' things ; and from the same motive the Laws of Motion are 
 divided into short statements which are considered separately. 
 
 It was at first intended that the book should be a new edition 
 of the small treatise on the Laws of Matter and Motion, issued 
 by the same publishers ; but examination soon showed that 
 most of the treatise could not fairly be brought into accordance 
 with modern scientific ideas. A few paragraphs of it have, 
 however, been made use of, with more or less alteration. 
 
 No originality whatever is claimed for this treatise ; its aim 
 being simply to put useful, thought-suggesting facts in a plain, 
 straightforward way. 
 
 H. G. MADAX. 
 
 ETON COLLEGE, August 1886. 
 
 It has been very reluctantly decided not to introduce the Metric 
 System into this issue of the book. An account of the system is given in 
 the Appendix, and any opinions as to the advisability of using it in a 
 future edition (if called for), will be gratefully received. 
 
CONTENTS. 
 
 CHAPTER I. GENERAL PROPERTIES OF MATTER. 
 
 Section 1. The Scope of Natural Science ... ... ... 5 
 
 ii 2. Extension and Form ... ... ... ... ... 8 
 
 ti 3. Impenetrability ... ... ... ... ... 8 
 
 ii 4. States of Aggregation ... ... ... ... ... 9 
 
 n 5. Divisibility. ... ... ... ... ... ... 11 
 
 ii 6. Indestructibility 13 
 
 7. Porosity 14 
 
 n 8. Attractability 15 
 
 n 9. Cohesion ... ... ... ... ... ... 16 
 
 n 10. Adhesion 21 
 
 n 11. Capillarity Diffusion Osmose 23 
 
 12. Gravitation 29 
 
 ii 13. Mass and Weight 35 
 
 n 14. Density ... ... ... ... ... ... 36 
 
 CHAPTER II. MOTION FORCE MOMENTUM. 
 
 Section 1. Motion and Velocity ... ... ... ... ... 38 
 
 2. Force 39 
 
 it 3. Momentum ... ... ... ... ... ... 41 
 
 n 4. Measurement of Forces ... ... ... ... 42 
 
 APPENDIX. Problems on Momentum ... ... 43 
 
 CHAPTER III. LAWS or MOTION. 
 
 Section 1 . The First Law of Motion 45 
 
 ii 2. Laws of Centrifugal Tendency ... ... ... 51 
 
 ii 3. Friction 57 
 
 n 4. The Second Law of Motion ... .. ... ... 65 
 
 ii 5. Composition of Forces ... ... .. ... 68 
 
 n 6. Resolution of Forces ... ... ... ... ... 73 
 
 APPENDIX. Action of the Eudder of a Ship ... 76 
 
 Section 7. The Third Law of Motion 77 
 
 n 8. Collision of Bodies SO 
 
 . 9. Laws of Reflexion S3 
 
 APPENDIX 85 
 
4 CONTENTS. 
 
 PAGE 
 
 CHAPTER IV. ACCELERATION. 
 
 Section 1. General Principles ... ... ... 87 
 
 ii 2. Gravitation as air Accelerating Force 88 
 
 CHAPTER V. CENTRE OF GRAVITY. 
 
 Sectionl. General Principles ... ... ... ... ... 95 
 
 ii 2. Equilibrium of Bodies ... ... ... ... ... 96 
 
 ii 3. Methods of finding the Centre of Gravity ... ... 106 
 
 APPENDIX. Centre of Percussion or of Inertia ... 110 
 
 CHAPTER VI. ENERGY AND WORK. 
 
 Sectionl. General Principles ... ... ... Ill 
 
 ti 2. Statical and Kinetic Energy 112 
 
 M 3. Conservation of Energy ... ... ... ... 115 
 
 ii 4. Measurement of Energy ... ... ... ... 116 
 
 APPENDIX A. Exact Valuation of the Energy 
 
 in a moving body ... ... ... ... 121 
 
 APPENDIX B. The Pendulum 122 
 
 CHAPTER VII. MACHINES. 
 
 Sectionl. General Principles ... ... ... ... ... 123 
 
 Mechanical Advantage 1 30 
 
 2. Pulleys 131 
 
 3. The Wheel and Axle 135 
 
 .. 4. The Lever 140 
 
 5. The Inclined Plane 148 
 
 .1 6. The Locomotive Engine as an illustration of the 
 
 Principles of Dynamics ... ... ... ... 154 
 
 APPENDIX The Metric System of Measures and Weights ... 158 
 
 QUESTIONS AND EXERCISES 104 
 
 INDEX 177 
 
 
LESSONS IN ELEMENTARY DYNAMICS. 
 
 CHAPTEK I. 
 GENERAL PROPERTIES OF MATTER. 
 
 SECTION 1. THE SCOPE OF NATURAL SCIENCE. 
 
 1. As we look around us at the vast crowd of objects which 
 our senses make us aware of, and which make up what we call 
 ' nature,' or the ' natural world,' we cannot help being struck by 
 two things ; firstly, the immense variety of these * phenomena ' 
 as they are termed (Greek ^am/uiya, appearances ) ; and secondly, 
 the ceaseless changes which they are undergoing. No one can 
 help asking what these objects really are, watching their various 
 actions on one another, and trying to find out the reasons of 
 what he sees. And no one can study nature carefully without 
 becoming convinced that the universe is not a mere collection of 
 things brought together by accident and ruled by chance, but 
 that there is an order throughout it ; that one thing happens 
 because something else has previously happened ; in a word, 
 that every particular change is the effect of some definite cause. 
 Thus, the boiling of water is an effect of which the cause is heat. 
 But we are at once led on to inquire what is the cause of the 
 heat itself ; and thus w r e trace back a phenomenon through a 
 long series of effects and causes, all connected together like the 
 links of a chain. 
 
 2. When we have found out an unchangeable link of con- 
 nection between two or more phenomena, we are said to have 
 
6 ELEMENTARY DYNAMICS. 
 
 discovered or established a law of nature. It is observed, for 
 instance, that whenever matter is heated, it becomes enlarged in 
 bulk; it is therefore recorded as a law of nature, that 'heat 
 expands bodies.' 
 
 3. When, again, we can show that some other phenomenon, 
 seemingly widely different, is really, though indirectly, caused 
 by the operation of the same law, we are said to explain that 
 phenomenon. Thus we explain the fact that a clock is apt to 
 go slower in summer than in winter, by first establishing that 
 a clock goes slower the longer the pendulum is, and then 
 inferring from the law of expansion by heat, that the pendulum 
 must be longer in summer than in winter ; so that the alteration 
 in rate is accounted for by the difference in temperature of the 
 seasons. The knowledge of these natural laws or rules according 
 to which the manifold changes going on in the universe seem 
 to occur, constitutes natural science in its widest sense. But 
 although we know, as yet, very little of what might be known 
 about the laws of nature, still the range of even our present 
 knowledge is so extensive that no one mind can take it all in. 
 Hence natural science is divided into a number of departments 
 or branches, each relating to a particular set of the phenomena 
 which occur in nature. 
 
 4. Some phenomena depend upon the peculiar kind of sub- 
 stance of which the body manifesting them is composed, and 
 consist in permanent changes in appearance and properties ; as 
 when sulphur, at a certain temperature, takes tire that is, 
 unites with the oxygen of the atmosphere, and forms a suffocating 
 gas, totally unlike either sulphur or oxygen. The study of facts 
 of this class forms the branch of science called Chemistry. 
 
 5. Organised bodies that is, plants and animals also manifest 
 a peculiar set of appearances which are summed up in the word 
 life. The consideration of vital phenomena belongs to the 
 department of science called Biology. 
 
 6. But there is a large and important class of phenomena of 
 a much less special kind, and which belong to matter in general 
 and to all bodies composed of it, whatever be their peculiar 
 constitution, and whether organic or inorganic. Thus, a stone, a 
 piece of sulphur, a plant, an animal, all fall to the earth if 
 unsupported, are all capable of being expanded by heat, 
 
GENERAL PROPERTIES OF MATTER. 7 
 
 all reflect more or less light, &c. It is the investigation of 
 universal laws of this kind, where no change of constitution is 
 concerned, that forms the branch of natural science called 
 Physics. 
 
 7. Of those physical phenomena, again, some have a higher 
 generality than others, and it is these most general laws of the 
 material world that naturally form the subject of this intro- 
 ductory treatise. Thus, every kind of matter is found to be 
 capable of being altered in shape, or moved, when a sufficient 
 amount of some power or force is applied to it The study of 
 the effect of forces in producing motion of masses of matter 
 constitutes the branch of Physics called Dynamics. 
 
 8. Matter (Lat. materies), which may be denned as the material 
 or stuff of which things are made, has certain properties; 
 \)y which is meant, that it has the power of making 
 certain impressions upon our senses, or of exciting in us 
 sensations. Through these sensations we are said to have a 
 perception of matter and bodies ; but as to what matter is in 
 itself, beyond its power of affecting our senses, we know nothing. 
 The something, whatever it is, in which this power is conceived 
 to reside, is called substance. Some philosophers deny the 
 existence of anything beyond the properties ; but though we 
 have no direct evidence of anything else, it is difficult, if not 
 impossible, to get rid of the notion, that there is a substance to 
 which the properties belong. So far as natural science is affected, 
 the question is of no moment ; what really concerns us is, how 
 matter appears and acts, and not what it is. 
 
 9. The more important of the properties of matter are 
 Extension, Impenetrability, Divisibility, Inertia, Porosity, 
 Attraction, States of Aggregation, &c. We shall describe and 
 illustrate them in succession, classing such qualities together 
 as seem to be naturally connected. Extension and Impenetra- 
 bility claim precedence as being essential to our very notion of 
 a body. We cannot conceive a body that does not extend over 
 or occupy a portion of space, however small, and that does not 
 exclude all other bodies from occupying the same space while it 
 is there. This quality of obstinately shutting out other portions 
 of matter from its own room, seems really what we chiefly mean 
 by substance. 
 
8 ELEMENTARY DYNAMICS. 
 
 SECTION 2. EXTENSION AND FORM. 
 
 10. Magnitude or size is one of those simple ideas that do not 
 require or admit of explanation, because there is nothing simpler 
 to explain them by. It is chiefly by their occupying a certain 
 amount of space that bodies make themselves known to our 
 senses ; and when we try to think of those minute particles of 
 matter that elude the senses, we must still conceive them as 
 being extended or having a certain magnitude. 
 
 11. Bodies are extended in three directions, or have three 
 dimensions namely, length, breadth, and depth. Width is the 
 same dimension as breadth ; and for depth we often use the 
 term height, and sometimes thickness. The way in which 
 these dimensions are bounded gives each body its peculiar form 
 or shape. This is equally true of a block of stone, a sheet of 
 paper, a hair, a particle of dust. 
 
 12. In a line we consider only length, or linear magnitude ; 
 and the quantity of it is expressed in numbers of some con- 
 venient unit, as an inch, a foot, a metre.* In a surface we con- 
 sider both length and breadth, or superficial magnitude. This, 
 which is sometimes called area, is expressed in square inches, 
 square feet, square metres, &c. Solid magnitude has length, 
 breadth, and depth. The quantity of solid magnitude in any 
 body makes its size, bulk, or volume ; and is expressed in cubic 
 inches, cubic metres, &c. 
 
 13. Surfaces are the boundaries of solid and liquid bodies, and 
 lines are the boundaries of surfaces. Thus, an ordinary box is 
 bounded by six plane surfaces, and these surfaces are bounded 
 or separated from one another by lines or edges. A sphere is 
 bounded by one curved surface. 
 
 SECTION 3. IMPENETRABILITY. 
 
 14. Impenetrability is that quality of bodies by virtue of 
 which each occupies a certain portion of space, to the exclusion 
 of all other bodies : it expresses the fact that two bodies cannot 
 be in the same place at the same time. The term Impenetra- 
 bility is not a happy one, though it is difficult to find a better. 
 
 * An account of the metric system of measures and weights is given at the end of 
 the book. 
 
GENERAL PROPERTIES OF MATTER. 9 
 
 In the popular sense of the word, matter is anything but 
 impenetrable. The hand can be thrust into water, a nail can 
 be driven into wood, and even the hardest substances are 
 pierced by others that are more or equally hard. But all these 
 are instances merely of displacement, or of removing part of one 
 body to make room for another. There is no wood where the 
 nail is, nor are the particles of the removed wood driven into 
 one another, so to speak ; they are merely forced closer together, 
 as those of a sponge are when squeezed. In some cases it might 
 seem at first sight that something like interpenetration of sub- 
 stances actually takes place. Water will rise in vapour, and yet 
 the portion of air in which it disappears may not occupy more 
 room than it did before. A measure of water and one of 
 sulphuric acid mixed together, occupy less space than the two 
 did separately. But in such cases we must conceive the 
 particles of the one substance as finding room in the intervals 
 . between the particles of the other, as will be more fully spoken 
 of under Divisibility and Porosity. 
 
 15. That the most movable and unsubstantial substances, when 
 displacement is prevented, occupy space as effectually as the 
 most solid, is seen in a blown bladder, or in an air-cushion. 
 This property of air is taken advantage of in the diving-bell. 
 An easy illustration is obtained by pressing a common glass 
 tumbler, mouth downwards, into a vessel of water. Though the 
 water ascends more or less according to the depth, the air makes 
 good its claim at ordinary depths to the greater part of the 
 space, and even though sunk to the bottom of the sea, the water 
 would never get quite to the top. If a small lighted taper, float- 
 ing on a bit of cork, be carried down with the tumbler, the 
 singular appearance may be beheld of a light burning under 
 water. 
 
 SECTION 4. STATES OF AGGREGATION. 
 
 16. Matter is found to exist in three states or conditions, 
 namely 
 
 Solid, as iron, stone, ice. 
 
 Uquid,* as water, spirit of wine, mercury. 
 
 Gaseous,* as air, steam, coal-gas, chlorine (a gas which is 
 
 * ' Fluid ' is a term which includes both liquids and gases. t 
 
10 ELEMENTARY DYNAMICS. 
 
 remarkable as not being colourless like most gases, but bright 
 yellowish green, and having a strong smell ). 
 
 17. We recognise these states by the following character- 
 istics. 
 
 (a) A solid has a definite shape and size, which it does not 
 change unless some force is used. All its surfaces are clearly 
 and sharply defined, as we see in the case of a brick, a bar of 
 iron, a piece of ice. This regularity of shape is beautifully 
 shown in crystals, such as those which are deposited when a 
 hot strong solution of alum is allowed to cool, and the various 
 kinds of spar which are found lining cavities in rocks. The 
 surfaces of these crystals are absolutely flat, and the angles at 
 which the surfaces are inclined to each other are mathematically 
 exact ; more so than any artificial process of grinding or cutting 
 could secure. 
 
 (&) A liquid shows little tendency to preserve any definite 
 shape under ordinary conditions (when, however, it is not in 
 contact with any solid, as a falling rain-drop, it shows a 
 spherical shape), but always moulds itself to the shape of 
 the solid vessel which contains it. It may, however, show a 
 distinct boundary or surface at the top, as water does in an 
 open basin or in a bottle which it only partially fills ; and 
 in this case the surface is always exactly level. 
 
 (c) A gas has no definite shape or size at all ; it does 
 not even show a level upper surface like a liquid. Matter 
 in the state of gas manifests an extraordinary power of 
 spreading through space, even when there is another gas 
 in the region already. Thus if only a little chlorine is 
 put into a bottle, it quickly spreads over the whole bottle, 
 and, if the mouth is open, it escapes from the bottle and* 
 can in a few minutes be detected by its smell in every 
 part of the room or house. No limit can be assigned to 
 this power of spreading (or 'diffusion,' as it is called) of a 
 gas ; and the presence of air in the bottle or room only 
 affects the rate at which the gas travels outwards, and not 
 the distance or amount of diffusion. The reasons for these 
 differences between solids, liquids, and gases, will be more 
 easily understood when we have considered some of the other 
 properties of matter. 
 
GENERAL PROPERTIES OF MATTER. 11 
 
 SECTION 5. DIVISIBILITY. 
 
 18. We are able, by mechanical means, such as cutting or 
 pounding, to divide masses qf matter into very small parts. A 
 chip of marble may be broken from a block, and that chip may 
 be crushed to powder. The smallest particle of this powder 
 discernible by the naked eye, when examined by the microscope, 
 is seen to be a block having all the qualities of the original 
 marble, and capable, by finer instruments, of being divided into 
 still smaller blocks, which may be again divided ; and so on, 
 with no other practical limit than the fineness of our senses and 
 instruments. 
 
 19. Gold can be beaten out into leaves which are not more 
 than -sTrfainr f an i ncn i n thickness, and which, when placed 
 edgeways, are quite invisible under the best microscopes hitherto 
 made. In preparing the gilt silver wire used in embroidery, a 
 rod of silver is covered with a thin layer of gold, and then drawn 
 out into a fine wire, in which the thickness of the gold covering 
 retains the same proportion to the silver as at first. A portion 
 of this wire, on which the layer of gold is less than 1 -millionth 
 of an inch in thickness, may be seen under the microscope to be 
 covered with a continuous coating of the metal, having all the 
 appearance of solid gold. 
 
 20. It is possible, however, to divide matter into even smaller 
 particles without difficulty. When a soap-bubble is blown, the 
 solution of soap can be extended until its thickness is not 
 greater than TSU^F^ of an inch before it breaks ; and yet there 
 must obviously be in this soap-film at least one row of particles 
 of soap lying side by side thus, oooooooo. Moreover, chemistry 
 teaches us that each particle of soap must contain more than 50 
 separate particles of carbon, hydrogen, oxygen, and sodium. 
 
 21. Still more minute must be the division when a substance 
 is dissolved in a liquid, or is heated until it assumes the state of 
 gas. Thus when water is, by the application of heat, turned 
 into steam, we separate it into particles far too small to be seen 
 under any microscope, and which there are good reasons for 
 believing to be at any rate not greater than ^injrrs-fftnnr of an inch 
 in diameter. 
 
12 ELEMENTARY DYNAMICS. 
 
 22. Divisibility thus extends far beyond the limits percep- 
 tible to the senses. Are we therefore to assume that it is 
 without limits that matter is infinitely divisible ? This would 
 be a rash assumption. On the contrary, there are many reasons 
 for believing that there is a limit somewhere, and that there are 
 ultimate particles, of a determinate size and shape, incapable of 
 further subdivision. Thus, in the case of steam above men- 
 tioned, it is pretty certain that we are dealing with the smallest 
 separate particles of water that can exist at all, retaining the 
 properties of water. These ultimate particles of matter are called 
 Molecules (Lat. molecula, a small mass). 
 
 23. The size and shape of these molecules can only be approxi- 
 mately judged of by our present means ; but several perfectly 
 independent lines of research point to the following conclusions : 
 
 (a) That the molecules of all kinds of matter are spherical 
 in shape and are all of the same size, but differ in other proper- 
 ties, for example, weight. 
 
 (6) That the diameter of a molecule is not greater than 
 aoooft<mnr of an inch, and not much less than OTTO^WTO of an 
 inch (roimnnF - innmnnnr of a millimetre). 
 
 24. Molecules are the smallest particles recognised and dealt 
 with in Physics ; and we shall see that a great many of the 
 peculiarities of matter can be explained by assuming their 
 existence. But even these molecules can generally be divided 
 by chemical means into still smaller particles, which seem to be 
 really the smallest particles of a substance which can show the 
 properties and affinities of the substance at all. These ultimate 
 particles of matter are called atoms (from a Greek word, 
 &<rep,o s , atomos, signifying 'indivisible'). For example, the 
 molecule of water can be undoubtedly separated into at least two 
 dissimilar parts, which we call the elements 'oxygen' and 
 ' hydrogen ' respectively ; such atoms of oxygen and hydrogen 
 being incapable of existing free and uncombined, but capable 
 of forming, with other atoms, molecules of chemical com- 
 pounds. 
 
 25. The distinction, then, between a molecule of matter and 
 an atom of matter must be carefully borne in mind. 
 
 A molecule is the smallest particle of matter which can exist 
 free and uncombined. 
 
GENERAL PROPERTIES 
 
 An atom is the smallest particle 
 part in a chemical combination. 
 
 SECTION 6. INDESTRUCTIBILITY. 
 
 26. Whatever views may be taken as to the size and shape of 
 the ultimate particles of matter, we know that human agency 
 can neither make nor destroy them. This is not, indeed, what a 
 first impression suggests, for nothing is more common than for 
 bodies to decay, dissolve, evaporate, and disappear. But it can 
 be proved that in no case is anything lost. The structure or 
 form is destroyed, the materials remain. Water, mercury, and 
 many other substances disappear in invisible vapour when 
 heated ; but if the vapour is carefully collected and cooled, the 
 water or mercury reappears without loss of weight. 
 
 27. When a piece of wood is heated in a close vessel 
 from which air is excluded, we obtain water, an acid, several 
 kinds of gas, and there remains a black, porous substance, 
 called charcoal. The wood is thus decomposed, and its 
 particles take a new arrangement and assume new forms ; 
 but that nothing is lost is proved by the fact, that if 
 the water, acid, gases, and charcoal be collected and weighed, 
 they will be found exactly as heavy as the wood was before 
 distillation. In the same manner, the substance of the coal 
 burnt in our fires is not annihilated ; it is only dispersed in 
 the form of smoke or particles of soot, gas, and ashes or dust. 
 Bones, flesh, and other animal substances may in the same 
 manner be made to assume new forms, without the destruc- 
 tion of a particle of the matter which they originally contained. 
 The decay of animal or vegetable bodies in the open air, or in 
 the ground, is only a process by which the atoms of which 
 they were composed change their places, and enter into new 
 combinations. 
 
 28. It is equally true that matter cannot be made or called 
 into existence by us. We talk of 'making' a chair, but this 
 really means only putting into a new shape matter which already 
 exists. Every particle of the matter of which we ourselves, our 
 flesh and bones, are made, existed already in the food we eat. 
 The plants on which we feed, and on which the animals thrive, 
 
 B 
 
14 ELEMENTARY DYNAMICS. 
 
 which we also use as food, derive their materials from the soil, 
 air, and water, which surround them ; and the decay and decom- 
 position of animal and vegetable matter after death supply the 
 materials for another generation of animals and plants. The 
 universe, in fact, so far as we know, contains precisely the same 
 number and kind of atoms of matter that it had when it was 
 created, and neither more nor fewer. 
 
 SECTION 7. POROSITY. 
 
 29. In common language, a pore (Greek vopos, a path) is a 
 small hollow space or interstice between the particles of a body, 
 large enough to be seen, or to admit the passage of liquids or 
 gases. In this sense, some substances, such as sponge, charcoal, 
 sugar, &c., are called porous, and others are contrasted with them 
 as solid. But experiment and reflection lead us to the conclu- 
 sion that all bodies are porous that is, that in no case do the 
 molecules fill the whole space occupied by the body, but that 
 there are, even in the most solid of solids, interstices of greater 
 or less size between them. 
 
 30. In the first place, liquids and gases are found to pass 
 through what seem the most compact solids. If a wooden cask 
 full of spirits of wine is sunk in water for a time, the cask will 
 be found filled with water, and the spirits gone. The spirits 
 escape, and the water enters through the pores of the wood. 
 A globe of silver filled with water, and closed with a screw, 
 was once submitted to great pressure ; the surface of the metal 
 became covered with dew, the water being forced through its 
 pores. In using Bramah's hydraulic presses, in which water is 
 forced under great pressure into cast-iron or steel cylinders four 
 or five inches thick, the water is often observed to pass through 
 the pores of the metal, and stand out like drops of dew on its 
 surface. Similarly gases, especially hydrogen gas, are found to 
 pass pretty readily through heated plates of platinum and iron. 
 
 31. In the next place, when a substance is placed under heavy 
 pressure, or cooled, it contracts in volume that is, its molecules 
 get nearer to each other ; and this could not occur unless there 
 were already spaces between them. Gases are thus shown to be 
 extremely porous. Air has been compressed into less than 
 
GENERAL PROPERTIES OF MATTER. 15 
 
 of the space it usually occupies, without reaching the limit of its 
 compressibility. Again, if water is allowed to evaporate in a 
 closed vessel 'filled' (as we usually say) with air, just as much 
 water-vapour is formed in the vessel as if there was no air 
 already in it. And if ether is now introduced into the vessel 
 and vaporised, just as much of its vapour is formed as if the 
 vessel contained no water-vapour or air already, the only differ- 
 ence being that a longer time is required, the fresh molecules 
 having to fight their way through an already somewhat crowded 
 space. We may go on putting into the same vessel other 
 vapours to an extent of which the limit has not yet been found ; 
 so large are the intervals between the molecules of gases at 
 ordinary temperatures and pressures.* 
 
 SECTION 8. ATTRACTABILITY. 
 
 32. The term * attraction/ by which we express simply the 
 tendency of one thing to approach another, is applied to a great 
 many phenomena which we must regard ,as due to different 
 causes. The tendency which is shown by a stone to fall to 
 the earth, is an example of an attraction due to the force of 
 Gravitation. The particles of the stone are held together by 
 an attraction called Cohesion; and mortar sticks to the stone 
 in consequence of a very similar attraction called Adhesion. 
 
 [ Other kinds of attraction are Magnetic attraction, which is 
 manifested between a piece of iron and a magnet, and between 
 certain parts of two magnets ; Electrical attraction, which occurs 
 between a piece of glass when rubbed with silk, for instance, 
 and other substances ; and Chemical attraction, which enables 
 atoms (and not merely molecules) to associate closely and form 
 definite, permanent chemical compounds, as when the elements 
 oxygen and hydrogen unite to form water. But the study of 
 these special attractive forces belongs to other branches of 
 natural science, and will not be further pursued here.] 
 
 * Of course, the addition of each successive gas increases the pressure on the inter- 
 nal surface of the vessel, as more and more molecules are crowded in. 
 
 It may be interesting to note that in one cubic inch of air, 
 at ordinary temperatures and pressures, there are probably 
 300,000,000,000,000,000,000 molecules of oxygen and nitrogen. In 
 one cubic centimetre, the size of one side of which is here given, 
 there are 2,000,000,000,000,000,000 molecules of a gas. 
 
16 ELEMENTARY DYNAMICS. 
 
 SECTION 9. COHESION. 
 
 33. This is the name given (from Lat. cohcerere, to stick 
 together) to the attractive force which holds together molecules 
 of the same kind, so as to form masses or bodies of matter. 
 Without such a force we should not have a compact mass of 
 sandstone, but a mere loose heap of sand ; steel would not be a 
 substance hard enough to cut most other things, but a mass as 
 weak and unstable as water. 
 
 34. Cohesion acts only when the molecules are at distances so 
 minute as to be insensible to us : at distances greater than 
 smTsv of an inch, it has no influence whatever ; and when the 
 molecules of a solid body are once separated, it is in most cases 
 impossible to bring them near enough again to make them 
 cohere. Two fresh cut surfaces of lead may be made to cohere 
 with some force ; but a slight film of rust or of grease will com- 
 pletely prevent the .necessary nearness of the metallic molecules. 
 Interrupted cohesion is easily restored when the body is in a 
 fluid or half-fluid state, owing to the mobility of the molecules ; 
 as when a broken stick of sealing-wax is mended by melting the 
 two ends and pressing them together, or two pieces of iron are 
 joined by welding. 
 
 35. The difference between the three states of matter, solid, 
 liquid, and gaseous, already treated of in par. 17, page 10, 
 can be explained as due to differences in the extent to which 
 cohesion is allowed to exert its power in holding the molecules 
 together. 
 
 36. In solids, cohesion shows itself strongly ; the molecules 
 are held firmly together, and can only be separated or altered in 
 relative position by a considerable force. Hence a solid is 
 observed to keep its shape. 
 
 37. In liquids, cohesion scarcely shows itself at all : the 
 molecules move easily past one another, and are easily separated 
 entirely. Hence the weight of the molecules is free to act, and 
 presses them into all the irregularities of shape of the vessel con- 
 taining them, as if into a mould : it also causes them all to get 
 as near the earth's centre as they can, a condition which can only 
 be satisfied by the upper surface of the liquid (that is, that 
 
GENERAL PROPERTIES OF MATTER. 17 
 
 farthest from the earth's centre) being perfectly level. Hence 
 also each drop of a liquid is perfectly spherical in shape. 
 
 38. In gases, no cohesion at all is shown : the molecules are 
 believed to be in extremely rapid motion, darting about in all 
 directions in straight lines, and knocking continually against one 
 another, and against whatever resists their onward course. Hence 
 there is a constant pressure against the sides of the vessel which 
 contains them, and a tendency of the gas to escape in whatever 
 direction the movements of the molecules are freest ; for 
 example, through the mouth of the bottle, in whatever direction 
 it may be turned. This view of the nature of gases is called the 
 Kinetic Theory (from a Greek word, *hwis t movement); and 
 it completely explains nearly all the observed facts.* 
 
 39. It is found that by simply applying heat to a substance, we 
 can change its state from solid to liquid and from liquid to 
 gaseous. Ice, for instance, when heated, becomes water ; and 
 when further heated, the water turns into steam. Similarly, by 
 taking away heat, and nothing else, from a gas, it assumes 
 successively the liquid and solid states. Hence we infer that 
 heat is the force which counteracts cohesion in liquids and 
 
 40. Some substances occur in a state which is intermediate 
 between solid and liquid for instance, oil, treacle, honey. In 
 such a state they are said to be 'viscous.' They show the 
 general characteristics of liquids, but imperfectly ; issuing in a 
 sluggish stream when poured out of a vessel, and remaining 
 heaped up for a certain time on the surface upon which they are 
 poured, though they eventually adapt themselves to its shape, 
 and show a level upper surface. In them cohesion shows itself 
 to a certain extent between the molecules, but not so much 
 as to enable them to retain a shape permanently as a solid 
 does. 
 
 41. The character and amount of cohesion varies greatly with 
 the nature of the particular substance. Hence arise the manifold 
 
 * The quickness of the motion of the molecules of a gas is surprisingly great. 
 The molecules of air, under ordinary conditions, are moving at the rate of 17 miles 
 a minute. The molecules of hydrogen gas are moving even faster that is, about 
 60 miles a minute, 60 times faster than an express train. This explains the ease 
 with which gases pass through fine tubes or cracks, or through pores such as exist 
 in unglazed earthenware ( like that of which flower-pots are made ). 
 
18 ELEMENTARY DYNAMICS. 
 
 degrees of hardness, tenacity, and elasticity, which must be more 
 fully considered. 
 
 42. Hardness. This means, resistance of the molecules of 
 a mass to any change in their relative positions. When, for 
 instance, we find that great force is required to cut into or 
 alter the shape of a mass, as in the case of steel or glass, we 
 call that substance 'hard.' It is the result of very strong 
 cohesion between the molecules ; and hence solids show it to 
 a much greater extent than fluids. But even in solids hardness 
 differs greatly (compare, for instance, glass with lead), and those 
 substances which easily yield to force we call 'soft,' meaning 
 only that they are less hard than the majority of substances. 
 Softness is, in fact, merely a comparative term ; just as in 
 calling a thing cold, we simply mean that it is less hot than 
 other things with which we compare it. 
 
 43. We can easily find out which of two things is the hardest 
 by trying which will make a scratch or groove in the other. The 
 one which does so that is, whrch displaces the molecules of the 
 other must have the stronger cohesion, and therefore must 
 be the harder. From results of experiments of this kind, it 
 is easy to make out such a list as the following, in which 
 the substances are arranged in order of hardness. Each is 
 scratched by those above it in the list, but will scratch all those 
 below it. 
 
 1. Diamond (the hardest sub- 
 
 stance known ). 
 
 2. Quartz (rock-crystal or flint). 
 
 3. Glass. 
 
 4. Steel.* 
 
 5. Iron. 
 
 6. Copper. 
 
 7. Lead. 
 
 8. Wax. 
 
 9. Butter. 
 
 The substances in the second column we should call, as a rule, 
 'soft' ; but it must be repeated that there is no definite line to 
 be drawn where hardness ends and softness begins. 
 
 44. A hard substance is generally very brittle that is, the 
 cohesion acts only through a very small distance, and its mole- 
 cules cannot be moved far from their positions without passing 
 beyond the range of their cohesive force, and breaking apart. A 
 
 * See par. 50, note. 
 
GENERAL PROPERTIES OF MATTER. 19 
 
 diamond can readily be crushed to powder by blows of a 
 hammer ; glass is an almost proverbial example of brittleness, 
 and a file of hardened steel sometimes snaps in two when 
 simply allowed to fall on a stone floor. 
 
 45. Tenacity (Lat. tenax, tenere, to hold). This means 
 resistance of the molecules to complete separation, even 
 when their places are greatly changed by force : cohesion 
 acting through a longer range than in the case of brittle sub- 
 stances. It is a quality shown chiefly by substances which are 
 only moderately hard, such as ordinary steel, iron, and brass. 
 Such substances can be bent, twisted, hammered out into thin 
 plates (malleability), pulled out into fine wire (ductility), and 
 will sustain great pressure (or 'stress,' as it is scientifically 
 termed) without breaking ; hence they are of great use for 
 engineering purposes. 
 
 46. Tenacity is usually measured by hanging weights from a 
 rod of the substance, and observing what force is required to tear 
 it in two. A table of the ' breaking stress,' as it is called, of a 
 few substances is given below 
 
 Breaking stress of a wire T V of an inch in diameter : 
 
 Steel 1000 Ibs. 
 
 Iron ( wrought) 550 1 1 
 
 Brass..., ....470 n 
 
 Copper. 300 Ibs. 
 
 Gold 150 
 
 Lead... ....25 .. 
 
 47. The malleability of some of the metals is remarkable. In 
 the manufacture of ' gold leaf ' for gilding purposes, a bar of gold 
 is first rolled out between polished rollers into a ribbon about 
 nnnr of an inch in thickness. This is cut into small squares, 
 which are placed between sheets of vellum and beaten out with 
 heavy hammers. The extended leaves are then cut into smaller 
 pieces, and the latter are beaten out between sheets of a thin 
 membrane called 'gold-beater's skin'; and the process is 
 repeated until leaves are obtained about 3^ inches square and 
 not more than s^Vinr of an inch (rrrjnr of a millimetre) thick ; 
 the cohesion of the molecules enabling them to hold together 
 even under this great extension. The leaves thus produced are 
 actually transparent, transmitting a beautiful green light. 
 Silver and platinum can be beaten out in a similar way into 
 leaves nearly but not quite as thin as gold. 
 
20 ELEMENTARY DYNAMICS. 
 
 48. The same metals also show great ductility. Wire is made 
 by pulling a thin rod of the metal through a conical hole in a 
 steel plate, the smaller end of the hole being rather less in 
 diameter than the rod. The metal is thus, partly by the pull 
 and partly by the compression as it passes through the hole, 
 made longer and thinner ; and by drawing it through smaller 
 and smaller holes in succession very thin wire can be obtained, 
 the thinnest gold wire being only T^ of an inch (-fa of a 
 millimetre) in diameter. 
 
 49. Elasticity (Greek 'ixa,fn, striving). This means the ten- 
 dency of the molecules of a body to go back to their original 
 positions when their places have been altered by force. For 
 instance, when a piece of india-rubber is stretched, its mole- 
 cules submit to be moved very far from their original 
 places without getting beyond the range of their cohe- 
 sion, although they resist this change with a force which 
 increases with the amount of displacement. But they do not 
 rest satisfied with their new places ; continued stress has to be 
 applied in order to keep the india-rubber stretched, and as soon 
 as this stress is taken off, the molecules come back to their 
 precise original places, and the piece of india-rubber resumes its 
 old form. A similar thing happens when a straight rod of 
 whalebone or steel is bent ; the molecules on the outside of the 
 curve are pulled farther apart, while those on the inside of the 
 
 curve are pushed nearer together 
 as shown in fig. 2. But if the 
 displacement has not exceeded 
 a certain amount, they all come 
 back to their original places 
 Fi &- 2< when the pressure is taken off. 
 
 (Compare this with the behaviour of a similar strip of lead 
 which remains bent after the removal of the pressure, the 
 molecules making no effort to recover their places. ) It is easy 
 to see why a thin rod may be bent farther than a thick one 
 without breaking. The thinner a rod is, the less difference in 
 length there is between the inside and the outside of the curve 
 when it is bent, and hence there is less displacement of the 
 molecules for a given amount of bending. In the rebound of an 
 elastic ball from a flat surface, a similar thing happens. The 
 
GENERAL PROPERTIES OP MATTER. 21 
 
 molecules of the ball are forced inwards at the moment of 
 contact, so that the ball is flattened there ; and the effort of the 
 molecules to recover their places drives the whole ball back. 
 
 50. The solids which show elasticity in the highest degree are 
 india-rubber, ivory, whalebone, steel (at a certain degree of hard- 
 ness)*, and glass. Glass is, until it actually breaks, the most 
 perfectly elastic solid known. A glass ball ('a solitaire' ball 
 will do) let fall on a smooth slab of marble,f rebounds very 
 nearly to the height from which it was let fall. A thin thread 
 of glass, such as ' spun glass/ may be tied into a knot, or one 
 end may be twisted 400 or 500 times round ; but it will resume 
 its exact original figure when allowed. 
 
 51. Liquids and gases are, so far as experiments have gone, 
 perfectly elastic. We can easily change their shape, but we 
 cannot permanently alter it by any force whatever. The com- 
 pressibility of air has been already alluded to in par. 31, p. 14, 
 and when it is released from pressure, however intense, it is 
 found to expand again to its exact original volume. 
 
 SECTION 10. ADHESION. 
 
 52. This is the name given (from Lat. adhcerere, to stick 
 to) to the attractive force which acts (at extremely small 
 distances only) between molecules of different kinds of matter ; 
 which causes, for instance, glass and sealing-wax to stick 
 together where their surfaces are in contact. It may eventu- 
 ally be found to be essentially the same force as cohesion, 
 but the effects of the two are so easily distinguishable, that it is 
 convenient to consider them separately, and give them different 
 names. 
 
 53. Particles of dust on an upright pane of glass, chalk-marks 
 
 * Steel can be obtained of very various degrees of hardness. When, in the pro- 
 cess of manufacture, it has been allowed to cool slowly from a red heat, it is scarcely 
 harder than iron, and shows comparatively little elasticity. If heated red-hot and 
 suddenly cooled by being plunged into cold water, it becomes ( owing to a change 
 in molecular structure ) even harder than glass. In this condition it is very brittle, 
 and its elasticity is only shown within narrow limits. But if this hard steel is care- 
 fully heated to a definite temperature ( 285 centigrade ), and allowed to cool slowly, 
 it is rendered less hard, and shows a very high degree of elasticity. In this state it 
 is used for watch-springs, &c. 
 
 t A slab of glass should not be used for this experiment. 
 
22 ELEMENTARY DYNAMICS. 
 
 on a wall, glue on wood or paper, paint on metals, &c., are all 
 instances of adhesion between dissimilar kinds of matter. The 
 use of cements depends upon the adhesion of the molecules of 
 the cementing substance to the surfaces between which it lies, 
 and also upon the cohesion between its own molecules. When 
 both are strong, as in the case of glue between surfaces of 
 wood, the joint is often the strongest part of the mass ; so 
 that if two pieces of wood are well glued together and then 
 broken, the fracture will take place anywhere rather than at 
 the joint.* 
 
 54. Glue has, on the contrary, very slight adhesion for metals 
 or glass, and hence it is of little use as a cement for such sub- 
 stances ; shellac, or some other gum-resin, must be used. 
 
 55. Adhesion of liquids to solids takes place much more readily 
 than that of solids to solids, because in the case of a liquid and 
 a solid the surfaces come into more complete contact. When 
 the hand or a rod of metal is dipped into water, a film of the 
 water adheres to the surface, and is borne up against its own 
 weight ; nor can any force shake it all off. Plunge a bit of gold, 
 or silver, or lead, into mercury, and a portion of the mercury 
 will in like manner adhere. Wherever we have wetting, we 
 have a case of adhesion of a liquid to a solid. It is from this 
 cause that in pouring water over the edge of a vessel, the water 
 is apt to run down the side of the vessel rather than fall perpen- 
 dicularly. To avoid this, jugs, &c., have lips or spouts; the 
 liquid, when it gets to the lower end of the lip, can only continue 
 in contact with the glass by running uphill, along the curved 
 under-surface of the lip, and its weight prevents it from doing 
 this, so it falls straight off. A temporary lip may be made by 
 
 * A good illustration of the relation of adhesion to cohesion is afforded by the 
 common experiment of splitting a sheet of paper in two. Take a piece of paper 
 such as a printed leaf of a book or newspaper, and two pieces of calico rather larger 
 than the paper. Lay the pieces of calico upon a board, and brush thin glue over 
 them, leaving one corner or edge untouched. Place the sheet of paper between 
 them, and press them into close contact with the paper in every part ; then hang 
 the whole up to dry. When dry, take hold of the unglued corners of the calico, and 
 tear the two pieces slowly and carefully apart. The cohesion of the calico and the 
 glue, and the adhesion of the glue to the calico and the paper, are both so much 
 greater than the cohesion of the paper, that the latter will split in two, half of the 
 thickness remaining attached to each piece of calico, from which it may be detached 
 by soaking for a few minutes in hot water. 
 
GENERAL PROPERTIES OP MATTER. 23 
 
 holding a wetted glass rod vertically against the edge of the jar 
 or basin. 
 
 56. But liquids do not always wet solids, or adhere to them. 
 A rod coated with grease, or the wing of a water-fowl, remains 
 dry when plunged in water. Mercury does not adhere to a 
 porcelain cup, or to a rod of iron or platinum. The explanation 
 is simple. There is probably in every case an attraction between 
 the solid surface and the liquid, but it is opposed by the attrac- 
 tion of the particles of the liquid for one another, and there can 
 be actual adhesion only when the first is stronger than the other. 
 When the adhesive force is able to overcome the attraction of 
 the liquid for its own particles, a film of it is separated and 
 carried off on the surface of the solid ; if the cohesion of the 
 liquid is the stronger of the two, there is 110 wetting of the 
 surface. 
 
 SECTION 11. CAPILLARITY DIFFUSION OSMOSE. 
 
 57. Cohesion shows itself at the surface of liquids in a striking 
 way. It causes all the molecules in the surface-layer to be in a 
 state of strain not unlike that of a stretched sheet of india- 
 rubber. This is called the surface tension of a liquid ; and it 
 is well shown in the strong tendency of a soap-bubble to con- 
 tract in size, the molecules of the surface-layers dragging at each 
 other so as to make the film shrink into the smallest possible 
 space. Thus, if after a bubble is blown at the end of a funnel,* 
 the mouth is withdrawn from the end of the tube before the 
 bubble is detached, the latter begins to contract with a force 
 sufficient to drive a strong stream of air out of the tube, as 
 may be proved by holding the flame of a candle close to its 
 end. 
 
 58. The surface tension of water is strong, but it is greatly 
 weakened by contact with some other liquids. If a clean f glass 
 plate is laid flat, and a little water is poured over it, and spread 
 
 * Thistle funnels with long stems, such as are used in chemical work, answer well 
 for blowing bubbles. A good solution is made by dissolving 5 grms. of sodium 
 oleate in 80 c.c. of distilled water, and adding 5 c.c. of glycerine, mixing the whole 
 thoroughly. 
 
 t.The plate should be cleaned by rubbing over it a little nitric acid with a tuft 
 of cotton wool, and then rinsing it with water. 
 
24 
 
 ELEMENTARY DYNAMICS. 
 
 with a glass rod so as to form a thin even layer, and if then 
 a drop of spirit of wine on the end of a glass rod is put in 
 the middle of the water, the surface tension is lessened at this 
 point, and the molecules shrink away on all sides, like a retreat- 
 ing army, so as to form a ring round the drop, separated from it 
 by a clear space. A drop of oil, placed on the surface of the 
 water in a similar way, follows up the retreating molecules and 
 spreads quickly over a large surface. Many different and 
 curious 'cohesion-figures' may be formed by dropping various 
 liquids, such as ether, turpentine, oil of cloves, upon a thin layer 
 of water as above described. 
 
 59. Capillary Attraction is only a particular effect of cohesion 
 and adhesion. A tube with a small bore, like a hair, is called a 
 capillary tube, from capillus, the Latin word for a hair. If the 
 end of such a glass tube is dipped in water, the water is seen to 
 
 rise in the tube above the 
 level of the rest of the sur- 
 face. In a series of tubes 
 of different diameters, the 
 liquid ascends highest in 
 the smallest ; or the heights 
 are inversely as the diame- 
 ters. Water will be seen to 
 rise in a similar way between 
 two glass plates placed as in 
 
 the figure, with two of the upright edges touching, and the other 
 two slightly apart. The sustained film rises higher as the 
 plates approach, assuming the form of a particular curve. The 
 fluid rises also slightly on the outside of the tubes and plates, 
 and the surface of the sustained column within the tube is seen 
 to be hollow like a cup. 
 
 60. But liquids do not always ascend in narrow tubes or 
 spaces ; it is only when they adhere to the solid substance that 
 they do so. If a greasy glass tube is dipped in water ; or, still 
 better, if a clean glass tube is dipped in mercury, the liquid 
 inside, instead of rising, sinks below the general level ; the 
 surface of the column, too, becomes convex instead of con- 
 cave. 
 
 61. The rise or the depression depends upon the adjustment 
 
 Fig. 3. 
 
GENERAL PROPERTIES 
 
 between the forces of adhesion and 
 
 wetting. When the liquid wets , ^^ , r u , N .r*.^ 
 
 surface have part of their weight suppor^ti by adhesion, and 
 thus a longer column is required to balance the pressure of the 
 rest of the fluid. In cases where the cohesive attraction of the 
 liquid particles within the tube for one another is too strong to 
 permit them to adhere to its surface, that cohesion tends to 
 draw them away from it, while the tube prevents them from 
 receiving the support they would have from the liquid particles 
 around them, if it were not there. Mathematicians have shown 
 that, if the adhesion between the solid and the liquid be equal to 
 half the cohesion of the particles of the fluid, the surface at the 
 point of contact will be neither elevated nor depressed ; if the 
 adhesion between the two be more than half the cohesion, 
 elevation will occur ; and if it be less than half, the surface will 
 be depressed and convex. 
 
 62. Capillary attraction is exemplified in many familiar 
 phenomena, and plays an important part in nature. Thus, the 
 rise of the sap from the roots into the branches and leaves of 
 plants is mainly due to capillary action in the numerous small 
 tubes of which vegetable tissue is in a great measure composed. 
 If a piece of sponge or a lump of sugar be placed so that its 
 lower corner touches some water, the fluid will rise up and wet 
 the whole mass. In the same mariner, the wick of a lamp will 
 carry up the oil to supply the flame, though the flame is several 
 inches above the level of the oil. The use of blotting-paper 
 depends on the tendency of the ink to rise by capillary attraction 
 through the pores of the paper. If one end of a towel happens 
 to be left in a basin of water, while the other hangs over below 
 the level of the water, the basin will be emptied of its contents; 
 and, on the same principle, when a dry wedge of wood is driven 
 into the crevice of a rock, and afterwards moistened with water, 
 it will absorb the water, swell, and sometimes split the rock. 
 
 63. A striking illustration of this subject is given by the 
 following experiment : Place a wine-glass on a book on the table, 
 and set another close by it, so as to be on a lower level. Pour 
 some water and some oil into the higher glass ; then moisten a 
 piece of cotton- wick in water, and drop an end of it into each 
 glass, so as to reach near the bottom and form a bridge between 
 
26 ELEMENTARY DYNAMICS. 
 
 them. The water which was below the oil in the one glass 
 will in an hour or two be found transferred to the other, leaving 
 the oil behind. If the wick be moistened with oil, the oil will 
 be transferred, leaving the water. 
 
 64. When two light bodies, such as two bits of cork, are left 
 to float on water near each other, they soon come together, 
 moving at last with a rush. This is owing to the attraction of 
 adhesion which we are considering. When the liquid wets the 
 floating bodies, it rises slightly all round them, and this sustained 
 liquid hangs as a weight on them on all sides. So long as it 
 rises equally, there is no motion ; but when the bodies come 
 near each other, the space between them becomes like part of 
 the inside of a capillary tube, the water rises higher than on 
 the opposite sides, and the bodies move towards the sides that 
 are most strongly pulled. When the floating bodies are not 
 wetted by the liquid, the effect is the same as if there was a 
 repulsion between them and it ; the surface between the two 
 bodies is depressed, and they are pushed together by the higher 
 column on the outside. If one of two bodies floating on water 
 is smeared with oil, so as to prevent the water from adhering, 
 instead of coming together, the two will recede from each other, 
 for reasons analogous to the above. 
 
 65. Adhesion between Solids and Gases. Every solid has, under 
 ordinary conditions, a film of air adhering to its surface, which 
 it is extremely difficult to get rid of. Dry iron-filings, and even 
 small needles, when gently laid on the surface of water, will 
 float, though eight times heavier than the water, because each 
 has a film of air adhering to it so strongly that even when it 
 does sink it carries a portion of the air along with it. In making 
 barometers, it is found that air adheres so firmly to the surface 
 of the glass, that the mercury must be boiled in the tube before 
 it can be expelled. Some porous solids, such as charcoal, absorb 
 air and other gases to an amount many times their own bulk, 
 the force of adhesion condensing the gases on the surface of 
 their molecules. When a lump of sugar is dropped into a cup 
 of tea, the film of air which surrounds the particles does not 
 quit them till they are dissolved ; bubbles are seen rising till all 
 the sugar has disappeared. 
 
 66. Diffusion. By this is meant the tendency shown by fluids 
 
GENERAL PROPERTIES OF MATTER. 27 
 
 to spread into one another until a perfectly uniform mixture is 
 obtained. It depends upon the freedom of the movements of 
 the molecules in such bodies (their cohesion having been, as 
 already explained, p. 17, nearly or entirely overcome by heat), 
 and the great porosity of matter in the fluid state. Thus, in 
 liquids, the molecules, in spite of a certain amount of cohesion 
 still showing itself, are constantly making excursions beyond 
 their natural boundaries, and thus spreading into a space even 
 when containing another liquid. If a jar is half filled with a 
 strong solution of sugar or salt coloured with ink or cochineal, 
 to render it visible, and water be poured upon the surface (very 
 slowly and carefully, to avoid mechanical mixing of the liquids*), 
 the boundary between the two will be pretty sharply defined for 
 a time, but gradually the coloured solution will rise, and finally 
 the whole mass will become equally coloured, and an equal 
 amount of sugar or salt will be found in every part of it. It is 
 found that the rate of diffusion varies very much with the 
 nature of the liquid ; and that solutions of substances which 
 readily form crystals, such as sugar or salt ('crystalloids'), 
 diffuse far quicker than solutions of such things as gum or glue, 
 or liquids such as oil and treacle, which have no tendency to 
 crystallise (' colloids,' from a Greek word meaning glue). 
 
 67. In gases, diffusion goes on much more quickly than in 
 liquids, as might be expected from the quickness of the move- 
 ment of the molecules of a gas (p. 17), and the large spaces 
 between them. The rapidity with which coal gas, escaping from 
 a leak in a pipe, diffuses through the whole room is a good 
 example. Heavy gases diffuse much more slowly than light 
 ones, since their molecules are in comparatively slow movement ; 
 but the ultimate result is the same in all cases, a perfectly 
 uniform mixture being obtained. If a jar of air is inverted 
 over a jar of chlorine, so that their open mouths are in 
 contact, the heavy chlorine can be seen by its colour to rise 
 into the upper jar ; and the air also, though much lighter 
 than chlorine, passes downwards, until in a short time the 
 
 * It is best to float a disc of paper, with a piece of thread attached to it, upon 
 the solution, and to pour the water gently down a glass rod or pencil upon the 
 paper. When the jar has been filled up, the paper can be gently withdrawn by 
 means of the thread. 
 
28 ELEMENTARY DYNAMICS. 
 
 proportions of chlorine and air are the same in every part of 
 the two jars. 
 
 68. Osmose. This term is applied to the tendency shown by 
 fluids to diffuse even through porous solids, such as paper, 
 unglazed earthenware, and parchment or other animal mem- 
 branes. The general character of the movement is much the 
 same as in ordinary diffusion, and crystalloids show it much 
 more than colloids, but the results are in many cases complicated 
 by differences of adhesion between the fluids and the surface of 
 the solid. 
 
 69. The distinction between colloids and crystalloids is even 
 more sharply marked in osmose than in the case of direct diffu- 
 sion. If, for instance, a glass cylinder (such as a lamp glass), 
 with a piece of parchment or bladder tied over one end, is 
 filled with a mixture of gum and sugar dissolved in water, and 
 hung in a basin of pure water, so that diffusion can take place 
 through the parchment, it is found that the sugar (a crystalloid) 
 soon passes through into the surrounding water, while practically 
 none of the gum (a colloid) passes through at all. At the same 
 time the water from the outer vessel passes through the mem- 
 brane more quickly than the solution of sugar, owing partly to 
 its greater adhesion to the animal tissue, and thus the liquid in 
 the cylinder becomes increased in volume, as shown by its 
 surface rising above its original level. This experiment illus- 
 trates a method called * Dialysis,' which is used to separate 
 mixtures of crystalloids and colloids. 
 
 70. In the case of gases, some curious results are obtained by 
 the different rates at which they diffuse through a porous sub- 
 stance. If a wide glass tube, or lamp cylinder, closed at one 
 end by a plug of well-dried plaster of Paris, is filled with hydro- 
 gen (or coal gas),* and placed with its open end dipping under 
 water in a basin, the light hydrogen will pass out through the 
 pores of the plaster so much quicker than the denser air enters, 
 that the volume of gas in the tube will for some time be dimin- 
 ished, and the level of the water inside will rapidly rise much 
 above the level of the water in the basin. If the mouth of a 
 tumbler is dipped into solution of soap, so as to form a flat 
 
 * While it is being filled, the end closed by the plug should be temporarily 
 covered by an india-rubber cap, to prevent premature diffusion. 
 
GENERAL PROPERTIES OF MATTER. 29 
 
 soap film adhering to its edge, and then placed in a jar of 
 carbon dioxide ('carbonic acid gas'), the soap film will soon 
 become convex and expand into a bubble, showing that the 
 carbon dioxide enters more rapidly than the air in the tumbler 
 gets out through the film. Now this is contrary to what we 
 should expect from the usual law, since carbon dioxide is much 
 denser than air ; but carbon dioxide is much more soluble in 
 water than air is, and hence it dissolves at the outside of the 
 soap-film, diffuses through the film in solution, and then diffuses 
 from the inner surface into the tumbler. 
 
 SECTION 12. GRAVITATION. 
 
 71. The attractive forces hitherto considered namely, cohesion 
 and adhesion act only between certain kinds of matter, and 
 through extremely small distances ; moreover, they differ in 
 amount according to the particular substance. But there is one 
 attractive force, called gravitation, which acts between all kinds 
 of matter, under all conditions, through very great distances, 
 and according to one simple universal law. It is this force 
 which causes things to show the property of ' weight/ by which 
 is meant pressure towards the earth's centre. If we hold up a 
 piece of iron in the hand, we observe that it presses against the 
 hand. in one definite direction, and we express this by saying 
 that it is ' heavy,' or ' has weight.' If the support of the hand is 
 taken away, the piece of iron moves, or 'falls,' towards the 
 surface of the earth until some obstacle prevents its further 
 movement. Evidently, however, its tendency to move does not 
 cease when it has reached this point. It will fall down a well 
 or shaft of a mine with scarcely diminished acceleration; and 
 we have every reason to believe that, if it could, it would go 
 on falling until it reached a point close to the centre of the 
 earth, and that there it would cease to show any tendency to 
 move in any particular direction; it would have no 'weight' 
 at all. 
 
 72. Sir Isaac Newton, in 1665 A.D., was the first who gave a 
 definite reason for the weight of matter. He was led, it is said, 
 by observing an apple fall from the tree, to ask himself why it 
 should move at all when detached from the tree ; why it should 
 
30 
 
 ELEMENTARY DYNAMICS. 
 
 move to the earth instead of from it : how quickly it fell ; 
 whether a large mass fell at the same rate as a small one, and so 
 on. He accounted for the fact of its fall by saying that there is 
 a universal attraction between the molecules of matter, however 
 great the distance between them ; and so every molecule of the 
 earth is attracting every molecule of the apple, and also every 
 molecule of the apple is attracting those of the earth. The 
 result of all these separate attractions is equivalent to one single 
 force tending to move the apple in the direction of the centre of 
 the earth ; and when it has reached this point, it will be 
 attracted equally in all directions by the earth's molecules, and 
 so will show no pressure or tendency to move in any particular 
 direction more than another. 
 
 73. But weight is only one case of the universal attraction of 
 gravitation. Sir Isaac Newton went on to prove that a force 
 acting according to the same laws holds the moon in its orbit 
 round the earth, and the earth and other planets in their orbits 
 round the sun. Even those almost immeasurably distant bodies, 
 the stars, are in many cases found to consist of pairs of bodies 
 ('double stars') revolving round each other, and held near each 
 other by the same force. 
 
 74. Direction and Magnitude of Gravitation. 
 The direction of this force, so far as relates to 
 its action between the earth and bodies within 
 its range, is very simply 
 
 shown by hanging a weight 
 to a string (fig. 4) ; the 
 weight stretches the string 
 in the direction in which it 
 would fall if it could, and 
 thus the line of the string 
 shows the direction of gra- 
 vitation at the particular 
 place (fig. 5). This instru- 
 ment is called a 'plumb- 
 line/ 
 
 75. It will now be easy to understand the 
 precise meaning of the common terms 'above,' 
 
 ' below,' 'higher,' 'lower.' In all cases reference is made to the 
 
 Fig. 4. 
 
GENERAL PROPERTIES OF MATTER. 31 
 
 "earth's centre, and not necessarily to any fixed distant point in 
 space, such as a star. When we say that a body is ' above,' or 
 * higher than ' another, we mean that it is farther from the earth's 
 centre, as measured along the direction of a plumb-line. When 
 we say that it is 'below,' or 'lower than,' another body, we 
 mean that it is nearer to the earth's centre as measured in 
 the same way. It is in this sense that a lamp is ' above ' the 
 floor, and a dog lies 'below 3 or 'under' a table. Things which 
 are at the same distance from the earth's centre are said to be at 
 the same ' level.' If we refer the position of bodies to a fixed 
 direction in space, such as a straight line drawn to a star, we 
 shall see that, in the case of two bodies which are at different 
 points along this line, the one of them which is nearest the star 
 would be said to be ' above ' the other in England, but ' below ' 
 the other in Australia, and 'on one side of the other near 
 the equator. But by reference to the earth's centre, the terms 
 become perfectly definite, and have the same significance all 
 over the world. 
 
 76. The process of weighing a thing consists in finding out 
 how much pressure it exerts towards the earth's centre. This is 
 usually stated by comparison with the pressure exerted by some 
 definite piece of matter which we call a ' weight,' such as a pound 
 or a gramme. Thus, when we say that a piece of lead weighs 
 two pounds, we mean that it presses towards the earth's centre 
 with twice as much force as the standard weight called a ' pound.' 
 This comparison of weights can be made in the two following 
 ways (among others) : 
 
 (1) By placing the bodies one at each end of a rod supported 
 
 at its exact centre so that it can p^-^-^> 
 
 move easily round the point of 
 
 support (fig. 6) ; that body which \ I 
 
 pulls down its end of the rod is 01 f% 
 
 clearly the heavier of the two, and 
 
 if the rod remains level, the bodies 
 
 are of equal weight. Such an apparatus is the ordinary balance 
 
 or pair of scales, a fuller account of which will be found on p. 143. 
 
 (2) By seeing how much each of the bodies will stretch a 
 spring ; the one which stretches it farthest is, of course, 
 the heaviest. This is the principle of the spring balance (fig. 7), 
 
32 
 
 ELEMENTARY DYNAMICS. 
 
 ill which a pointer is attached to the spring, and marks are 
 made on a scale over which the pointer 
 moves, to show the extent to which the 
 spring is stretched, by weights of 1 lb., 2 Ibs., 
 &c. 
 
 77. Gravitation acts at all distances, but 
 varies in intensity according to the distance 
 between the attracting molecules ; getting less 
 very quickly when the distance is made 
 greater, and getting greater very quickly when 
 the distance is lessened. A force which does 
 this is said to vary inversely with the distance. 
 
 78. But the above is not an exact statement 
 of the law. In scientific work we ought not 
 to be satisfied with knowing in a general way 
 that the force varies when the distance varies ; 
 we should endeavour to find out precisely how 
 much it varies for a given change of the 
 distance. Sir Isaac Newton was the first to 
 discover the simple and exact law of the 
 variation, which will next be explained. 
 
 79. Suppose that A and B are two bodies 
 attracting each other at a distance of 1 mile 
 
 Fig. 7.* with a force equivalent to 1 grain. Then if 
 
 B is removed to twice the distance that is, 
 2 miles the force of attraction is found to be J of a grain. At 
 a distance of 3 miles it is found to be only ^ of a grain ; at 4 
 miles, T V of a grain. If the distance is lessened to half a mile, 
 the attraction becomes as great as 4 grains ; and at ^ of a mile, 
 it is 9 grains. The result of experiments may be expressed in 
 the following way : 
 
 Distance, i 1 2 3 4 miles. 
 
 A 00 B 
 
 Force, 941 \ -J T \ grains. 
 
 We have to examine whether there is any definite relation 
 
 between the numbers denoting the distances and those denoting 
 
 the amounts of attraction. Now, starting with the facts that the 
 
 * Part of the front plate has been removed to show the spiral spring inside. 
 
GENERAL PROPERTIES OF MATTER. 33 
 
 square of a number is the product of the number multiplied by 
 itself, and that the inverse or reciprocal of a number is obtained 
 by expressing the number as a fraction (if not already one), and 
 inverting it that is, making the numerator denominator, and 
 the denominator numerator (thus, 2 = f, inverted = -|), it 
 is easy to see that the numbers in one of the above rows are 
 the inverses or reciprocals of the squares of the corresponding 
 numbers in the other row. Thus ^ squared = -J, inverted = f or 
 9 : 2 squared = 4 = f, inverted = j, and so on. 
 
 80. The law, then, deduced from such experiments as the 
 above may be stated as follows : 
 
 THE ATTRACTION OF GRAVITATION VARIES IN THE SAME PRO- 
 PORTION AS THE INVERSE OP THE SQUARE OF THE DISTANCE 
 BETWEEN THE BODIES WHICH ARE ATTRACTING EACH OTHER. 
 
 Or more shortly 
 
 GRAVITATION VARIES INVERSELY WITH THE SQUARE OF THE 
 
 DISTANCE.* 
 
 81. Thus, in order to calculate the change in the force of 
 gravitation for any given change of distance, it is only requisite 
 to work out the following proportion sum : 
 
 Inverse of square . Inverse of square . . Force at . Force at 
 of orig. distance ' of changed distance ' ' orig. distance " changed distance. 
 
 82. It is clear, from what is said above, that the weight of 
 bodies is liable to variation : for instance, things outside the 
 earth will weigh less in proportion as they are taken farther 
 from the earth's centre. We cannot find this out by weighing 
 them in a pair of scales, since the pieces of metal used as weights 
 would have their attraction altered just as much as the things 
 which were being weighed. Some other method, such as a 
 spring-balance, must be used ; the attraction of cohesion which 
 causes the elasticity of a spring does not vary according to the 
 same law as gravitation.! 
 
 * It may be noted that this same law holds good in many other cases besides 
 gravitation ; it applies, in fact, to all phenomena which are due to an influence 
 radiating in all directions from a centre, such as magnetic attraction, intensity of 
 radiant heat and light, &c. 
 
 t An extremely accurate method of measuring small variations in the force of 
 gravitation consists in observing the number of swings made in a given time by the 
 same pendulum at different places ; the greater the attractive force, the quicker the 
 pendulum swings. A fuller account of the pendulum will be found on p. 122. 
 
34 ELEMENTARY DYNAMICS. 
 
 83. Anything which weighs exactly 1 Ib. on the ground will, 
 if taken to the top of a house 30 feet high, weigh ^ of a grain 
 less than a pound, a quantity easily measurable. If taken in a 
 balloon to a height of 4 miles above the ground, it will weigh 
 14 grains less, losing, in Jact, -g-J^ of its original weight. If it 
 was taken to a place as far off as the moon, we can calculate 
 what its weight would be by applying the law explained above. 
 Thus, the distance from the earth's surface to its centre is 4000 
 miles (nearly) ; the distance from the moon to the earth's centre 
 is 240,000 miles (nearly). These numbers are in the proportion 
 of 1 : 60, so that the 1 Ib. weight, when at the distance of the 
 moon, will be 60 times as far from the earth's centre as it was at 
 first. Then, from par. 81, 
 
 Ib. ib. 
 
 1 ! 1 _1 
 
 I 2 ' GO 3 3600 
 
 so that the body would weigh only 2-16 grains, and be 'as light 
 as a feather.' 
 
 84. Even at different parts of the earth's surface things do not 
 weigh the same. Anything which weighs 1 Ib. at the poles is 
 found to weigh 36 grains less at the equator : so that, for 
 instance, we should (if a spring-balance were used) get more 
 sugar in a pound at the equator than at the poles. There 
 are two main reasons for this variation in weight. 
 
 (a) Because the earth is not quite round, but bulges out a 
 little at the equator ; its shape being, in fact, a spheroid 
 (roughly speaking, like that of an orange), and not an exact 
 sphere. Hence anything on the surface at the equator is 13 
 miles farther from the centre than it would be at the poles, and 
 therefore the attraction of gravitation is less. 
 
 (6) Because anything which is revolving round a centre shows a 
 tendency to get farther from the centre ( which is called ' centri- 
 fugal tendency,' and is more fully explained on p. 51) ; and the 
 quicker the body is moving, the more strongly this tendency is 
 shown. Now, bodies at the poles, which are the extremities of 
 the axis of rotation of the earth, are not moving relatively to 
 this axis, while bodies on the surface of the earth at the equator 
 are moving at the rate of 1000 miles an hour, and the centri- 
 fugal tendency due to this enormous velocity partly overcomes 
 
GENERAL PROPERTIES OF MATTER. 35 
 
 gravitation. If the earth turned round seventeen times faster 
 than it actually does, the centrifugal tendency would increase 
 so much that it would just balance the attraction of gravitation 
 at the equator, and things on the surface there would show no 
 weight at all. 
 
 SECTION 13. MASS AND WEIGHT. 
 
 85. It is necessary to insist on the fundamental distinction 
 between these two terms, which are often used in common 
 language as if they meant the same thing. In weighing different 
 things (at the same place) we notice that some are much heavier 
 than others. Two principal reasons may be assigned for this 
 difference in weight. 
 
 (a) There may be more molecules in the substance to be 
 attracted and to attract. Thus, two cubic inches of lead weigh 
 more than one cubic inch of lead, obviously because there are 
 more molecules in the former than in the latter. 
 
 (6) Each molecule of one substance may have more of the 
 property of gravitation (just as it may have more or less of the 
 property of cohesion) than each molecule of the other substance. 
 Thus a cubic inch of lead weighs more than a cubic inch of iron 
 (although there are probably about the same number of mole- 
 cules in each*), because with each molecule of lead there can be 
 associated more of the attractive force of gravitation than with 
 each molecule of iron. We express this by saying that there is 
 more ' matter ' in a lead-molecule than in an iron-molecule, or 
 that its 'mass' is greater; meaning by 'matter' here simply 
 something which can be associated with, and acted on by 
 forces : the more there is of this, the more force can reside in 
 it and act. 
 
 86. Mass, then, means strictly the quantity of matter in a body, 
 while weight means the amount of its attraction towards the earth's 
 centre. And although we commonly express the masses of things 
 in terms of that particular result of the quantity of matter in 
 them which we call their weight, and may truly and accurately 
 compare the masses of two bodies by comparing their weights 
 
 * It is pretty certain, at any rate, that equal volumes of gases, whatever their 
 nature, contain (when measured under the same conditions of temperature and 
 pressure ) the same number of molecules. 
 
36 ELEMENTARY DYNAMICS. 
 
 under the same conditions, yet the following considerations will 
 show that the two terms are perfectly distinct. 
 
 87. The weight of a body varies with the place where it is, as 
 already seen. At the centre of the earth it has no weight: at 
 the distance of the moon it has comparatively little weight. 
 But the mass of it, or the quantity of matter in it, is obviously 
 the same, whatever may be its position. 
 
 88. The force expended in kicking a football along the ground 
 depends upon the mass of the ball, and not directly upon its 
 weight : the same effort would be required to set it in motion 
 with the same speed, wherever it was in the universe. But the 
 force expended in lifting it from the ground depends upon its 
 weight as well as its mass. 
 
 SECTION 14. DENSITY. 
 
 89. In speaking of the ' density ' of a substance, we take 
 into account its size as well as its mass. Suppose that 2 cubic 
 feet of air are compressed into a volume of 1 cubic foot. Then 
 the mass of the air is, of course, in no way altered, but there is 
 twice as much matter in the 1 cubic foot of compressed air as in 
 1 cubic foot of air under ordinary conditions. The compressed 
 air is then said to have twice the ' density' of ordinary air. 
 
 90. Density, then, means the mass of, or quantity of matter in, 
 a given volume of a substance. In comparing the densities of 
 different things, some particular substance, such as water in the 
 case of liquids and solids, is taken as the standard of comparison, 
 and equal volumes of it and other substances have their masses 
 compared, usually by weighing them under the same conditions. 
 The number which expresses how much heavier or lighter a 
 certain volume of any substance is than the same volume of 
 water, is called the specific gravity, or (since mass is found, 
 under the same conditions, to be proportional to weight) the 
 relative density of the substance. Thus, in saying that the 
 density of lead is 11^, we imply that a certain volume of lead 
 (such as a cubic inch or cubic centimetre) weighs 11^ times as 
 much as the same volume of water. 
 
 91. The following table shows the comparative densities of a 
 few of the more familiar substances : 
 
GENERAL PROPERTIES OF MATTER. 
 
 37 
 
 Table of Densities, or Specific Gravities, of Liquids and Solids. 
 
 Water 1-00 
 
 Platinum 21-5 
 
 Gold 19-3 
 
 Mercury 13-6 
 
 Lead 11-5 
 
 Silver 10-5 
 
 Copper 8-9 
 
 Iron 7-8 
 
 Diamond... .. 3-5 
 
 Glass (flint) 3-3 
 
 Marble 2-8 
 
 Aluminium 2-6 
 
 Porcelain 2-4 
 
 Sulphur 2-0 
 
 Boxwood 1-3 
 
 Ice 0-9 
 
 Alcohol 0-8 
 
 Cork... ....0-3 
 
 92. In the case of gases, they are all (under ordinary pressures) 
 so much less dense than water, that air is usually taken as the 
 standard of comparison. It is preferable, however, for many 
 'reasons, to take hydrogen (the least dense of all known kinds of 
 matter) as the standard. Fractions are thus in a great measure 
 avoided, and (as explained more fully in the text-book on 
 Chemistry) the comparative mass of the molecules of each 
 substance is indicated with some certainty. 
 
 Table of Densities, or Specific Gravities, of Gases. 
 
 Air = i -oo. Hydrogen = i-oo. 
 
 Chlorine 247 35-5 
 
 Carbon Dioxide 1-53 22 
 
 ('Carbonic Acid') 
 
 Oxygen 1-10 16 
 
 Nitrogen 0-97 14 
 
 Steam 0-62 9 
 
 Hydrogen 0-07 1 
 
38 ELEMENTARY DYNAMICS. 
 
 CHAPTER IT. 
 
 MOTION FORCE MOMENTUM. 
 
 SECTION 1. MOTION AND VELOCITY. 
 
 93. Motion means change of place. When anything which 
 has been observed to be in one place, is found after an interval 
 of time to be in another place, we say that it has 'moved.' We 
 find out, for instance, whether a clock is going by noting whether 
 the hands point, as time goes on, to different figures on the dial. 
 We ascertain whether a distant ship is sailing on, or lying at 
 anchor, by observing whether or not it is seen at successive 
 intervals on different points of the horizon, or in different posi- 
 tions with regard to other ships. But here there often arises a 
 considerable difficulty. How are we to know whether these 
 ships are not themselves moving, while the one we are watching 
 is really motionless ? 
 
 94. When two trains are standing at a station, and one of 
 them begins to move, it is at first not easy for a passenger to tell 
 with certainty whether it is the train in which he is, or the other 
 train, which is in motion, without a reference to something 
 which is considered most likely to be fixed, such as the walls of 
 the station. But these walls are undoubtedly themselves in 
 rapid motion with the surface of the earth on which they stand ; 
 only we do not perceive their motion because they are moving 
 in every respect similarly to the earth's surface. Moreover, the 
 earth itself is all the time in still swifter motion round the sun, 
 and the sun with all the planets is moving through space, we 
 know not whither. Scientific investigations point to the fact 
 that nothing, not even a single molecule of the hardest solid, is 
 absolutely at rest. In short, all that we can distinguish and 
 judge of is relative motion that is, whether a body is moving at 
 a different rate, or in a different direction to another body with 
 which we compare it. If a boat sails against a stream exactly as 
 fast as the stream flows, it is at rest relatively to the bottom 
 
MOTION FORCE MOMEN 
 
 and banks, but in motion as regards the \vt 
 means change of place with regard to space 
 no means of marking a fixed point in space, 
 can never observe such a motion. 
 
 95. Velocity means rate of motion. It is generally expressed 
 by stating how far the body would move in a certain time, such 
 as one second, if the rate of motion continued uniform. Thus, 
 in saying that a cannon-ball moves with a velocity of 1400 feet 
 per second, we mean that if it went on at the same rate, it would 
 at the end of one second be 1400 feet from the place where 
 it was at the beginning of the second. In this sense a train is 
 said, with perfect correctness, to be travelling with a velocity of 
 60 miles an hour, although very few trains ever cover 60 miles 
 iri that space of time. The velocity is uniform, when equal 
 spaces are always passed over in equal times ; it is accelerated, 
 when gradually increased, and retarded, when gradually 
 diminished. If the increase or diminution is equal in equal 
 times, the motion is said to be uniformly accelerated or uniformly 
 retarded. 
 
 SECTION 2. FORCE. 
 
 96. Matter cannot set itself in motion. A ball placed on a 
 level table remains where it is put, and will remain there so 
 long as the conditions are unaltered. Some power or influence 
 must act on it to make it move ; and any power which does this 
 is called a 'force.' It is equally true that (as will be explained 
 more fully in Chapter III. ), when the ball is once for all set in 
 motion, it will not stop or turn aside from a straight course 
 unless some force is applied. 
 
 97. Force, then, may be denned as that which produces motion, 
 or changes motion, or destroys motion. No one has ever seen 
 forces ; they have none of the properties of ordinary matter ; in 
 fact, they are not forms of matter at all, they are influences 
 which give life, as it were, to the dead materials of the universe. 
 We only recognise them by their effects. When we observe a 
 body moving, we know that some force must have acted on it ; 
 if it increases its speed, we know that some force must be acting 
 on it still ; if it swerves aside or stops, we know that another 
 force has been at work. 
 
40 ELEMENTARY DYNAMICS. 
 
 98. The following will serve as examples of the more 
 commonly occurring forces : 
 
 (a) Gravitation : which makes a stone fall ; keeps most 
 ordinary clocks going ; moves a train down an incline ; changes 
 the motion of the earth from a straight line to a curve round 
 the sun. 
 
 (6) Cohesion : which is acting in an elastic spring, such as 
 that which keeps a watch going ; and when a falling stone 
 touches any hard surface, destroys the motion which gravitation 
 has caused. 
 
 (c) Muscular action : which enables us to raise a weight, move 
 from place to place, hit or throw a cricket-ball. 
 
 99. In considering the action of forces, three things have to be 
 taken into account : (1) the point at which the force is applied ;* 
 (2) its direction that is, the line in which it tends to make the 
 body move ; (3) its magnitude that is, the amount of it as 
 determined by the effect it produces. Both the direction and 
 the magnitude of a force can be conveniently and accurately 
 represented by drawing a line of a definite length. The direction 
 of the line will denote, of course, the direction in which the 
 force acts, and the length of the line, adjusted to any definite 
 
 A scale such as inches or centi- 
 
 2ozC 1 ["^1 1 D2oz metres, may indicate the niag- 
 
 nitude of the force. 
 
 [ I "J 
 
 100. Suppose, for instance, 
 .. i that a weight of 4 oz. is resting 
 
 on a table, and is pulled in 
 .,2 opposite directions by two 
 
 forces, each equivalent to 2 oz. 
 These conditions can be repre- 
 sented by the following dia- 
 z gram, fig. 8, in which A is the 
 
 Fig. 8. body ; AB, 4 units long (on 
 
 any convenient scale, such as 
 inches or centimetres), will denote the magnitude and direction 
 
 * As will be seen later on, p. no, however large or irregular in shape a body may 
 be, a single point can always be found in it (called the ' centre of inertia ' ), at which 
 a force may be considered to be applied, and will produce the same effect as if it 
 acted on every separate molecule of the mass. 
 
MOTION FORCE MOMENTUM. 41 
 
 of the force of gravitation ; AC and AD, each 2 units long on 
 the same scale, will represent the forces pulling it to the right 
 and left respectively. 
 
 101. Equilibrium of Forces. The case represented in the above 
 diagram affords a good illustration of the fact that a force does 
 not necessarily actually produce motion. The weight A is 
 really acted on by four forces : (1) gravitation, pulling it down- 
 ward ; (2) the cohesion of the molecules of the table, which 
 holds them together against the pressure of the weight, and 
 supplies a force which just counteracts the force of gravitation ; 
 (3) the force AC ; (4) the force AD. The last two, being equal 
 in magnitude and opposite in direction, obviously just counter- 
 act each other, like the first two ; and so the weight A does not 
 move at all. When forces balance each other in this way, so 
 that the body acted on does not move, they are said to be ' in 
 equilibrium.' 
 
 SECTION 3. MOMENTUM. 
 
 102. A moving body clearly has force associated with it. It 
 can set other things in motion. A cricket-ball can knock down 
 the wicket ; a cannon-ball, a train, an iceberg, will overthrow 
 obstacles which are so unfortunate as to come in their way. 
 Even particles so light as those of air may have force enough to 
 produce great effects, as proved by the destructive power of 
 hurricanes. 
 
 103. Momentum is the term used to express the force with 
 which anything is moving. It is found to be proportional to (a) 
 the velocity, (6) the mass of the moving body. The heavier a 
 thing is, and the quicker it is moving, the greater is the momen- 
 
 _tum which it has. The amount of momentum can be con- 
 veniently expressed by the number obtained by multiplying the 
 mass of a body (stated in pounds, grammes, &c.) by its velocity 
 in feet or centimetres per second. The product of these numbers 
 is called the ' momentum ' of the body. For instance, if a cannon- 
 ball weighing 9 Ibs. is moving with a velocity of 500 feet per 
 second, its momentum is said to be (9 x 500 =) 4500 (in Ibs. 
 ft. sec.). If another ball weighing 3 Ibs. is moving 1500 feet per 
 second, its momentum is (3 X 1500 =) 4500. 
 
 104. We observe that the momenta of the two balls are equal, 
 
42 ELEMENTARY DYNAMICS. 
 
 although one is so much lighter than the other. In fact, in 
 order that a body may move with great force that is, have a 
 great momentum it is sufficient that either its mass or its velocity 
 should be great. Thus a rifle-bail, though very light, has a high 
 momentum because its velocity is so great. Hailstones, though 
 very small, do much damage because they reach the earth with 
 great velocity. An iceberg, though it moves very slowly, has 
 great momentum because its mass is enormous, so that it will 
 slowly but surely crush a ship. 
 
 SECTION 4. MEASUREMENT OF FORCES. 
 
 105. If the power of a locomotive engine had to be found, we 
 might estimate it either (a) by seeing how many horses pulling 
 against it would just keep it from moving, or (6) by seeing with 
 what velocity it would move a train of known weight. Thus 
 we might compare it with other engines or motive powers. 
 
 106. Speaking generally, there are two principal ways of 
 finding the magnitude of a force. 
 
 (a) We may find out how much of some more easily measur- 
 able force is required to balance it, so as to get equilibrium. 
 
 (6) We may find out how much momentum it produces in a 
 certain time, such as 1 second. 
 
 For example, the magnitude of gravitation may be measured : 
 
 (a) By observing how far a certain mass, such as 1 lb., when 
 acted on by it, would stretch a spring ; thus balancing gravita- 
 tion against cohesion. 
 
 (6) By allowing it to act on a certain mass, such as 1 lb., for 
 1 second, and observing what velocity it produces in the body ; 
 since from these data the momentum can be calculated, as 
 already explained. 
 
 107. The subject of the measurement of forces will be more 
 fully considered in the next chapter. It is mentioned here in 
 order to make clear the distinction between the two branches 
 into which Dynamics is divided namely, Statics and Kinetics. 
 These differ mainly in the methods employed in them for the 
 measurement of forces. 
 
 In Statics (<rra-nxoj, that which makes to stop), the magnitude 
 of a force is measured by the first of the two methods described 
 
MOTION FORCE MOMENTUM. 43 
 
 above that is, by balancing it against another force, so that no 
 motion is produced in the body acted on. 
 
 In Kinetics ('W< ?} movement), the magnitude of a force is 
 measured by the second method namely, by observing what 
 momentum it produces. 
 
 108. In what follows, the action of forces will be treated 
 mainly on the principles of kinetics. It is obviously best to 
 observe the direct effect of a force as shown by the motion it 
 produces, instead of complicating matters by introducing other, 
 and possibly less understood, agencies to balance it. 
 
 APPENDIX. 
 Problems on Momentum. 
 
 In questions relating to the momentum of bodies, there are three 
 quantities to be taken account of 
 
 1. Mass. 2. Velocity. 3. Momentum. 
 
 If we know any two of these three quantities, a very simple arith- 
 metical process will enable us to find the third. 
 
 1. When mass and velocity are known. Then, as above explained, 
 mass x velocity = momentum. 
 
 EXAMPLE. A bird weighing 3 Ibs. is observed to be flying at the 
 rate of 50 feet per second. Hence its momentum will be, 
 3 x 50 = 150. 
 
 2. When momentum and mass are known. Then it may be 
 
 ,,, dl , momentum . .. 
 
 proved* that = velocity. 
 
 mass 
 
 EXAMPLE. A cannon-ball weighing 12 Ibs. has a momentum of 
 10^800 : what must be its velocity 1 
 
 * 10,800 ftnn , , 
 L- = 900 feet per second. 
 
 3. When momentum and velocity are known. Then 
 momentum 
 
 velocity 
 
 * Suppose a, b, and c are three quantities so related that a, X b = c. 
 
 Then to find b when a and c are known, divide both sides of the equation by a. 
 
 = - . That is, cancelling a on left side, b = . 
 a a (i 
 
 Similarly, to find a, divide both sides by b. 
 
 ab c c 
 
 J = r That is,* =y 
 
44 ELEMENTARY DYNAMICS. 
 
 EXAMPLK A boat which is being rowed at the rate of 20 feet 
 per second has a momentum of 36,000- Then its weight must be 
 
 36,000 
 
 = 1800 Ibs. 
 
 CHAPTER III. 
 
 LAWS OF MOTION. 
 
 109. The chief facts observed respecting the movements of 
 bodies have been summed up in the form of a few general pro- 
 positions, which were first put into shape by Sir Isaac Newton 
 under the name of laws of motion. They may be considered as 
 statements of universal rules by which the action of forces on 
 matter appears to be guided. We shall first give them nearly 
 in the form in which they were laid down by Sir Isaac Newton 
 himself, and then proceed to a fuller explanation of each. 
 
 Law I. A body, if it is at rest, will continue at rest ; and if 
 it is in motion, will continue in motion in a straight line with 
 uniform velocity, unless some force acts on it. 
 
 This law teaches us how we can recognise the existence of a 
 force namely, by observing its effect in changing the state of 
 rest or motion of a body. 
 
 Law II. The momentum produced by a force is exactly propor- 
 tional to the magnitude of the force : and when several forces act 
 on a body, each produces motion in its own direction, just as if it 
 was the only force acting. 
 
 From this law we learn (1) how the magnitude of a force may 
 be accurately measured namely, by observing how much 
 momentum it produces ; (2) how we can estimate the joint 
 effect of several forces acting all at once upon a body ; the 
 direction, for instance, a football will take when kicked by 
 two players simultaneously. 
 
 Law III. The action of a force is always accompanied by a 
 reaction of the body to which it is applied. This reaction is 
 equal in magnitude, but opposite in direction to the original 
 force. 
 
LAWS OF MOTION. 45 
 
 In this law we consider the effect of the communication of 
 force, not merely on the body to which it is imparted, but 
 also on the body which imparts it. 
 
 SECTION 1. THE FIRST LAW OF MOTION. 
 
 A body, if it is at rest, will continue at rest ; and if it is in 
 motion, will continue in motion in a straight line with uniform 
 velocity, unless some force acts on it. 
 
 110. This law is little else than a full definition of the 
 property of matter which is called 'Inertia' (Lat. inertia, 
 inactivity) namely, the inability of a body to change of its 
 own accord its state of rest or of movement (alluded to already 
 in par. 96, p. 39). It will be convenient to divide the law into 
 three distinct statements, and examine each of them separately. 
 
 (A) Matter cannot set itself in motion. 
 
 111. Even the smallest particle of dust will remain passively 
 where it has settled on a shelf or carpet until a brush or a 
 breeze (like a member of the police 'force') makes it 'move 
 on.' A heavy train requires a great force to start it ; in fact, 
 the inertia, or passive resistance to change, of a body is found 
 to be exactly proportional to its mass. 
 
 112. If there are several things resting close together or 
 loosely joined, and a force is applied to one of them, that one 
 alone will move, and break away from the rest, which will 
 remain still because of 
 
 their inertia. A simple 
 experiment will illus- 
 trate this statement. 
 Let a card be balanced 
 upon the tip of the 
 finger, and a shilling (or 
 other bit of metal about 
 the same in size) be laid 
 upon it (fig. 9). Let 
 the card then be smartly Fig. 9. 
 
 struck or filliped away ; 
 
 it will fly from beneath the coin, leaving the latter supported 
 on the finger. The inertia of the metal causes it to retain its 
 
 D 
 
46 ELEMENTARY DYNAMICS. 
 
 place ; while the card, to which alone force was applied, passes 
 from beneath it. For the same reason, when a horse starts 
 unexpectedly, or bolts, the rider may lose his seat, remaining 
 where he was, while the horse shoots on. When a train 
 or a boat starts with a jerk, passengers who happen to be 
 standing up are liable to be 'thrown down' (as we incor- 
 rectly say), owing to their inability to take up at once the 
 motion, .of the surface on which they are standing. So also 
 the difficulty of carrying along a glass full of water with- 
 out spilling any, arises from the inertia of the particles of 
 water keeping them in their places when the glass begins to 
 move, or changes the direction or rate of its movement from 
 unsteadiness of the hand which holds it, or irregularity of the 
 pace in walking. 
 
 113. The process of beating a carpet, or dusting a book, rests 
 on the same principle as the experiment with the shilling and 
 the card. The carpet being struck, is suddenly put in motion, 
 while the particles of dust remain where they were, their inertia 
 being sufficient to overcome the slight force with which they 
 adhere to the surface of the carpet. When a dusty book is 
 struck against a table, the book and the dust are first brought 
 into rapid motion together, and the book being then arrested 
 by the table, the dust continues in motion by its inertia, and is 
 thus detached. Skaters are olten able to glide over dangerously 
 thin ice without its breaking, if they move quickly ; because 
 the pressure is taken off each successive particle of the ice 
 before there is time to overcome its inertia and break it away 
 from the rest. Many a train has escaped a serious accident 
 owing to the high speed at which it has passed over a loose 
 rail or an unsafe bridge ; the inertia of the rail, &c., keeping 
 them in place while the train shoots over them. The common 
 feat of breaking a stick resting on two wine-glasses depends on 
 the suddenness of the blow ; the stick is broken before the 
 pressure upon the glass supports has had time to overcome 
 the inertia of their molecules and beat them down. A candle 
 fired from a gun will make a clear hole through a board 
 half an inch thick ; the inertia of the particles of tallow 
 keeping them nearly iu their places while the wood is pene- 
 trated. 
 
LAWS OF MOTION. 47 
 
 (B) Matter cannot stop itself when moving. 
 
 114. That a body at rest will continue at rest, unless acted on 
 by something, seems so obvious that no one perhaps ever thought 
 of asserting the contrary. Why do we not as readily assent to 
 the statement that a body, once in motion, will never stop 
 unless something stops it ? The reason is, that in all ordinary 
 cases we observe that things moving, such as a cannon-ball, a 
 spinning top, do soon come to a standstill. But, in fact, there 
 are forces acting on these to lessen, and finally destroy their 
 motion ; such as friction and inertia of the particles of air which 
 are flung aside by the moving body. A ball rolled along rough 
 ground soon stops ; on a smooth pavement it continues longer 
 in motion ; and on smooth ice, longer still. A common top 
 continues to spin for a greater length of time than it usually 
 does, when placed in a space from which the air has been 
 extracted by an air-pump. Thus we learn that, as the obstruc- 
 tions to its motion are lessened or removed, a body will go on 
 moving longer and longer ; and we may infer that if all obstruc- 
 tions could be removed, the motion would go on for ever. 
 
 115. We cannot realise such conditions in our experiments ; 
 but we have in the earth and the other planets and heavenly 
 bodies, excellent examples of free, unobstructed motion, and 
 here the most accurate observations have failed to detect any 
 sensible slackening of speed.* The length of the day of 24 
 hours (which, of course, depends upon the rotation of the earth 
 on its axis) has probably not varied one- tenth of a second since 
 the earliest astronomical observations were made about 2000 
 years ago. Scientific reasoning, then, leads us to the conclusion 
 that motion, if unobstructed, no more requires to be kept up by 
 the continued action of a force than rest, and that matter retains 
 a state of motion no less easily than a state of rest. 
 
 116. Many illustrations can be given of the results and appli- 
 cations of the inertia of bodies which are moving. In jumping, 
 it is found of great advantage to take a short run before the 
 
 * It is true that in the case of one small comet a perceptible retardation of motion 
 has been observed ; but the amount is no greater than can be fairly attributed to the 
 resistance of the small amount of matter which undoubtedly exists throughout all 
 space, and which may in the course of millions of years produce a measurable effect 
 on the motions of the earth and planets. 
 
48 
 
 ELEMENTARY DYNAMICS. 
 
 actual jump, because the forward motion thus gained continues 
 after the feet leave the ground, and thus not only is the body 
 carried farther, as in the 'long jump,' but also in the 'high 
 jump' the muscles have only the work of raising the body to 
 the height of the bar, the forward motion necessary to clear it 
 being given by the inertia due to the run. Riders in a circus, 
 when jumping through hoops from the back of a galloping horse, 
 soon learn that they must spring upward only, and not forward, 
 if they wish to alight on the horse's back after passing through 
 the hoop. A man jumping from a moving carriage or train will 
 certainly fall prostrate on the ground, if he leaps down as if he 
 were descending from a carriage which is standing still ; for 
 when he makes the attempt, his body has the same motion as 
 the carriage, and when his feet touch the ground, their motion 
 is arrested while the velocity of the upper part of the body con- 
 tinues ; and thus he finds himself thrown down. 
 
 117. Coursing, or hare-hunting, affords a striking illustration 
 of inertia. In that field-sport, the hare seems to possess an 
 instinctive consciousness of the exist- 
 ence of this law of motion. When pur- 
 sued by the greyhound, it does not run 
 in a straight line to the cover ; but in 
 a zigzag one. It doubles that is, sud- 
 denly changes the direction of its course, 
 and turns back at an acute angle with 
 the direction in which it had been run- 
 ning. The greyhound being unprepared 
 to make the turn, and therefore unable 
 to resist the tendency to persevere in 
 the rapid motion which it has acquired, 
 is impelled a considerable distance for- 
 ward before it can check its speed and 
 But, in the meantime, the hare has been 
 enabled to shoot far ahead in the other direction ; and although 
 a hare is much less fleet than a greyhound, by this scientific 
 mancEuvririg it often escapes its pursuer. Those who have 
 witnessed horse-racing may have observed that the horses shoot 
 far past the winning-post before their speed can be arrested. 
 This is also owing to the inertia of their bodies. 
 
 Fig. 10. 
 return to the pursuit. 
 
LAWS OF MOTION. 49 
 
 118. When several bodies are moving close together or loosely 
 joined, and one of them is stopped by a force, the others still go 
 on moving because of their inertia. For instance, in throwing a 
 cricket-ball, or a stone, the hand moves with the ball at first, 
 but is checked at the proper moment, and the ball goes on 
 moving with the same speed, and in the same direction, as the 
 hand was moving at the moment when it let go the ball. The 
 whole art of taking a good shot at the wicket, or other object, 
 depends on letting go the ball or stone precisely at the moment 
 when it and the hand are moving in the direction and with the 
 speed required to hit the object. 
 
 119. When a horse suddenly stops, as in refusing a jump, the 
 rider has a tendency to go on moving, and thus he may be, in 
 the correct sense of the word, ' thrown ' from his seat over the 
 head of the horse. Passengers in a train which comes into 
 collision with another train, or any such obstacle, continue 
 moving forward with the same speed as the carriage was moving 
 before it was checked by the collision, and are thus thrown 
 violently against the side of the carriage and 'severely shaken.' 
 
 (C) A single force can only produce motion in a straight line. 
 
 120. For example, a stone, when acted on by gravitation alone, 
 falls in a straight line towards the earth's centre : a billiard ball, 
 when struck, moves in a straight line over a level table. Hence, 
 whenever we observe a moving body deviate from a straight line, 
 we may be sure that some force must be acting on it to make it 
 do so. A cannon-ball, as soon as it leaves the mouth of the gun, 
 does not continue to move straight onward, but deviates in a 
 curve downwards, because gravitation is acting freely upon it, 
 and changes the direction of its original motion. 
 
 121. If a falling ball has a string attached to it, which is tied 
 to a fixed point on one side, it will move in a curve, because the 
 cohesion of the string supplies a force which makes it bend 
 more and more from the straight line in which gravitation alone 
 would make it move. If the end of the string is held in the 
 hand, and the ball is whirled round, there are two forces acting 
 on it, namely (1) the force applied by the hand, which moves 
 it onward against the resistance of the air ; (2) a force also 
 applied by the hand, and acting through the string, which keeps 
 
50 
 
 ELEMENTARY DYNAMICS. 
 
 Fig. 11. 
 
 it from moving straight on as it naturally would do, and pulls it 
 inward so as to keep it at a fixed distance from the centre round 
 which it is whirled. 
 
 122. While it is moving in this circle, it shows a tendency to 
 
 get farther from the centre, which is 
 called centrifugal tendency (Lat cen- 
 trum, a centre, and fugio, I fly). This 
 is simply a consequence of the inertia of 
 >B the ball. Let A, fig. 11, represent the 
 ball. When set in motion, it would 
 naturally, as above stated, go on in the 
 straight line, AB ; force, therefore, has 
 to be exerted through the string, AC, 
 in order to pull it away from the 
 straight line, and owing to its inertia 
 there is resistance to this force. Thus 
 centrifugal tendency is not a real force 
 itself, as it was formerly considered 
 to be, urging the body directly away from the centre. 
 
 123. That it is not so is sufficiently proved by observing the 
 result of letting go, or cutting, the string at the moment the ball 
 reaches the point A. If any real centrifugal force was being 
 exerted, then (since a force causes motion in its own direction) 
 the ball would move in the direction AD. But it does not do 
 so ; it simply goes on in the direction in which it was moving at 
 the moment it was set free, namely, in the line AB. 
 
 124. The same is true at whatever point in the circle the ball 
 
 is released. Thus, if released at E, fig. 
 12, it takes the direction EF ; if at G, 
 the direction GH, and so on. Now, at 
 any point in a circle, the direction of 
 the curve is that of a straight line 
 drawn through the point at right 
 angles to the radius. Such a straight 
 line is called a ' tangent/ and hence 
 we can express generally the direction 
 which the ball takes when set free 
 from the constraining force, by saying 
 that it flies off at a ' tangent.' 
 
LAWS OP MOTION. 51 
 
 SECTION 2. LAWS OF CENTRIFUGAL TENDENCY. 
 
 [For illustrating the facts and laws of centrifugal tendency, a so- 
 called ' whirling table ' (fig. 13) is extremely convenient. It consists 
 
 Fig. 13. 
 
 of a firm base-board, near one end of which is fitted an upright 
 spindle carrying a small pulley. On the same board a large grooved 
 wheel is fitted, and connected by a cord with the small pulley, so 
 that by turning the wheel the vertical spindle may be made to 
 rotate very quickly. This spindle has a screw nose at the top, to 
 which various pieces of apparatus can be fitted. ] 
 
 125. The centrifugal tendency of a body moving in a circle 
 varies in amount according to the following laws : 
 
 I. It increases with the mass of the body. 
 
 A ball weighing 2 Ibs. resists the bending from its straight 
 course with twice as much force as a ball weighing 1 lb., under 
 the same conditions of velocity, &c. For the inertia of a body 
 varies with the mass (par. Ill), and centrifugal tendency is, as 
 above explained, a consequence of inertia. 
 
 II. It varies (for a given velocity) inversely with the radius of 
 the circle in which the body moves; getting less in proportion as 
 the radius is greater, and greater as the radius is less. 
 
 126. For the smaller the size of the circle in which the body 
 is compelled to move, the sharper is the curve described by 
 it, and hence the greater is the distance through which it has 
 to be dragged out of its straight course in a given time. Thus, 
 if ACX, fig. 14, is a circle of 2 feet radius, and AB is the 
 distance through which the body, A, would move in 1 second, 
 if it went on in a straight line, then it has to be dragged 
 
52 
 
 ELEMENTARY DYNAMICS. 
 
 through the distance BC in 1 second in order to keep it in 
 
 the circumference of the circle. 
 And if A'C'X' represents another 
 circle with a radius of only 1 
 foot, it is clear that a body 
 A' having the same mass and the 
 same velocity, will have to be 
 dragged through the distance B'C' 
 in 1 second. Now B'C' can easily 
 be proved (for small arcs) to be 
 twice BC; thus the inertia of A' 
 has to be overcome through twice 
 as great a distance as in the case 
 of A, so that A' shows twice as 
 much centrifugal tendency. 
 
 III. It increases with the square of the velocity. 
 
 127. Thus, a body moving round a centre with a velocity 
 of 1 foot per second, shows a certain amount of centrifugal 
 tendency. If its velocity in the same circle is increased to 2 feet 
 per second, it shows (not twice, but) 4 times, that is, 2 x 2 (or 
 2 2 ) times, as much centrifugal tendency. The reason will be 
 plain if we consider that the body, when it is travelling twice as 
 fast as it originally was, ( 1 ) has to be dragged out of its straight 
 course through twice the distance in a given time ;* (2) has to 
 be moved through this double distance at twice the rate, in 
 order to keep it in the circle. Thus, altogether 2x2 times the 
 force has to be used ; and the centrifugal tendency is measured 
 by the force required to overcome it. 
 
 Similarly, if the velocity of the body is increased to 3 feet per 
 second, the centrifugal tendency becomes 9 times that is, 3 x 3 
 times, its original amount. 
 
 128. If a solid body is pierced by a straight rod, and made to 
 turn on it as on an axis (as, for example, a wheel or a grind- 
 
 * This will be plain by reference to fig. 12, p. 50. Suppose that, to begin with, 
 the body has such a speed that it travels in one second from E to G, in the circle, 
 and therefore has to be deflected from F to G in that time. Then, if its speed is 
 doubled, it will reach I in i second, and so will have to be deflected from H to I, 
 as well as from F to G, in the time. Hence, it will undergo two deflections instead 
 of one only that is, will be deflected through twice the distance in i second. 
 
LAWS OF MOTION. 53 
 
 stone), every particle of the body describes a circle round this 
 axis. All these circles are described in the same time ; and the 
 larger they are, the quicker the particle must move in order to 
 complete the circle in the time. In fact, the velocity increases 
 in the same proportion as the radius of the circle, so that 
 particles which are 2 feet from the axis, move twice as fast as 
 those which are 1 foot from the axis. Hence, it is clear that the 
 centrifugal tendency of the outer particles will be much greater 
 than that of the particles near the axis ; although the increased 
 centrifugal tendency due to increase of velocity (Law III.), is 
 partly compensated by the decrease caused by the greater size of 
 the circle described ( Law II. ). Thus, the fourth law of centri- 
 fugal tendency may be stated as follows : 
 
 IV. When a number of particles describe circles of different 
 sizes in the same time, the centrifugal tendency of each is in 
 direct proportion to the radius of the circle described. 
 
 We proceed in the next place to give some illustrations and 
 practical applications of centrifugal tendency. 
 
 129. In slinging a stone, the latter is whirled in the sling 
 round the hand as a centre, until a great velocity, and therefore, 
 a great centrifugal tendency, is attained, and it is let go at the 
 moment when it has reached a point in the circle, the tangent to 
 
 \_which is in the direction of the object aimed at. The same is, 
 of course, true of the athletic sport of ' throwing the hammer ' ; 
 the long handle of the hammer serving the purpose of a sling. 
 The ordinary process of throwing a stone and bowling a cricket- 
 ball is of much the same nature, but the hand then only 
 describes a portion of a circle of which the shoulder is the 
 centre. In all such cases, the main difficulty is to judge 
 correctly the proper moment at which the ball or the stone is 
 to be set free to follow the direction in which its inertia will 
 keep it moving. 
 
 130. In rapidly-moving parts of machinery there is often 
 great danger of breakage, owing to the high centrifugal tendency 
 of those portions which are at a great distance from the axis of 
 rotation. Thus, large fly-wheels have been known to break to 
 pieces when the engine has from any cause begun to move faster 
 than usual. Similarly, the massive grindstones used in cutlery 
 
54 ELEMENTARY DYNAMICS. 
 
 works sometimes break up without any warning, and the flying 
 fragments cause great damage owing to their mass and velocity, 
 like stones from an immense sling.* 
 
 131. Wet mops are easily and quickly dried by twirling them 
 rapidly round on their handles as axes of rotation ; the particles 
 of water travel to the extremities of the woollen strands, their 
 centrifugal tendency increasing as they get farther from the centre 
 (Law IV.), and then they fly off in well-marked tangent lines. 
 Mud is thrown off from a quickly -moving carriage wheel in a 
 similar way. 
 
 132. In sugar manufactories, the crystallised sugar is separated 
 from the liquid syrup by placing the whole in a cylindrical 
 vessel, the sides of which are perforated with small holes like a 
 sieve. This is turned quickly round at a rate of 1500 or 2000 
 revolutions a minute, and the liquid portions fly out through 
 the holes into a surrounding vessel, leaving the sugar nearly 
 
 dry. Clothes are dried in many laundries in 
 a very similar machine called a 'centrifugal 
 extractor.' 
 
 [ This can be illustrated by attaching to the upright 
 spindle of the whirling table a large glass cup with 
 rim turned inward (shaped like the bowls in which 
 Fig. 15. goldfish are kept), placing in this a little water 
 coloured with ink or indigo, and making it rotate 
 quickly. The water spreads outwards, and, as the velocity of rota- 
 tion increases, leaves the bottom of the glass and forms a broad 
 coloured band at the widest part of the cup (fig. 15), getting as far 
 from the axis of rotation as it can. ] 
 
 133. Centrifugal tendency is also usefully employed in the 
 manufacture of 'crown' glass, or window glass. A lump of 
 melted glass is attached to the end of an iron tube, and blown 
 into a hollow globe (a, fig. 16). This is, while still soft, opened 
 out into a cup, 6 ; and this cup is held in front of a furnace, and 
 rapidly twirled round on the iron tube as an axis. The edges of 
 the cup soon widen out owing to centrifugal tendency, assuming 
 
 * It is interesting to notice that the very same agency, centrifugal tendency, is 
 employed in an apparatus invented by Watt, the great engineer, for enabling the 
 engine to regulate its own speed, and thus to prevent or render unlikely such 
 accidents as those mentioned above. A description of this ' centrifugal governor ' 
 will be found in any treatise on the steam-engine. 
 
LAWS OF MOTION. 55 
 
 the shape c; and finally the whole flashes out into a thin, uni- 
 form, flat sheet, d. 
 
 I] 
 d 
 c 
 
 Fig. 16. 
 
 134. In equestrian performances in a circus, both rider and 
 horse incline their bodies inwards to just such an extent that 
 the tendency to fall inwards, due to gravitation, counteracts 
 their centrifugal tendency ; which would, if they went along 
 upright, make it impossible for them to keep in the circle. In 
 running round a corner, and in describing a sharp curve in 
 skating, the same thing is done, and for a similar reason. 
 When a curve has to be made on a railway, the rail on the 
 outside of the curve is always raised a little above the level of 
 the inner rail, so as to tilt the train inwards, and thus lessen 
 the risk of its running off the rails from its tendency to per- 
 severe in a straight course. 
 
 135. We may also trace the influence of centrifugal tendency 
 a very large scale in the shape and motions of the bodies 
 
 which make up the solar system. The earth rotates, as if on a 
 spindle or axis the extremities of which are at the N. and S. 
 poles, once in 24 hours. Now, since its circumference is approxi- 
 mately 24,000 miles, the parts of its surface at the equator must 
 move through about 24,000 miles a day, or 1000 miles an hour, 
 while the surface in the latitude of England has a much less 
 rapid motion, about 640 miles an hour, and this velocity becomes 
 less and less as the poles are approached. It follows from this 
 that the centrifugal tendency of the particles which compose the 
 earth must be much greater near the equator than near the 
 poles ; a fact which has two important results. 
 
 (1) There is no reason to doubt that the earth was at one time 
 a large drop of liquid, and if it had been at rest its shape would 
 (as explained in par. 17 b, p. 10) have been exactly spherical. 
 But owing to its quick rotation, the parts of it near the equator 
 spread outwards through their great centrifugal tendency ; thus 
 
56 ELEMENTARY DYNAMICS. 
 
 the whole mass became spheroidal in shape, like an orange ; 
 and when it became solid it kept this shape. Hence it is 
 that the diameter of the earth is 26 miles greater at the equator 
 than at the poles. The planet Jupiter rotates more than twice 
 as quickly as the earth, and it is found, as we should expect, 
 to bulge out at the equator even more than our earth, so that 
 it appears distinctly oval in a telescope. 
 
 [The influence of centrifugal tendency in causing this change of 
 shape may be illustrated by rotating a large india-rubber ball (5 in., 
 or more, in diameter ) on a vertical axis attached to the spindle of 
 the whirling table. The ball should be attached to this axis at the v 
 top only, the lower hole through which the axis passes being large 
 enough to admit of free motion on the axis as the equatorial parts of 
 the ball spread out. Three or four cuts in the india-rubber should 
 be made across the equator in the direction of the meridians on a 
 globe, reaching to about 30 from each pole : these will increase the 
 flexibility of the india-rubber. If the ball is rotated at a moderate 
 speed, it will assume a spheroidal shape. ] 
 
 (2) All bodies on the surface of the earth are, of course, being 
 whirled round with it, and consequently show a tendency to fly 
 off from the surface, which partly counteracts the force of gravi- 
 tation : in other words, lessens their weight (as has been already 
 mentioned in par. 84 &, p. 34). It can be calculated from the 
 laws above explained how much the weight of a body at the 
 equator is, owing to its centrifugal tendency, less than what it 
 would be if the earth was still ; and it is found that a mass 
 weighing 1 Ib. at the poles weighs 24 grs. less than a pound at 
 the equator loses, in fact, ^ of its weight owing to centrifugal 
 tendency alone. Further, it can be shown that if the earth were 
 to rotate seventeen times faster than it does at present, the 
 centrifugal tendency at the equator would be so greatly increased 
 as to balance the whole force of gravitation ; and a very small 
 further increase of velocity would cause bodies to leave the 
 surface of the earth altogether. 
 
 136. The earth is moving in its orbit round the sun with 
 a velocity of 1080 miles a minute, or nearly 65,000 miles 
 an hour : and the centrifugal tendency due to this enormous 
 speed just balances the sun's force of gravitation, and main- 
 tains the earth at a definite distance from the sun. If its 
 
LAWS OF MOTIO 
 
 onward velocity were to cease, it wajhld fall straigl 
 the sun. If its velocity were to be increased about one-half, 
 the increased centrifugal tendency would overcome the sun's 
 attraction altogether, and the earth would get farther and 
 farther from the sun, and never come near it again. Each 
 of the planets and their satellites, every comet, and each 
 component of a 'double star/ is found to be moving in just 
 such a path as enables its centrifugal tendency to balance 
 the force of gravitation to which it is subjected. 
 
 SECTION 3. FRICTION. 
 
 137. It has been already stated that, although there is no 
 doubt of the truth of the first Law of Motion, we cannot prove 
 by experiment that moving bodies, if left to themselves, go on 
 for ever with unaltered speed ; and friction was mentioned as 
 one great cause of this failure. We find, in fact, that whenever 
 two surfaces are moving in contact with each other, their motion 
 is resisted, each ' putting the drag/ as it were, on the other. 
 This resistance to the motion of surfaces which are in contact 
 is called ' Friction/ and is a good example of the action of forces 
 in checking, instead of causing motion. 
 "~"l38. Friction is due mainly to two causes : 
 
 (1) No surface can ever be made perfectly smooth; little 
 inequalities are visible through a microscope in even the 
 hardest and most carefully polished surface ; and so the rough 
 projecting parts of one surface catch against the rough parts of 
 the other (like a couple of files laid one on the other) and 
 hinder it in moving along. 
 
 (2) There is always some adhesion or cohesion between the 
 molecules of the surfaces when they are brought pretty close 
 together (par. 34, p. 16), and careful smoothing and fitting 
 only tends to increase this, since it brings more molecules within 
 range of each other's attraction. 
 
 Laws of Friction. 
 
 139. To investigate these it is generally most convenient to 
 use a 'statical' method (par. 107, p. 42) that is, to balance 
 friction against some easily measurable force, and see how much 
 of the latter is required for this purpose, or rather to keep the 
 
58 
 
 ELEMENTARY DYNAMICS. 
 
 surfaces just moving in spite of their friction. Gravitation i? 
 
 the best force to use, and 
 a very simple apparatus, 
 shown in fig. 17, will serve 
 to illustrate these laws. It 
 consists of a strip of well- 
 seasoned, straight-grained 
 oak, about 2 feet long and 
 5 inches broad, supported 
 on a stand, or placed on a 
 table so that one end pro- 
 Fig. 17. jects over the edge of the 
 table. One surface of the 
 
 strip should be planed or scraped as flat and smooth as 
 possible.* At one end of the block a pulley is attached, with 
 a cord passing over it, to one end of which blocks of various 
 shapes and materials may be connected, while from the other 
 end may be hung weights or a scale-pan in which weights may 
 be put. 
 
 Law I. The friction between two surfaces is greater when they 
 are resting in contact than when they are moving in contact with 
 one another. 
 
 140. To show this, a block of oak about 4 in. square x 2 
 in. thick is placed on the board ; and weights are hung from 
 the cord, until the amount required to set the block in motion 
 along the board is found. It will be noticed that the block, 
 when once started, begins to move very quickly, and that if 
 contact between it and the board is disturbed by a slight lateral 
 push, or even by tapping the board with the finger, a decidedly 
 less weight is required to keep it moving steadily and slowly 
 along the board. Hence, in subsequent experiments, the block 
 should always be started by lightly tapping the board, in order 
 to obtain the true moving friction. 
 
 Law II. Friction does not vary much with the velocity with 
 which the surfaces are moving. 
 
 141. There is about the same resistance to motion between a 
 
 * No glass-paper should be used in smoothing it, since the particles of glass get 
 imbedded in the wood : and of course no varnish, polish, or oil should be applied. 
 
LAWS OF MOTION. 59 
 
 carriage wheel and its axle, or between a sledge and the ground, 
 or between a skate and the ice, whether the motion be fast or 
 slow. It was formerly thought that friction was quite 
 independent of the velocity, but recent experiments have 
 shown that, when bearings are well lubricated, there is, 
 fortunately for engineers, decidedly less friction in proportion 
 at high speeds than at low ones. 
 
 Law III. Friction varies with the material of the surfaces, even 
 when they are equally smooth and equally pressed together. 
 
 142. Thus there is more friction between two pieces of wood 
 than between two pieces of metal under similar conditions, and 
 still more friction between wood and metal. This may easily 
 be illustrated by experiments with the apparatus already 
 described, and a table showing the amount of friction between 
 different substances is given on p. 61. We learn from such 
 experiments that wood is one of the best materials for brake- 
 blocks, and one of the worst for bearings of axles or runners 
 for sliding seats in boats. 
 
 Law IV. Friction varies directly with the pressure between 
 the surfaces. 
 
 143. Twice the pressure causes twice the friction ; three times 
 the pressure, three times the friction, and so on. This may be 
 illustrated by arranging a block of wood on the board, as in the 
 previous experiments, noting the weight required to keep it just 
 gently moving, and then putting on it another block of the 
 same weight There must then, of course, be twice the original 
 pressure between the lower surface of the block and the board ; 
 and it will be found that twice as much weight will now be 
 required to keep the block moving. A third block may then 
 be placed on the other two, and the resulting friction observed 
 in like manner. Hence, the heavier an engine is, the greater is 
 the hold which the driving-wheels have on the rail ; the tighter 
 a rope or a bat is gripped, the less likely it is to slip through 
 the hand. 
 
 Law V. Friction does not vary with the size of the surfaces 
 when the total pressure between them is unaltered. 
 
 144. At first sight we should hardly expect this to be true ; it 
 
60 ELEMENTARY DYNAMICS. 
 
 would seem, for instance, that a sledge must go more heavily on 
 broad runners than on narrow ones. An experiment to test the 
 correctness of the law may be made as follows. A wooden 
 block, 4 inches square and 2 inches thick, 
 has a deep broad groove cut out of one side 
 so as to give it the section shpwn in fig. 18. 
 The portions C, C, left on each side of the 
 groove may be in. broad. Thus we can place 
 three different-sized surfaces on the board 
 namely, A, the area of which is, of course, 
 (4 in. x 4 in. =) 16 square inches ; B, having an area of (4 in. 
 x 2 in. = ) 8 sq. inches ; or C, C, having each an area of 
 (| in. x 4 in. = ) 1 sq. in., or altogether 2 sq. inches ; the total 
 pressure (which is, of course, the weight of the block) being in 
 all three cases the same. The amount of friction between 
 the block and the board may be determined in the usual way, 
 and it will be found to be practically the same whether the 
 block is resting upon A, or upon B, or upon C, C. 
 
 145. The reason will be plain, if we consider (1) that friction 
 varies directly with the pressure (that is, increases and decreases 
 in exact proportion as the pressure increases and decreases ) ; ( 2 ) 
 that when the surface is large the pressure is distributed over 
 many points, and so there is not much of it on each point, while, 
 when the surface is small (the total pressure remaining un- 
 altered), the whole pressure is concentrated on few points, and 
 therefore there is more of it between each point of the surfaces 
 in contact. Suppose, for instance, that the block weighs 1 Ib. ; 
 then, when it rests on A (16 sq. in.), the pressure on each sq. 
 in. of the surface will be T V lb-j or 1 oz. ; when it rests on B 
 (8 sq. in.), all the pressure is collected on half the surface, and 
 therefore there will be twice as much, or 2 oz., on each sq. inch. 
 So that the advantage apparently gained by making the surface 
 smaller is lost on account of the greater pressure between each 
 point, and the total friction remains the same. 
 
 146. We learn from this that (so far as friction is concerned) 
 there is no advantage in narrow runners over broad ones for a 
 sledge or a skate ; or in broad rims for the driving-wheel of a 
 locomotive engine, where the object is to have as much friction 
 on the rail as possible. A box requires as much force to drag 
 
LAWS OF MOTION. 61 
 
 it along a floor, whether it is lying flat or resting on one corner 
 only. 
 
 Coefficient of Friction. 
 
 147. From the experiments made to illustrate Law IV., it 
 will be seen that the force required to overcome the friction 
 between two surfaces is always the same fraction of the pressure 
 existing between the surfaces, whatever that pressure is. Thus, 
 if a wooden box weighs 10 Ibs., and if 3 Ibs. are required to 
 keep it moving over a wooden floor, then we may state the 
 friction as being & of the pressure. If the box weighs 20 Ibs., 
 a force of 6 Ibs. will be required to keep it moving ; but 6 Ibs. 
 is -fa or T \ of the pressure that is, the same fraction as in the 
 former case. Hence we can conveniently and accurately express 
 the amount of friction between two surfaces by stating once 
 for all what fraction of the whole pressure between the surfaces 
 is required to overcome it. This fraction is called the ' Co- 
 efficient of Friction/ and its approximate value for different 
 substances is given in the table below. 
 
 Table of Coefficients of Friction. 
 
 [The figure means the fraction of the total pressure between the surfaces which 
 is required to overcome the friction when they are moving over one another. ] 
 
 Iron and Sandstone 
 Wood and Metal 
 Wood and Wood 
 
 Metal and Metal 
 Grease and Grease (less 
 
 Methods of lessening Friction. 
 
 148. For many purposes we want to get rid of friction as 
 much as possible, since force has to be spent in overcoming it, 
 and various means may be employed for this object. 
 
 (1) The surfaces should be made of those materials between 
 which the least friction is found to exist. 
 
 (2) They should be made as smooth as possible. Thus, 
 highly polished surfaces of steel working on bearings of brass or 
 bronze are employed in machinery. In watches the delicate 
 steel pivots work in polished holes cut in rubies or sapphires, 
 or similar extremely hard 'jewels.' 
 
 (3) They should be covered with oil or grease. Then, whatever 
 
 E 
 
62 ELEMENTARY DYNAMICS. 
 
 the material below may be, the friction is between surfaces 
 of oil only, and this (as the above table shows) is very small. 
 This is one of the most general and effective methods of pre- 
 venting friction, and is employed in addition to the other 
 means above mentioned. The bearings of machinery usually 
 have oil-cups fitted to them, from which the lubricating substance 
 is constantly supplied to the moving parts. More than a gallon 
 of oil is thus consumed in every locomotive engine in running 
 200 miles, while no less than 100 gallons of lubricants are 
 required per diem in the powerful machinery of the large 
 steamers which cross the Atlantic.* 
 
 (4) The surfaces may be made to roll over one another 
 instead of sliding. Thus, in moving large blocks of stone along 
 
 the floor of a quarry, a couple 
 of rollers are put under the 
 stone, as shown in fig. 19, 
 and the heavy mass is then 
 easily pushed forwards. As 
 -p. 19 one of the rollers comes out 
 
 behind in the course of the 
 
 movement, another is put in front ; and thus immense blocks 
 can be transferred from place to place by the use of a very 
 small force. In fact, friction (strictly so called) may by this 
 expedient be abolished altogether. 
 
 149. The employment of wheels for carriages depends upon 
 the same principle, the force required to draw a carriage being 
 far less than that which would be wanted for a sledge of the 
 same weight. Thus the coefficient of friction (as it may still be 
 called, for convenience) for a carriage on a well-made level road 
 is inr ; while on a railway, or a good tramway, it is only ^ for 
 moderate speeds. For example, a carriage weighing 1 ton 
 requires, to drag it at a moderate speed along a good road, -^ of 
 a ton, or 74 Ibs. ; to drag it along a railway only ^ of a ton, 
 or 8 Ibs. are required.! 
 
 * These figures are given on excellent authority, and show what is required even 
 with rigid economy. The expense of merely lessening friction by lubrication in a 
 passenger engine amounts to nearly 20 a year. 
 
 t At high speeds the resistance of the air becomes a serious obstacle, so that the 
 coefficient is ? T ff or more (about 30 Ibs. per ton) for a train running 60 miles an 
 hour. 
 
LAWS OF MOTION, 63 
 
 150. The reasons why, after inertia has been overcome, force 
 has to be continually supplied in order to keep up the motion 
 of a wheel carriage, are mainly the following : 
 
 (11) Adhesion or cohesion between the edge of the wheel and 
 the road. This has to be overcome when each point of the wheel 
 or roller is lifted up from the road as it moves on. 
 
 (2) Imperfect hardness of the road or ivheel, or both. The effect 
 of this is that the surface gives way more or less under the 
 pressure, so that the wheel is always in a hollow, and therefore 
 as it moves on it has continually to push down the higher part 
 in front of it ; and force must, of course, be spent in doing this. 
 It is easy to observe how much even a massive steel rail yields 
 under the pressure of a passing train ; and the ruts in a road 
 are a permanent record of the expenditure of force in the above 
 way. 
 
 (3) Sliding friction at the axle of the wheel. The whole weight 
 of the carriage acts between the axle and its bearing, and 
 between the edge of the wheel and the road. As the carriage 
 moves on, each point in the edge of the wheel is held by the 
 friction on the road, so that the whole wheel moves round and 
 the bearing slides upon the axle. The friction between these 
 latter surfaces is the full amount due to the weight of the carriage, 
 but the force which overcomes it is applied at the edge of the 
 wheel that is, at the end of the long arm of a lever formed 
 by each radius or spoke of the wheel ; and so this force acts 
 with greater advantage (as will be more fully explained under 
 the head of LEVERS) than if it was applied at the point where 
 the bearing and the axle are in contact. Thus, the effect of the 
 sliding friction at the axle in resisting the motion of the 
 carriage, is much less than it would be if the axle rested on the 
 road ; and it is less in proportion as the arm of the lever that 
 is, the length of the radius or spoke of the wheel is longer. 
 Hence we see why carriages with large wheels are easier to draw 
 than those with smaller ones. 
 
 151. It is possible to lessen materially, and almost abolish, 
 sliding friction at the axle in the following ways : 
 
 (1) Two wheels are placed close to each other in a frame, 
 as shown in fig. 20, with their edges overlapping. The axle 
 rests on their edges, and turns them round as it moves, rolling 
 
64 ELEMENTARY DYNAMICS. 
 
 instead of sliding on their surfaces. Thus the sliding friction 
 is only that of the supporting wheels on their axles ; and this 
 
 is comparatively small, as above 
 explained. The figure shows 
 how this principle is applied in 
 supporting the pulley of an 
 apparatus called Attwood's 
 machine, used in experiments 
 on the Laws of Motion ; and 
 large grindstones are often 
 mounted in a similar way. 
 Such pairs of wheels are called 
 'friction wheels.' 
 
 (2) A number of accurately-made hard steel balls are placed 
 in a circular recess surrounding the axle, which rests on their 
 surfaces, and rolls them round in the box as it rotates. Such 
 arrangements are called 'ball-bearings/ and are extensively 
 used in bicycles. 
 
 152. Before passing on, we may consider for a moment the 
 advantages and disadvantages of friction. It is a phenomenon 
 which meets us everywhere. Whenever we try to make a body 
 move, not only has its inertia to be overcome, but some force has 
 always and continuously to be expended in overcoming friction ; 
 so that we can never, however perfect a machine may be, get 
 the full theoretical eifect of a force in useful work. Every 
 movement of our bodies implies friction at each joint, although 
 nature has supplied a wonderfully efficient lubricating apparatus 
 and material wherever it is needed. f In rowing, we find 
 friction at the rowlock, and a considerable amount of it at the 
 sliding-seat. 
 
 153. Not only is force thus wasted, but the surfaces in con- 
 tact soon wear away, particles being constantly torn off owing to 
 the friction. Hence, bearings of machinery soon work loose 
 and have to be renewed ; boots and clothes wear out ; and a 
 little dust becomes an interesting study under the microscope, 
 from the variety of materials which frict'.on has contributed to it. 
 
 * From Prof. Balfour Stewart's Elementary Physics. 
 
 \ Any injury to these 'synovial membranes," as they are called, results in a 
 permanent stiffness of the joint. 
 
LAWS OP MOTION. 65 
 
 154. Yet we must not think that friction is universally harm- 
 ful, or even useless. We could not, in the most literal sense of 
 the words, ' get on ' without it. The friction between our feet 
 and the ground enables us to walk or run onward, as any one 
 will soon find out if he tries to walk or run on ice with skates 
 on. Window-blinds are raised and lowered by the friction of 
 the cord upon the pulley at the end of the blind-roller ; and 
 machinery in a mill is often worked by leather straps passing 
 with great friction over pulleys on a shaft turned by the engine. 
 A railway train is moved on by the friction between the driving- 
 wheels of the engine and the rails ; it is stopped by the friction 
 between the brake and the wheels. If the rails are from any 
 cause greasy, the driver soon finds that the engine is powerless 
 to draw the train owing to the wheels slipping ; sand is then 
 scattered on the rails, which increases the friction until the 
 driving-wheels again ' bite ' on the rail. 
 
 155. Friction, again, contributes immensely to the stability of 
 things. Without it, furniture, books, &c. would be continually 
 slipping about, like dead leaves before a breath of wind. India- 
 rubber is chosen for the soles of racquet- and fives- shoes because 
 of the high coefficient of friction between it and stone. It is 
 owing to friction that nails and screws stick so firmly in wood, 
 and that secure knots can be tied in string or rope. Even the 
 wear and tear produced by it are made to serve a useful purpose. 
 Most cleaning and polishing operations depend on friction ; 
 knives, &c. are sharpened on a grindstone ; corn is ground to 
 flour in a mill. 
 
 SECTION 4. THE SECOND LAW OF MOTION. , 
 
 The momentum produced by a force is exactly proportional to 
 the magnitude of the force ; and when several forces act on a 
 body, each produces motion in its own direction, just as if it was 
 the only force acting. 
 
 156. We have in this law two distinct statements as to the 
 action of forces, which must be considered separately. 
 
 A. Momentum varies exactly with the force which produces it. 
 Thus, if a cricket-ball weighing ^ Ib. is thrown with a force 
 which gives it a velocity of 12 feet per second, its momentum 
 
66 ELEMENTARY DYNAMICS. 
 
 will, of course, be (^ x 12 =) 4 ; now, if we find that the ball 
 has twice this momentum, we know that exactly twice as much 
 1'orce must have been exerted in throwing it ; similarly, three 
 times the force will give it three times the momentum, and 
 so on. 
 
 157. This fact is of great use in the exact comparison and 
 measurement of forces ; for if we want to find out how much 
 one force is greater than another, we have only to make each of 
 them act for the same time upon the same body, and observe 
 how much momentum each produces. If one force gives ten 
 times as much momentum as the other, then although we can- 
 not catch a glimpse of the forces themselves, we are sure 
 (from the above law) that the former force must be of ten times 
 the magnitude. 
 
 158. In expressing the magnitude of forces we must begin by 
 choosing some particular amount of force, to be taken as the 
 'unit' or standard ; just as a gramme or a pound is fixed upon 
 as a unit of weight, and a metre or a foot as the unit of length. 
 Then we can always give an exact idea of the magnitude of any 
 given force by saying that it is so many units. The unit of 
 force adopted in England is called a ' poundal,' and is 
 
 That amount of force which, acting for 1 second on a mass of 
 1 pound, gives it a velocity of 1 foot per second. 
 
 159. It is easily seen from the above definition that a poundal 
 of force, when applied to a body for 1 second of time, gives 
 it a momentum of 1 in terms of Ibs. ft. sec. ; so that, by 
 finding the momentum of a body in terms of these units, we 
 at once learn what impulse in poundals has been applied to it. 
 Suppose, for instance, that a football weighing \ Ib. is moving, 
 directly after being kicked, with a speed of 20 feet per second. 
 Then its momentum is, of course (\ x 20 = ) 10, and hence the 
 amount of impulse given it by the kick was equivalent to 10 
 poundals of force. Again, if we allow a stone which has a mass 
 of 1 Ib. to fall freely for exactly 1 second, we find that at the 
 end of the second it has acquired a velocity of 32 feet per second 
 (nearly). Hence its momentum is (1 x 32 =) 32 ; and there- 
 fore the magnitude of gravitation-force in England is 32 
 
LAWS OF MOTION. 67 
 
 poundals. Similarly, supposing that an eight-oar racing-boat 
 weighs with the crew on board 1800 Ibs., and that 1 second after 
 starting it is found to be moving at the rate of 8 feet per second. 
 Then its momentum is (8 x 1800=) 14,400; and the crew 
 must have exerted 14,400 poundals of force in the first second 
 after the start. 
 
 B. Every force acts independently of others. 
 
 160. A force is, as it were, an absolutely selfish but conscien- 
 tious individual, which doggedly does its own proper share of 
 work, and neither more nor less, totally regardless of the inter- 
 ruptions or solicitations of other forces. For instance, the earth 
 and all things upon it are (as already mentioned, p. 38) in rapid 
 motion, and yet we find that we can move about and move 
 other things about just as well as if the earth was not moving 
 at all ; in fact, the forces which we apply act quite independently 
 of the earth's motions. 
 
 161. If a ball is let fall from the top of the mast of a ship, 
 it ^will strike the deck at the bottom of the mast in just the 
 same place whether the ship t is motionless at anchor, or is in 
 swift but steady motion. In the latter case there are, at the 
 moment when it is let fall, two forces acting upon the ball : ( 1 ) 
 gravitation pulling it downwards ; (2) the force of the wind or 
 the steam, which is moving it, and everything else connected 
 with the ship, onwards. After it has begun to fall, the latter 
 force ceases to act, but the onward motion which it has com- 
 municated to the ball continues (according to Law I.) ; so that 
 by the time the ball reaches the deck it has gone as far forward 
 as the deck itself has. So, again, a juggler finds that in tossing 
 up balls and catching them again, he has to move his hands in 
 precisely the same direction whether he is standing still or is on 
 the back of a galloping horse ; for in the latter case the necessary 
 onward motion of the balls is supplied without any effort on his 
 part, so that they keep up with him and fall into his hands 
 again just as if he was standing still. When a bar is held in his 
 way above the horse, he does not require to leap forward in sur- 
 mounting it ; he springs directly upwards, and this upward 
 motion, combined with the forward motion he has in common 
 with the horse, results in carrying him in a curve over the 
 
68 ELEMENTARY DYNAMICS. 
 
 obstacle, and planting him on the very spot of the horse's back 
 he sprang from. 
 
 162. The above principle is of extreme importance in enabling 
 us to find the exact direction in which a body will move when 
 (not one, but) several forces act on it. This would be a most 
 complicated problem if we had to allow in every case for the 
 influence of one force upon another in modifying its effect ; but 
 as it is, we can consider the velocity and direction imparted by 
 each force quite irrespective of the others, with perfect certainty 
 that this velocity and direction will appear unaltered in the 
 final result. 
 
 SECTION 5. COMPOSITION OF FORCES. 
 
 163. This means the examination of the combined effect of 
 several forces which act on a point in the same body, in order 
 to find a single force which will produce the same effect as all 
 of them. Resultant is the term applied to such a single force 
 which is equivalent to several others ; and the separate forces 
 to which it is equivalent are called its components. We shall 
 consider three separate cases of the composition of forces, and 
 show how the principle of Law II. is applied in each of 
 them. 
 
 164. It may be noted here that, since the forces are considered 
 in their action on the same body, the mass of which is unaltered 
 during their action, their relative magnitudes may be quite 
 correctly expressed by simply stating the velocity (and not the 
 momentum) which each would produce in the body. Thus, if a 
 cricket-ball is hit by one player with a force which makes it 
 move 10 feet per second, and the same ball is hit by another 
 player with a force which makes it move 30 feet per second, 
 we know that the latter force must be three times as great as 
 the former, without regarding the weight of the ball. So in 
 many subsequent examples the forces will be expressed by 
 stating the velocity they produce in the body on which they 
 act. 
 
 A. When the forces act in the same direction on a point in a 
 body. 
 
 165. Here, since each force produces its own proper effect, 
 and neither more nor less, we have only to add the magnitudes 
 
LAWS OF MOTION. 69 
 
 of the forces together. The sum of these magnitudes is the 
 required resultant, and it acts, of course, in the same direction 
 as each and all of its components. 
 
 166. For example, when a team of horses is drawing a waggon, 
 the resultant force applied to the waggon is the sum of the 
 separate efforts of each horse. When a barge is towed by a 
 steam-tug along a river with the stream, the resultant speed of 
 the barge is that which would be communicated to it by the 
 stream alone added to that which the steamer itself imparts. 
 
 B. When two forces act in opposite directions on a point in a 
 body. 
 
 167. In this case we must subtract the magnitude of the 
 smaller force from that of the greater ; the remainder will be 
 the magnitude of the resultant, and it will act in the direction 
 of the greater force. 
 
 168 ; Suppose, for instance, that a steam-tug is towing a barge 
 against the stream of a river. Then the resultant force on the 
 barge, as shown by the speed it goes, will be the difference 
 between the force of the stream and the force of the tug. If the 
 stream alone would carry on the barge at the rate of 2 miles 
 per hour, while the steamer alone would drag it along at the 
 rate of 7 miles per hour on a canal or lake, the actual resultant 
 speed of the barge will be (7 2=) 5 miles per hour up the 
 river. 
 
 169. The same applies, of course, to rowing on a river : the 
 rate of progress of a boat up-stream is the difference between the 
 speed due to the current and that due to the force applied by 
 the rowers.* Again, in football, if two players urge the ball in 
 opposite directions, the actual effect on the ball is the difference 
 between the forces they apply respectively. 
 
 170. A good example of both the above cases of the composi- 
 tion of forces is afforded by the game called the ' tug of war,' 
 or ' French and English.' In it two sets of players endeavour to 
 drag a rope in opposite directions ; and the resultant force with 
 which the rope moves is found by ( 1 ) adding together the forces 
 
 * It is said to be harder to row against the stream, not necessarily because more 
 labour is expended by the rowers in each stroke, but because more strokes are 
 needed to carry the boat through a given distance. 
 
70 ELEMENTARY DYNAMICS. 
 
 of the separate players on each side, (2) subtracting the smaller 
 sum from the greater ; the remainder expresses the resultant 
 force with which the rope and the weaker set of players clinging 
 to it will be dragged over the ' scratch ' line. 
 
 C. When two forces act in directions which make an angle 
 (other than 180) with each other. 
 
 171. This is a rather more difficult case. Suppose a ball is 
 rolling along a table or a cricket-field, and it is hit sideways, in 
 what direction will it thenceforwards move ? 
 
 172. It is necessary in the first place to attend to the dis- 
 tinction between motion in the same direction and motion in the 
 same straight line. In a regiment of soldiers on the march, each 
 man is moving in the same direction, northwards for instance, 
 though he is not moving in the same straight line as those on 
 his right and left. Now, when a force has produced motion in 
 its own direction, it has done its proper work, whether the move- 
 ment be in the same straight line or not. In fig. 21 the ball B 
 
 is moving in the same direction, 
 
 "R Ji 
 
 whether it move in the line AC 
 or in any line parallel to AC, as 
 gf or ED ; in any of these cases it 
 is equally approaching the line 
 CD. In the same way a motion 
 Fig. 21. from B towards E, or from h 
 
 towards i, or from C towards D, 
 is still in the same direction, because these lines are parallel. 
 
 173. Now, let the ball be moving along the line AC with a 
 velocity that would carry it from B to C in two seconds, and 
 when at B let it receive a blow that would carry it from B to E 
 in the same time ; the question is, How will the ball now move ? 
 This is best understood by supposing it placed, not on a plane 
 surface, but in a groove in the upper side of a movable bar 
 lying on a table. The ball being now set rolling at the same 
 rate as before along a groove in the bar AC, let the bar be made 
 at the same time to slide across the table, keeping parallel to 
 itself, and carrying the ball along with it, so as to arrive at the 
 position ED in two seconds. The common motion of the bar 
 and the ball will not in any way interfere with the motion of 
 
LAWS OF MOTION. 71 
 
 the ball in the groove, any more than the common motion of a 
 ship and a man on board of it interferes with the man in 
 walking across the deck. The ball will be at the end of the 
 groove at the end of the two seconds, just as if the bar had been 
 at rest ; it will therefore, as a result of the two movements, be 
 found at the point D. 
 
 174. If the position of the ball on the table is observed at the 
 intermediate points, it will be found to describe a straight line 
 from B to D ; for since we have supposed both motions uniform, 
 the bar will, at the end of the first second, be in the position gf, 
 midway between EC and ED, and the ball will at the same 
 instant be half-way from g to /, at k; and it can be proved 
 (Euclid, VI. 26) that k is in a straight line between B and D. 
 The same could be shown as to any intermediate stage. When 
 both motions are not uniform, the body moves in a curve, as 
 is the case with projectiles. 
 
 175/The movable groove is introduced to make the effect of 
 two movements conjoined more readily conceived ; to show 
 palpably, as it were, that a body may be moving in two 
 directions at one and the same time. But if it receive the 
 second impulse by a blow while rolling freely on the table, it 
 will still arrive at D by the same path. 
 
 176. Now, since ED is parallel to BC, and CD is parallel to 
 BE, the figure BODE is a parallelogram (Euclid, I. Definitions), 
 two adjacent sides of which, such as BE and BC, represent 
 respectively the two forces which act on the body A. Further, 
 a line drawn from B to D is called a 'diagonal' of the parallelo- 
 gram ; arid it has been shown to represent accurately, both in 
 magnitude and direction, the resultant of the two forces BC and 
 BE. Such a figure is called a parallelogram of forces. 
 
 177. It will now be seen how easily we can, by drawing such 
 a figure correctly to scale, find out quite accurately the magni- 
 tude and direction of any two forces which act at any angle 
 upon a point in a body. We must do as follows : 
 
 Through the point at which the forces act draw lines, AB, AC, 
 fig. 22 (next page), to represent each force in direction and 
 magnitude ; taking the lengths on any convenient scale of equal 
 parts, such as centimetres or inches. 
 
 Through the outer ends of each, B and C, draw a line parallel 
 
72 
 
 ELEMENTARY DYNAMICS. 
 
 to the other force, namely, BD and CD, so as to make a parallelo- 
 gram. 
 
 Through the point at which the forces act draw a diagonal, 
 AD, of the parallelogram. 
 
 B A 
 
 Then this diagonal will represent the resultant of the two forces, 
 both in direction and magnitude. 
 
 [The student should notice how the value of the resultant 
 changes with the .angle between the component forces. In the 
 above figures the components have the same value namely, AB 
 = 4 units, AC = 3 units, but the magnitude and direction of the 
 resultant is very different. In fig. 23, for instance, it is 6 units, 
 while in fig. 24 it is only 3 units, owing to the wide angle between 
 the directions of the components. The wider this angle, the more 
 nearly the forces act against one another, and when it is 180 the 
 resultant is only equal to the difference between the forces, as already 
 stated. ] 
 
 178. We may next take some practical examples of the com- 
 position of forces ; and for simplicity we shall consider only 
 cases where there are two forces acting at right angles,* and we 
 shall take the velocity produced by the forces to express their 
 respective magnitudes, as explained in par. 164. 
 
 179. Suppose that a ferry-boat is being rowed across a river 
 with a force which gives it a speed of 4 miles an hour ; and 
 meanwhile the current is carrying it down the river at the rate 
 of 3 miles an hour. It is plain that the boat will not go straight 
 across the river, but in a slanting direction, so as to reach the 
 opposite side at a place farther down the river than the point 
 directly opposite its starting-place. The exact direction and 
 
 * If there are more than two forces concerned, first the resultant of any two of 
 them is found as above, and then this resultant is combined with another of the 
 forces, and the resultant of this pair is found, and so on. 
 
LAWS OF MOT 
 
 speed of the boat may be found by th 
 thus : Draw (as in fig. 25) a line, AB, 
 such as one of centimetres or __ 
 inches) to represent the force of - 
 the rowers. Through A draw a 
 line AC, 3 units long (on the 
 same scale) to represent the force 
 of the current. Complete the 
 parallelogram ACDB, and 
 through A draw a diagonal AD. 
 Then this diagonal will repre- 
 sent the direction in which the -^^i 
 boat will actually move ; and ""* 
 it will be found to be just 5 
 units long. Therefore the boat will move 
 
 Fig. 25. 
 
 in the direction A 
 to ITwith a force which gives it a speed of 5 miles an hour. 
 
 180. Again, suppose that a cricket-ball, bowled with a force of 
 50 poundals, is hit in the direction of square-leg with a force 
 of 120 poundals. Then, drawing a parallelogram as above 
 described, with sides of 50 and 120 units (a millimetre scale 
 may be used), we shall find that the diagonal is 130 units long. 
 Thus the ball will travel more nearly in the direction of long- 
 leg, and with a force of 130 poundals. 
 
 181. Many other examples of the composition of forces will 
 suggest themselves ; such as, a ball thrown at the wicket by a 
 player while running, a shot fired from a moving ship at a 
 battery on shore, a boat towed along a river by two men, one on 
 each bank. 
 
 SECTION 6. RESOLUTION OF FORCES. 
 
 182. This is the exact con- 
 verse of the composition of 
 forces, and means the division 
 of one force into several others, 
 called its 'components/ which, 
 taken together, are equivalent 
 to it. 
 
 183. For instance, if a man 
 
 Fig. 26. 
 is dragging a block of stone along the ground by a rope slanting 
 
74 
 
 ELEMENTARY DYNAMICS. 
 
 upwards from it to his hand, as shown in fig. 26, the single 
 force, AB, which he applies along the rope and in its direction, 
 is really equivalent to, and might be replaced by, two forces, AC 
 and AD, one of which tends to raise the block of stone directly 
 from the ground, while the other alone is effective in dragging 
 it along the ground. 
 
 184. The magnitude of these two components can easily be 
 found by constructing a parallelogram of forces. To do this, we 
 must consider the single original force as a diagonal, round 
 which we have, as it were, to fit a parallelogram ; and the 
 adjacent sides of this will represent the component forces 
 required. 
 
 185. It is evident that we must know the directions which 
 the two components are to have.* Suppose that the total force 
 applied to the rope is 8 poundals, and that we want to find how 
 much of it is spent in dragging the stone along, and how much 
 is spent in merely raising it from the ground. The directions of 
 the components will then be at right angles to each other, as in 
 the figure. 
 
 186. Make the line AB, fig. 27, 8 units long, and through A 
 
 draw two lines of any 
 length, AX horizontal, 
 and AY vertical ; these 
 lines will then represent 
 the directions of the re- 
 quired components, and 
 we have to find what 
 lengths must be cut off 
 from them to represent 
 
 Fig. 27. 
 
 correctly the magnitudes. Through B draw BC parallel to AD, 
 and also BD parallel to AC. Then ACBD is a parallelogram, for 
 its opposite sides are parallel ; and two of its adjacent sides, 
 AC and AD, represent correctly two forces, which are together 
 equivalent to the single force AB. If these two sides are 
 measured, AC will be found almost exactly 7 units, and AD 
 4 units of the scale. Hence we learn that 7 poundals of force 
 
 * Or, of course, their magnitudes, in which case their directions may be found ; 
 or the magnitude and direction of one of them, in which case the magnitude and 
 direction of the other may be found. 
 
LAWS OF MOTION. 75 
 
 are being spent in moving the stone onward, and 4 poundals in 
 raising it upward, or at any rate lessening the pressure between 
 it and the ground. 
 
 [ Several similar figures should be drawn, with the direction of the 
 rope that is, the diagonal of the parallelogram more or less inclined 
 to the horizontal Hue of the ground ; and the effect on the relative 
 magnitudes of the components should be noted. ] 
 
 187. When a barge is being towed along a river or canal by a 
 horse on the bank, the force exerted through the tow-rope is 
 not all effective in moving the barge onwards. It may, in fact, 
 as shown in fig. 28, be - 
 
 resolved into two com- ^^ == ^_ A o 
 
 ponents, AC and AD, 
 at right angles to one 
 
 another; one of which, 
 AC, "moves the boat 
 
 straight along the river, while the other, AD, only tends^to 
 pull it towards the bank. Suppose, for instance, that the horse 
 is pulling with a force of 85 poundals in the direction AB. 
 Make AB 85 units long ; then, on constructing the parallelogram, 
 it will be found that AC represents a force of 84 poundals, and 
 AD one of 13 poundals. In order to counteract this latter com- 
 ponent, and prevent the boat being dragged into the bank, the 
 rudder must be used ; and its action supplies another rather 
 more complicated example of the resolution of forces, which 
 will be further explained in the appendix to this chapter. 
 
 188. Other good examples of the resolution of a force in such 
 a way that it seems to act in a direction other than its own, are 
 afforded by the action of a horizontally-blowing wind on a kite 
 so as to cause it to rise vertically in the air ; the action of a 
 wind blowing across the course of a ship on sails set obliquely 
 to the keel ( a north wind, for instance, impelling the ship along 
 a westerly or even north-westerly course) ; the effect of a wind 
 on the sails of a windmill. In all these cases two distinct pro- 
 cesses of resolution have to be performed ; the final result being, 
 that a certain amount of the force appears as a component 
 which acts in a direction making a right angle, or even a 
 greater angle, with that of the original force. 
 
76 
 
 ELEMENTARY DYNAMICS. 
 
 APPENDIX. 
 Action of the rudder in altering the course of a ship. 
 
 189. The rudder is a flat plate, hinged vertically to the stern of 
 the ship, so that it can swing from side to side like a door. When 
 its surface, or * plane,' is in the same line as the keel, it merely acts 
 as a portion of the keel, and steadies the ship in the course in which 
 she is going. But when it is turned at all obliquely to the right or 
 left of the line of the keel, it and the sternpost to which it is 
 hinged undergo a pressure in the opposite direction. Thus in 
 fig. 28, the rudder has been moved to the left of the line of the keel ; 
 hence the stern of the ship is pushed to the right hand, and there- 
 fore the bow points to the left of its former course, and so the 
 whole vessel proceeds in that direction. The question is, how does 
 the rudder get this pressure sideways ? 
 
 As the ship moves on, the inertia of the water-particles which 
 the rudder meets causes them to press against its surface in 
 the direction AB, fig. 29. Now, AB may be resolved into two 
 
 Fig. 29. 
 
 Fig. 30. 
 
 components at right angles to each other namely, CB, which is 
 parallel to the surface of the rudder, and therefore has no 
 effect on it at all, and DB, which presses at right angles to the 
 surface. 
 
 190. Let us next take this latter component DB as a distinct 
 force (see fig. 30). It may be resolved into two components at 
 right angles to each other namely, EB, which simply presses the 
 rudder in the contrary direction to that in which the ship is going, 
 and thus retards the whole ship ; and FB, which pushes the rudder 
 sideways, and therefore the stern of the ship also. It is this latter 
 
LAWS OF MOTION. 77 
 
 component alone which is effective in changing the course of the 
 ship, which it does in the manner above explained. 
 
 191. The calculation of the action of the wind on sails of a ship, 
 or on a kite, referred to in par. 188, p. 75, is made on exactly the 
 same principle as above described namely, to resolve the force of 
 the wind into two components, one of which is at right angles to 
 the sail or kite, and then to resolve this latter force into two others, 
 one of which urges the ship on, or raises the kite upwards. It will 
 be a useful exercise to work out these problems more fully from the 
 hints just given. 
 
 SECTION 7. THE THIRD LAW OF MOTION. 
 The action of a force is always accompanied by a reaction in the 
 body to which it is applied. This reaction is equal to the force 
 in magnitude, and is in the opposite direction. 
 
 192. When a football is kicked, it presses against the foot 
 with just as much force as the foot presses against it. When 
 one stone is dashed against another stone at rest, the moving 
 stone is hit as hard, and is as likely to break, as the one at rest ; 
 and when one person knocks his head against his neighbour's, 
 it is difficult to say which is most hurt. The hand pressed 
 against a fixed body is equally pressed in its turn. If a man 
 standing in a boat attempts to push off another boat of the same 
 weight that is alongside, both boats will recede equally from 
 each other ; if he pulls the other boat towards him, his own boat 
 advances half-way to meet it. A magnet draws a piece of iron 
 towards it ; but the magnet is also drawn towards the iron, as 
 is seen when they are both suspended so as to move freely. In 
 all these cases we see that the body which we consider as acting 
 upon the other, is itself acted upon in turn, and in the opposite 
 direction : this is what is meant by reaction. But to determine 
 more exactly the equality of the action and reaction in all 
 cases, it is necessary to advert to the way in which action is 
 measured. 
 
 193. In all cases of the action of a force there are two portions 
 of matter concerned (A) the one in which the force is con- 
 sidered to reside ; (B) the one on which it is considered to act. 
 Thus, in the action between a magnet and a piece of iron, the 
 force of attraction is considered to reside in the magnet, and the 
 
 F 
 
78 ELEMENTARY DYNAMICS. 
 
 piece of iron is usually only considered as being attracted. Hither- 
 to we have hardly considered A except as the vehicle, as it were, 
 of the force ; but in point of fact, as the above examples show, 
 just as much effect is produced on it during the action of the 
 force, as on B. Now, as we have seen already, p. 41, the effect 
 of a force is estimated, not alone by the velocity it produces, but 
 by taking into consideration also the mass on which it acts. 
 Mass multiplied by velocity that is, the momentum produced 
 is the true measure of the force which has acted on a body ; and 
 what the third law of motion asserts is this : That in any case 
 of the action of a force, just as much momentum is produced 
 in the body in which it is considered to reside, as in the 
 body on which it acts ; but this momentum is in the opposite 
 direction. 
 
 194. To recur to the examples of reaction formerly cited. If 
 the magnet and the piece of iron are of the same weight, they 
 move to meet each other with equal velocities, for thus only 
 can the momentum be the same in both cases. If the magnet 
 is three times the weight of the piece of iron, the iron must 
 move with three times the velocity of the magnet to make the 
 momentum the same ; and so it is found to do. In the case of 
 the boats, suppose the one in which the man is seated to be ten 
 times the weight of the other, then for every ten feet that the 
 light one moves off, the heavy one will recede one foot ; so 
 that the two will have the same momentum. 
 
 195. In the last case, both motions would still be visible. 
 But let a boat of a ton weight be pushed away from the side of 
 a ship of one thousand tons weight, and then only one seems to 
 move ; for while the boat moves off a yard, the ship recedes 
 only the thousandth part of a yard, which it would require 
 minute observation and measurement to render apparent. From 
 this we can pass to the extreme case of a boat pushed off from 
 shore. Where is the evidence of reaction here ? We see none, 
 it is true ; still, the consideration of the cases already adduced, 
 and of a thousand similar, lead us irresistibly to believe that 
 the shore, if it is free to move, must recede from the boat. But 
 the shore can move only by carrying the earth with it ; and 
 considering the vast mass of the earth compared with that of the 
 boat, the space moved over would defy measurement, even if 
 
LAWS OF MOTION. 79 
 
 we had any fixed mark to count from. We cannot help believ- 
 ing, then, that when a stone falls in other words, when the 
 earth draws a stone towards it the earth is itself drawn, or falls, 
 towards the stone. 
 
 19C. Other examples of the equality of action and reaction 
 are the following : When a spring is compressed, although the 
 compressing force is only applied at one end, yet there is pro- 
 duced in every part of the spring a strain which shows itself as 
 a force acting in two opposite directions ; so that the spring may 
 be -used to propel a bullet either in the direction of the compress- 
 ing force, or in the opposite direction. 
 
 197. Similarly, when a gun is fired, the gases suddenly pro- 
 duced in the breech act like a strongly compressed spring, and 
 exercise pressure, not only against the bullet, but also against the 
 closed end of the gun. The result is not only that the bullet 
 is driven out with great velocity, but also that the whole mass 
 of the gun is driven in the opposite direction with an exactly 
 equal momentum. This explains the recoil or ' kick' of the gun 
 against the shoulder. For instance, the weight of an army rifle 
 is 10 Ibs., and the bullet weighs about 1^- oz. ( T V lb. ) ; that is, 
 the gun is 120 times as heavy as the bullet. Now, suppose that 
 the bullet is driven out with a velocity of 1200 feet per second ; 
 then its momentum is ( 1200 x T V Ibs. =) 100. And the gun will 
 recoil with a momentum equal to this ; but since its weight is 
 120 times that of the bullet, its velocity will be 120 times less ; 
 that is (jfff of 1200 = ) 10 feet per second. In order to avoid the 
 consequences of this recoil, the carriages of large guns are made 
 very massive, and are allowed to run back up an incline and 
 checked by ropes or hydraulic buffers. Still, cases have been 
 known of a gun being fairly shot away from its carriage, and 
 doing much damage in its backward course. 
 
 To illustrate roughly some of the facts of action and reaction, an 
 apparatus made on the principle of a common toy spring-gun ( or one 
 of the actual toys slightly altered) is convenient. The spring, when 
 compressed, should be held by a loop of thread hung over a hook 
 attached to the gun ; a pencil or a bit of iron rod being put into the 
 barrel as a projectile. The gun is then laid on a smooth table, and 
 the spring released by burning the thread with a lighted match. 
 As an extreme case, the projectile may be made of the same weight 
 
80 ELEMENTARY DYNAMICS. 
 
 as the gun, and the distance travelled by each from the starting- 
 point (that is, the point where the projectile is in contact with the 
 spring within the barrel) may be measured.* 
 
 198. Another interesting example of reaction is a rocket. 
 Here the gun is actually used as the projectile ; a constant 
 stream of particles of gas is rushing with immense velocity from 
 the mouth of the rocket, and the latter recoils with an equal 
 momentum. 
 
 199. In rowing, the feet always rest against a cross-piece or 
 1 stretcher/ firmly attached to the boat. But they do not merely 
 rest against it ; they press against it with the same force as 
 that which is applied to the oar ; and for the following reason : 
 When the oar is pulled by the hands, it pulls (owing to reaction) 
 against the hands with an equal force ; and the body would be 
 pulled up to the oar, if it were not that the feet, set rigidly 
 against the stretcher, prevent this movement, so that the force 
 of the muscles is only operative in pulling the oar up to the 
 body. It is well known in rowing that, unless a rower is * feel- 
 ing his stretcher ' that is, is exerting consciously a pressure 
 against it he cannot be doing any useful work. 
 
 200. In the case of a carriage being run away with, persons 
 riding in it have been known to lay hold of the sides to hold it 
 back : they forget that, while pulling back with their hands, 
 they are pushing forwards with their feet, and that the action 
 and reaction, being equal and contrary, destroy each other's 
 effects. 
 
 SECTION 8. COLLISION OF BODIES. 
 
 201. When one moving body strikes against another body, 
 which either is at rest, or is moving in a different direction, or 
 is moving in the same direction but with a different velocity, 
 the two are said to 'come into collision.' Such common examples 
 will suggest themselves as collisions between two trains or ships ; 
 the striking of a cricket-ball against a bat, of a fives-ball against 
 the walls or floor of the court, of a billiard-ball against another 
 ball, or against the cushion of the table. In all such cases there 
 
 * It should, of course, be observed that these distances express energy, and not 
 merely momentum, as will be more fully explained in a later chapter. 
 
LAWS OF MOTION. 81 
 
 is invariably some change in the motion of the bodies ; and the 
 nature of the change will be easily understood if we bear in 
 mind the principle (to be more fully explained in the chapter 
 on Energy and Work), that no force whatever is under any circum- 
 stances actually created or destroyed in the universe, so far as we 
 know it ; so that, when momentum is communicated from one 
 body to another, the original amount of momentum, and neither 
 more nor less, remains in the whole mass affected, though its 
 distribution may be different. This is the general law, but the 
 results actually observable are dependent on the amount of 
 elasticity of the substances concerned that is, upon the extent 
 to which their molecules endeavour to recover their original 
 positions when a pressure or strain has been put upon 
 them. 
 
 (A) Collision of non- elastic bodies. 
 
 202. Suppose that a piece of soft moist clay or of lead (bodies 
 of which the elasticity is very slight) weighing 12 Ibs., is driven, 
 with a velocity of 10 feet per sec., against a similar piece weigh- 
 ing 4 Ibs., which is at rest. The momentum of the first mass is 
 obviously (12 x 10 =) 120. Now, the first effect of the action 
 and reaction is to compress the molecules of both pieces until 
 their cohesion balances the force, and in this way some of the 
 force associated with the moving body is spent, which we may 
 express by a loss of momentum of, say, 20. The next effect is, 
 that the masses move on in contact with a momentum which is 
 equal to all that remains of the original momentum of the 
 striking body, that is, (120 - 20 =) 100. But the velocity of the 
 united mass must clearly be less than 10 feet per sec., since the 
 mass in motion is now (12 + 4 =} 16 Ibs., instead of only 
 12 Ibs. The velocity in feet per sec. will, in fact, be repre- 
 sented by such a number as will, when multiplied by 16, give 
 a product of 100 ; and this is, of course, the quotient of 100 
 divided by 16 namely, 6J. So that the observed effect will 
 be, that the two masses will move on together with a velocity 
 of 6| feet per sec. in the same direction as the heavier body was 
 originally moving. 
 
 203. If the bodies which come into collision are both in 
 motion but in opposite directions for instance, two football 
 
C2 ELEMENTARY DYNAMICS. 
 
 players charging each other the result will depend upon their 
 relative momenta. If their momenta are equal, the reaction of 
 each destroys the motion of the other, and they are both 
 brought to rest. If their momenta are different, then they 
 move on in contact after the collision, in the direction in which 
 that body was moving which had the greater momentum. 
 
 204. Suppose, for example, that a player weighing 140 Ibs., 
 and moving 10 feet per sec., charges another weighing 100 Ibs., 
 and moving 9 ft. per sec. Then the momentum of the first is 
 (140 x 10 =) 1400 ; and that of the second is (100 x 9 =) 900. 
 When they come in contact, the momentum of the lighter 
 pla} T er, namely, 900, counteracts an equivalent amount of the 
 momentum of the heavier one ; so that the remaining momentum 
 is (1400 - 900 =) 500. Of this we may suppose 20 lost in com- 
 pression of the molecules, &c., leaving 480 remaining. This 
 480 is the momentum of the whole mass of (140 + 100 =) 240 Ibs. 
 So the velocity will be only (ff# =) 2 feet per sec., and the 
 motion will be in the direction in which the first player was 
 running ; in fact he will overpower the other and press him 
 back. 
 
 (B) Collision of elastic bodies. 
 
 205. This is by far the most usual case ; few, if any, substances 
 being quite destitute of elasticity. Suppose, for instance, that 
 an ivory ball strikes another similar ball of the same weight. 
 The first effect of the collision is, as already stated, to compress 
 the molecules of both balls, and alter their position against the 
 force of their cohesion. In this, no force is lost as mechanical 
 force (as with non-elastic bodies, in which the cohesion has no 
 power to bring back the molecules to their places), but is stored 
 up as in a compressed spring. In the next place, this force of 
 cohesion exerts itself, and the balls are pushed apart with equal 
 momenta in opposite directions. The result is, ( 1 ) that the ball 
 which was originally moving, being met by an equal and 
 opposite force, has its motion stopped entirely; (2) that the 
 other ball is set in motion with the same velocity ( since it is of 
 equal weight ) as that of the ball which struck it. 
 
 [ To show this, two ivory billiard-balls ( stone balls will answer, 
 but not so well) may be hung by strings from a frame, fig. 31, so 
 
LAWS OF MOTION. 83 
 
 as to be just in contact. One of them is then drawn aside, and 
 allowed to swing against the other, which immediately moves 
 onward (the first ball remaining stationary), and 
 may be caught before it swings back. ] 
 
 206. This peculiar action of elastic bodies 
 appears when a number of ivory balls are 
 placed close in a row, and the 
 outermost at one end is smartly 
 struck against the next ; none of 
 them move sensibly from their 
 places, except the outermost at the 
 other end of the row. Each ball 
 2 in turn receives the whole motion 
 
 from the one that precedes it, and 
 gives it away entire to the next. The last becomes 
 thus the vehicle of the whole motion. Instead of placing the 
 balls on a table, they may be suspended as in fig. 32. 
 
 207. If the striking ball is heavier than the other, thu 
 momentum generated during the spring-action above explained 
 is not sufficient to stop its onward motion, but only to lessen it. 
 This is what usually occurs when a cricket-bat or a golf-club 
 hits the ball. The bat or club still moves onward after the ball 
 is hit,* but a comparatively small effort on the part of the 
 striker is sufficient to stop it. If the ball is missed, the conse- 
 quences are unpleasantly felt : the bat moves on with undimin- 
 ished momentum, and may be flung, as it were, out of the 
 player's hand. 
 
 SECTION 9. LAWS OF REFLEXION. 
 
 208. When a body strikes a fixed surface, if both are completely 
 inelastic, its motion is destroyed and it remains on the surface. 
 But this is true only of soft masses ; all hard solids have more or 
 less elasticity, and rebound or are reflected from the surface, and 
 this reflexion follows a regular law of direction. If an ivory 
 ball, for instance, be dropped, as from L, fig. 33, on a level 
 marble slab at K, it will rebound in the same perpendicular 
 
 * Unless, of course, the ball itself is bowled so swiftly against the bat as to supply 
 a sufficient momentum on its own account. 
 
84 ELEMENTARY DYNAMICS. 
 
 line, and, being almost perfectly elastic, will rise again nearly 
 to L. But if the ball is thrown obliquely in the direction H 
 
 to K, the action and reaction 
 drive it back in a direction KI, 
 which makes the same angle 
 with the perpendicular KL, 
 drawn to the surface where the 
 ball hits it, as the original direc- 
 tion of the ball made with this 
 perpendicular. In fact, the 
 angle HKL, which is called the 
 * angle of incidence,' is always found to be equal to the angle 
 LKI, which is called the 'angle of reflexion.' Moreover, the 
 direction of incidence, HK ; the direction of reflexion, KI ; 
 and the perpendicular, KL, always lie in the same plane. 
 
 209. Thus the two laws of reflexion for perfectly elastic bodies 
 may be stated as follows : 
 
 I. The angles of incidence and reflexion are equal to one another 
 
 II. The directions of incidence and reflexion lie in the same 
 plane, which also includes a perpendicular drawn to the surface 
 through the point where the body strikes it. 
 
 [A proof of these laws is given in the Appendix, p. 85.] 
 
 210. Practical examples of these laws are found in many 
 games. Thus, in the games of fives and racquets, successful play 
 greatly depends upon correctness in mentally judging the angles 
 of incidence and reflexion of the ball, so as to drive it in a 
 direction which makes it difficult for the opposite player to 
 ' return it.' The method of calculating the precise direction in 
 which a ball must be hit in order that, after striking the wall, it 
 may proceed to a given point, is given in the Appendix to this 
 chapter ; but of course, in practice, allowance has to be made 
 for the want of perfect elasticity (or 'deadness') of the ball and 
 of the plaster wall of the court. 
 
 211. In the game of billiards, both the ivory balls and the 
 india-rubber cushions of the table fulfil much more nearly the 
 conditions of perfect elasticity, and a ball rebounds from the 
 cushion, or from another ball, very nearly indeed in the 
 
LAWS OF MOTION. 
 
 85 
 
 theoretical direction. The whole game is full of practical 
 illustrations of the Laws of Motion. In cricket, the position 
 of the surface against which the ball strikes that is, of the 
 bat, is changed instead of the direction of the ball ; and thus 
 when a ball is to be 'sent into the slips,' for instance, the 
 bat is held obliquely, so that the ball strikes it at such an 
 angle as to rebound in the required direction. 
 
 APPENDIX A. 
 
 212. Proof of the law that when a perfectly elastic body strikes 
 on a perfectly elastic surface, the angle of reflexion is equal to the 
 angle of incidence. 
 
 Let AB (fig. 34) represent the direction and magnitude of the 
 force with which 
 
 C 
 E 
 
 the body strikes the 
 surface at B. 
 
 Through B draw 
 BC perpendicular to 
 the surface. 
 
 Through A draw _____ F \*/ / D 
 
 AD parallel to BC, W^^^a 
 meeting the surface Fig. 34. 
 
 at D, and AE 
 
 parallel to the surface, meeting BC in E. Thus AEBD is a 
 parallelogram. 
 
 Then the force AB may be resolved into two components 
 namely, DB parallel to the surface, and EB perpendicular to it. 
 
 Now, the component DB is not affected by the collision of the 
 ball with the surface ; but the component EB is met by an equal 
 and opposite force, or reaction. So that, after the ball has struck 
 the surface, it is acted on by two forces namely, BF, equal to DB, 
 and BE, equal and opposite to EB. 
 
 We have to find the resultant of these forces. 
 
 Complete the parallelogram BFGE, and draw the diagonal BG. 
 
 Then BG represents the direction and magnitude of the resultant 
 force acting on the ball after it has struck the surface. 
 
 It is required to prove that BG makes the same angle with BC 
 that AB does. 
 
 Since AE = BD = BF = EG, therefore AE = EG. 
 
 And EB is common to the two triangles AEB and GEB. 
 
8G 
 
 ELEMENTARY DYNAMICS. 
 
 Also, since AG was drawn parallel to the surface, and BC per- 
 pendicular to it, therefore the angle AEB = GEB. 
 
 Therefore the triangles AEB and GEB have two sides and one 
 angle equal. 
 
 Therefore these triangles are equal. 
 
 Therefore the angle GBE is equal to the angle EBA. 
 
 Q. E. D. 
 
 APPENDIX B. 
 
 213. Problem. AC and BC, fig. 35, are two of the walls of a 
 
 fives-court : the ball is at D, 
 and the player wishes to 
 strike it so that, after rebound- 
 ing from BC, it may hit the 
 point A. In what direction 
 must he hit it ? 
 
 From D let fall a perpen- 
 dicular DE upon BC, and 
 produce it to F, making 
 EF = DE. 
 
 Join FA, cutting BC in G, 
 and join DG. 
 
 Then DG is the direction in 
 which the ball must be hit. 
 
 Proof. In the two triangles 
 
 DEG, FEG, the side DE = EF, and GE is common to the two 
 triangles. 
 
 Also the angle DEG = FEG, since both are right angles, DF 
 having been drawn perpendicular to BC. 
 Therefore these triangles are equal. 
 Therefore the angle DGE = FGE. 
 
 But FGE is equal to the vertically opposite angle CGA ; there- 
 fore the angle DGE = CGA. 
 
 Through G draw GH perpendicular to BC. 
 
 Then the angles CGH, HGE are equal, being right angles, and 
 the parts of them, CGA, EGD have been proved equal ; therefore 
 the remaining angles, HGA, HGD, are equal. 
 
 And these are the angles of incidence and reflexion respectively. 
 Therefore the ball, if hit in the direction DG, will rebound in the 
 direction GA. 
 
 Fig. 35. 
 
87 
 
 CHAPTER IV. 
 ACCELERATION. 
 
 SECTION 1. GENERAL PRINCIPLES. 
 
 214. Acceleration means the quickening of speed caused by tJie 
 continued action of a force. 
 
 Hitherto the action of forces has, for the most part, been con- 
 sidered as if they acted for an instant only, like the kick given 
 to a football, or the blow of a hammer on a nail. But, strictly 
 speaking, forces seldom do this : their action lasts for an 
 appreciable time, as for instance, the force exerted by gunpowder 
 on a bullet all the time that it is in the barrel of the gun, of an 
 engine in moving a train, of gravitation on a falling stone. 
 Clearly, the longer a force acts on a body, the greater will be 
 the velocity which it imparts to the body ; and a little reflection 
 will show that the velocity must be increased in exact proportion 
 to the time during which the force acts. Thus, if a force of 
 1 poundal (par. 158, p. 66) acts continuously for several seconds 
 on a mass of 1 pound, at the end of 
 
 1 second, the velocity produced will be 1 foot per second. 
 
 2 seconds, ,, n 2 feet 
 
 30 
 II n n II O II n 
 
 &c. &c. 
 
 So that, in order to find out the speed with which a body is 
 moving under the action of a uniform unimpeded force, we have 
 only to multiply the velocity produced in the first second by the 
 number of seconds during which the force has acted. 
 
 215. Suppose, for example, that a train is moving out of a 
 station, and that 1 second after the start it is found to be moving 
 at the rate of 1| feet per second. Then (if it were not for such 
 impediments as the resistance of the air, &c.), at the end of 
 
 10 seconds, it will be moving (10 x 1^ feet = ) 15 feet per second ; 
 20 n n (20 x Hfeet = ) 30 , 
 
 and so on.* 
 
 * Practically, of course, in the case of a train, the acceleration only goes on up 
 to a certain point, when friction of various kinds just balances the force of the 
 engine. Then there can be no more increase of speed, the whole of the force being 
 expended in overcoming friction, and the continuance of the speed being due to the 
 inertia of the train. 
 
88 ELEMENTARY DYNAMICS. 
 
 216. The same is true when a force is applied to a moving 
 body in the opposite direction to that in which it is moving ; for 
 instance, when a brake is applied to a moving train. If the force 
 applied through the brake is such as would cause a velocity of 
 5 feet per second, then 5 feet per second will be subtracted from 
 the speed of the train during every second that the brake acts, 
 until the train comes to a stop. 
 
 SECTION 2. ACTION OF GRAVITATION. 
 
 217. Gravitation is perhaps the best example of a practically 
 uniform accelerating or retarding force at the earth's surface. As 
 already partly explained (p. 30), it makes everything tend to 
 move or ' fall ' towards the earth's centre, and it is constantly 
 acting on them while they are doing so. We may now consider 
 more fully what are the results of this continuous accelerating 
 force. 
 
 218. We have to find out two things : 
 
 1. What is the effect of (a) the material, (6) the mass of a 
 body upon the rate at which it falls ? 
 
 2. What is the velocity produced in a falling body by gravita- 
 tion acting on it for 1 second ? 
 
 Here are questions which cannot be answered by any amount 
 of mere reasoning. We must try experiments, many and varied 
 in character, closely observe the results, and then draw the 
 proper logical inferences from these results. We may accept 
 such deductions as solid scientific truths (until, at any rate, they 
 are disproved by more reliable experiments), whether they 
 agree with our preconceived notions or not. 
 
 219. The experiments may be of the following character, 
 (a) Pieces of different materials such as lead, ivory, glass, 
 india-rubber, a fives-ball, &c., but of the same weight, may be 
 allowed to fall, starting simultaneously, from a height above 
 the floor ; and the order in which they reach the floor may be 
 noted. 
 
 [A small box, the bottom of which is hinged like a trap-door, is 
 useful for this experiment. The trap-door should be held by a catch ; 
 two or three of the substances mentioned should be put into the 
 box. The latter should then be drawn up to a height above the 
 
ACCELERATION. 
 
 floor, a box or tray filled with sand being placed on the floor to 
 receive the falling weights, and the catch may be released by a 
 string or a simple electro-magnetic arrangement.] 
 
 220. It will be found, as the result of many trials, that all the 
 bodies, whatever they may be made of, if let fall simultaneously 
 from a height, reach the floor at the same moment.* And 
 similar experiments made with a large number of substances 
 show that the material of which a body is made has no influence 
 whatever upon the rate of its fall. 
 
 221. (&) Pieces of the same material, such as lead, but of very 
 different masses (for example, a large bullet and a small one ; 
 or a large stone and a piece broken off from it), may be dropped 
 from the box in the manner already described. It will be found 
 that (allowing for the resistance of the air, as explained in the 
 note below) these very different masses of stone or lead reach the 
 floor at the same moment. 
 
 222. By such experiments it has been established that the 
 mass of a body has absolutely no influence upon the rate of its 
 fall. This may seem at first sight rather surprising ; especially 
 since, if the pieces of lead are lifted, the large piece certainly 
 presses against the hand with greater force than the small piece, 
 and it would appear, therefore, that the former ought to fall 
 quicker. But the result is easily explained if we consider that 
 gravitation has a great deal more work to do in moving the 
 large mass than in moving the small one. Suppose that one 
 body, A, has 10 times as much matter in it as another, B. Then 
 the action of gravitation on A will be 10 times as great as its 
 action on B, but it has 10 times the amount of matter to move, 
 so that it cannot move A any quicker than B.f Thus we may 
 state generally that, the greater the mass of a body is, the greater 
 is the power of gravitation upon it, but the greater also is the 
 
 * That is, if allowance is made for the fact that, owing to their unequal size, the 
 resistance of the air retards the larger bodies more than the smaller ones ; and thus 
 a piece of wood will in its fall lag a little behind a piece of lead of the same weight. 
 In a vacuum the statement is absolutely true. 
 
 t As a rough illustration, suppose that a truck drawn by one horse is coupled to 
 another truck of the same weight, also drawn by one horse. Then, though two 
 horses are employed, the trucks will not move any quicker than either separately, 
 because there is twice the weight to move. 
 
90 ELEMENTARY DYNAMICS. 
 
 work to be done in moving it, so that the body does not fall any 
 quicker. 
 
 223. The next point to be considered is What is the accelera- 
 tion produced by gravitation in a given time, such as 1 second ? 
 The following experiment may be made in order to obtain an 
 answer to this question. Let the box used in the previous 
 experiments, containing a weight such as a brass or ivory ball 
 (the actual weight and material have been shown to have no 
 influence on the result), be drawn up to a height of 16 feet 
 1 inch above the surface of the sand in the tray below. Let some 
 arrangement for marking seconds, such as a loud-ticking pendu- 
 lum,* be set working, and let the weight in the box be made to 
 commence its fall at one tick of the pendulum. Then it will be 
 found that the weight reaches the sand just as the second tick 
 sounds. The experiment may be repeated several times to make 
 
 sure of the result. 
 
 224. It will thus be proved that a body, when allowed to fall 
 freely under the action of gravitation, passes through 16 feet 
 1 inch in the first second of its fall. 
 
 [In what follows, the distance through which a body falls in 
 1 second will be considered for simplicity as 16 feet.] 
 
 225. From this fact we can deduce the velocity with which 
 it was moving at the end of the second that is, the acceleration 
 produced by gravitation in 1 second as follows : The body 
 must, at the end of the second, be moving quicker than 16 feet 
 per second, because it started with no velocity at all, and yet it 
 got through 16 feet in the time. (For example, a runner will do 
 100 yards in 10 seconds, but at the end of the time he must be 
 running more than 10 yards per second, since he started from 
 rest and moved slowly at first.) In fact, since the acceleration 
 is uniform, at the end of the second it must be moving as much 
 faster than 16 feet per second as it was moving slower than that 
 rate at the beginning of it, thus : 
 
 * An ordinary metronome may be made to answer; but the best plan is to place 
 the electro-magnet which releases the trap-door in the same circuit as a single-stroke 
 electric bell, and make a seconds pendulum complete the circuit in the middle of its 
 swing. 
 
ACCELERATION. 91 
 
 Velocity at the beginning of the second ft. per sec. 
 ii middle n n = 1G ft. n 
 
 end ,, = 32 ft. 
 
 Since, then, at the end of the second the body must be moving 
 at the rate of 32 feet per second ; and since gravitation (for 
 moderately small distances, at any rate) acts uniformly and 
 continuously upon bodies, we may take it as proved that the 
 force of gravitation is one which, at the earth's surface, causes a 
 velocity of 32 feet per second during every second that it acts 
 that is, that the acceleration produced by gravitation is 32 feet 
 per second.* 
 
 226. We have now ascertained the space fallen through, and 
 the velocity attained by a body at the end of the first second of 
 its fall. A little reflection will enable us to see what space a 
 body will fall through, and what velocity it will have, in 
 succeeding seconds.! 
 
 227. The velocity acquired by the body at the end of the first 
 second is 32 feet per second, and if gravitation were to cease at 
 that moment, the body would (by the first Law of Motion) move 
 through 32 feet in the next second. But gravitation goes on 
 acting upon the body, and thus will make it fall another 16 feet 
 in addition to the 32 feet that is, through (32 + 16 =) 48 feet in 
 all ; and also will add another 32 feet per second to its velocity, 
 so that at the end of the second second it will be moving at the 
 rate of (32 -f 32 =) 64 feet per second. Similarly, in the third 
 second it will fall through, not merely 64 feet due to its velocity 
 at the beginning of the second, but (64 + 16=) 80 feet; 
 
 * The exact value in London is 32 feet 2-3 inches. It varies, as already 
 explained ( par. 84, p. 34 ), with the latitude. The exact acceleration produced by 
 gravitation at the earth's surface in a few latitudes is given below : 
 
 Latitude. Acceleration. 
 
 ft. in. 
 
 o { Equator ) 32 i 
 
 45 (Bordeaux) 32 2 
 
 5 1 30' ( London ) 32 2-3 
 
 60 (Stockholm) 32 2-6 
 
 90 ( Pole ) *. 32 3 
 
 t The spaces and velocities are here so great, that direct experiments on the 
 subject would be difficult to make. But a very ingenious apparatus, called 
 Attwood's machine, has been invented, in which the action of gravitation is so far 
 diluted (as it were) as to bring it within reasonable bounds. 
 
1 second, 
 
 32 ft. per sec. 
 
 16 feet 
 
 2 seconds, 
 
 64 
 
 48 H 
 
 3 
 
 96 ,, 
 
 80 
 
 4 ., 
 
 128 
 
 112 
 
 5 
 
 160 
 
 144 
 
 92 ELEMENTARY DYNAMICS. 
 
 and it will at the end of the second have a velocity of 
 (64 + 32 =) 96 feet per second. By the same course of reason- 
 ing, we can calculate its progress during succeeding seconds. 
 
 228. In order to find the total space fallen through in a given 
 number of seconds, we have only to add together the spaces 
 fallen through in each second. Thus the total distance through 
 which a stone falls in two seconds is 16 feet + 48 feet, or 
 64 feet. 
 
 The following table will show the results already arrived at : 
 
 Time of fall. Velocity at end of Space fallen through Total space fallen 
 
 time. in the second. through. 
 
 16 feet. 
 
 64 
 144 
 256 
 400 .. 
 
 229. From the above table it is easy to see that there is a 
 very simple relation between the time of fall and the total space 
 passed through in the time. Thus : 
 
 In 1 second the space passed through is 16 feet. 
 In 2 seconds ,. 64 ,. = 16 x 4 (or 22). 
 
 In 3 M it H 144 I. = 16 x 9(or3 a ). 
 
 In 4 it ii ii 256 H = 16 x 16 (or4 2 ). 
 
 That is, the total space fallen through increases with the square 
 of the time of fall. Thus we get the following simple rule for 
 finding how far a body will fall in a given time : 
 
 Take the square of the number of seconds, and multiply 16 feet 
 by it ; the product is the distance fallen through. 
 
 For example : A bag of sand was let fall from a balloon, and 
 reached the ground in 8 seconds. The square of 8 is 64 ; and 
 16 x 64 = 1024 feet. Hence the distance of the balloon from 
 the earth was 1024 feet. 
 
 230. The above example illustrates a practical application of 
 the laws of gravitation which have been explained namely, a 
 method for finding approximately the height of a tower or cliff, 
 or the depth of a well. "We have only to let a stone drop, and 
 observe accurately how many seconds elapse before it touches 
 
ACCELERATION. 93 
 
 the ground (or the water in a well), and then apply the above 
 rule. Thus, if a stone dropped from the top of a cliff took 
 5 seconds to reach the base, then 5 2 = 25, and 16 x 25 = 400 ; 
 thus the cliff was 400 feet high. It would really be rather less 
 than this, since the resistance of the air checks the speed of the 
 falling stone, so that in the 5 seconds it really fell less than 
 400 feet. 
 
 231. The acceleration produced by gravitation explains why 
 hailstones do so much damage although they are so small. 
 They have fallen from a great height, and thus have acquired a 
 very high velocity ; hence their momentum is considerable in 
 spite of their small mass. We also see why, when water falls 
 from a height, as in a waterfall, it breaks into drops before it 
 has gone far. The lower part of the descending mass of water 
 has a much higher speed .than the upper part, because it has 
 been falling longer ; hence it breaks off from the rest, and 
 separates into drops, which separate more and more as they 
 descend. When a viscid liquid, like treacle, is poured out from 
 a height, the bulky sluggish stream becomes gradually rapid 
 
 "and smaller, and is at last reduced to a thread ; but wherever a 
 vessel is held into the stream, it fills equally fast. 
 
 232. We may next consider what happens when a body such 
 as a cricket-ball is thrown straight up into the air. At the 
 moment it leaves the hand it has a certain velocity, and if 
 nothing occurred to stop it, it would (according to the first Law 
 of Motion) go on rising continually with undiminished speed. 
 But gravitation acts upon it quite irrespectively of any motion 
 it may have from the action of other forces (according to the 
 second Law of Motion ), and, by pulling it downwards, gradually 
 lessens its upward motion until it comes to rest. But it does 
 not stop there ; gravitation is still acting on it, and it begins to 
 fall with accelerated motion in the usual way. Now, it is 
 pretty easy to see that, since gravitation acts upon it through 
 the same space during its fall as during its rise, the force will 
 produce in the ball the same velocity, by the time it reaches 
 its starting-point, as it had when it originally started. For 
 example, suppose that the action of gravitation stopped its 
 upward motion in 3 seconds. Then since we know (see table 
 in par. 228) that gravitation is a force capable of producing in 
 
 G 
 
94 ELEMENTARY DYNAMICS. 
 
 3 seconds a velocity of 96 feet per second, and since the force 
 required to destroy a given motion must be equal to the force 
 which has produced that motion, therefore the ball, when it left 
 the hand, must have been moving at the rate of 96 feet per 
 second, and when it returns to its starting-point, will have the 
 same velocity. 
 
 233. Hence it follows : 
 
 (1) That a body thrown or shot upwards takes just as long to 
 fall as it does to rise. 
 
 (2) That the height to which it rises is equal to the space 
 through which it would fall by gravitation in the observed time 
 of its rise. Thus, if the body is 3 seconds in the air before it 
 stops rising, it must (see table, p. 92) have risen to a height of 
 144 feet. 
 
 (3) That the force with which it strikes any obstacle placed 
 at the same level as its starting-point (such as a hand held out 
 to catch it), is equal to the force which was originally used to 
 propel it upwards. 
 
 234. We thus learn : 
 
 (1) How to calculate approximately the height to which a 
 ball or arrow, projected upwards, has ascended. We have only 
 to observe the number of seconds which elapse between the 
 moment of its start and the moment it returns to its starting- 
 point again. Half of this time will have been spent in falling, 
 and we can calculate, by the rule already given (p. 92), what 
 space it must have fallen through in the known time. This 
 space must, of course, be equal to the height to which it has 
 risen. Thus, supposing that an arrow shot upwards takes 12 
 seconds to return to the level of its starting-point. It will have 
 occupied 6 seconds in falling, and therefore must have passed 
 through (6 2 x 16 =) 576 feet. Hence it must have risen to a 
 height of 576 feet. 
 
 235. This has been practically applied to determine the height 
 to which stones shot upwards from volcanoes have ascended. In 
 a recent eruption of Vesuvius, rocks were projected out of the 
 crater, which were observed to be 10 seconds in the air before 
 falling into the crater again. These rocks must have been 
 5 seconds in falling, and hence they must have risen to a height 
 of (5 2 x 16 =) 400 feet. 
 
CENTRE OF GRAVITY. 
 
 95 
 
 (2) The reason why it is so dangerous to hit racquet-balls 
 away at random, and fire bullets up into the air. The force 
 with which such a ball hits any object in its fall is equal to the 
 force with which it was originally projected upwards (except- 
 ing, of course, the loss due to resistance of the air) ; and several 
 fatal accidents have happened from this cause. 
 
 CHAPTER Y. 
 
 CENTRE OF GRAVITY. 
 
 SECTION 1. GENERAL PRINCIPLES. 
 
 236. In examining the laws of falling bodies, we have simply 
 considered the earth, as a whole, to attract a body, as a whole, 
 towards itself. But every body is made up of a very large 
 number of molecules, and the force of gravitation acts between 
 separate molecule and every other one. In the case of 
 a solid, the molecules, being fast bound together by the force 
 of cohesion, must necessarily move all together, like well- 
 drilled soldiers, when they move 
 at all ; and we do not notice the 
 separate attractions of the individual 
 molecules. But in fact each mole- 
 cule of a falling stone is being pulled 
 sideways, right and left, as well as 
 downwards, by the earth's molecules, 
 as indicated in fig. 36. Now, on the 
 principle of the Composition of 
 Forces already explained (p. 68), all 
 these separate forces can be shown 
 to be equivalent to one resultant 
 force acting between a certain point 
 
 in the earth and a certain point in the stone ; and these 
 points are called the centres of gravity of the earth and the 
 stone respectively. 
 
 The centre of gravity of a body, then, may be defined as 
 tlie place where tlie resultant of all the attracting forces 
 
96 ELEMENTARY DYNAMICS. 
 
 between the separate molecules of the body is considered to 
 be applied.* 
 
 237. It follows from this that, if we want to support any 
 solid body that is, to counteract the effect of gravitation in 
 making it fall we need not put props under every part of 
 it (though the earth is attracting every part), but only one 
 prop at the centre of gravity, or else directly above or directly 
 below the centre of gravity ; because one force applied there 
 in the opposite direction will counteract the resultant of all 
 the earth's forces. Thus one man can carry a long ladder 011 
 one of his shoulders, and a ruler or a stick of uniform thick- 
 ness may be poised on the tip of one finger, if supported just at 
 its middle point, where (as will be shown, p. 106) the centre of 
 gravity lies. 
 
 SECTION 2. EQUILIBRIUM OF BODIES. 
 
 238. When anything rests on a support without showing 
 a tendency to move of its own accord, it is said to be 
 in equilibrium (Lat. cequus, equal; librare, to balance); because 
 the attraction of gravitation on any one part of the body, 
 tending to pull that part downwards, is balanced by an equal 
 attraction on some other part of the body on the opposite side 
 of the support, so that the body has no tendency to tumble 
 off its support in one direction more than another. 
 
 239. It is a matter of common -experience that most bodies 
 will rest much more steadily in some positions than in others : 
 a book, for instance, rests on a table much more steadily when 
 laid on its side than when standing on its edge ; an egg, when 
 placed on its side, will remain in that position, but we find 
 great difficulty in balancing it on either end. It is important, 
 then, to examine what are the true conditions of equilibrium, and 
 
 * It must be noted that the point we call the ' centre of gravity ' of a body is not 
 in all cases in the same position. The direction in which a stone falls at any 
 particular place on the earth's surface passes through a point in the earth which is 
 the 'centre of gravity' for bodies at that place : but the directions in which things 
 fall at other places do not, as a rule, pass through precisely the same point, although 
 the variation is small and may usually be neglected. In fact, comparatively few 
 bodies have (like spheres) a 'centre' of gravity (as defined in the text) which is 
 invariable in position under all conditions. 
 
Fig. 37. 
 
 3 LIB; -^^V 
 
 CENTRE OF 'GRAVITY/)? THf 
 
 !{ IINJVEBBITY 
 
 a few simple experiments of th|MolIawing kind may 
 to illustrate them. ^S 
 
 Take a flat circular piece of wood, aboucsixor eight inches in 
 diameter and half an inch thick,* and bore holes in different 
 parts of it, such as a, b, c, d, fig. 37 ; one of 
 these, a, being in the exact centre, which is 
 (as will be presently shown) the position of 
 the centre of gravity. Support a piece of 
 thick wire (a small pencil, or the thin end of 
 a penholder, will do) horizontally, and hang 
 the board on it, trying the various holes in 
 succession, and noting carefully under what 
 conditions the board remains in equilibrium, 
 and when it rests most steadily and least steadily on the 
 support. 
 
 240. It will be found : 
 
 (1) That the board will only remain in equilibrium that is, 
 without moving when the hand is taken away when The 
 support is either at the centre of gravity, or directly above the 
 centre of gravity, or else directly below the centre of gravity. 
 A plumb-line (fig. 4, p. 30) should be held close to the board in 
 order to define the exact positions called 'above' and 'below' 
 (see p. 31). 
 
 (2) That when the support is at the centre of gravity, the 
 board will rest in any position indifferently, and will require 
 very little force to move it from one position to another. 
 
 (3) That the board rests much more steadily when the point 
 of support is above the centre of gravity than when it is below 
 the centre of gravity ; so much so that, if the board is pushed 
 away from this position, it tends to come back thither of its 
 own accord ; whereas, if the support is below the centre of 
 gravity, there is some little difficulty in placing the board in 
 equilibrium at all, and a very slight touch makes it leave this 
 position and swing round until it finally settles in such a posi- 
 tion that the centre of gravity is as low as it can get. 
 
 241. These facts show that the steadiness of anything depends 
 upon where the support is placed with reference to the centre 
 
 A piece of very thick cardboard will answer, but not so well. 
 
98 ELEMENTARY DYNAMICS. 
 
 of gravity, and whether the centre of gravity is likely to be 
 raised or lowered when the body is put into some other posi- 
 tion. Thus we are able to distinguish three different kinds of 
 equilibrium, called stable, neutral, and unstable equilibrium 
 respectively. 
 
 (1) Stable equilibrium, 
 
 242. This is when the body rests steadily and requires some 
 force to move it ; and, when it is moved, it tries to get back to 
 its former position. 
 
 Examples of bodies in stable equilibrium are a table or chair, 
 a book lying flat on the table, a man standing on both feet. 
 
 In this kind of equilibrium the support is so placed that the 
 centre of gravity would be raised by altering the position of the 
 body. Thus, if the book be raised from its flat position, turning 
 on one of its edges as if on a hinge, it is evident that its centre 
 of gravity (which is nearly its middle point) will be raised 
 higher than it was before. 
 
 (2) Unstable equilibrium. 
 
 243. This is when the body is easily moved, and if moved a 
 little way, tends to go on moving farther from its original posi- 
 tion until the centre of gravity is as low as it can get. That is, 
 the body is ' top-heavy/ or tends to overturn. 
 
 As examples of this may be taken a chair balanced on one 
 leg, a top balanced on its point, a cricket-bat standing on its 
 handle, a man walking on stilts. 
 
 In this case, the support is so placed that the centre of gravity 
 would get lower when the body is moved from its position. If, 
 for instance, the cricket-bat resting on its handle was pushed 
 sideways, its centre of gravity (which is not far from the centre 
 of the blade) would get lower and lower until it lay flat on the 
 ground. 
 
 (3) Neutral equilibrium. 
 
 244. This is when the body is easily moved, but will rest in 
 any position indifferently. 
 
 For example, a carriage wheel whether supported at its centre 
 or at its edge, a roller, a cricket-ball, resting on a horizontal 
 surface. 
 
 In this case the support is so placed that the centre of gravity 
 
CENTRE OF GRAVITY. 
 
 is not raised or lowered by altering the position of the body. 
 In a wheel, for instance, the centre of gravity of which is at its 
 centre, this point is not raised or lowered when it turns on an 
 axle ; and if it rolls along a horizontal road, the centre of gravity 
 remains at the same height above the road (since all radii of a 
 circle are equal), and the point where the wheel is supported by 
 the road is always vertically below the centre of gravity. 
 
 245. Many bodies can, 
 from their shape, be sup- 
 ported in such positions 
 as to illustrate all the 
 three kinds of equilibrium. 
 Thus, a cone, fig. 38, when 
 resting on its base a, is in 
 stable equilibrium, since, 
 if it is tilted on any part of its edge in order to push it over, the 
 
 Fig. 38. 
 
 Fig. 39. 
 
 centre of gravity is raised, passing along the curve g h, fig. 39, and 
 this is resisted at first by the whole effect of the weight of the 
 body. When placed on its point, as shown in b, it is in unstable 
 equilibrium, since the slightest lateral push causes the centre of 
 gravity to move along the descending curve shown in the figure, 
 and thus a portion of the weight of the body aids in pulling it 
 over. When the cone is laid on its side, as shown in c, it is in 
 neutral equilibrium, and will remain in any position on a 
 horizontal surface, since the centre of gravity cannot be made 
 higher or lower by moving the body over the surface, as 
 explained with reference to the carriage wheel. An egg, when 
 resting on its side, is in stable equilibrium or in neutral equili- 
 brium, according to the direction in which it is moved ; when 
 resting on its end, it is in unstable equilibrium. The reason will 
 be sufficiently obvious from what has been already said. 
 
100 
 
 ELEMENTARY DYNAMICS. 
 
 General Laws of Equilibrium. 
 
 Law I. A body is in stable equilibrium as long as a perpen- 
 dicular line drawn through the centre of gravity falls some way 
 within the base on which it rests. 
 
 246. The following experiment will illustrate the truth of 
 this law. Take a block of wood about the size and shape of a 
 brick (or a brick itself will do), and drive a small nail into the 
 exact centre of one of its sides (which, as will be shortly proved, 
 defines the position of the centre of gravity of the block). Place 
 the block on one end at the edge of a table, and hang from the 
 nail a small plumb-line, long enough to reach a little way below 
 the edge of the table, fig. 40. Apply pressure near the top of 
 
 Fig. 40. 
 
 the block so as to tilt it a little way on one of its edges. Con- 
 siderable force will be required at first to move it, and it will 
 come back again to its former steady position when the pressure 
 is removed. It is, in fact, in stable equilibrium. Now tilt it 
 farther by degrees, noting the position of the plumb-line with 
 regard to the edge on which the block is being tilted. The line 
 will gradually approach this edge ; but so long as it is well 
 within the edge, the block will show stable equilibrium, although 
 less and less force will be required to move it. As soon as it is 
 tilted so far that the line passes in the least degree beyond the 
 edge of the base, the block will go on moving in the same 
 direction, and will topple over. The same result will be 
 obtained if the block is similarly tilted on the other edge. 
 
 247. The reason of the law is easily seen. As long as a 
 
CENTRE OF GRAVITY. 101 
 
 perpendicular line (defined by the plumb-line) falls within the 
 base on which a body rests, some part of the support is verti- 
 cally below the centre of gravity : and this, as has been shown 
 in par. 237, p. 96, is all that is necessary to keep the whole mass 
 up. Moreover, if we observe the curve described by the centre 
 of gravity as the body is tilted on one edge (as if on a hinge or 
 axis), it will be seen to be an ascending curve, as in the case of 
 the cone, fig. 39, a. Hence the centre of gravity must be raised 
 when the body is thus tilted, and force enough to overcome 
 nearly its whole weight must be at first applied. 
 
 248. Thus we see that the farther a perpendicular drawn 
 through the centre of gravity lies within the base, the more 
 stable is the equilibrium of the body. A pyramidal or conical 
 building is the steadiest and firmest of structures, because 
 the base is so large and the centre of gravity so low that the 
 latter would have to be raised almost vertically upwards by any 
 force tending to overthrow the structure, and a considerable 
 change of position would be required before a perpendicular 
 drawn through the centre of gravity would reach the edge of 
 the base. The legs of a table or a tripod are often spread out- 
 wards, so as to increase the size of the base. A wall, tower, or 
 chimney stands steadiest when quite upright, and masons take 
 great pains to insure perfect uprightness of the building by 
 constant use of the plumb-line. 
 
 249. A tower may, however, lean some way from the perpen- 
 dicular, and overhang its base, without actually falling, although 
 its stability is lessened, as already shown. Such an inclination 
 is sometimes produced by the foundations giving way on one 
 side, and the precise extent to which it may proceed without 
 rendering the building actually unsafe may be calculated on the 
 principles already explained. The most famous example is the 
 Leaning Tower of Pisa in North Italy (fig. 41, next page), which 
 is 180 feet high and 52 feet in diameter. It leans so far that a 
 plumb-line let down from the top touches the ground 14 feet 
 from the base of the building ; but it might, as can easily be 
 proved, lean more than twice as much without actually toppling 
 over. 
 
 250. A man stands firmly when resting upright on both feet, 
 and his steadiness is increased by placing the feet wide apart as 
 
102 ELEMENTARY DYNAMICS. 
 
 sailors do. In resting on one foot, the centre of gravity of the 
 body must be thrown over that foot ; hence, in walking, the 
 body is almost unconsciously swayed slightly from side to side. 
 The fact is forced into notice when two men walk close together, 
 but do not keep step ; one putting forward his right foot and the 
 
 Fig. 41. Leaning Tower of Pisa. 
 
 other his left foot at the same moment. They then jostle each 
 other, one leaning to the right and the other to the left at the 
 same time. 
 
 251. In dancing, walking on stilts, and skating, we have 
 examples of still more refined series of experiments on keeping 
 the centre of gravity over the base, or ' preserving one's balance,' 
 as it is commonly called. The narrower the base, such as the 
 end of a stilt or the edge of a skate, the less is the lateral move- 
 ment required to throw the perpendicular drawn through the 
 centre of gravity outside it that is, the more unstable is the 
 equilibrium.* A performer on the tight rope holds a long pole 
 
 * In skating, the principles of inertia and centrifugal tendency ( Chapter III., 
 Sect. 2) are called in to aid us. Forward progress on the edge of the skate is 
 made in a series of curves, the skater not caring always to keep his centre of 
 
CENTRE OF GRAVITY. 103 
 
 horizontally, and when in his movements the perpendicular 
 through his centre of gravity falls outside the rope, he brings it 
 back again by quickly shifting the pole a little to the opposite 
 side. 
 
 Law II. The lower the centre of gravity is, the more stable is 
 the equilibrium of the body. 
 
 252. The truth of this law may be shown by the following 
 method. Take a flat wooden rod, about 12 inches long, 2 inches 
 broad, and ^ inch thick, fitted with a leaden weight which is 
 capable of sliding along it, but clasps it so tightly as to remain 
 in any required position, fig. 42. Bore a hole through 
 the centre of the piece of wood, and hang it on a 
 horizontal wire or small peg. Adjust the sliding 
 weight until the rod will remain indifferently in any 
 position. Then it is in neutral equilibrium, and we / \ 
 know that the centre of gravity must be at the point 
 of support. Slide the weight a short way along the 
 rod ; the latter will now take a position of stable 
 equilibrium, in which the centre of gravity is below 
 the support ; but a very slight pressure will be suffi- 
 
 cient to move it on one side, and it will swing back- Fig. 42. 
 wards and forwards slowly until it regains its first 
 position. Next lower the weight, and therefore also the centre 
 of gravity, still farther : the rod will now require a greater 
 pressure to move it, and after a few quick swings it will settle 
 decidedly into its original position. If, finally, the weight is 
 put quite close to the end of the rod, the equilibrium will be 
 found to be still more stable. 
 
 253. The reason of the law is plain : for the lower the centre 
 of gravity is already, the more likely it is to be raised by any 
 change in the position of the body. Moreover, if the body is 
 supported on a broad base, and the centre of gravity is low (as 
 in fig. 39 a), any change in the position of the body effected by 
 tilting it on one edge of its base must raise the centre of gravity 
 nearly vertically upwards, and this is resisted by the whole 
 
 gravity directly over his skate, but preserving himself from falling by balancing 
 his centrifugal tendency against gravitation. The theory of the use of the 
 bicycle is of a very similar character. 
 
104 
 
 ELEMENTARY DYNAMICS. 
 
 Fig. 43. 
 
 weight of the body. Even when the base is narrow, it is easy 
 to see that the centre of gravity, when it is low down, has to be 
 moved through a greater distance and along a more sharply- 
 ascending curve before a perpendicular through it falls outside 
 the base, than when it is high up in the body. 
 
 254. A cart loaded with hay is, when tilted by one wheel 
 
 passing over a heap of stones, as in 
 fig. 43, much more likely to upset 
 than when loaded with the same 
 weight of stones or iron. For the 
 load of stones would only fill the 
 cart up to the top of the side- 
 boards, so that its centre of gravity 
 would be near C, and a perpen- 
 dicular drawn through it would 
 still fall a little within the wheel- 
 base ; but the load of hay would 
 be piled up much higher, and the 
 
 centre of gravity would be near C', the perpendicular through 
 which would fall outside the wheel. Similarly, a coach with 
 luggage packed low is much less likely to be overturned than 
 when passengers and luggage are on the top. 
 
 255. The safety of a ship depends on many tons of 'ballast' 
 being put as low as possible in the hold, so that when the ship 
 rolls in a heavy sea, the centre of gravity may always be raised, 
 and then in its descent it will tend to bring the whole mass 
 back to an upright position. A boat is more liable to upset if 
 passengers stand up in it, because the centre of gravity is raised 
 so high that the whole may be put into unstable equilibrium. 
 The same principle explains why there is so much risk in 
 tossing oars in a light boat, and why it is so difficult at first to 
 manage an outrigged boat, although when the oars are tied into 
 the rowlocks a much broader base is gained while they rest on 
 or in the water. 
 
 256. A cone may be made to rest steadily on its point by 
 fixing weights to it below, as shown in fig. 44, because the centre 
 of gravity of the whole mass (cone and weights) can be thus 
 brought down below the point of support. A coin may in a 
 similar way be balanced on the point of a needle, by being 
 
CENTRE OF GRAVITY. 105 
 
 affixed to a cork on each side of which is stuck a fork or a 
 
 Fig. 44. 
 
 Fig. 45. 
 
 pocket-knife sloping downwards. Many toys are constructed 
 on a similar principle, one of which is shown in fig. 45. 
 
 General illustrations of the Laws of Equilibrium. 
 
 257. Only a few of these can be mentioned here ; many others 
 will suggest themselves to those who think over the subject. 
 
 (1) The necessity for balancing the parts of quickly-moving 
 machinery. If, for instance, a wheel is heavier in one part 
 than another, its centre of gravity will not coincide with the 
 centre on which it turns, and will therefore be swaying from 
 side to side at each revolution ; thus from the inertia of the 
 mass and its reaction against the force which swings it, the 
 whole framework is made to vibrate and strained. 
 
 (2) The method of setting a swing in motion. This depends 
 upon quick changes in position, so made as to shift the centre 
 of gravity from one side of the vertical position of the swing 
 to the other ; this sets up a swinging motion which is increased 
 by properly timed movements of the same kind. 
 
 (3) The reason why an umbrella, or a cricket-bat, is more 
 
106 ELEMENTARY DYNAMICS. 
 
 easily balanced on its heaviest end. The centre of gravity in 
 this case is comparatively low down, and only moves through a 
 small distance round the point of support while the whole 
 object sways through a considerable arc, so that the supporting 
 finger is more easily by a slight movement kept exactly below 
 the centre of gravity. 
 
 258. It will be useful to think over and explain, on the 
 principles above given, such simple problems as the reasons why 
 it would be unsafe to add much to the height of, or to hang a 
 set of heavy bells near the top of the leaning tower of Pisa 
 why a candlestick is more liable to upset when a long candle is 
 in it than when the candle has burned down to the socket why 
 a man with a load on his back leans forward ; if he is carrying 
 a box in one hand he leans to the opposite side why a man 
 standing against a wall cannot stoop forward to pick up any- 
 thing while both his heels are against the wall why if his 
 side touches the wall he cannot lift the outside foot from the 
 ground without falling why he leans forward in rising from 
 a chair. 
 
 SECTION 3. METHODS OF FINDING THE CENTRE OF GRAVITY. 
 
 259. In cases where the body is uniform in structure through- 
 out, and symmetrical in shape, the centre of gravity can be 
 found by simple measurement ; since all that we have to do is 
 to find its exact middle point round which all the molecules are 
 regularly arranged, so that there are just as many of them on 
 one side of this point as on the other. 
 
 (1) To find the centre of gravity of a very thin straight rod, 
 which is uniform that is, alike in every part. 
 
 Rule. Find the exact middle point of its length : the centre of 
 gravity will be at this point. 
 
 This may be proved as follows : 
 
 AC x P B The resultant of the forces exerted 
 
 |J by gravitation on the two end- 
 molecules A and B, fig. 46, must 
 Fig. 46. be midway between them that 
 
 is, at X. The resultant of the 
 forces on the next two, C and D, must be also midway between 
 
CENTRE OF GRAVITY. 107 
 
 them that is, at X ; and so on for the rest. Therefore the 
 resultant of all the attractions will be at the middle point, X, of 
 the rod ; and this will be the centre of gravity. 
 
 260. To illustrate this, a thin straight piece of steel wire, 
 about 16 or 18 inches long, may be taken, and its exact middle 
 point found by measurement, and marked. If it is hung up by 
 a piece of string tied to it just at this point, it will be found to 
 balance horizontally ; which could only occur if the centre of 
 gravity was just where the support is.* 
 
 (2) To find the centre of gravity of a thin uniform plate, 
 shaped like a parallelogram. 
 
 Eule. Find the middle point of its length, and through this 
 point draw a line across the plate parallel to the ends. Find also 
 the middle point of its breadth, and through this point draw a line 
 along the plate parallel to its sides. 
 
 Then the point where these lines cross will be the centre of 
 gravity of the plate. 
 
 Proof of this : The plate may be regarded as made up of a 
 number of thin rods side by side, such as AB, CD, &c., fig. 47 ; 
 and the centre of gravity of each of these will be in the middle 
 of its length. Therefore the centre of gravity of the whole will 
 be somewhere along the line WX. 
 
 Similarly, the plate may be considered to be made up of a 
 number of rods EF, GH, &c., fig. 48 ; and the centre of gravity 
 
 E G 
 
 X F H 
 
 Fig. 47. Fig. 48. 
 
 of each of these will be in the middle of its length ; so that the 
 centre of gravity of the whole must be somewhere along the line 
 YZ. 
 
 And since it has already been proved to be in the line WX, it 
 must be at the point where these two lines cross. 
 
 261. To prove this experimentally, take a piece of thick card- 
 
 * Strictly speaking, it should balance in any position ; but the stiffness of the 
 string prevents this. 
 
108 ELEMENTARY DYNAMICS. 
 
 board, about 12 inches long, and 2 inches broad, and draw on it the 
 lines as above directed : bore a hole (best with a sharp leather- 
 punch) exactly at the point where the lines intersect, pass a 
 piece of string through the hole, and bring it up on each side of 
 the cardboard, so as to suspend the latter in a wide loop. If the 
 work has been carefully done, the cardboard will hang in neutral 
 equilibrium, showing that the centre of gravity is where the 
 support is. Also the cardboard may be hung by a piece of string 
 passed through the hole and knotted below : it will be found to 
 hang horizontally. 
 
 262. It should be observed that in the above cases what has 
 really been found is the centre of gravity of the surface-layer. 
 If the rod or plate is very thin, this will be nearly the true 
 centre of gravity of the whole mass : but if it has a sensible 
 thickness, the centre of gravity will be midway between the 
 centres of gravity of its two surfaces. 
 
 263. On the same principle namely, by regarding the body as 
 made up of a number of thin rods the centre of gravity of any 
 regularly-shaped body may be proved to lie at the exact centre 
 of its figure. For example, the position of the centre of gravity 
 of a round disc, such as a wheel ; of a cylinder, such as a round 
 ruler ; and of a sphere, such as a cricket- ball, is known exactly ; 
 although we cannot always practically get at it. 
 
 264. The position of the centre of gravity of a triangular- 
 
 shaped plate is not quite so 
 obvious, but it may readily be 
 found by the following rule : 
 Bisect any two of the sides of the 
 triangle, and draw lines from the 
 points thus found, D, E, fig. 49, 
 to the opposite angles at A and 
 C. The centre of gravity lies 
 at the point where these lines cross, F. 
 
 The truth of this rule depends on the same principle as that 
 above explained : the small rods being supposed to diminish 
 gradually in length up to a point of the triangle. It may be 
 useful to remember that (as can be easily proved from Euclid) 
 the centre of gravity of a triangle is one-third of the way up 
 from the base to the apex. 
 
CENTRE OF GRAVITY. 109 
 
 Method of finding the Centre of Gravity by Experiment. 
 
 265. This method is applicable to all bodies, even though 
 their shape is irregular and their density unequal in different 
 parts. The principle of it is this that when anything, 
 supported at one point, is in stable equilibrium, the centre of 
 gravity must be somewhere along a perpendicular line drawn 
 through, or let fall from, the point of support (as has been 
 already shown in par. 246, p. 100). 
 
 266. The experiment is performed thus : 
 
 (1) Hang the thing up by a point near the edge, and let it take 
 up a position of stable equilibrium. 
 
 (2) Hang a plumb-line from the same point, and mark the 
 direction of the perpendicular line on the body. Then the 
 centre of gravity must be somewhere along this line. 
 
 (3) Hang the object up by another point, and mark the 
 perpendicular line through this point, in the same way as before. 
 Then the centre of gravity must be in this line, as well as the 
 first line. 
 
 Therefore it must be where the two lines cross. 
 
 267. In illustration of the method, take an irregular piece of 
 cardboard, fig. 50, bore a hole near the edge at any part, as at A 
 (the hole must not be so large as shown 
 
 in the figure), and hang it on a 
 horizontal pin. Hang a heavy plumb- 
 line from the same pin, and mark with 
 a pencil the lowest point, B, where the 
 line is over the cardboard (the two 
 should hang just clear of each other); 
 then remove the plumb-line, and draw 
 a straight line from the centre of the 
 hole to the point just marked. Bore a 
 hole in another part of the cardboard, 
 as at C, and mark the direction of the 
 perpendicular in the same way. To insure accuracy, it will be 
 as well to bore a third hole, as at D, and mark the perpendicular 
 through it as before. If proper care has been taken, all the 
 three perpendiculars should cross in the same point ; and the 
 centre of gravity may be proved to lie at this point by hanging 
 
 H 
 
110 ELEMENTARY DYNAMICS. 
 
 the card from this point by a string, and observing that it is 
 in a condition of neutral equilibrium. 
 
 268. The centre of gravity of many bodies does not lie in the 
 
 substance of the 
 body itself, but 
 in the space out- 
 side its surface. 
 Thus the centre 
 of gravity of a 
 chair, as deter- 
 mined by the 
 method above 
 described, see 
 fig. 51, is in the 
 space below the 
 seat ; and the 
 Fig. 51. centre of gravity 
 
 of a ring is in 
 the centre of the space \vithin the circle of the ring. 
 
 APPENDIX. 
 Centre of Percussion, or of Inertia. 
 
 269. Although, in considering the point called the centre of 
 gravity, we have hitherto referred to the force of gravitation only, 
 yet it must be observed that the same point is the point of applica- 
 tion of the resultant of any set of parallel forces acting on the body, 
 or of the force exerted by the body itself when its molecules are all 
 moving in parallel lines at the same rate, as one mass. Thus the 
 force with which a cricket-ball strikes the bat acts as if it was all 
 collected at the centre of gravity of the ball But it must not be 
 supposed that the centre of gravity of the bat is the proper place to 
 strike the ball ; because, as the bat is swung in the hand, the 
 different parts of it are not moving at the same rate, those farther 
 from the hand moving quicker and therefore having more momentum 
 than those nearer to it. There is, however, a point in the bat which 
 has a momentum which is the average of the momenta of all the 
 different parts, and this is the point where the ball ought to be 
 struck. It is called the centre of percussion or centre of 
 inertia ; and when the ball is struck there, the bat acts as if all the 
 
ENERGY AND 
 
 force applied to it was concentrated at tlfa po&t in the 
 which the ball is to be hit, and the bat \hnves ' 
 is not struck by this point, the reaction 
 whole bat round on the true centre of percussion as if on a pivot, 
 and the hand is jarred (the bat 'stings,' as the expression is), so 
 that sometimes the bat is lost hold of. The centre of percussion is 
 nearer to the end of a bat than the centre of gravity ; in a stick of 
 uniform thickness ( like that used for ' rounders ' ) it is, when the 
 stick is swung round one end as a pivot, one- third of the whole 
 length from the outer end, as may be proved by hitting a rail 
 with the stick, and noting when least vibration is felt by the 
 hand. 
 
 CHAPTER VI. 
 
 ENERGY AND WORK 
 
 SECTION 1. GENERAL PRINCIPLES. 
 
 270. When we say of a man that he 'has energy,' we mean 
 that he shows great power of overcoming difficulties, that he is 
 an active man of business, that he is capable of doing, and 
 ready to do, a large amount of work. Similarly in natural 
 science, when we observe a body, such as a moving cannon-ball, 
 to be capable of doing mechanical work, such as knocking down 
 a wall or piercing a hole through an iron plate, we say that 
 there is ' energy ' in it. 
 
 271. Work means the overcoming of obstacles, such as the 
 setting a body in motion against the resistance which inertia, 
 cohesion, &c. oppose to motion. For example, the cannon-ball 
 does work when it knocks aside the heavy stones of the wall, or 
 drives before it the molecules of the iron armour-plate in spite 
 of the toughness of the material. A locomotive engine does 
 work when it overcomes the inertia of the train and forces its 
 way through the resisting air. Energy, then, may be denned as 
 the condition of a body which makes it capable of doing work. 
 
 272. We have already spoken of a moving cannon-ball as 
 having momentum (p. 41), which was explained to mean the 
 force with which it is moving ; but practically we are much 
 more closely concerned with the work which such a cannon-ball 
 will do before its motion is stopped, than with speculations on 
 
112 ELEMENTARY DYNAMICS. 
 
 the supposed force which is in it during its flight ; and it is 
 satisfactory to be able to turn from the rather shadowy con- 
 ception of 'quantity of motion/ and consider the actual tangible 
 results in the shape of work which can be got out of a body 
 having energy in it. 
 
 273. In the first place, it is found that work can only be done 
 while energy is being transferred from one piece of matter to 
 another. A cannon-ball does no work while it is moving 
 through the air (except the comparatively slight amount done 
 in knocking aside the particles of air in its way) ; it is only 
 when it comes to something which can and will take some of the 
 energy out of it some resistance, in fact that we observe work 
 to be done. Then energy passes out of the ball into the 
 particles of stone or iron which it displaces (the splinters would, 
 if collected, do as much damage as the ball*), and the ball itself 
 comes to rest, having lost the energy which was imparted to it 
 in the gun. Similarly, a moving billiard-ball loses little energy 
 in travelling over the table, but when it strikes against another 
 ball, its energy is transferred to that ball, which immediately 
 moves on at -the same rate (as we have seen, p. 82) as the first 
 ball, while the latter stops dead owing to loss of its energy,f and 
 can do no more work until struck by the cue. 
 
 SECTION 2. STATICAL AND KINETIC ENERGY. 
 
 274. In the next place it is observable that a body may have a 
 great deal of energy in it even though it is not moving at all.l 
 An energetic man is not always displaying his energy by doing 
 work, although he may 'have it in him' (as the phrase is). 
 Similarly, there may be a great deal of mechanical energy 
 stored up in a body ready for transfer and work, but giving no 
 
 * Except, of course, that some of the energy passes into the form of heat. 
 
 f It may be noticed that this may possibly indicate a simple explanation of the 
 fact of an action being always met by a reaction (as stated in Law III., p. 77). 
 The truth is, not that there is any real force developed anew in the reaction, but 
 that the moving body, after striking another body, has less energy by precisely the 
 amount which it gave up in striking that other body, and therefore is so much the 
 less capable of moving on against resistance. Thus exactly the same effect in 
 lessening or stopping its motion is produced, as if an equivalent of new force had 
 been applied in the opposite direction to the body. 
 
 t Observe the distinction here between momentum and energy. A body can only 
 have momentum when it is moving ; it may have energy when motionless. 
 
ENERGY AND WORK. 113 
 
 sign of its presence. For example, place an iron 1 Ib. weight 
 upon a piece of glass laid on the table. The weight presses 011 
 the glass, but has not energy enough to break it. Now raise the 
 weight, and hang it by a piece of string about 2 feet above the 
 glass. The weight now, although motionless, has more energy 
 in it than before ; as may be proved by cutting the string, when 
 the weight will do the work of breaking the glass to pieces, 
 which it could not do before. The energy in a body which 
 is not actually doing work is called Statical Energy (<r<ra.Ttxos, 
 1 at rest'), or sometimes Potential Energy (potentia, power). 
 On the other hand, energy while in the act of being transferred 
 and doing work is called Kinetic Energy (x/>^<r/*oj, 'fit for 
 causing motion ' ). 
 
 275. In illustration, we may trace the changes of energy in 
 the weight of a clock. While it is being wound up, it gains 
 energy, which is put into it as kinetic energy by the muscular 
 power which raises it against the attraction of gravitation. 
 Before it begins to fall, this energy remains as statical energy, 
 depending on the position of the weight as raised above the 
 floor. While it is falling, it gives out the energy as kinetic 
 energy in turning the wheels of the clock. When it has reached 
 the point from which it was raised, all the energy which it had 
 gained in being wound up is spent ; and if it can fall no farther, 
 no more work can be got out of it. So also in a watch, when 
 the spring is coiled up, energy is transferred to it from the 
 muscles of the hand which winds it, and this is stored up in 
 it as statical energy, and given out gradually as kinetic energy 
 in making the watch go.* When a train is going up an incline, 
 some of the power of the steam is being stored up as statical 
 energy, which carries the train down the next incline without 
 much help from the engine. Water in a river or lake at a 
 higher level than the sea possesses statical energy, which it 
 gives out while falling to the sea-level, as kinetic energy in 
 turning the water-wheels of mills, carrying boats down 
 stream, &c. 
 
 * Thus we are in fact ourselves working our own watches and clocks ; storing up 
 at intervals in the powerless machinery sufficient statical energy to keep it moving, 
 while being gradually giveri out as kinetic energy, for a day or a week without 
 further supply. 
 
114 ELEMENTARY DYNAMICS. 
 
 276. Let us consider what happens to the water when it has 
 got down to the sea-level. Gravitation can move it no farther, 
 and (like a clock weight which has run down) it can do no more 
 useful work. But the heat of the sun now supplies it with 
 energy enough to enable it to rise in the form of vapour, against 
 the force of gravitation, high in the air above the earth's surface. 
 There, as it loses heat, it condenses into mist or cloud, and then 
 begins its downward course as drops of rain. Some of it falls 
 into the sea again, but much of it falls on hills and high levels, 
 and collects in streams and lakes, from which its course was 
 traced above. 
 
 277. It is worth especial notice how very much of the work 
 done on the earth is due to energy supplied by the sun. All 
 the power gained from water is, as we have just seen, due to 
 this cause. Winds also, which work mills and propel ships, 
 are simply currents of air caused by the sun's heat. But steam 
 is now extensively used instead of wind or water power for 
 driving mills, &c. Now, the energy of the steam which drives 
 the engine is put into it by the heat produced by the coal burnt 
 under the boiler. This coal is the remains of trees and plants 
 which grew on the earth many ages ago, and the energy 
 necessary for their growth was supplied to them by the sun's 
 rays, and these only. As long as plants are in light they 
 flourish, and no longer.* Therefore the work done by our 
 modern engines is due to the sun's energy stored up in a statical 
 form on the earth in very early times. 
 
 278. So again with regard to the living machine of the human 
 body. We gain the power of doing work from the food we eat, 
 as any one may prove by trying to play cricket, or row, or run, 
 or read, without breakfast or dinner. No\v, our food is partly 
 vegetable, such as bread, tea, potatoes ; partly animal, suck 
 as meat, milk, butter. The corn, potatoes, &c. derive their 
 power of growth from the sun, as has just been explained. 
 The animals whose flesh, &c. we use as food, live and 
 
 * What actually happens is this : Plants feed upon carbon dioxide ( ' carbonic 
 acid ' ), a compound of carbon and oxygen which exists in air. While light falls 
 upon them, they have the power of decomposing this substance, combining with its 
 carbon, and giving out its oxygen to the air. When they are burnt, their carbon 
 combines a,qain with oxygen, and in doing this a great deal of energy is made avail- 
 able in the form of heat. 
 
ENERGY AND WORK. 115 
 
 thrive on grass and other plants, which themselves live by 
 the sun. 
 
 279. Thus the sun is really working our trains, grinding as 
 well as growing our corn, weaving our clothes, supporting our 
 lives. No one who is able to think at all can help going a 
 step farther in thought, and meditating on, if not yet fully 
 comprehending, the Great Source of the original energy of 
 the sun itself. 
 
 SECTION 3. CONSERVATION OF ENERGY. 
 
 280. It appears certain, from modern research, that not a trace 
 of energy has ever been produced or destroyed in the universe 
 (so far as we know it) by human means. Energy, in fact, is as 
 indestructible as matter. A certain amount of energy was 
 associated with the matter of the universe at its creation ; and 
 this precise amount, and neither more nor less, is present now, 
 although its form may have been changed many times. The 
 energy of sunlight may pass into vital energy, vital energy may 
 pass into heat, heat may be transformed (as in the steam- 
 engine) into mechanical motion, and mechanical motion may 
 be reconverted into heat, as seen in the sparks which appear 
 when the brake is put on to stop a train ; but absolutely 
 nothing is lost in the transfer, and nothing is gained. 
 
 281. It might seem that energy actually disappears when a 
 cricket-ball is stopped dead by the bat ; but it is found that 
 both the ball and the bat become hotter than before. The 
 energy of the ball is, in fact, converted into an exact equivalent 
 of heat-energy, which is soon diffused through the air, but not 
 destroyed. The same change into heat takes place when a 
 bullet is stopped by a target which it does not pierce : the heat 
 evolved is enough to melt the lead, as may often be seen from 
 the shape of the flattened bullet. 
 
 282. It might seem, again, that energy was lost when a stone 
 is thrown up to the roof of a house and lodges there ; but all 
 the energy which was imparted to the stone when it was thrown 
 up is present in it as statical energy, and remains there until 
 it is dislodged from the roof. When it falls it gives out this 
 energy as kinetic energy in one form or another ; for instance, 
 as heat (as in the case of the bullet) when it strikes the ground. 
 
116 ELEMENTARY DYNAMICS. 
 
 It might seem that energy was created when gunpowder is fired 
 in a gun ; but in point of fact the energy was already present in 
 the materials of the gunpowder, put into them in the process of 
 making them, and simply given out when they act on each 
 other. 
 
 283. Many attempts have been made to invent a ' perpetual 
 motion ' machine a clock which will go without being wound 
 up; a water-wheel which will pump up its own supply of water; 
 a steam-engine which will do its work without fuel, or some 
 other source of energy. The strongest proof of the doctrine of 
 the conservation of energy is that all such attempts have been 
 utter failures ; they are as futile as the attempt of a man 
 standing in a basket to raise himself from the ground by pulling 
 upward at the handles. 
 
 284. In fact, all that has been said about the action of forces, 
 in explanation of the Laws of Motion, might be comprehended 
 in the general statement 
 
 * Energy cannot be created or destroyed. We can get out 
 of a body the energy which has been put into it, and 
 neither more nor less.' 
 
 SECTION 4. MEASUREMENT OF ENERGY. 
 
 285. "We cannot see Energy, any more than we can see a mind ; 
 but we can see and measure the work done in consequence of it. 
 A man's bodily strength and resolution can be fairly gauged by 
 observing how many miles he can walk, or run, or row ; how 
 many pounds-weight he can lift from the ground with his 
 hands ; or (as in the strength-testing machines often seen at 
 railway stations) how far he can compress a spring by a blow. 
 On a similar principle the amount of available energy in a body 
 is generally measured by observing the amount of mechanical 
 work which is done during its transfer to other bodies. 
 
 286. Thus if one cannon-ball will pierce four plates of iron, 
 each 1 inch thick, while another ball will only pierce two such 
 plates, the former ball has twice as much energy as the latter. 
 Again, when a mass of 1 Ib. is raised 1 foot from the ground, a 
 certain definite amount of work is done in overcoming the force 
 of gravitation. If a mass of 3 Ibs. is lifted 1 foot high, 3 times 
 
ENERGY AND WORK. 117 
 
 as much work is done as in the first case. Also, if 1 Ib. is 
 lifted 3 feet high, 3 times as much work is done as in the first 
 case ; for it is the same thing as raising 1 Ib. through 3 succes- 
 sive stages of 1 foot each. 
 
 287. If a cricket-ball is thrown straight up 16 feet by A, 
 64 feet (or 4 times as far) by B, and 144 feet (or 9 times as far) 
 by C, then B does 4 times as much work with his muscles as A, 
 and C 9 times as much. 
 
 288. From these examples it can be seen that the amount of 
 work done in overcoming a resistance varies, 
 
 (1) with the mass moved ; 
 
 (2) with the space through which it is moved ; 
 
 and that if we multiply the number expressing the mass 
 moved by the number expressing the space through which it 
 is moved, the product will give us a measure of the energy 
 employed in effecting the work. 
 
 289. It is convenient to fix upon some definite quantity of 
 work as a unit or standard to compare other quantities with. 
 The unit of work commonly adopted in this country is called a 
 'foot-pound/ and is 
 
 That amount of work which is done in raising a mass of 1 Ib. 
 through 1 foot against the attraction of gravitation. 
 
 290. As explained above, we have only to multiply the mass 
 of a body expressed in pounds by the height in feet through 
 which it is raised, in order to find the number of foot-pounds of 
 work which have been done, and hence to get a measure of the 
 energy which has been used. 
 
 For example Ten books, each weighing 1 Ib., lying on the 
 floor, are raised to a shelf 6 feet above the floor. Here the whole 
 mass moved is 10 Ibs. and the height is 6 feet ; therefore the 
 work done is ( 10 x 6 = ) 60 foot-pounds. Again, a box 
 weighing 80 Ibs. is carried from the ground to a room 20 feet 
 above the ground. Then 80 x 20 = 1600 foot-pounds of work 
 are done by the person who carries it up-stairs. 
 
 291. In walking, the whole body is raised through about 4 
 inches at each step ; so that any one who takes 2000 steps in 
 walking a mile, raises his body through (4 x 2000 = ) 8000 
 inches, or C67 feet (nearly) in walking that distance. If he 
 
118 ELEMENTARY DYNAMICS. 
 
 weighs 140 Ibs. (10 stone) he does (140 x 667 = ) 93,380 foot- 
 pounds of work. 
 
 Relation of Energy to Velocity. 
 
 292. We have seen that the amount of work done depends 
 upon the space a body is moved through : we have next to 
 inquire how it depends upon the velocity of movement. Has a 
 locomotive which is moving 2 feet per second twice as much 
 work-power as when it is moving 1 foot per second ? The 
 following considerations will show that it has really much more 
 than twice the energy. 
 
 293. Suppose that the engine is moving 1 foot per second, 
 and that the work it is doing consists in knocking away lumps 
 of snow from the line. Then, if it is made to move 2 feet per 
 second, 
 
 (1) It meets twice as many pieces of snow hi 1 second, and 
 hence does twice as much work : 
 
 (2) It meets each piece with twice the velocity, and therefore 
 knocks it farther away. Here again it does twice the work. 
 
 So that altogether it does (2 x 2 =) 4 times the work if it 
 moves with twice the velocity. 
 
 Similarly, if it moves with 3 times the velocity, it does 
 (3 x 3 = ) 9 times the work, and therefore must have 9 times as 
 much energy as when it was moving 1 foot per second. 
 
 Hence we may state it as a law that, 
 
 The energy in a body depends upon the square of the velocity 
 with which it is moving. 
 
 294. For example A golf-ball is hit by a club moving at the 
 rate of 8 feet per second, and is driven to a distance of 40 yards. 
 If at another stroke the club hits it with a velocity of 16 feet 
 per second, how far will the ball go ? The velocity in the last 
 case is twice as great as before, and therefore the ball will be 
 hit (2 2 =) 4 times the distance; that is, (4 x 40=) 160 
 yards. 
 
 A hammer falling on a nail with a velocity of 3 feet per 
 second drives it inch into a board. How far will the nail be 
 driven in if the hammer meets it with a velocity of 9 feet per 
 second ? In the last case the hammer moves with 3 times the 
 
ENERGY AND WORK. 119 
 
 velocity ; it therefore has ( 3 2 = ) 9 times the energy, and the 
 nail will be driven in (9 x & = ) 1^ of an inch. 
 
 295. This very rapid increase of energy \vitli the speed of a 
 body explains several commonly-observed facts ; for example, 
 the great damage done in a collision by an express train com- 
 pared with that done by a slower train of the same weight. A 
 train moving 60 miles an hour has 9 times the energy of one 
 moving 20 miles an hour, and all this ninefold energy must be 
 got out of it before it stops, either by powerful brakes or by 
 collision with something else. 
 
 296. We see also why a cannon-ball has so much greater 
 penetrating power than the gun out of which it comes. The 
 mere momentum of the one is, as we have seen (p. 79), exactly 
 equal to that of the other, but the energy of the quickly-moving 
 ball is far greater than that of the gun. Suppose that, as in the 
 example given on p. 79, the ball is moving with 120 times the 
 velocity that the gun recoils. Then its energy will, so far as 
 its velocity is concerned, be (120 2 =) 14,400 times that of the 
 gun, and hence, although it is much lighter, it has a much 
 greater destructive power. The recoil of a rifle can be borne by 
 the shoulder with no worse effect than an occasional bruise ; 
 but the result would be very different if the bullet was driven 
 (by the same charge of powder) against the shoulder, the gun 
 being held with the stock outwards, and allowed to go as far as 
 the recoil would carry it. 
 
 Work done by falling Bodies. 
 
 297. On the principle of the Conservation of Energy (p. 115), a 
 mass of 1 Ib. will, in falling through the distance of 1 foot, do 
 just as much work as was required to raise it 1 foot that is, it 
 will do 1 foot-pound of work. If it has fallen through 2 feet, 
 it will do 2 foot-pounds of work, and so on. 
 
 298. This is well illustrated in the machine used for driving 
 posts or * piles ' into the ground for foundations of bridges, &c. 
 This ' pile-driver ' consists of an upright frame in which a heavy 
 iron weight is made to slide up and down (fig. 52). The frame 
 is placed so that the weight rests on the head of the post to 
 be driven in ; but the mere weight of the iron is quite 
 insufficient to drive the post into the ground, and all the 
 
120 
 
 ELEMENTARY DYNAMICS. 
 
 strength of several men could not push the post down. The 
 men, however, by pulling at a rope attached to the iron mass 
 
 and passing over a pulley, 
 raise the weight 8 or 10 
 feet above the post, and in 
 doing so a certain amount 
 of work is done, and a 
 corresponding amount of 
 energy is transferred from 
 their muscles to the mass, 
 and stored up in it as statical 
 energy. Then the weight is 
 let fall, and all the energy 
 collected in it comes out, with 
 Bfc " : the result of work, when it 
 reaches the head of the post 
 
 Fig 52 
 
 1 pile ' is driven some way into 
 
 the ground, as a nail by a gigantic hammer. Suppose that the 
 block of iron weighs 50 Ibs., and that it is raised 10 feet above 
 the head of the pile. Then (50 x 10 =) 500 foot-pounds of 
 work must be done in raising it, and it will do 500 foot-pounds 
 of work in driving down the pile when it falls upon the latter. 
 
 Relation of Work to Time. 
 
 299. The value of work done depends, not only upon how much 
 is done, but how quickly it is done. A horse is rated more highly 
 as a worker than a donkey or a dog, because it will do the same 
 piece of work much more quickly than either of the latter ; 
 though even they would get through the work if we gave them 
 time enough. 
 
 300. It is estimated, from experiments, that a first-rate horse 
 can raise 33,000 Ibs. (nearly 15 tons) 1 foot high in a minute 
 that is, can do 33,000 foot-pounds of work per minute. This 
 amount of work done in a minute is called one horse-power ; an 
 expression constantly used in stating the power of engines, &c. 
 When a steam-engine, for instance, is said to be of * 8 horse- 
 power,' it is implied that the machine can do, during every 
 minute that it works, (8 x 33,000 = ) 264,000 foot-pounds. 
 
ENERGY AND WORK. 121 
 
 301. The following table will give an idea of the amount of 
 energy involved in some of the more commonly occurring kinds 
 of work : 
 
 Work done per Minute. 
 
 ft.-lbs. 
 
 In walking, about 2,500 
 
 n rowing 3,600 
 
 it ir at racing speed 6,000 
 
 it working a bicycle 2,300 
 
 By an ordinary labourer 2,000 
 
 it ahorse (average) 25,000 
 
 ii an express engine, when taking a load of ) 
 150 tons at the rate of 60 miles an hour J " 
 
 APPENDIX A. 
 Exact valuation of the energy in a moving body. 
 
 302. As already stated (p. 112), when a moving body has its 
 motion stopped or changed, a definite amount of energy cornea-out 
 of it and goes into the object which stops it ; and work is done on 
 the latter. 
 
 Thus the energy of a hammer falling on a nail is shown by the 
 extent to which it drives in the nail. This work has been explained 
 in general terms (pp. 117, 118) to vary (a) with the mass, (b) with 
 the square of the velocity of the moving body ; and if the work could 
 be done in an instant, the energy would be exactly expressed by the 
 product of the mass x velocity 2 . But it takes time to do the work, 
 and during this time the velocity is gradually getting less and less, 
 until there is none left, and then there can be no more work done. 
 Hence, in the first half of the time, more of the work will be done 
 than in the last half of the time. And the whole amount of work 
 which a moving body can do in the time during which its motion 
 is being stopped, will correspond to the average or mean amount of 
 energy between that which it has at the beginning of the time, and 
 that which it has at the end of the time. Now, 
 
 Its energy at the beginning = mass x velocity 2 . 
 end = 0. 
 
 And the average or mean of these two quantities is half the sum of 
 them ; that is, it 
 
 = | | ( mass x velocity 2 ) + } 
 or 5 (mass x velocity 2 ). 
 
122 ELEMENTARY DYNAMICS. 
 
 Hence the rule for finding the value of the energy in a moving body 
 may be stated thus 
 
 Multiply the mass by the square of the velocity, and divide ths 
 
 product by 2. 
 
 For example : A fives-ball weighing 1| oz., and moving 20 feet per 
 second, and a racquet-ball weighing \ oz., and moving 30 feet per 
 second, strike 0*1 a heap of sand ; which will go deepest into it ? 
 
 Energy of fives-ball = (1| x 20 2 ) -7- 2 or -i-*^* = 250. 
 
 1 v Q(V) 
 
 Energy of racquet-ball = (i x 30 2 ) -f- 2 or * = 225. 
 
 21 
 
 So the fives-ball will penetrate the deepest. 
 
 303. Since, on the principle of the conservation of energy (p. 115), 
 just as much energy must go into a body to set it moving, as can be 
 got out of it when its motion is stopped, the above numbers express 
 also the amount of energy employed in throwing the fives-ball and the 
 racquet-ball against the bank. 
 
 A cannon-ball weighing 10 Ibs. is to have a velocity of 1200 feet 
 per second when it leaves the gun. How much energy is required 
 for this ? 
 
 (10 x 1200 2 ) 4- 2 = 10 x 1,440,000 = 7)200)000 foot-pounds. 
 
 APPENDIX B. 
 The Pendulum. 
 
 304. The simple pendulum is a weight hung from a support by a 
 string or rod (having no appreciable weight itself), so that it can 
 swing easily from side to side. Fig. 53 represents a pendulum of 
 the most common construction. A is the axis 
 or point of suspension ; B is the rod ; C is the 
 bob; consisting of a ball, or a round flattMi 
 piece of metal, which is fastened to the rod 
 by a screw behind, by which screw it can be 
 raised or lowered on the rod ; DD' is the path 
 D or arc which the ball traverses in swinging. 
 ..--''' When the pendulum is at rest, it hangs per- 
 Fi~53 pendicularly, as here represented. 
 
 305. If the ball is now drawn to one side, 
 as to D, and then let go, gravity urges it downwards, while the 
 
ENERGY AND WORK. 123 
 
 cohesion of the string or rod keeps it at a fixed distance from A. 
 The composition of the two forces makes it describe a descending 
 curve to C, where gravity is completely counteracted by the 
 opposite pull of the rod. But the ball does not stop here ; it goes 
 on, by the law of inertia, in the ascending curve CD', until gravity 
 destroys its acquired motion. If friction and the resistance of the 
 air did not interfere, it would take as long to destroy the motion as 
 to generate it, and the ball would reach the same elevation as it 
 started from. But these causes gradually render the ascent less arid 
 less, and at last bring the pendulum, when it has no maintaining 
 power, to a state of rest. 
 
 306. One sweep of a pendulum from D to D' is called an oscilla- 
 tion ; and the path it describes, being part of a circle, is called its 
 arc. The size or amplitude of the oscillation is measured by the 
 number of degrees in the arc, each degree being a 360th part of the 
 whole circle. For reasons which will presently appear, pendulums 
 are generally made to describe small arcs, not exceeding 5 or 6. 
 
 307. The pendulum affords an excellent illustration of statical 
 and kinetic energy. When it is drawn to one side, it is raised a 
 little farther from the centre of the earth, and energy is transferred 
 to it and stored up as statical energy. As it swings back to the 
 perpendicular position, this energy becomes available as kinetic 
 energy, just as in a falling weight ( par. 275, p. 1 13 ). When it reaches 
 the perpendicular, all the energy imparted to it is in the condition 
 of kinetic energy ; but after it has passed this point, the energy 
 available in a kinetic form becomes less and less, being converted 
 by degrees into the statical form. When it pauses at the end of its 
 swing, all the energy is statical once more ; and then, on its com- 
 mencing to move downwards, the change into the kinetic form 
 begins anew. 
 
 308. The most remarkable property of the oscillations of the 
 pendulum, and that on which its use as a regulator of movement 
 depends, is that, whether long or short, they are all performed in 
 very nearly the same time. If we observe the vibrations of any 
 body set swinging, we find that though the arc it describes is 
 continually diminishing, there is no sensible shortening of the time 
 in which the single swings are accomplished. As the space to be 
 passed over becomes less, so does the velocity. Galileo, the great 
 astr-onomer, discovered and investigated this important fact, in 
 1582 A.D., when only eighteen years old ; being led to think on 
 the subject, it is said, by observing the regular swinging of lamps 
 hung from the roof of the cathedral at Pisa. 
 
124 
 
 ELEMENTARY DYNAMICS. 
 
 Do 
 
 I w 
 
 309. The cause is easily understood. The wider the sweep of the 
 ball, the steeper is its descent at the beginning, and this gives 
 it a greater velocity, and enables it to go over the longer journey 
 in the same time as over a shorter. If we take a moderately 
 short arc, such as CD, fig. 53, the steepness of descent at D is 
 almost exactly double of what it is at E, its middle point; so 
 that a ball beginning its motion at D, moves on an average twice 
 as fast as one starting from E, and thus both arrive at C at the 
 same time. 
 
 310. But it is only short oscillations that are thus 
 isochronous, as it is called ; when the arcs are large, 
 the steepness does not increase in exact proportion 
 to the length, and therefore the isochronism is not 
 perfect;- Accordingly, pendulums are made to swing 
 in short arcs ; and then, though no practical contriv- 
 ance could make the extent of the oscillations exactly 
 uniform, the times are virtually equal. 
 
 311. But though the time of oscillation is indepen- 
 dent of the largeness of the arc, it is greatly affected 
 by the length of the pendulum itself. Long pendu- 
 lums vibrate more slowly than short ones. Though 
 the balls B and D, fig. 54, have the same amplitude 
 
 of vibration, or go over corresponding arcs, the journey of the one is 
 longer than that of the other. But the steepness of descent, or 
 
 inclination of the path, is the same in both ; 
 
 therefore B must take longer time to perform its 
 journey than D. We must not, however, conclude 
 that when the length of the pendulum is doubled, 
 the time of oscillation is also doubled. The motion 
 of the pendulum is an accelerated motion ; and, 
 as in all other uniformly accelerated motions ( see 
 par. 229, p. 92), the spaces described are as the 
 squares of the times. To give double the time of 
 vibration, then, requires the pendulum to be four 
 times as long ; treble the time, nine times as long ; 
 and so on. 
 
 312. The truth of this is easily proved by ex- 
 periment. Suspend three leaden balls on double 
 threads, as in fig. 55, so that the lengths measured 
 Fig. 55. in the dotted line may be as 1, 4, and 9. While 
 
 the lowest ball makes one oscillation, the highest 
 will be found to make three, and the middle ball two. 
 
 Fig. 54. 
 
ENERGY AND WORK. 125 
 
 Length of the Seconds Pendulum. 
 
 313. A pendulum of a little more than 39 inches in length (almost 
 exactly one metre), beats once in a second, and one of one-fourth 
 that length beats half-seconds. As a pendulum that beats seconds 
 must always be of the same length, it has been proposed to make it 
 a fixed standard of measure, which can be found again when 
 artificial standards are lost. 
 
 314. When we say that the seconds pendulum is always of the 
 same length, we must be understood to speak of the same place. 
 In different places, its length varies. It will be readily conceived 
 that as the motion of the pendulum is caused by gravitation, any 
 alteration in the force of gravitation must alter the rate of its 
 movement. Now the force of gravitation is different in different 
 parts of the earth, being least at the equator, and greatest at the 
 poles, and therefore a pendulum which is to make one swing in a 
 second must be rather shorter at the equator than at the poles. In 
 fact its length must increase as the latitude increases. 
 
 Length of a pendulum which makes one swing in a second, 
 at different places : 
 
 Spitzbergen 79 49' 58" N. Lat. , 39'2146 inches. 
 
 Edinburgh 55 58 40 .. 391554 .. 
 
 London 51 31 8 n 391390 
 
 Jamaica 17 56 7 39'0350 
 
 Sierra Leone 8 2928 .. 39'0195 ,, 
 
 315. How is the length of a pendulum measured ? Is it from the 
 point of suspension to the bottom of the ball, or to what point ? 
 The answer to this will be plain from what has been said on p. 110 
 regarding the 'centre of percussion.' We have as yet considered 
 gravitation as acting only on the ball, because the greater part of 
 the matter is concentrated there. But the matter composing the 
 rod is concerned in the movement, and if there were no ball, the 
 rod would form a pendulum of itself. Now, if each particle of 
 matter in the swinging body were at liberty to swing separately, 
 those particles that are nearer to the axis would move more rapidly 
 than those that are farther off. But as the whole cohere in one 
 mass, this tendency is checked ; the motion of nearer particles is 
 retarded by the more remote, and the motion of the more remote is 
 accelerated by the nearer. There must, however, be a point divid- 
 ing the particles that are moving slower than their natural rate 
 
 I 
 
126 ELEMENTARY DYNAMICS. 
 
 from those that are moving faster than their natural rate a point 
 which is moving exactly as a particle there situated would move if 
 free to vibrate alone. This point is . the centre of oscillation. 
 It is evident that the whole of the matter composing a swing- 
 ing body may be considered as acting at this point ; and if it were 
 all concentrated there, the rate of vibration would be the same. 
 The length of a pendulum, then, is measured from the centre of 
 suspension to the centre of oscillation. 
 
 316. We may find pretty nearly the centre of oscillation of a 
 common pendulum, or of any swinging body, by suspending in 
 front of it, and from the same axis, a small ball of lead attached to 
 a fine thread. This ball and thread form what is called a simple 
 pendulum, because the weight of the thread may be reckoned as 
 nothing, and the matter is as nearly as possible collected in one 
 mass that is, the ball. Both bodies are now made to swing, 
 and the thread is lengthened or shortened till the two vibrate 
 in exactly the same time ; when they come to rest, the centre of 
 the spot on the swinging body covered by the ball is the centre of 
 oscillation. 
 
 317. A bar of wood or metal of uniform thickness, may be 
 suspended as a pendulum, and its centre of oscillation determined 
 in the way described. If its position is now reversed, and the 
 centre of oscillation made the centre of suspension, it is found 
 to vibrate in exactly the same time as before. This is expressed 
 by saying that the ' centres of oscillation and suspension are inter- 
 changeable.' 
 
 318. By making the centre of oscillation the centre of suspension, 
 we shorten the pendulum, and might therefore expect that the time 
 of vibration would be shortened. But in this arrangement a 
 portion of the matter is thrown above the axis, and this acts as a 
 check, and proportionally retards the movement. 
 
 319. A pendulum alone, without wheel-work, would form a time- 
 keeper, if we took the trouble to observe and count its vibrations, 
 and if friction and the resistance of the air did not, after a time, 
 bring its motion to an end. The use of the wheel-work in a clock 
 is to answer these two ends to count and record the swings 
 of the pendulum, and to act as a maintaining power that is, to 
 supply to the pendulum fresh motion in place of what is con- 
 stantly being destroyed. It is still the pendulum that measures 
 the time. 
 
 320. In ordinary clockwork (see fig. 56), the wheels are put in 
 motion by a heavy weight which is attached to a cord wound round 
 
ENERGY AND WORK. 
 
 127 
 
 a barrel. One of the wheels is so placed that a piece of metal, 
 
 fixed on the top of the pendulum, and shaped something like the 
 
 claw of an anchor, projects its ends, as the pendulum swings, 
 
 between two teeth of the wheel alternately, first 
 
 on one side, then on the other. Thus the wheel is 
 
 held in check, and only one tooth allowed to pass 
 
 at each swing ; and it is evident that an index or 
 
 seconds hand fixed to the axis of this wheel will 
 
 record the vibrations of the pendulum. But while 
 
 the pendulum thus regulates the rate at which the 
 
 wheel is allowed to move, the teeth of the wheel 
 
 in contact with the anchor of the pendulum, give 
 
 it a push as they are disengaged from its ends, 
 
 and so communicate just as much moving-power 
 
 to the swinging part as it is losing by friction. 
 
 Such a contrivance for bringing the pendulum 
 
 into connection with the wheel- work, is called an 
 
 escapement, of which there are several varieties. 
 
 321. For adjusting the length of the pendulum, 
 the ball is made to slide on the rod by means of a 
 fine screw. A difference in length of the 1000th 
 
 part of an inch causes an error of about a second a day. Since all 
 siibstances are expanded by heat and contracted by cold, changes of 
 temperature must affect the rate of clocks, making them go slower 
 in summer than in winter. Compensation pendulums, accordingly, 
 have been contrived, in which expansion in one part is made to 
 counteract expansion in another part. 
 
 322. To save space, timepieces are often regulated by pendulums 
 one-fourth the ordinary length, and therefore beating half- 
 seconds. But a long pendulum, with a heavy bob or ball, is 
 desirable where evenness of rate is the object. A pendulum 
 beating two seconds keeps time much more accurately than one 
 beating single seconds. 
 
 Fig. 56. 
 
128 ELEMENTARY DYNAMICS. 
 
 CHAPTER VII. 
 MACHINES. 
 
 SECTION 1. GENERAL PRINCIPLES OP MACHINES. 
 
 323. In very many cases it is not convenient or practicable to 
 apply energy directly to do a piece of work. When a load of 
 bricks is to be raised to a scaffolding, we do not force them up 
 by the direct pressure of steam, or by exploding a charge of 
 gunpowder under them. Either a man carries them up a few 
 at a time, or the whole load is raised at once by pulle} T s and 
 ropes worked by the man, or by an engine. Some apparatus, 
 in fact, is contrived by which the energy necessary to raise the 
 bricks is applied in an indirect but more convenient manner. 
 Such an apparatus is called a ' machine.' 
 
 324. A machine (from pyix,*, a device) may be denned as 
 a contrivance for transferring energy from one point to another in 
 order to do some particular kind of work conveniently and advan- 
 tageously. For example : A clock is a machine in which the 
 force of gravitation pulling a weight downwards is made to turn 
 the hands slowly and uniformly in a circle. A poker is a 
 machine by which a small force directed downward is made to 
 lift a heavy piece of coal upward. 
 
 325. In considering the action of machines, the source from 
 which the energy comes is connected with some one part of the 
 apparatus, and is called the Power. The work to be done is 
 connected with some other part of the apparatus, and is called 
 the Resistance. Thus in a clock the Power is gravitation act- 
 ing on a weight : the Resistance is the inertia and friction of the 
 wheels and hands. In an engine, the Power is the pressure of 
 the steam applied to the piston ; the Resistance is the inertia, 
 friction, &c. of the machinery it drives or the carriages it 
 draws. 
 
 Advantage gained in using Machines. 
 
 326. It must be borne in mind, to begin with, that a machine 
 cannot really do more work than what corresponds to the energy 
 
MACHINES. 129 
 
 which is employed on it. It cannot create energy : it can only 
 apply energy in a convenient way for particular purposes. 
 
 327. To take a simple case : Suppose that a mass of 3 Ibs. has 
 to be raised 1 foot. Then 3 foot-lbs. of work must be done in 
 some way or other ; for instance : 
 
 (a) By raising the whole mass at once with the hands 
 through 1 foot. 
 
 (6) By dividing it into several parts (say 3 of 1 Ib. each), and 
 raising them one by one. 
 
 aft. 
 
 Fig. 57. 
 
 (c) By using a machine, such as a lever (fig. 57), of such a 
 kind that the point where the Power is applied moves through 
 a greater distance than the point to which the Resistance, that 
 is, the 3 Ibs., is attached. Suppose that the point P of the lever 
 moves through 3 times the space that the point R does, when 
 the lever is moved ; then a Power of 1 Ib. applied at P will 
 move through 3 feet, and therefore do 3 foot-lbs. of work while 
 R is raised through 1 foot. 
 
 328. The advantage of using the method last described is very 
 evident in cases where an extremely large resistance has to be 
 overcome. In fact, a machine, such as a lever or a set of pulleys, 
 will enable a man to lift a block of stone far heavier than he 
 could lift directly with his hands. But yet he will have to 
 move his hands in working the machine through as large a 
 space, and thus do just as much work as if the stone was cut 
 into several pieces, and he lifted them one by one. 
 
 329. Most machines are simply contrivances for enabling a 
 small Power or source of energy to move through a great space 
 during its transfer, and thus effect a great deal of work without 
 losing its hold on the Resistance (as it does when the portions 
 of stone are lifted one by one). 
 
130 ELEMENTARY DYNAMICS. 
 
 Mechanical Advantage. 
 
 330. This term means the advantage gained in being able to 
 move a great Resistance by means of a small Power applied in 
 the machine through a large space. In order to estimate it for 
 any given machine, we have only to observe how much space 
 the Power moves through, and compare it with the space through 
 which the Resistance is moved, when the machine is working. 
 The mechanical advantage is represented, in fact, by the number 
 which expresses how many times the space through which the 
 Power moves is greater than the space through which the Resist- 
 ance is moved. 
 
 For example: in the lever above mentioned (fig. 57), the 
 hand applied at P moves through 3 feet, while the weight at R is 
 raised 1 foot that is, the Power moves through 3 times as much 
 space as the Resistance. Hence, the * mechanical advantage ' 
 of this lever is said to be 3 ; and a man could with it raise a 
 weight three times as great as he could lift with his hands 
 alone. 
 
 Rule for finding the mechanical advantage of any machine. 
 Divide the space through which the Power moves by the space 
 through which the Resistance is moved ; the quotient is the 
 mechanical advantage. 
 
 For example : In using a poker, the end where the hand is 
 applied moves through 1 foot, while the other end placed under 
 the coal moves through 2 inches. Then the mechanical advan- 
 tage is ( m ' = J 6 ; and a pressure of 1 Ib. will lift a piece of 
 
 coal 6 Ibs. in weight. 
 
 331. Sometimes it is more convenient to have the Power 
 move through a small space, while the Resistance is made 
 to move through a large space. Thus, in a clock, the weight 
 only moves a few feet downwards in turning the hands many 
 times round. In such machines there is no mechanical advan- 
 tage, strictly so called, but the reverse ; and the Power must be 
 greater than the Resistance. 
 
 332. It is an unfortunate fact that no machine whatever will 
 do all that it ought, by its principle, to do. The power required 
 for working it is always somewhat greater than it should be, as 
 
M ACHINEf [ ft >T T T?U r "* 
 
 \V ' J V EjR.S f'j-y ' 
 
 calculated from the strict mechanical Mvafijfepgp r The reason is, 
 that in all the moving parts of a machine" -tEere is some kfertia 
 and friction to overcome, and often a great deal ; so that more 
 or less additional energy has to be spent in overcoming this, 
 apart from what is used in doing the actual work for which the 
 machine was intended. 
 
 333. Machines are usually arranged under four heads, named 
 after the simplest typical machine of each class. They are 
 
 1. The Pulley. 
 
 2. The Wheel and Axle. 
 
 3. The Lever. 
 
 4. The Inclined Plane. 
 
 SECTION 2. PULLEYS. 
 
 334. A pulley, fig. 58, is a wheel with a groove in its 
 turning easily in a frame called 
 the 'block.' A cord is placed 
 in the groove, and the Power 
 is attached to one end of this 
 cord. 
 
 A pulley is said to be fixed 
 when it is attached to a steady ||||| [jjii'f .rVl 
 support, such as a beam. It is 
 said to be movable when it is 
 attached to the Resistance. 
 
 Fig. 58. 
 
 Fixed Pulleys. 
 
 335. These are used for chang- 
 ing the direction of a force, and give no mechanical advantage, 
 since the Power always moves through the same space as the 
 Resistance ; as will be plain from fig. 59, in which A is a 
 pulley attached to the beam B. If a weight of 1 Ib. (with a 
 small additional weight to overcome friction, &c., as explained 
 in par. 332 above) is attached as the Power to one end, P, 
 of the cord, it will, in falling through 1 foot, raise a Resistance 
 of 1 Ib. attached to the other end of the cord, but it will not 
 raise more than 1 Ib. ; and it will raise this to a height of 
 1 foot. 
 
132 
 
 ELEMENTARY DYNAMICS. 
 
 1 
 
 336. Such pulleys are to be met with in every house. In 
 sash-windows there is a fixed pulley in the 
 frame at each side ; a cord passes round it, 
 one end of which is fixed to the window 
 sash, and the other end to a weight sliding 
 up and down within the frame ; so that 
 the sash can be raised and lowered with 
 slight effort. They are also used for draw- 
 ing curtains, and for opening and shutting 
 windows which are placed too high to be 
 
 Qw 
 
 Fig. 59. 
 
 reached with the hand. 
 
 Single Movable Pulley. 
 
 337. This is a pulley attached to the Resistance, with a cord 
 passing round it, one end of which is tied to a fixed support ; the 
 other end is attached to the Power. The simplest arrangement 
 is shown in fig. 60, but the cord is often led over a fixed pulley, 
 as in fig. 61, simply in order to enable the Power to act in a 
 more convenient direction. 
 
 2 
 
 Fig. 60. 
 
 Fig. 61. 
 
 338. The mechanical advantage of such a pulley is estimated 
 as follows : The Resistance is obviously held up by two cords, 
 A and B, fig. 60, and in order that it may be raised up 1 inch, 
 each of them must be shortened 1 inch. But since they are 
 really parts of the same cord which goes round the pulley, 
 both can be thus shortened by simply pulling B through 2 
 inches ; or, what comes to the same thing, by the power at P, 
 fig. 61 (which is part of the same cord), moving through 2 
 
MACHINES. 133 
 
 inches. Hence the Power moves through 2 inches, while the 
 Resistance is moved through 1 inch; so there is a mechanical 
 advantage of (f ) 2- That is, a Power of 1 oz. (with a slight 
 additional weight to overcome friction) will raise a Resistance of 
 2 oz. 
 
 Example. A log of wood weighing 1 cwt. .is to be raised by 
 using a movable pulley ; how much power is required ? 
 
 Since the mechanical advantage is 2, the resistance will be 
 twice the Power ; that is, the Power will be half the Resistance ; 
 and x 112 Ibs. = 56 Ibs. 
 
 Therefore a little more than 56 Ibs. will do the work. 
 
 339. The movable pulley is extensively used in ships for 
 tightening the rigging and moving the yards ; it is also often 
 attached to cranes for lifting heavy weights. In clocks the 
 weight is frequently attached to a movable pulley ; in this case 
 the weight being the power, there is no mechanical advantage, 
 but the reverse, and a weight twice as heavy as usual is required. 
 But the wheels make twice as many revolutions for a given fall 
 of the weight ; so such clocks have the advantage of going for a 
 longer time without being wound up. 
 
 Systems of Pulleys. 
 
 340. Pulleys are often combined together in various ways, in 
 order to increase the mechanical advantage. Only 
 
 two of these systems will be described here. 
 
 System I. In this, as shown in fig. 62, (a) all 
 the pulleys are movable. (6) Each has a 
 separate cord, one end of which is fastened to 
 a fixed support ; the other end is tied to the 
 block of the next pulley, (c) The cord of the 
 last pulley, C, is attached to the Power ; the 
 block of the first pulley, A, is attached to the 
 Resistance. 
 
 341. To calculate the mechanical advantage of 
 this system, we must consider that it acts like a 
 number of single movable pulleys, each of which 
 gives (as already explained) a mechanical advan- 
 tage of 2. Thus, Fig. 62. 
 
134 
 
 ELEMENTARY DYNAMICS. 
 
 The mechanical advantage at A = 2. 
 
 B = 2 x 2 = 4. 
 C = 4 x 2 = 8. 
 
 The Power, then, applied at P, will move a Besistauce 8 times 
 as great as itself ; but it will only move it through ^ of the 
 space. 
 
 Rule for finding the mechanical advantage. Set down 2 as 
 many times as there are movable pulleys, and multiply the 
 numbers together. The product is the mechanical advantage. 
 Example. A man capable of exerting a power of 1 cwt. has 
 to raise blocks of stone by the help of 4 movable 
 pulleys ; what is the heaviest block he can lift ? 
 
 2x2x2x2= 16. 
 
 Hence he can raise a block weighing slightly 
 less than 16 cwt. 
 
 342. The disadvantage of this system is, that 
 when it is in action, some of the pulleys soon 
 come up close to one another, and then no more 
 work can be done. Hence it is chiefly used 
 where a great resistance has only to be moved 
 through a small distance ; as in tightening the 
 rigging of ships. 
 
 System II. In this, as shown in fig. 63, (a) 
 some of the pulleys are in one block, A, which 
 is fixed ; the rest are in another block, B, which 
 is attached to the Resistance. 
 
 (6) There is only one cord, which goes round 
 all the pulleys ; one end of it is tied to one of 
 the blocks, the other end is attached to the 
 Power. 
 
 343. To calculate the mechanical advantage of this system, 
 which is the most useful and most generally employed of all, 
 we proceed on the same principle as before, of seeing how many 
 cords are employed in holding the resistance, and considering 
 that each must be shortened in some way or other, in order that 
 the resistance may be moved. 
 
 In the arrangement shown in the figure, there is one pulley in 
 the block to which the resistance is connected, and one end of 
 
 Fig. 63. 
 
MACHINES. 
 
 135 
 
 the cord is also tied to this block. Thus, the resistance is held 
 by 3 cords, and each of these must be shortened 1 inch when 
 the resistance is moved through 1 inch. Now, since the cord 
 going round the pulleys is all in one piece, this shortening of 
 the 3 parts of it can be effected by simply pulling the loose end 
 of it through 3 inches. Hence, if the power is attached to this 
 end, it will move through 3 inches while the resistance is moved 
 through 1 inch. That is, the mechanical advantage is 3. Thus 
 a man capable of lifting 1 cwt. directly, would, by pulling at P, 
 be able to lift a mass of 3 cwt. (nearly). 
 
 Rule for finding the mechanical advantage of a set of pulleys 
 arranged on the second system. Count the number of cords at the 
 block to which the resistance is attached ; this number will 
 express the mechanical advantage. 
 
 Example. In the set of pulleys shown in fig. 64, it is easy to 
 see that there are 7 cords at the block B. There- 
 fore the mechanical advantage will be 7. 
 
 344. This system of pulleys, although it does 
 not give as high a mechanical advantage as the 
 same number of pulley ' sheaves,' or wheels 
 arranged on the first system (as can be easily 
 seen by comparing the examples already given), 
 and although the loss of energy by friction is 
 very great, has yet the important advantage that 
 the resistance can be moved through a consider- 
 able distance ; in fact, until the two blocks come 
 in contact, or nearly so. It may be seen in use 
 almost wherever heavy masses have to be dealt 
 with, and moved about ; in quarries, in house- 
 building, in engine and other machinery shops, 
 in dockyards, in ships. In a large vessel, for instance, more 
 than a thousand pulley-blocks are required. 
 
 SECTION 3. THE WHEEL AND AXLE. 
 
 345. This machine consists of two cylinders of different sizes 
 fixed on the same spindle or axis, so that they turn together. 
 The larger one is called the wheel (see fig. 65), and generally 
 has a cord round it, to the end of which the Power is attached. 
 
136 ELEMENTARY DYNAMICS. 
 
 The smaller one is called the axle, and generally has a cord 
 round it, to which the Resistance is attached. The cords are 
 wound in opposite directions, so that when 
 \, x the machine is worked, one cord is wound 
 
 on, while the other is unwound from, its 
 cylinder. 
 
 346. The mode in which the relative dis- 
 tances moved through by the Power and 
 the Resistance are estimated, will be under- 
 stood from the following considerations : 
 Suppose that the machine is turned once 
 Fig. 65. round, so as to unwind some cord from the 
 
 wheel. Then the length of cord which is 
 thus removed from the wheel is equal to its circumference ; and 
 this expresses the distance moved through by the power. Also, 
 the length of cord which is at the same time wound up on the 
 axle, is equal to its circumference ; and this expresses the dis- 
 tance moved through by the resistance. If, for instance, the 
 circumference of the wheel is 12 inches, and the circumference 
 of the axle is 6 inches, then the power moves through J ^-, or 
 twice as much space as the resistance. Therefore the mechanical 
 advantage of such a wheel and axle would be 2 ; and a power 
 of 1 Ib. applied at P, would raise a resistance of 2 Ibs. (nearly) 
 applied at W. 
 
 347. We may find the mechanical advantage in an even 
 simpler way than by measuring the circumferences of the wheel 
 and of the axle. For, since the circumference of a circle is 
 always 3f times its diameter, or 6f times its radius, it will be 
 sufficient to measure the diameters or the radii of the wheel and 
 axle ; and the proportion between these diameters or radii will 
 just as truly represent the proportionate distances passed through 
 by the power and the resistance respectively. Thus, if the 
 diameter of the wheel is 4 times that of the axle, its circumfer- 
 ence will be 4 times as great also, and the mechanical advantage 
 of such a machine will be 4- 
 
 Rule for finding the mechanical advantage of a wheel and axle. 
 Divide the diameter of the wheel by the diameter of the axle; 
 or divide the radius of the wheel by the radius of the axle. 
 The quotient is the mechanical advantage. 
 
MACHINES. 
 
 137 
 
 Example. The diameter of the steering-wheel of a ship's 
 rudder is 3 feet ; the diameter of the axle is 4 inches. Here, the 
 diameter of the Wheel of the machine is 36 inches, and that of 
 the Axle is 4 inches ; hence * = 9 = the mechanical advantage, 
 so that a power of 1 Ib. applied at the rim to turn the wheel 
 round will give a strain of 9 Ibs. (nearly) on the ropes or chains 
 attached to the tiller. 
 
 348. The form of wheel and axle shown in fig. 65 is often very 
 widely departed from in actual practice ; so much so, that it is 
 sometimes hard to recognise a machine as really belonging to this 
 class. Perhaps the commonest example of a wheel and axle is a 
 window-blind, in which the blind is 
 raised by unwinding a string from a 
 deep-grooved pulley-wheel fixed at 
 one end of the blind-roller. Here, 
 the roller is an Axle, and the pulley is 
 a Wheel ; the resistance is the weight 
 of the blind. 
 
 Another common form is the cap- 
 stan, fig. 66, used on ships for pulling 
 up the anchor, tightening the moor- Fig. 66. 
 
 ing ropes, &c. In this, the Wheel is 
 
 not a complete one, all the rim being wanting, while the spokes 
 are there in the shape of wooden bars called ' handspikes,' 
 against the ends of which men push, instead of the power being 
 applied by a cord. 
 
 In the ordinary windlass, fig. 67, only one spoke, BC, of the 
 wheel survives, as it were. 
 To the end of this a handle, 
 D, is fitted, to which the 
 power is applied by alter- 
 nately pushing and pulling 
 it backwards and forwards, 
 up and down, 
 machine is used 
 buckets of water from deep 
 wells ; in raising coal and ore 
 from mines ; and, in conjunc- 
 tion with pulleys, for lifting materials to the top of scaffolding. 
 
 Such a 
 in raising 
 
 Fig. 67. 
 
138 
 
 ELEMENTARY DYNAMICS. 
 
 In the bicycle, fig. 68, the power is applied to a treadle, which 
 corresponds to the axle or smaller cylinder of fig. 65. Thus, 
 
 there is no mechanical 
 advantage ; the power of 
 the muscles is, in fact, 
 applied at a disadvantage, 
 and hence there is much 
 difficulty in working such 
 a machine along anything 
 but a hard level road. 
 But it has the practical 
 convenience that, by mov- 
 ing the foot through a 
 small distance, the edge 
 of the large wheel is made 
 to travel over a much 
 greater distance of road. 
 The crank of an engine, 
 fig. 69, or a lathe, or a 
 sewing-machine, acts on 
 the same principle. Other 
 common examples of the 
 -pj 68 application of the wheel 
 
 and axle are : A water- 
 wheel (fig. 70), a paddle-wheel of a steamer, a treadmill, a door 
 handle ; the ' rack and pinion ' used in raising the sluices of 
 locks, or raising the wick of a lamp, or working an air-pump, 
 
 I 
 
 Fig. 69. 
 
 Fig. 70. 
 
 fig. 71. In this form, the circumference of the axle is cut into 
 teeth or ' cogs ; ; it is then called a pinion, and these cogs catch 
 
MACHINES. 
 
 139 
 
 against similar cogs cut in a straight rod called the rack, which 
 
 takes the place of the more usual cord attached to the resistance. 
 
 349. To obtain greater mechanical advantage, several wheels 
 
 and axles are sometimes connected together by cutting cogs in 
 
 Fig. 71. 
 
 Fig. 72. 
 
 the circumference of the first axle, so as to work or 'gear' into 
 cogs cut in the circumference of the next wheel. The axle of 
 this latter may be made similarly to gear into another wheel, 
 
 Fig. 73. 
 
 and so on, as shown in fig. 72. Such an arrangement is called 
 a ' train of wheels,' and is used for two distinct purposes. 
 
 (1) To gain power : for which purpose the power is applied to 
 the wheel, A, of the first axle, and the resistance is connected 
 
140 ELEMENT ARY DYNAMICS. 
 
 with the axle, E, of the last wheel. The powerful windlass 
 called a crab, fig. 73, is an example of this. 
 
 (2) To gain speed. In this case, as shown in fig. 72, the power 
 is applied to the axle, E, of the last wheel, and the resistance 
 is connected with the wheel, A, of the first axle. The wheel- 
 work of clocks and watches affords an excellent illustration of 
 this principle. In an ordinary watch, the barrel containing the 
 spring only makes 3| turns in 24 hours, while the seconds hand 
 makes 1440 turns. 
 
 350. Wheels and axles may also be connected by an endless 
 
 cord or strap, passing round 
 both, and tightened until 
 there is sufficient friction to 
 prevent its slipping. Lathes 
 and sewing-machines are 
 worked in this way, and the 
 various machines in a mill 
 are driven by straps passing 
 round pulleys or ' drums/ on 
 a long shaft worked by the engine, fig. 74. 
 
 SECTION 4. THE LEVER. 
 
 351. There are many cases in which we only want to move a 
 body through a short distance ; as when a large paving-stone 
 has to be lifted a few inches for street repairs. It would be a 
 cumbrous expedient to set up pulleys or a windlass for such a 
 purpose, and a much simpler machine called a lever (Lat. levare, 
 ' to raise') is generally used. It may be looked upon as essen- 
 tially a wheel and axle intended only to turn a little way 
 round. 
 
 352. A lever (see figs. 75, 76, 77) is a strong, stiff rod, mov- 
 able at one point round a firm support, called a fulcrum (Lat. 
 fulcire, to support). The Power and the Resistance are applied 
 at two different points on this rod. 
 
 353. There are three different orders of levers, the distinction 
 between them depending on whether the fulcrum, the Resistance, 
 or the Power, is put between the two ends of the rod. Thus we 
 have 
 
MACHINES. 
 
 In the First Order, fig. 75 : 
 
 Power at one end ; 
 Resistance at the other end ; 
 Fulcrum between them. 
 
 141 
 
 Fig. 75. 
 
 In the Second Order, fig. 76 : 
 
 Power at one end ; 
 Fulcrum at the other end ; 
 Resistance between them. 
 
 In the Third Order, fig. 77 : 
 
 Resistance at one end ; 
 Fulcrum at the other end ; 
 Power between them. 
 
 Fig. 77. 
 
 354. The lever is considered as made up of two parts which 
 are called the arms : the length of these being in all cases 
 measured from the fulcrum. Thus, 
 
 The distance from the Power to the fulcrum is one arm. 
 ti it it Resistance n u the other arm. 
 
 [In the following examples, the lever will be assumed in all cases 
 to have its arms equally balanced before any work is done ; so that 
 the weight of the rod itself may be neglected.] 
 
 355. The mechanical advantage of the lever is calculated on 
 the same principle as that of the wheel and axle ; one arm of the 
 lever being considered as the radius of the wheel, and the other 
 arm as the radius of the axle. Thus, if the Power-arm is twice 
 
 j 
 
142 
 
 ELEMENTARY DYNAMICS. 
 
 Fig. 78. 
 
 as long as the Resistance-arm, as in fig. 78, the Power will 
 move through twice as much space as the Resistance, when 
 the lever is being used ; and there will be a mechanical advan- 
 tage of 2. 
 
 Rule for finding the mechani- 
 cal advantage of any lever. 
 Divide the length of the Power- 
 arm by the length of the Resist- 
 ance-arm ; the quotient is the 
 mechanical advantage of the 
 lever. 
 
 Example^ In a lever of the first order, the power-arm is 30 
 inches long, the resistance- arm is 6 inches long ; what work 
 could be done with a power of 2 Ibs. ? Dividing 30 by 6, we 
 have a quotient of 5 as the mechanical advantage : so that a 
 resistance of (5 x 2 Ibs. = ) 10 Ibs. could be moved. 
 
 356. It must be observed that the ends of the arms of the lever 
 move in the circumferences of circles of which the fulcrum is the 
 
 centre, as indicated in fig. 
 79 ; and that, when the 
 resistance is simply a 
 weight to be lifted straight 
 up, most of the work is 
 done along a perpendicular 
 line, AB, passing through 
 its centre of gravity. In 
 such cases it is easily seen 
 that the real acting arm of 
 the lever is not FR, but the 
 length of the shortest line 
 Fig. 79. that can be drawn from the 
 
 fulcrum to the perpen- 
 
 dicular AB that is, the line FB/ at right angles to AB. This line 
 is generally shorter than FR, and so the mechanical advantage is 
 greater in this position of the lever. As the lever comes up to the 
 horizontal position, the length of the acting arm FR' increases until 
 it becomes equal to FR : if the lever is moved farther up, FR' 
 diminishes again. The same is true, of course, of the other arm, 
 when the power can only act in one direction ; but when the power 
 is applied by the hand, it is continually changing its line of action, 
 
MACHINES. 
 
 143 
 
 as when a windlass handle is worked, and then the acting length of 
 the power-arm remains the same. 
 
 Moment of a Force. 
 
 357. When a force is used to turn a body round some point as a 
 centre, the length of such, a line as FR/ in fig. 79 that is, of the 
 shortest line which can be drawn from the centre of rotation to the 
 line of action of the force expresses what is called the moment of the 
 force ; that is, its effect in turning the body round the given centre. 
 
 Levers of the First Order. 
 
 358. There are many common examples of levers of the first 
 order. For instance, a poker ; 
 
 the Power being the muscular 
 
 energy of the arm and hand, the 
 
 fulcrum being the bar of the 
 
 grate, while the Kesistance is the 
 
 piece of coal to be lifted ; a 
 
 crowbar, when used as shown in 
 
 fig. 80, to lift heavy weights ; a spade, when used to detach a 
 portion of earth from the main mass 
 by forcing back the handle ; a pair 
 of scissors, or pincers, in which we 
 
 Fig. 80. 
 
 Fig. 81. 
 
 Fig. 82. 
 
 have a double lever, the common fulcrum being the joint on 
 which the blades turn ; a pump handle, fig. 81. 
 
 359. In the common balance, or * pair of scales,' fig. 82, we 
 
144 ELEMENTARY DYNAMICS. 
 
 have a lever with arms of equal length, so that there is no 
 mechanical advantage. This lever is called the beam, and from 
 its ends, at precisely equal distances from the middle point 
 or fulcrum, are hung pans or scales. It is used in 'weighing' 
 a thing that is, in finding out how much the earth attracts it, 
 by balancing it against some standard weight, such as a pound. 
 For when the thing to be weighed is put into one scale, there 
 can only be equilibrium that is, the beam can only remain 
 horizontal when there is an equal weight in the other scale, 
 because the arms of the lever from which the scales are hung 
 are made exactly equal in length. 
 
 360. In the 'steelyard,' fig. 83, we have a lever in which one 
 of the arras can be made to vary in length. Only one weight 
 
 is used, which is made 
 to slide along the beam ; 
 and a substance is 
 weighed by putting it 
 into the scale "W, and 
 seeing where the weight 
 P has to be placed on 
 the beam in order to 
 balance it. Suppose that 
 the weight P is 1 Ib. ; 
 83. then if it is put just as 
 
 far from C, the fulcrum, 
 
 as A is from C on the other side, the arms of the lever are of 
 equal length, and if the substance put into the scale produces 
 equilibrium, it must weigh 1 Ib. If the substance just balances 
 P when the latter is placed on the beam twice as far from C as 
 C is from A, it must weigh 2 Ibs., for the power-arm PC is now 
 twice as long as the resistance-arm CA, and there is a mechanical 
 advantage of 2. The beam is graduated as shown in the figure, 
 and stamped with numbers which show the actual weight of the 
 substance in the scale-pan corresponding to different positions of 
 P. Letter- weighing machines are often made on the same 
 
 principle. 
 
 Levers of the Second Order. 
 
 361. In these, as is obvious from fig. 84, there must always be 
 a mechanical advantage, because the Power is always farther 
 
MACHINES. 
 
 145 
 
 from the fulcrum than the Eesistance is. The amount of 
 mechanical advantage is calculated in the same way as for 
 levers of the first order ; it being borne 
 in mind that the power-arm is the dis- 
 tance from P to F, and the resistance- 
 arm is the distance from R to F. 
 
 For example, in one of such levers, 
 the distance from the power to the 
 fulcrum is 30 inches, and the distance 
 from the resistance to the fulcrum is 
 6 inches. Then, 
 
 Fig. 84. 
 
 Length of power-arm _ 
 Length of resistance-arm 
 
 _ 3 _ 
 
 ~~ ~^~ ~ 
 
 So the mechanical advantage is 5 ; and a man who exerted a 
 force which would lift 1 cwt. directly, could by this lever raise 
 a weight of 5 cwt. (nearly). ^ 
 
 362. As examples of levers of the second order, the following 
 may be taken : A crowbar, when 
 used as shown in fig. ^85. An oar ; 
 in which the fulcrum is the water 
 against which the blade of the oar 
 presses, the power is applied by the 
 hand at the inner end of the oar, 
 and the resistance is the inertia and 
 
 Fig. 85. 
 
 friction of the boat (and in some 
 cases, of course, the force of the 
 stream) applied at the rowlock. A 
 wheel-barrow, fig. 86 ; in which the 
 
 fulcrum is the 
 
 axis of the wheel, 
 
 the power is ap- 
 -p. 87 plied near the end 
 
 of the handle, and 
 the resistance is the weight of the barrow and the load in it. A 
 door or gate ; in which the fulcrum is the hinges, the power 
 is applied in opening or shutting the door, and the resistance 
 is the inertia and friction of the door (gravitation, except in 
 the case of a trap-door, not being directly concerned). A pair 
 
 Fig. 86. 
 
146 
 
 ELEMENTARY DYNAMICS. 
 
 of nutcrackers, fig. 87, which, as is easily seen, are a double 
 lever ; the fulcrum being the joint which connects the arms, 
 the power is applied by the hand pressing the arms together, 
 and the resistance is the cohesion of the nut. 
 
 363. The action of this kind of lever is also shown when 
 a load is being carried on a 
 pole by two men, each bearing 
 one end of the pole, fig. 88. Each 
 man acts as the power in lifting 
 the weight, and at the same time 
 the shoulder of each serves as a 
 fulcrum for the lever worked by 
 the other. If the weight hangs 
 fairly from the middle of the pole, 
 each man will bear just half the 
 burden ; but if the weight is slipped along towards one 
 end, then the man to whom it is nearest supports a greater 
 load than the other. 
 
 Levers of the Third Order. 
 
 364 In these, as shown in fig. 89, there obviously can never 
 be any mechanical advantage at all, since 
 the resistance, being at the end, is always 
 farther from the fulcrum than the power, 
 and therefore it must move through a 
 greater space than the power. Hence the 
 power must always necessarily be greater 
 than the resistance, and this kind of lever 
 is chiefly used where there is plenty of 
 energy available, and it is required to 
 move anything through a large space. 
 The { mechanical disadvantage/ as it may be called, is found 
 
 in the usual way ; the power-arm being the distance from 
 
 P to F, and the resistance-arm the distance from R to F. 
 
 Suppose, for instance, that the length from P to F is 4 inches, 
 
 and from R to F 20 inches, then, 
 Power-arm 
 
 Fig. 89. 
 
 Resistance -arm 
 
- Ji B &: U7 
 
 So that the resistance can only He TJJsbf \t5$ 'power, or the 
 power must be 5 times the resistance. , put this resistance 
 will be moved through 5 times as nitteh. space as the power 
 passes through. 
 
 365. Nearly every joint in the human body affords an 
 illustration of this kind of lever. 
 Thus the principle of the action 
 of the elbow-joint, when the arm 
 is bent, is shown in fig. 90. The 
 fulcrum is the elbow-joint itself, 
 at F ; the power is supplied by 
 the forcible contraction of a strong 
 muscle (the biceps), one end of 
 which is attached by tendons to a 
 bone of the forearm, B, near the 
 joint, while its other end is 
 attached to the shoulder at S (in 
 
 Jft 
 
 Fig. 90. 
 
 the figure a weight is shown, passing over a pulley at S, 
 merely to illus- 
 trate the mode of 
 action); the resist- 
 ance is anything 
 to be moved by 
 the hand. The 
 actual arrange- 
 ment of bones and 
 muscles of the arm 
 is shown in fig. 
 91. The work of 
 straightening out 
 the arm when 
 bent is done by a 
 
 muscle lying behind the elbow-joint, and passing over it, as 
 over an axle : so the extension of the arm is effected by a 
 machine on the principle of the wheel and axle. 
 
 Since nearly all the muscles are attached very near the 
 joints of the limbs they move, the power-arms of these 
 human levers are very short as compared with the resistance 
 arms: and hence the energy actually concerned in doing 
 
 Fig. 91. 
 
148 ELEMENTARY DYNAMICS. 
 
 such work as the strength of a man enables him to do, must 
 be enormous. Take, for instance, the work implied in lifting 
 1 cwt. from the ground by bending the arm at the elbow. 
 The distance from the middle of the hand to the elbow- 
 joint may be taken as about 15 inches : the biceps muscle 
 is attached about 1 inch from the joint. Hence the mechanical 
 disadvantage is & that is, the muscle in contracting must 
 exert a force which would raise 15 cwt. if applied directly. 
 
 On the other hand, since the space moved through by the 
 resistance is comparatively very great, we gain from this 
 arrangement of the limbs and muscles the real and important 
 advantage of being able by a small movement of the muscles 
 to move the limbs far and quickly ; as in hitting, throwing, 
 running, playing the piano, &c. 
 
 366. A cricket-bat is another example of a lever of the third 
 order ; acting, in fact, in some cases like a mere prolongation of 
 the arm ; though more often, as in blocking a ball, the left hand 
 is held steady as the fulcrum, and the power is applied chiefly 
 by the right hand pressing on the bat farther down the handle. 
 In the use of a spade for throwing up soil, and of a pitchfork 
 for lifting hay, a similar action is observable, one hand serving 
 as the fulcrum while the power is applied mainly by the other. 
 In practice, however, both hands move a little in all the above 
 cases, so we have an action of levers of both the first and the 
 third order. In using a pencil or pen, and a pair of sugar-tongs 
 or ordinary tongs, force is applied to levers of a similar kind. 
 
 SECTION 5. THE INCLINED PLANE. 
 
 367. This may be described as a smooth, flat surface, placed 
 
 obliquely to the direction in 
 which the* resistance has to be 
 moved. Thus, to take the 
 commonest case, that of a 
 weight such as W, fig. 92, to 
 be lifted vertically upwards, 
 Fig. 92. I represents an inclined plane 
 
 making some angle less than 
 
 right angle with a vertical direction. 
 
 Suppose that the resistance is a weight of 1 lb., and that it 
 
MACHINES. 149 
 
 has to be raised to a level 1 foot above the place where it is. 
 Then 1 foot-lb. of work is required ; and this may be done 
 either (a) by a power sufficient to lift it straight up ; or (6) by 
 a smaller power moving it up a long slope to the required level. 
 If the slope is 2 feet long, then the power moves through 2 feet 
 instead of 1 foot in raising the weight ; so that a mechanical 
 advantage of 2 is gained. 
 
 368. It is clear that the longer the plane is, compared with 
 the distance through which the resistance is moved in the 
 required direction, the greater will be the mechanical advantage 
 gained ; and the way of calculating its exact value will be seen 
 from what follows. 
 
 Let A, fig. 93, be the end of the plane where the resistance is 
 placed. Through the other 
 end, B, draw a line, CB, in 
 the direction in which the 
 resistance is to be moved ; and A 
 from A draw AC perpendi- 
 cular to BC. Then BC is the 
 distance through which the re- Fig. 93. 
 
 sistance is actually moved in 
 
 the required direction by the power. It is generally called the 
 'height' of the inclined plane (although, of course, the machine 
 will act in any direction, and not merely in lifting bodies). 
 
 Now, when the power acts along the plane as in fig. 92, the 
 length of the plane is the distance through which it moves, and 
 the 'height' of the plane is the distance through which the 
 resistance is moved in the required direction. Hence we get 
 the following rule : 
 
 Rule for finding the mechanical advantage of an inclined plane, 
 when the power acts in the direction of the length of the plane. 
 Divide the length of the plane by its height ; the quotient is the 
 mechanical advantage. 
 
 Example. An inclined plane is 18 inches long, and 6 inches 
 high ; what will be the mechanical advantage ? 
 
 Length of plane _ j a. _ 3 
 Height of plane ~~ 
 
 Hence the mechanical advantage will be 3. 
 
150 
 
 ELEMENTARY DYNAMICS. 
 
 Fig. 94. 
 
 369. The commonest example of an inclined plane is a road 
 
 leading up a hill. 
 A carriage drawn 
 along it is raised 
 more and more 
 from the centre 
 of the earth, and 
 hence a greater 
 power must be 
 used than is 
 necessary to draw 
 it along a level 
 road. Sometimes 
 the road is made 
 to wind as in 
 fig. 94, instead of 
 going straight up 
 
 the hill, in order to make the inclined plane longer, and thus 
 increase the mechanical advantage : so that a single horse may 
 be able to draw a heavily-loaded waggon up it. 
 
 The slope of the road is generally expressed by saying 
 how much it rises in a certain length. For example an 
 incline of ' I in 100 ' means that if the plane were 100 feet 
 long, the top would be 1 foot above the level of the lower 
 end ; and in this case the mechanical advantage would be 
 100, whatever the actual length of the plane might be. In- 
 clines on a railway seldom exceed a ' gradient 'of 1 in 60 ; 
 but on ordinary roads the gradient is sometimes as steep as 
 1 in 7 or 8. 
 
 370. In order to raise a heavy cask into a cart, an inclined 
 plane is often extemporised by laying planks sloping upwards 
 from the ground to the bottom of the cart, up which the cask 
 is rolled. The immense stones of which the Pyramids are built, 
 were probably raised up to their places by a contrivance of this 
 kind. 
 
 371. Sometimes the power is applied, not along the plane, 
 but in a direction at right angles to that in which the resistance 
 is to be moved that is, in the direction of the line AC, fig. 93, 
 which is often referred to as the ' base ' of the plane. Then, of 
 
MACHINES. 
 
 151 
 
 Fig. 95. 
 
 course, the power moves through the distance AC in moving the 
 resistance through the distance CB, and, 
 
 Base of the plane , . 
 
 - . ,. ,-,, the mechanical advantage. 
 Height of the plane 
 
 A good example of this is afforded by the ' wedge,' fig. 95, 
 which really consists of two inclined planes 'put 
 base to base. The resistance is put at the thinnest 
 part of the wedge, A, and the power is applied 
 in the direction of the base CA. It moves the 
 resistance by shifting the plane along it ; so the 
 wedge is simply a movable inclined plane. 
 
 372. To find the mechanical advantage of a 
 wedge, we must observe that the resistance is 
 moved through the space BB', while the power 
 moves through the space CA. Hence 
 
 Rule for finding the mechanical advantage of 
 a wedge. Divide the length of the wedge, measured 
 along its middle line, by its thickness at the blunt end ; the 
 quotient is the mechanical advantage. 
 
 Example. If the length of a wedge is 6 inches, and its 
 greatest thickness is 1 inch, then f = 6. So the mechanical 
 advantage is 6. 
 
 Usually the wedge is made very long in proportion to its 
 thickness, so the mechanical advantage gained is theoretically 
 very great ; but a large amount of energy is spent in overcoming 
 the friction between the wedge and the material 
 into which it is being driven,* so that practically 
 not more than half the proper mechanical advantage 
 is gained. 
 
 373. A common use of the wedge is for splitting 
 timber, as illustrated in fig. 96 ; also for detaching 
 blocks of stone in quarries, a series of wedges being 
 driven in along the line in which the stone is to be 
 split off from the main mass. Ships are raised in 
 docks by driving in wedges under them. 
 
 374. But there are still commoner instances of its 
 
 Fig. 96. 
 
 * This friction, however, serves a useful purpose, since the wedge thus holds its 
 place in the material, so that it can be driven farther and farther in. 
 
152 ELEMENTARY DYNAMICS. 
 
 employment, for a little observation will show that all cutting 
 and boring instruments, ploughs, axes, chisels, saws, knives, 
 scissors, &c., act on the principle of this simple ' machine ' ; the 
 finest surgeon's lancet no less than the roughest pickaxe. 
 Indeed, the wedge not only cuts our materials into shape, 
 but also holds them together when shaped ; for nails 
 and pins are only square or round wedges, driven in and 
 holding their places by friction. 
 
 375. Another modification of the inclined plane is the 
 screw, fig. 97, which consists of a cylinder with a raised 
 ridge, called the * thread,' running round it in a spiral 
 Fig. 97. line. It works in a hollow cylinder called the nut, 
 with spiral grooves cut in it which fit the threads of the 
 screw. When either the nut or the screw is turned (the other 
 being fixed), the nut travels slowly from end to end of the 
 screw. 
 
 376. The screw is really an inclined plane wound round a 
 cylinder (somewhat like the road in fig. 94), as indicated in 
 fig. 98. If the distance between one thread and the next 
 
 Fig. 98. 
 
 (which is called the 'pitch' of the screw) is small, the slope 
 of the plane is very gradual, and a great mechanical advantage 
 is gained. To calculate this, we must notice that the * base ' 
 of the inclined plane is the end of the 
 cylinder on which the screw is formed, or 
 any surface at right angles to the length of 
 the cylinder. Thus the power which turns 
 the screw or the nut is applied parallel to 
 the base of the plane. Now suppose that the 
 nut is attached to the resistance, and pre- 
 vented from turning while the power is 
 applied to the screw. When the screw has 
 been turned once round, the nut (and with it the resistance) 
 
MACHINES. 
 
 153 
 
 has been moved through a distance equal to that between one 
 thread and the next (called the 'pitch' of the screw), while 
 the power (if it acts at the edge of the cylinder) has moved 
 through a space equal to the circumference of the cylinder. 
 But the power is, in practice, applied, not to any point on the 
 cylinder, whether in the nut or screw, but to the edge of a 
 wheel, or the end of a handle fixed to the screw or nut, as in 
 fig. 99 ; and so the circumference of this wheel, or of the circle 
 in which the end of the handle moves, must be taken into 
 account in calculating the mechanical advantage. 
 
 Rule for finding the mechanical advantage of the screw. 
 Divide the length of the circumference of the circle in which 
 the power moves by the distance between one thread of the 
 screw and the next thread; the quotient is the mechanical 
 advantage. 
 
 Example. In a screw-press, such as that shown in fig. 100 
 below, the length of the handle is 21 inches, and the distance 
 between two threads of the screw that is, the 'pitch' of the 
 screw is 1 inch : find the mechanical advantage. 
 
 Here the radius of the circle in which the power moves 
 is 21 inches ; therefore the circumference of the circle will 
 be 6?- times this, or (6f x 21 = ) 132 inches. Then, 
 
 Circumference of circle _ 13a _ __ 
 Pitch of screw T 
 
 So the mechanical advantage is 132. 
 
 Fig. 100. 
 
 Fig. 101. 
 
 By making the pitch of the screw very fine, and applying 
 the power at the end of a long lever, an enormous mechanical 
 
154 ELEMENTARY DYNAMICS. 
 
 advantage can be obtained ; at the same time, however, nearly 
 one-half of this is lost by friction. 
 
 377. One useful application of the screw is seen in the 
 'screw-press,' fig. 100, in which the nut, H, is fixed in a strong 
 frame, and the screw, A, is turned round by a lever placed in 
 holes shown at B. As the screw descends, its lower end presses 
 upon a thick plate, C, below which are placed the books or 
 packets of paper which are to be compressed. A similar 
 kind of press is used in cheese-making and cider-making. 
 Coins are stamped by the more powerful press shown in 
 fig. 101, in which heavy weights, L, L, are attached to the end 
 of the levers which work the screw. These levers are whirled 
 quickly round, and the energy thus accumulated in the 
 mass is expended in one powerful blow upon the dies 
 between which the blank disc of metal is compressed. 
 
 378. Simpler illustrations of the screw are 
 found in the 'bolts and nuts,' fig. 102, used 
 for clamping together the beams of a roof 
 and the various parts of a machine. About 
 800 of these are used in the construction of 
 a single locomotive engine. In the common 
 F'w~i02 ' wood-screw,' fig. 103, the threads are thin andjv^ -.QO 
 
 ' sharp, so that they cut into the wood like a 
 knife, and make grooves for themselves as the screw is 
 turned round by the screw-driver. 
 
 SECTION 6. 
 
 379. We may, in conclusion, turn our attention for a moment 
 to one of the finest of modern machines, the locomotive engine 
 (see frontispiece), and see how it illustrates in its construction 
 and work the properties of matter, the laws of motion, and the 
 principles of machines, which have been treated of in the fore- 
 going pages. 
 
 380. The three states of matter are illustrated in the solid 
 framework (the skeleton, as it were) of the engine, the water in 
 the boiler, and the steam into which it is converted by the heat 
 of the fire. 
 
 381. The various kinds of cohesion of matter are shown in 
 
MACHINES. 155 
 
 the strong steel piston rods, the tough iron framework, the 
 elastic springs, the perfectly elastic steam. 
 
 382. The weight of modern engines is enormous. While 
 Stephenson's original engine, the 'Kocket,' only weighed four 
 tons and a quarter, an express engine, as now made, weighs forty 
 or even fifty tons. A single look will show how even the 
 strongest steel rails, on the most solidly constructed road, bend 
 under its weight. 
 
 383. Owing to this great mass, the momentum of an engine, 
 especially when travelling at a high velocity, is very great 
 indeed. Engines have been known to start off of their own 
 accord, through some negligence, and go right through the 
 brick wall of the engine-shed with perfect ease, although the 
 speed attained could not have been high. 
 
 384. Yet an engine cannot of itself move a single inch. 
 Energy must be supplied in some form ; and this is done by the 
 burning of the fuel that is, its combination with the oxygen of 
 the air. Energy is thus made available in the form of heat, 
 which is transferred to the water, turning it into steam, putting 
 its molecules, in fact, into such rapid motion that they press 
 strongly against the sides of the boiler and against the pistons 
 of the engine. These latter are driven to and fro, and through 
 the medium of the piston rods and connecting rods turn the 
 cranks round, and with them the driving-wheels with which 
 they are connected. Thus the straight to-and-fro motion 'caused 
 by the steam is converted first into rotatory motion of the wheels, 
 and then into continuous motion in one direction of the engine 
 and its load along the line. 
 
 385. The force thus continuously acting causes a velocity 
 which would, if it were not for opposing forces, increase continu- 
 ously (uniformly accelerated motion) ; and when once the inertia 
 of the mass is overcome, the whole would go on moving, 
 according to the first Law of Motion, even though the steam 
 was shut off. But owing to resistance of the air and friction 
 of the numerous moving parts, the velocity is not accelerated 
 beyond a certain point, where the various forces are in equili- 
 brium. It then becomes uniform ; and if the steam is shut off, 
 it is retarded and ultimately ceases. If more friction is brought 
 into action by means of the brake, the energy of the moving 
 
156 ELEMENTARY DYNAMICS. 
 
 mass passes still more quickly into the form of heat, as the sparks 
 flying from the brake-blocks abundantly testify. 
 
 386. The laws of Centrifugal Tendency are illustrated by the 
 pressure against the outside rail when the engine goes round a 
 curve ; a pressure which, at high velocities, may become so great 
 as to displace the rail or cause the wheel to override it. Hence 
 this rail is raised so as to tilt the engine and throw the centre of 
 gravity so far inwards that one of the components into which 
 the force of gravitation may be then resolved, pulls the 
 whole mass inwards sufficiently to counteract the centrifugal 
 tendency. 
 
 387. The Composition of Forces is also illustrated in cases 
 where the engine is going up or down an incline. Here gravita- 
 tion, which primarily is a single force pulling the mass straight 
 downwards, is resolved into two components, one of which acts 
 along the plane and either accelerates or retards the motion of 
 the train. 
 
 388. The third Law of Motion is illustrated by the action of 
 the steam in the cylinder. It presses just as strongly against the 
 ends of the cylinder as against the piston, and only moves the 
 latter because it is most easily moved. When the engine runs 
 up against an obstacle, such as a mass of earth or brick, or 
 another train, it suffers damage itself as well as the obstacle, and 
 the comparative masses and momenta of the two decide which 
 suffers most. If the obstacle is relatively light, such as a block 
 of stone, or a gate, or a cow, it is usually cleared out of the way 
 by the massive engine. 
 
 389. In order that the engine may run steadily and safely, it is 
 essential that the wheel-base should be broad and the centre of 
 gravity as low as possible. Hence the boiler is made in the 
 form of a long horizontal cylinder, not placed over the furnace 
 as usual, but nearly on the same level with it ; the water being 
 heated by a multitude of tubes within the boiler through 
 which the flame passes. The machinery is all packed below the 
 boiler, as near the rails as safety will allow. Thus the equili- 
 brium is extremely stable, and the engine will run through a 
 hurricane without being blown over. 
 
 390. Every kind of Mechanical Power, except perhaps the 
 Pulley, is fully employed in the engine. The Wheel and Axle 
 
MACHINES. 157 
 
 appear in the cranks and wheels (the crank being technically 
 the Axle, and the driving-wheel the Wheel of the machine) ; 
 also in the handles of the numerous stopcocks employed. The 
 Lever appears in the starting lever, the reversing lever, the 
 1 link motion ' which works the slide valves for admitting steam 
 to the cylinders, and the long arm by which the safety valve is 
 held down. The Inclined Plane is seen in the wedges or 
 ' cotters ' which tighten up the bearings ; and in the multitude 
 of screws employed throughout the machine. 
 
 391. As an illustration of the principles of Energy and Work, 
 the locomotive has been already often referred to. It has been 
 shown that the energy stored up in a statical form in the coal and 
 the oxygen of the air becomes kinetic when they combine ; at 
 first appearing in the form of heat, then as swift mechanical 
 motion of the whole mass. While the engine is running, energy 
 is constantly being transferred to the molecules of the air, of the 
 rails, and of the machinery ; appearing, in fact, eventually as 
 heat, which unfortunately is lost to us as far as useful work is 
 concerned. 
 
 392. It is disappointing to reflect how very imperfect a 
 machine even the best steam-engine is. Out of the whole 
 energy set free by the combustion of the coal, not more than one- 
 sixteenth is, in the most scientifically constructed engine, con- 
 vertible into useful mechanical work. All the rest is dissipated 
 and lost (to us) in the form of heat. 
 
 393. It is, in fact, easy to convert energy of mechanical motion 
 entirely into the form of heat ; but it is impossible, with our 
 present means, to convert heat-energy entirely into mechanical 
 motion. Thus, during all movements, whether of the human 
 body or of inanimate machines, some energy is continually pass- 
 ing into a less available form, namely, heat ; and the investiga- 
 tions of science teach us that, if the present course of things 
 continues, we must look forward to a time when so much of the 
 energy present on the earth will have been degraded, as it were, 
 into the form of heat, that all things will be at the same 
 extremely high temperature, and life and work will be 
 impossible. 
 
158 ELEMENTARY DYNAMICS. 
 
 APPENDIX. 
 
 THE METRIC SYSTEM OF MEASURES AND WEIGHTS. 
 
 Measure means the space over which anything extends. 
 
 Weight means the pressure of bodies towards the centre of 
 the earth caused by the force of gravitation. 
 
 The sizes and weights of things are usually expressed in terms of 
 some ' unit ' or standard amount of space or pressure, such as a 
 'foot,' a 'yard,' a 'metre,' a 'pound,' or a 'gramme.' Thus, in 
 saying that a rod is six feet long, we mean that it extends in length 
 over six times the space of the unit of length which we call a 
 ' foot.' Again, in saying that a piece of lead weighs two pounds, we 
 mean that it presses towards the earth's centre with twice as much 
 force as a particular piece of matter which we call a 'pound- 
 weight.' 
 
 In selecting a unit for practical purposes, we are mainly guided 
 by three considerations : 
 
 1. The unit must be of such a kind that another exactly similar 
 one could be easily obtained, if the original unit was lost or 
 damaged. 
 
 2. It must not be very large or very small ; otherwise in common 
 use we should constantly have to deal with awkward fractions or 
 inconveniently large multiples of it. 
 
 3. It must have other measures and weights derived from it by 
 the simplest possible methods of multiplication and division. 
 
APPENDIX. 159 
 
 The unit, or starting point, of the metric system (which is now 
 almost universally employed in scientific work) is the METRE, 
 which is a length of one forty- millionth part of the circumference 
 of the earth, measured under the meridian of Paris. 
 
 The actual metre is a flat bar of platinum, about 39-4 inches long, 
 each end of which is exactly at right angles to the length of the 
 bar ; the distance between the ends at the temperature of freezing 
 water is defined to be one metre. Several extremely accurate copies 
 of this have been made, and it is probable that scientific men will 
 be content with these copies, and other copies of them, without 
 again deriving the unit from an actual measurement of the earth. 
 
 MEASURES OF LENGTH. 
 
 In deriving other measures of length from the metre, only the 
 number 10 and its multiples are employed ; and names are selected 
 which denote the relation of the particular measure to the unit. 
 
 Thus, the next larger measure is a length ten times that of the 
 metre, and is called a decametre (Gr. Sex, deca, ten). The 
 next larger measure is a length 100 (that is, 10 x 10) times that of 
 the metre, called a hectometre (Gr. 'lx,a<rov, hecaton, a hundred). 
 The largest measure practically used, is a length of 1000 (that is, 
 10 x 10 x 10) metres, called a kilometre (Gr. #/A/a, chilia, a 
 thousand). 
 
 Similarly, for the smaller measures, we have a length of T V of a 
 metre, called a decimetre (Lat. decem, ten) ; a length of ^ of a 
 metre, called a centimetre (Lat. centum, a hundred) ; and a length 
 f TTnnr of a metre, called a millimetre (Lat. millc, a thousand). 
 
 Thus, the names of all the measures larger than the unit, are con- 
 structed by adding to the name of the unit a prefix derived from 
 a Greek numeral ; the names of the measures smaller than the unit, 
 are obtained by adding to the name of the unit a prefix derived from 
 a Latin numeral. 
 
 A complete table of the measures of Length is given below, and 
 fig. 104 (next page) shows the actual length of a decimetre, which 
 is divided into centimetres and millimetres ; a scale of English 
 inches is added for comparison, by which it is seen that a deci- 
 metre is very nearly equal to four inches. 
 
160 
 
 ELEMENTARY DYNAMICS. 
 
 I 
 
 TABLE OF THE MEASURES OF LENGTH. 
 ( The usual abbreviations are put in brackets. ) 
 
 Kilometre = lOOOmetres. 
 
 Hectometre = 100 .. 
 
 Decametre = 10 ,, 
 
 METRE (m.) = 1 metre. 
 
 Decimetre = 0-1 
 
 Centimetre (cm.) = 0-01 
 
 Millimetre (mm.) = 0-001 .. 
 
 The table may also be put in the following form : 
 
 10 millimetres = 1 centimetre. 
 
 10 centimetres = 1 decimetre. 
 
 10 decimetres = 1 metre. 
 
 10 metres = 1 decametre. 
 
 10 decametres = 1 hectometre. 
 
 10 hectometres. = 1 kilometre. 
 
 MEASURES OF VOLUME. 
 The unit of Volume or capacity, is a cube, each 
 side of which measures 1 decimetre ( in other words, 
 ' one cubic decimetre ' ). It is called a LITRE ; 
 and from it the larger and smaller measures of 
 volume are derived in precisely the same way as 
 those of length are derived from the metre. Their 
 names are also given on a similar principle (Greek 
 prefixes being used for the larger, Latin prefixes 
 for the smaller measures ) ; and the following table 
 hardly requires further explanation. 
 
 TABLE OF THE MEASURES OF VOLUME. 
 ( The usual abbreviations are put in brackets. ) 
 
 Kilolitre = 1000 litres. 
 
 Hectolitre = 100 n 
 
 Decalitre = 10 
 
 LITRE = 1 litre. 
 
 Decilitre = 0-1 .. 
 
 Centilitre = 0-01 it 
 
 Mfflilitre ( or cubic centimetre ) ( c. c. ) . . . = 0-001 
 
 It should be noted : 
 
 1. That the name 'cubic centimetre' is almost universally used 
 instead of ' millilitre ' ( a cubic centimetre being readily demonstrated 
 to be T^Vo- of a cubic decimetre, or litre). 
 
 Fig. 104. 
 
APPENDIX. 161 
 
 2. That quantities smaller than the litre are usually expressed in 
 cubic centimetres. Thus three-fourths of a litre would be expressed, 
 not as 7 decilitres 5 centilitres, but as 750 cubic centimetres. 
 
 MEASURES OF WEIGHT. 
 
 The unit of Weight is the weight of one cubic centimetre ( milli- 
 litre ) of water, at the temperature of 4 centigrade.* It is called a 
 GRAMME, and from it the larger and smaller weights, and their 
 names are derived in exactly the same way as in the case of the 
 measures of length and volume. 
 
 TABLE OF WEIGHTS. 
 (The usual abbreviations are given in brackets.) 
 
 Kilogramme = 1000 grammes. 
 
 Hectogramme = 100 n 
 
 Decagramme = 10 u 
 
 GKAMME (grm.) = 1 gramme. 
 
 Decigramme = 0-1 M 
 
 Centigramme = 0-01 " 
 
 Milligramme = 0-001 it 
 
 RULES FOR REDUCTION. 
 
 (These apply to all the tables given above.) 
 
 I. To reduce the larger and smaller measures to the unit, and 
 vice versa. (Principle. The name of each measure expresses 
 what multiple of the unit it is.) 
 (a) To reduce a given larger measure to the unit, or a given 
 
 unit to one of the smaller measures. 
 
 Multiply by the number expressed in the name of the 
 measure. 
 
 Examples : 
 
 Reduce 18 Mometres to metres. 18 x 1000 = 18,000 m. 
 Reduce 6 grammes to centigrammes. 6 x 100 = 600 grms. 
 (6) To reduce a given smaller measure to the unit, or a given 
 
 unit to one of the larger measures. 
 Divide by the number expressed in the name of the measure. 
 
 Examples : 
 
 Reduce 1885 centimetres to metres. 1885 -f 100 = 18-85 m. 
 Reduce 1724 litres to decalitres. 1724-j- 10 = 172-4 decalitres. 
 
 * The reason why this particular temperature is specified in defining the gramme 
 is as follows : A given mass of water alters in bulk as its temperature changes (as is 
 more fully explained in treatises on Heat ) ; but at 4 C. it occupies the smallest 
 space that it ever occupies while in the liquid state. Hence a cubic centimetre of 
 water has more matter in it, and therefore weighs more, at 4 C. than at any other 
 temperature. 
 
162 ELEMENTARY DYNAMICS. 
 
 II. To reduce any given measure to the next larger or the next 
 smaller measure. ( Principle. Each measure is ten times the 
 next smaller one, and one- tenth of the next larger one. ) 
 
 (a) To reduce a measure to the next larger oue. 
 Divide the number by 10. 
 
 Example : Reduce 152 centigrammes to decigrammes. 
 152 -f 10 = 15-2 decigrammes. 
 
 (b) To reduce a measure to the next smaller one. 
 Multiply the number by 10. 
 
 Example: Reduce 16-2 kilometres to hectometres. 
 16-2 x 10 = 162 hectometres. 
 
 It is obvious that, since our system of numeration is, like the 
 metric system itself, a decimal system, that is, is based on the 
 number 10, all the processes of multiplication and division required 
 by the above rules are extremely simple. The actual figures have 
 not to be changed at all ; their value is altered simply by changing 
 their place in reference to the units-figure. This is, of course, done 
 by altering the position of the decimal point; the latter being 
 always considered to exist, even if not actually expressed, immedi- 
 ately after (that is, to the right of) the units-figure. 
 Thus, to multiply a number by 
 
 10, shift the decimal point one place to the right. 
 100, it ii two places. 
 
 1000, it ti three places, &c. 
 
 Again, to divide a number by 
 
 10, shift the decimal point one place to the left. 
 100, ii ii two places. 
 
 1000, it ii three places, &c. 
 
 Cyphers being put in, if necessary, to fill up the interval between 
 the decimal point (implied or expressed), and the first figure of the 
 number which is being dealt with. 
 
 It will be useful to bear in mind the following points in connec- 
 tion with the metric system : 
 
 1. That the number of cubic decimetres which expresses the 
 size of a body, also expresses the volume of the body in litres ( since 
 a cubic decimetre is, by definition, 1 litre). 
 
 Thus, if a cistern is 6 decims. long, 4 decims. broad, and 3 decims. 
 deep internally, its size will be ( 6 x 4 x 3 = ) 72 cubic decimetres ; 
 and its capacity is known at once to be 72 litres. 
 
 2. That the number which expresses the volume of a given quan- 
 
APPENDIX. 163 
 
 tity of water in cubic centimetres, also expresses (very nearly) 
 its weight in grammes, at ordinary temperature (since the 
 gramme is, by definition, the weight of 1 c.c. of water at 4 C.). 
 
 Thus 1000 c.c. of water is known at once to weigh (neglecting the 
 small correction for temperature ) 1000 grammes, or 1 kilogramme. 
 
 Conversely, of course, a given weight of water in grammes will 
 measure ( very nearly ) the same number of cubic centimetres. 
 
 So that, for instance, if 100 c.c. of water are required for an 
 experiment, and no measures are at hand, it will only be necessary 
 to weigh out 100 grammes in a counterpoised beaker, in order to 
 obtain the quantity required. 
 
 Again, a very useful measure may be made from a jar or stout 
 test-tube, by counterpoising it, and weighing into it 1, 5, 10, &c. 
 grammes of water, marking the level of the surface of the liquid 
 with a file or diamond, at each successive weighing. 
 
 What has just been said applies to the case of other liquids than 
 water, if a correction is made for any difference in density ( for an 
 explanation of this term, see par. 90, p. 36). Thus, if a liquid is 
 twice as heavy as the same volume of water, it is obvious that 
 twice as much of it, that is, 2 grammes, must be weighed out in 
 order to obtain a volume of 1 c.c. 
 
 UNITS OF FORCE AND WORK. 
 
 The quantities adopted in defining the above units are : 
 
 The centimetre as unit of length. 
 The gramme n n mass. 
 The second M u time. 
 
 The unit of jForce is called a Dyne ( Gr. Ivvupts, force ). It is That 
 amount of Force which, acting for 1 second on a mass of 
 1 gramme, gives it a velocity of 1 centimetre pdr second. 
 
 For example Suppose that a force applied for 1 second to a 
 billiard-ball weighing 110 grms., makes it move 80 cm. per sec. 
 Then the value of this force is ( 110 x 80 = ) 8800 dynes. 
 
 The unit of Work is called an Erg (Gr. spyov, work). It is 
 That amount of work which is done by 1 dyne of force 
 acting through a space of 1 centimetre. 
 
 For example A skater is pushing a chair before him on the ice 
 with a force of 200 dynes. Then for every metre ( = 100 cm. ) that 
 the chair moves through, he does ( 100 x 200 = ) 20,000 ergs of work. 
 
 (Examples and Exercises on the Metric System are given on p. 175.) 
 
164 ELEMENTARY DYNAMICS. 
 
 QUESTIONS AND EXERCISES. 
 
 CHAPTER I. 
 
 1. What is the subject of natural science generally, and of those branches 
 
 of it called Chemistry and Dynamics in particular? What pro- 
 perties, for instance, of a piece of iron would be examined under 
 these branches, respectively? 
 
 2. What is a ' molecule ' ? What evidence have we that molecules are 
 
 extremely minute less than one-millionth of an inch in diameter ; 
 How does an ' atom ' differ from a molecule ? 
 
 3. State the chief points of distinction between a solid, a liquid, and a 
 
 gas. Give examples of each, and show how the same thing can be 
 changed from one state to another. How can we explain the 
 difference between the three states of aggregation of matter ? 
 
 4. What is meant by saying that matter is ' porous ' ? Give examples 
 
 showing, for instance, how liquids and gases can be proved to be 
 porous. 
 
 5. Steel is hard, tenacious, and elastic. Explain what these properties 
 
 mean, and how they may be tested by experiment. Compare 
 glass and lead with steel in respect of the above properties. 
 
 6. Define ' force.' How can you tell whether a force has acted or is still 
 
 acting upon a body? Mention all the forces which are acting 
 (a) on a cricket-ball resting on a table, (6) on a boat rowed against 
 a stream, (c) on a ball thrown upwards by a rider in a circus. 
 
 7. Distinguish between the forces of cohesion, adhesion, and gravita- 
 
 tion ; mentioning cases in which they are all three acting. 
 
 8. What is meant by ' capillary attraction ' ? Mention some common 
 
 cases of its action, and explain the reason of it. 
 
 9. State the law of the variation of gravitation-force with distance. 
 
 10. A piece of rock weighs 9 Ibs. on the earth's surface that is, 4000 
 
 miles from the centre ; if it was taken to a place 12,000 miles 
 from the centre, what would it weigh ? 
 
 11. A leaden ball is at a distance of 4 yards from another ball, and the 
 
 attraction of gravitation between them is represented by 8 grains. 
 What will be the amount of attraction when they are placed 
 ( a ) 16 yards apart, (6)2 yards apart ? 
 
QUESTIONS AND EXERCISES. 165 
 
 12. Distinguish between the ' weight ' and the ' mass ' of a body, stating 
 
 how each is measured. Which of them varies with the place 
 where the body is ? Give reasons for this variation in amount. 
 
 CHAPTER II. 
 
 13. How can we represent the direction and magnitude of a force in a 
 
 drawing ? A weight of 3 oz. is acted on by a force of 4 oz. pulling 
 it to the left, and by another force of 1 oz. pulling it to the right. 
 Represent these forces accurately in a drawing ; and state what 
 forces will be required to produce equilibrium, and in what direc- 
 tions they must be applied. 
 
 14. A moving cannon-ball is said to have momentum. "What does this 
 
 mean, and what must be known in order that it may be calculated ? 
 How would you give a racquet-ball the same momentum as a 
 cricket-ball 8 times as heavy ? 
 
 15. A cannon-ball weighing 2 Ibs. is moving at the rate of 1200 feet per 
 
 second ; and a hammer weighing 10 Ibs. is thrown with a velocity 
 of 480 feet per second. Find the momentum of each, and compare 
 them. 
 
 16. A cricket-ball weighing i Ib. is bowled with a velocity of 12 feet per 
 
 second, and hit with a velocity of 21 feet per second. Compare 
 its momentum in the two cases. 
 
 17. A football weighing ^ Ib. is kicked with a force which makes it move 
 
 24 feet per second : find its momentum. What velocity must a 
 fives-ball weighing 1 oz. have, in order that its momentum may be 
 the same as that of the football ? 
 
 18. A man running at the rate of 10 yards per second has a momentum 
 
 of 4500 (in Ibs., feet, and seconds) ; what must be his weight? A 
 fish weighing 8 Ibs. has a momentum of 480 (in Ibs., feet, and 
 seconds ) ; what must be its velocity ? 
 
 19. Mention the most important forces in nature, giving examples of 
 
 their action. Explain the different methods of ascertaining the 
 magnitude of a force. 
 
 20. Distinguish between 'mass,' 'weight,' and 'density.' The mass of 
 
 the planet Saturn is known to be very much greater than that of 
 our earth, and yet its density is less. Explain this. 
 
 CHAPTER III. 
 
 21. Write down clearly the first Law of Motion, and give the three 
 
 statements into which it may be divided, with one example in 
 illustration of each. 
 
 22. A greyhound weighing 30 Ibs., and running 35 feet per second, is 
 
 chasing a hare weighing 7 Ibs., and running 30 feet per second. 
 
166 ELEMENTARY DYNAMICS. 
 
 Compare the momenta of the two, and explain why the hare, 
 although running slower, has a good chance of escape. 
 
 23. Explain (a) why it is an advantage to run before a jump; (6) 
 
 why it is safest to skate quickly over thin ice; (c) why the 
 rider may lose his seat when a horse either bolts or refuses a 
 jump. 
 
 24. Explain what is meant by 'centrifugal tendency.' Show that it is a 
 
 consequence of the first Law of Motion, and give reasons for not 
 calling it a force. 
 
 25. Centrifugal tendency gets less as the size of the circle in which the 
 
 body is moving gets larger. "What condition is omitted in this 
 statement? Explain why, nevertheless, when a weight placed 
 half-way along the spoke of a wheel is moved to the rim, its centri- 
 fugal tendency increases. 
 
 26. A stone is being whirled in a sling with a velocity which gives a 
 
 centrifugal tendency equivalent to 5 Ibs. If the velocity is 
 increased to 6 times the rate, how many Ibs. must the sling be 
 able to bear, that it may not be broken ? 
 
 27. What is meant by ' friction,' and what are its causes ? By what 
 
 experiments can it be shown (a) that friction varies with the 
 pressure between the surfaces, ( b ) that it does not vary with the 
 size of the surfaces if the total pressure between them is unaltered ? 
 Can you give any reason for the latter law ? 
 
 28. The coefficient of friction of iron on iron is &. Explain exactly what 
 
 this means. A brake-van weighs 5 tons ; if all the wheels were 
 prevented from moving, what force would be required to drag it 
 along the rails ? 
 
 29. A wooden box weighing 20 Ibs. is resting on a board, the coefficient 
 
 of friction between them being &. Two weights, one of 3 Ibs. and 
 the other of 5 Ibs., are pulling it eastward ; and three weights, of 
 6 Ibs., 5 Ibs., and 4 Ibs. respectively, are pulling it westward. In 
 which direction will it move, and with what force ? 
 
 30. An engine weighs 30 tons, and all its wheels are coupled together and 
 
 worked by the steam. How many trucks, each weighing 8 tons, 
 could it draw along without the wheels slipping ? 
 
 31. An iron armour-plate is found to require a force of 7 tons to drag it 
 
 along the iron floor of the foundry ; what must be its weight ? 
 Suggest methods for moving it more easily, and give reasons for 
 the advantage gained by each method. 
 
 32. In making a brake for a railway train, is it best to make the block of 
 
 wood or of iron, broad or narrow ? If the block is of wood, why 
 does the brake act less powerfully when the wheels are gripped so 
 tightly that they cannot turn round ? 
 
QUESTIONS AND EXERCISES. 167 
 
 33. Will it be easier for a horse to draw a cart weighing 1| ton on a 
 
 common road, or 2 trucks, each weighing 6 tons, on a railway ? 
 
 34. Explain exactly what is meant by a ' poundal.' 
 
 35. A four-oar weighs, with crew, 1000 Ibs. How many poundals of 
 
 force must each of the crew exert ( supposing them all to be of 
 equal strength), in order that 1 second after the start the boat 
 may have a speed of 5 feet per second ? 
 
 36. A man weighing 160 Ibs., sculling in a boat weighing 40 Ibs., is 
 
 exerting a force of 2000 poundals. If 400 poundals are required 
 to overcome friction, how many feet per second will the boat be 
 moving 1 second after the start ? 
 
 37. Define ' composition of forces,' ' resolution of forces,' ' resultant,' 
 
 'component.' 
 
 38. In a game of the ' tug-of-war ' there were 20 men on one side, each 
 
 exerting a force of 800 poundals ; and 18 men on the other side, 
 each exerting a force of 900 poundals. In which direction will the 
 rope move, and with what force ? 
 
 39. Explain what is meant by the parallelogram of forces. A cricket- 
 
 ball is bowled with a force of 4 poundals ; it is hit by the bat in a 
 direction at right angles to its course with a force of 8 poundals. 
 Explain fully, with a diagram, the direction it will take, and the 
 force with which it will move. 
 
 40. Two forces, of 8 and 15 poundals respectively, act upon a body in 
 
 directions which are at right angles to one another. Find the 
 magnitude and direction of a third force which would just prevent 
 the body from moving. 
 
 41. Two forces, of 9 and 12 poundals respectively, act upon a body. 
 
 Find their resultant, (a) when they act together in the same 
 direction, (b) when they act in opposite directions, (c) when they 
 act at right angles to one another. 
 
 42. A steamer is going westward with a velocity of 12 miles an hour ; a 
 
 north wind is blowing with a force which gives it a speed of 
 5 miles an hour. Explain with a diagram the actual course and 
 velocity of the ship. 
 
 43. A ship, sailing along the coast at the rate of 14 miles an hour (= 20 
 
 feet per second ), is firing at a town on the shore. The gun is 
 pointed in a direction at right angles to the ship's course, and the 
 shot leaves the gun with a velocity of 800 feet per second. Draw 
 a diagram showing the course it takes. 
 
 44. A player at cover-point, running parallel to the line between wickets 
 
 at the rate of 20 feet per second, fields the ball, and throws it at 
 the wicket with a force which gives it a velocity of 50 feet per 
 second. Draw a diagram showing the direction in which he must 
 aim in order to hit the wicket. 
 
168 
 
 ELEMENTARY DYNAMICS. 
 
 We have here given the magnitude and direction of one force, the 
 magnitude but not the direction of the other, and the direction but not the 
 magnitude of the resultant. Let AB (fig. 105), 20 
 scale-units long, represent the force applied to the 
 
 ball by the motion of the player in running. Let wickets 
 
 AD, of indefinite length, represent the direction of 
 the resultant that is, the line in which the ball must 
 travel in order to hit the wicket. Round the centre 
 B, with a radius of 50 scale-units, describe an arc of a 
 circle cutting AD in D. Join BD. Through D 
 draw DC parallel to AB, and through A draw AC 
 parallel to BD. Then, if A is the point where the ball 
 leaves the hand, AC is the direction in which it must 
 be thrown in order to hit the wicket. 
 
 PROOF. ABDC is, by construction, a parallelo- 
 gram ; hence the diagonal AD represents correctly a 
 resultant force equivalent to the two components AC 
 and AB. Now AB, 20 units long, represents the force 
 applied to the ball in consequence of the motion of 
 the player in running. And AC = BD, which latter 
 (being a radius of the circle) is 50 units long. Hence AC represents 
 correctly in magnitude and direction the force with v/hich the player 
 must throw the ball, in order that it may move in the resultant direc- 
 tion AD. 
 
 45. Show that, when a boat is towed along a canal by a horse on the 
 
 bank, it is an advantage to have the towing-rope as long as 
 possible. If the rope is 60 feet long, and the distance from the 
 mast (to which it is fastened) to the bank is 11 feet, calculate what 
 proportion of the force exerted by the horse is effective in dragging 
 the boat straight along the canal. 
 
 46. A box weighing 120 Ibs. is being drawn along a floor by a rope 13 
 
 feet long, attached to it close to the floor, and sloping upwards 
 to the shoulder, 5 feet above the floor. Draw a diagram showing 
 how much of the force applied along the rope is effective in moving 
 the box along the floor. If the coefficient of friction is vh, calculate 
 how much force must be used to drag the box along. 
 
 47- Apply the principles of the second Law of Motion to explain (a) the 
 apparent slanting direction of vertically-falling rain-drops when 
 seen from a moving carriage ; ( 6 ) the spinning of an artificial 
 minnow when dragged through the water ; (c) the onward motion 
 of a fish caused by the lateral movement of its tail; (d) the 
 lessening of the useful effect of an oar when it is not at right 
 angles to the length of the boat, or when the blade is not held 
 square to the water j ( e ) why a windmill works when the axis to 
 which the sails are attached points in the direction of the wind, 
 but not when the axis is at right angles to the wind. 
 
 48. What is meant by the ' reaction ' which occurs in the application of 
 
QUESTIONS AND EXERCISES. 169 
 
 a force ? Mention some cases of it which you have observed, other 
 than those described in the book. 
 
 49. A cannon weighs 600 Ibs., and a ball weighing 4 Ibs. is shot out of it 
 
 with a velocity of 1500 feet per second. With what velocity will 
 the gun recoil ? 
 
 50. A ball is shot out from a cannon with a velocity of 900 feet per 
 
 second. The cannon weighs 360 Ibs., and its velocity at the com- 
 mencement of the recoil was 5 feet per second. What was the 
 weight of the ball ? 
 
 51. A fishing-boat weighing 4 tons is lying-to (unanchored) 40 yards 
 
 from the shore ; and a man in it is hauling a cask weighing 
 2 cwt. from the shore. How far will the cask be from the 
 shore when it is close to the boat ? 
 
 52. An arrow weighing 4 oz., and moving 176 feet per second, kills a 
 
 sitting bird weighing 40 oz. With what velocity will the two 
 bodies move on together ? 
 
 53. An engine weighing 40 tons, and moving 12 miles an hour, strikes a 
 
 truck weighing 8 tons which is standing on the line. With what 
 velocity will the two move after the collision ? 
 
 54. A loaded truck, weighing 12 tons, running down an incline, strikes 
 
 some carriages weighing 18 tons, which are standing at a station. 
 The whole move on at the rate of 12 miles an hour. At what 
 rate was the truck moving when it struck the carriages ? 
 
 55. A football player, weighing 12 stone and running 10 feet per second, 
 
 charges another player weighing 10 stone and running in the 
 opposite direction at the rate of 14 feet per second. What will 
 be the result ? 
 
 56. A fives-court is 26 feet long and 16 feet broad, and a player hits a 
 
 ball from the middle of the court, in such a direction that it strikes 
 the front wall 4 feet from the left-hand corner. Draw a diagram 
 to scale, showing where the ball will strike the side wall. 
 
 CHAPTER IV. 
 
 57. A train starts from a station, and at the end of 1 second is found to 
 
 be moving at the rate of 2 feet per second. If the acceleration 
 was uniform, what would be its velocity at the end of 8, 12, 16, 
 and 24 seconds ? 
 
 58. What is meant by saying that gravitation is a force of 32 poundals ? 
 
 Explain fully why it causes all things to fall equally quickly. 
 
 59. A bag of sand is let fall from a balloon. Assuming that it falls 16 
 
 feet in the first second, show how we can find (a) its velocity at 
 the end of the first second, (6) the space through which it falls 
 
170 ELEMENTARY DYNAMICS. 
 
 during the third second, (c) the total space it will have fallen 
 through in 7 seconds. 
 
 GO. A waroocket is fired point-blank at some men at a distance. It 
 moves through 4 feet in the first second, and reaches its destina- 
 tion in 6 seconds. How far were the men off, and what velocity 
 (approximately) had the rocket when it reached them ? 
 
 61. A carriage is ' slipped ' from a train when it is going 30 miles an hour, 
 
 and brought to a stop in 2 minutes. How far will it have gone 
 from the place where it was detached from the train ? 
 
 Since the carriage was going 30 miles an hour when it was detached, 
 the force applied to it by the engine must be such as to give it a velocity 
 of i mile in 2 minutes ( taking 2 minutes as the unit of time ), if nothing 
 occurred to stop its movement. Now the force necessary to destroy this 
 motion must be equal to the force which produced it. But a force which, 
 applied for 2 minutes, produces a velocity of i mile per 2 minutes will (as 
 explained in par. 225, p. 90 ) make the body move through a space correspond- 
 ing to the average velocity during the time ; which is /tatfihe final velocity, 
 that is, half a mile per 2 minutes. So the carriage would, under the action 
 of such a force, pass through half a mile in the 2 minutes ; and this is, 
 therefore, the distance it will traverse before it is stopped. 
 
 62. A piece of coal let fall down the shaft of a mine reached the bottom 
 
 in 9 seconds. How deep was the mine ? 
 
 63. Explain the cause of the upward motion of an ordinary rocket. 
 
 Why is it necessary that a stick should be attached to it ? 
 
 64. A rocket weighing 2 Ibs. is driven upwards by a continuous force 
 
 of 72 poundals. How high will it rise in 4 seconds ? 
 
 65. The speed of a railway train increases uniformly after starting ; and 
 
 at the end of a minute it has gone 200 yards. Find what velocity 
 in yards per minute it has now gained, and how many yards it 
 will go in the next minute, if the speed goes on increasing at the 
 same rate. 
 
 CHAPTER V. 
 
 66. What is meant by the 'centre of gravity'? How does the steadiness 
 
 with which anything rests on a support depend on the position of 
 the centre of gravity ? Give several examples in illustration. 
 
 67. A square pillar is 60 feet high and 2 feet square. How would you 
 
 find out where its centre of gravity was ? How would you examine 
 whether it was upright or not ? If it was not upright, how would 
 you ascertain whether it was in danger of toppling over ? 
 
 68. Explain how the centre of gravity of a piece of cardboard may be 
 
 found (a) if it is triangular, (6) if it is quite irregular in shape. 
 How would you prove the point found to be the true centre of 
 gravity ? 
 
QUESTIONS AND 
 
 69. Explain the exact meaning of the common -^expressions, * above^/ 
 
 ' below,' ' keeping one's balance,' ' a 'well- trimmed boat,' ' top- 
 heavy.' 
 
 70. A cylindrical factory chimney, 100 feet high and 3 feet in diameter, 
 
 leans 2 feet from the perpendicular that is, so that a plumb-line 
 let fall from the edge of the top on one side touches the ground 
 2 feet from the edge of the base on that side; how much 
 farther might it lean without actually falling? 
 
 Let fig. 106 represent the chimney in such a position that it would be in 
 unstable equilibrium. Since it is cylindrical and 
 uniform in structure, its centre of gravity will be at 
 the middle of its figure, X. From X let fall the 
 perpendicular XA; this will, as explained in par. 
 246, p. 100, pass through the edge of the base at A. 
 Through X draw XB parallel to the sides of the 
 chimney; B will be the centre of the base, and since 
 the chimney is 3 feet diameter, the distance BA 
 will be ij feet. 
 
 From the edge of the top at C let fall the perpen- 
 dicular CD, touching the ground at D. We have J ' " 
 to find how far D is from A, the edge of the base. 
 
 Now, the triangles XBA and CAD are equi- Fig. 106. 
 
 angular, for XB is parallel to CA, and XA to CD, 
 
 and BA and AD are in the same straight line. Hence (by Eucl. vi. 4) 
 CA : XB : : AD : BA. But CA is twice XB, for X is the middle point of 
 the chimney, therefore AD is twice BA. 
 
 But BA is i foot, .'. AD is 3 feet. 
 
 Therefore a plumb-line let fall from C would touch the ground 3 feet 
 from the base of the chimney, when the latter was on the point of toppling 
 over. 
 
 71. Explain by the Laws of Equilibrium, ( a ) why any one in danger of 
 
 falling stretches out an arm or a leg in the opposite direction ; ( 6 ) 
 why any one stooping forward advances one foot ; (c) why weights 
 are put between the centre and rim of the driving-wheel of an 
 engine, and what must be considered in fixing their mass and 
 position; (d) why a cask will roll down a slope on which a box 
 would rest pretty steadily. 
 
 CHAPTER VI. 
 
 72. Define energy, and distinguish between the two forms which it may 
 
 assume, giving examples of both. 
 
 73. "What is a 'foot-pound' ? 
 
 74. A trunk weighing 1^ cwt. is to be carried up-stairs to a room 30 feet 
 
 above the ground ; how many foot-pounds are required ? 
 
 75. A cricket-ball weighing 5J oz. is thrown 20 yards straight up ; how 
 
 many foot-pounds are required? 
 
172 ELEMENTARY DYNAMICS. 
 
 76. How far could a fives-ball weighing 1 oz. be thrown up by the same 
 
 muscular power ? 
 
 77. A cricket-ball weighing $ Ib. is thrown straight up, and 5 seconds 
 
 elapse before it returns to the hand. How high did it go, and how 
 many foot-pounds of energy were used in throwing it ? 
 
 78. The weight used in a pile-driver (see fig. 52, p. 120) is 2 cwt. ; and it 
 
 is raised 6 feet by the efforts of 3 men. How much work does 
 each of them do every time that it is lifted ? 
 
 79. A battering-ram weighing 1 cwt. is driven against a gate with a 
 
 velocity of 60 feet per second. What velocity must a cannon-ball 
 weighing 5 Ibs. have, in order to have the same energy ? 
 
 80. A bullet weighing 1 oz. is fired vertically upwards, and rises for 11 
 
 seconds before it stops. How high will it rise (neglecting the 
 resistance of the air), and how many foot-pounds of work will it 
 do if it falls on the roof of a house 36 feet high? 
 
 81. A train of 40 trucks, each weighing 6 tons, is moving 15 miles an 
 
 hour ; compare its momentum with that of a train of 9 carriages, 
 each weighing 4 tons, moving 50 miles an hour. Compare also the 
 amount of damage each would do in a collision before its motion 
 was stopped. 
 
 82. Define ' horse-power,' showing how it differs from work. 
 
 83. An engine of 2 horse-power is intended to work a hammer which 
 
 has to be raised 2 feet high 40 times a minute. What is the 
 utmost weight the hammer can have, allowing 60 foot-pounds 
 for friction, &c. ? 
 
 84. A machine can raise 12 tons to a height of 10 feet in 2 minutes. Of 
 
 how many horse-power is it ? 
 
 85. The depth of a coal-pit is 120 fathoms ( a fathom = 6 feet ) ; how 
 
 many tons of coal will an engine of 3 horse-power raise from the 
 bottom of it per hour ? 
 
 CHAPTEB VII. 
 
 86. What is meant by a ' machine '? A machine cannot do more work 
 
 than that amount which is applied to it ; explain why, neverthe- 
 less, a power of 1 Ib. can, by means of pulleys, raise a resistance of 
 8 Ibs. (nearly). How many movable pulleys, arranged on the 
 first system, would be required to do this ? Give a drawing of the 
 arrangement. 
 
 87. Explain the meaning of the term * mechanical advantage,' and illus- 
 
 trate it by reference to a windlass for raising stones from a quarry. 
 If the mechanical advantage of the windlass is 6, what weight of 
 stone ought a man, exerting a force of 40 Ibs. , to be able to raise ? 
 
QUESTIONS AND EXERCISES. 173 
 
 Why can he not, in practice, raise quite so much as this by the 
 machine ? 
 
 88. A bell weighing 1 ton has to be raised to a belfry 30 yards above the 
 
 ground. How many foot-pounds of work will have to be done ? 
 What machine or machines may be used for the purpose ? Give a 
 sketch and explanation of an arrangement by which two men, each 
 exerting a force of 120 Ibs., can do the work. 
 
 89. A bucket of water, weighing 60 Ibs., has to be raised from the bottom 
 
 of a well 20 yards deep by the help of a single movable pulley. 
 What power will be required, and what length of rope will have 
 passed through the man's hands by the time the bucket reaches 
 the top of the well ? 
 
 90. A man weighing 150 pounds is pulling up a weight of 7 cwt. by means 
 
 of 3 movable pulleys, arranged on the first system. What is his 
 pressure upon the floor on which he stands ? 
 
 91. In what respects does the second system of pulleys differ from the 
 
 first system ? Describe the usual form of the second system, and 
 show how the mechanical advantage can be estimated. 
 
 92. Four pulleys are given you, one of which is simply for use as a fixed 
 
 pulley to change the direction in which the power moves. Explain 
 how you would arrange the others, ( a ) on the first system, ( 6 ) on 
 the second system, so as to get the greatest mechanical advantage. 
 Compare the mechanical advantage obtainable by each arrange- 
 ment. 
 
 93. In a model of a wheel and axle, the diameter of the axle is 3 inches, 
 
 and a weight of 15 oz. is suspended from it ; the diameter of the 
 wheel is 8 inches, and a weight of 6 oz. is suspended from it. 
 Which weight will descend, and with what force will it press upon 
 the floor after reaching it ? 
 
 94. The handle of a windlass is 2 feet long ; the axle is 6 inches diameter. 
 
 A bucket of water, weighing 80 Ibs., has to be raised by means of 
 it ; what power is required ? 
 
 95. A windlass is used for lifting the weight of a ' pile-driver ' ( fig. 52, 
 
 p. 120) ; the handle is 21 inches long, the diameter of the axle is 7 
 inches. The power available is one of 60 Ibs. ; how heavy may the 
 weight be ? How many turns of the handle will be necessary to 
 raise it to a height of 11 feet ? 
 
 96. The diameter of a capstan is 10 inches ; the length of each capstan- 
 
 bar measured from the centre is 5 feet ; ten men are employed, 
 each exerting a force of 40 Ibs. If one-fifth of the whole force is 
 lost by friction, how heavy an anchor can be raised ? 
 
 97. The wheel of a bicycle is 50 inches diameter; the radius of the 
 
 treadle is 5 inches ; the weight of the rider together with that of 
 L 
 
174 ELEMENTARY DYNAMICS. 
 
 the machine is 240 Ibs. What power will be required to move 
 the bicycle along a level road ? ( Coefficient of friction = A of 
 load.) 
 
 98. Point out the connection between the wheel and axle and the lever, 
 
 and show how the mechanical advantage of a lever can be 
 estimated. 
 
 99. In a lever of the first order, the power-arm is 30 inches long, the 
 
 resistance-arm is 2 inches long. What resistance could be moved 
 by a power of 20 Ibs. ? 
 
 100. A fives-ball, hung from one end of a stick 2 feet 6 inches long, is 
 
 found to balance a cricket-ball hung from the other end, when 
 the stick is supported on a finger placed 5 inches from the 
 cricket-ball How much heavier is the cricket-ball than the 
 fives-ball? 
 
 101. A man wishes to carry two boxes, one weighing 60 Ibs., the other 
 
 weighing 40 Ibs., placed one at each end of a pole 5 feet long across 
 his shoulder. At what point must his shoulder be placed ? 
 
 102. Two men, A and B, are carrying a cask weighing 123 Ibs. on a pole 
 
 8 feet long laid across their shoulders, the load being 6 inches 
 nearer to A than to B. What share of the burden does each 
 bear? 
 
 103. Of what kind of lever is a pair of nut-crackers an example ? 
 
 104. A nut can just resist a force of 4 Ibs., and the outer ends of a pair 
 
 of nut-crackers 6 inches long are compressed by the hand with a 
 force of 3 Ibs. At what point must the nut be placed so as just 
 to crack ? 
 
 105. A ladder, uniform in breadth and structure, is being raised from a 
 
 horizontal to a vertical position, the lower end being held steadily 
 down. (1) Find the position and path of its centre of gravity. 
 (2) State what orders of levers it represents during its rise. (3) 
 Explain why the work becomes easier when the ladder is nearly 
 vertical. 
 
 106. The distance from the middle of the blade of an oar (where the 
 
 fulcrum may be considered to be applied ) to the rowlock is 8 feet ; 
 the distance from the rowlock to the point where the rower's 
 force is applied is 3 feet. The rower exerts a force of 56 poundals ; 
 what is the actual force urging the boat on ? Why does this force 
 vary in different parts of the stroke, and in what position of the 
 oar is it the greatest ? 
 
 107. A carriage weighing 2 tons has to be drawn by a horse up an 
 
 incline of 1 in 40. What power is required? (Coefficient of 
 friction = &.) 
 
QUESTIONS AND EXERCISES. 175 
 
 108. A train of 28 trucks, each weighing 8 tons, has to be dragged up an 
 
 incline of 1 in 90, by an engine capable of exerting a tractive 
 force of 3 tons. Can the engine do the work ? ( Coefficient of 
 friction =Tri<i.) 
 
 109. The pitch of the screw of a screw-press is 1 inch ; the length of 
 
 the handle by which it is turned is 21 inches. If a power of 
 10 Ibs. is applied to the end of the handle, what force will be 
 available for compression ( 20 per cent, being allowed for loss by 
 friction ) ? 
 
 EXERCISES ON THE METRIC SYSTEM. 
 
 1. Reduce 1886 millimetres to the decimal fraction of a metre. 
 
 2. How many centimetres are there in 15'35 metres ? 
 
 3. How many metres are there in 39,371 millimetres ? 
 
 4. Express 48 kilometres, 3 hectometres, 1 decametre, in metres. 
 
 5. How many kilometres are there in 3,675,824 millimetres? 
 
 6. The length of a cricket-bat is 86 centimetres ; the distance between 
 
 wickets is 20 '21 metres. How many bats' lengths is this ? 
 
 7. A ladder is 6'6 metres long ; the distance between the rungs is 22 
 
 centimetres ; how many rungs are there ? 
 
 8. The driving-wheel of a locomotive is 5 metres 2 decimetres in circum- 
 
 ference. How many kilometres will it pass over in turning round 
 500 times ? 
 
 9. How many litres are there in a cubic metre ? 
 
 10. A cistern is 8 decimetres long, 5 decimetres broad, and 6 decimetres 
 
 deep. How many litres will it hold ? 
 
 11. A trough is 2 '4 metres long, 6 decimetres broad, and 120 millimetres 
 
 deep. How many litres will it hold ? 
 
 12. A lecture room is 9 metres long, 7 metres broad, and 5 metres high. 
 
 How many litres of air does it contain ? 
 
 13. What is the weight of ( a ) a litre of water ; ( 6 ) a cubic metre of 
 
 water, in kilogrammes, and in hectogrammes ? 
 
 14. How many litres will 8 kilogrammes, 3 hectogrammes, of water 
 
 measure ? 
 
 15. A cistern is 2'2 metres long, 90 centimetres broad, and 1'2 metres 
 
 deep. How many litres of water will it hold, and what will the 
 water weigh ? 
 
 16. 24 centilitres of water are required for an experiment, but no 
 
 measures are at hand. What quantity of water must be weighed 
 out in order to get this volume ? 
 
176 ELEMENTARY DYNAMICS. 
 
 17. If you had a long narrow tube closed at one end (a 'test-tube' ), 
 
 scales and weights, and some water, explain how you would 
 graduate the tube as a measure of cubic centimetres. 
 
 18. A rifle bullet, weighing 30 grammes, is shot with a velocity of 400 
 
 metres per second out of a rifle weighing 4*2 kilogrammes. Find 
 the momentum of the bullet (in centimetre-gramme-seconds), 
 and the velocity in centimetres per second with which the rifle 
 recoils. 
 
 19. When a mass of 1 gramme is allowed to fall freely under the action 
 
 of gravitation, at the end of 1 second it has fallen through 490 
 centimetres, and has acquired a velocity of 980 centimetres. 
 What is the value of gravitation-force in dynes, and how many 
 ergs of work will the mass do if it strikes the ground at the end of 
 the first second? 
 
 20. A racquet-ball weighing 18 grammes is driven with a force which 
 
 gives it a velocity of 40 metres per second. Express its kinetic 
 energy in ergs. 
 
177 
 
 
 INDEX. 
 
 PAGE 
 
 ' Above,' meaning of term 31 
 
 Acceleration, meaning of term 87 
 
 ii ti produced by Gravitation. 88 
 
 Adhesion, meaning of. 21 
 
 1 Arms ' of a lever, meaning of. 141 
 
 Atoms, account of 12 
 
 Attraction, various kinds of 15 
 
 Balance, or pair of scales, descrip- 
 tion of 143 
 
 Ballast in a ship, use of 104 
 
 Barge, action of forces upon a 74 
 
 ' Below ' meaning of term 31 
 
 Bicycle, description of 138 
 
 Bicycles, ball-bearings for 64 
 
 Biology, definition of 6 
 
 Brittleness, meaning of 18 
 
 Bullet, danger of firing upwards a. . . .95 
 ti energy of, compared with 
 that of the rifle 119 
 
 Cannon, recoil of 79 
 
 ii ball, energy of 119 
 
 ir ii momentum of 41 
 
 Capillarity, meaning of 24 
 
 Capstan, description of 137 
 
 Cause and Effect 5 
 
 Centre of Gravity, meaning of term. ..95 
 ii ii methods of find- 
 ing 106-110 
 
 Centre of Inertia no 
 
 Centre of Percussion i ic 
 
 Centrifugal extractor 54 
 
 Centrifugal tendency, effect of, on 
 
 weight 34, 56 
 
 Centrifugal tendency, illustrations of.. 53 
 ii ii laws of 51 
 
 PAGE 
 
 Centrifugal tendency, meaning of. 50 
 
 n ii orbits of the 
 
 planets influenced by 56 
 
 Chemistry, definition of. 6 
 
 Cohesion-figures 24 
 
 Cohesion, general account of 16 
 
 n overcome by heat 17 
 
 Coining Press, the 154 
 
 Collision of bodies 80 
 
 Colloids 27 
 
 Component forces, meaning of 68 
 
 Composition of Forces 68 
 
 ! Cone, made to balance on its point . . 104 
 
 j Conservation of Energy 115 
 
 I Cricket-ball, action of forces upon a . . 72 
 Cricket-bat, as a lever of the Third 
 
 Order 148 
 
 Cricket-bat, centre of percussion of a. in 
 n n stinging of a, explained., ii i 
 
 Crown glass, manufacture of. 54 
 
 Crystalloids 27 
 
 ' Degradation ' of Energy 157 
 
 Densities, relative, table of. 37 
 
 Density, meaning of. 36 
 
 Dialysis, operation of. 28 
 
 Diffusion of gases 27 
 
 n of liquids 27 
 
 Divisibility of matter n 
 
 ' Doubling ' by hares, principle of 48 
 
 Ductility 19 
 
 Dynamics, definition of 7 
 
 ' Dyne,' the, defined 163 
 
 Earth, attraction of 29, 95 
 
 n movements of the 55, 56 
 
 n reasons for shape of the 56 
 
178 
 
 ELEMENTARY DYNAMICS. 
 
 PAGE 
 
 Elastic bodies, collision of 82 
 
 Elasticity, meaning of 20 
 
 Elbow-joint, description of the 147 
 
 Energy of the Sun 114 
 
 it conservation of 115 
 
 it ' degradation 'of 157 
 
 ii exact valuation of 121 
 
 it meaning of. in 
 
 it measurement of 1 16 
 
 ii relation of, to velocity 1 18 
 
 ii statical and kinetic 112-114 
 
 it transference of 112 
 
 Equilibrium, different kinds of 98 
 
 it examples of 99 
 
 it experiments illustrating.. 97 
 
 n laws of 100-104 
 
 ii offorces 41 
 
 n of bodies. 96 
 
 ' Erg,' the, defined 163 
 
 Extension and Form 8 
 
 Falling bodies, experiments illustrat- 
 ing laws of 88-90 
 
 Falling bodies, table of spaces passed 
 through by 92 
 
 Falling bodies, work done by 119 
 
 Ferry-boat, action of forces upon a ... 72 
 
 First Law of Motion 45 
 
 Fives-ball, mode of calculating the 
 direction in which to hit a 86 
 
 ' Foot-pound,' meaning of. 117 
 
 Force, meaning of term 39 
 
 n metric units of 163 
 
 n moment of a 143 
 
 11 unit of 66 
 
 Forces, composition of. 68 
 
 ii continued action of 87 
 
 n direction of, mode of express- 
 ing 40 
 
 Forces, equilibrium of. 41 
 
 n examples of. 40 
 
 it independent action of 67 
 
 n magnitude of, mode of ex- 
 pressing 40 
 
 Forces, measurement of 42 
 
 n parallelogram of 71 
 
 n resolution of. 73 
 
 Friction, advantages and disadvant- 
 ages of. 64 
 
 Friction, causes of. 57 
 
 it coefficient of 61 
 
 PAGE 
 
 Friction, coefficient of, on roads and 
 railways 62 
 
 Friction, laws of 57~6i 
 
 n meaning of. 57 
 
 it methods of lessening 61 
 
 it rolling, nature of 63 
 
 ii table of coefficients of 61 
 
 ii variation of, with pressure . . 59 
 
 Friction- wheels 64 
 
 Fulcrum, meaning of. 141 
 
 Gaseous state, definition of. 10 
 
 Gases, diffusion of. 27 
 
 ti elasticity of 21 
 
 ii kinetic theory of 17 
 
 it porosity of 14 
 
 n table of densities of. 37 
 
 Glass, elasticity of. 21 
 
 'Gradient' of a road, explained 150 
 
 Gravitation, general account of 29 
 
 n action of, in acceleration.. 88 
 n law of variation with 
 
 distance 33 
 
 Gravity, centre of, meaning of term. . .95 
 
 Hardness, list of substances in order 
 
 of 18 
 
 Hardness, meaning of. 18 
 
 Heaviness, meaning of. 29 
 
 Height of a tower, method of finding.. 92 
 n to which a ball has risen, 
 
 method of finding. 94 
 
 ' Horse-power,' meaning of term 120 
 
 Impenetrability 8 
 
 Incidence, angle of 84 
 
 Inclined Plane 148-154 
 
 Indestructibility of matter 13 
 
 Inertia, centre of no 
 
 it examples of 45~49 
 
 n meaning of 45 
 
 Irregular bodies, method of finding 
 
 the centre of gravity of. 109 
 
 Isochronism of the pendulum 123 
 
 Kinetic Energy 113 
 
 ii theory of gases 17 
 
 Kinetics, methods employed in 43 
 
 Law of nature 6 
 
 Laws of centrifugal tendency 5i~54 
 
INDEX. 
 
 179 
 
 PAGE 
 
 Laws of equilibrium 100-104 
 
 n falling bodies 88-90 
 
 it friction 57-60 
 
 n motion 44 
 
 it reflexion 83 
 
 Leaning towers 101 
 
 Levers 140-148 
 
 it different orders of 141 
 
 Liquid state, definition of 10 
 
 Liquids, table of densities of. 37 
 
 Locomotive engine, the 154-157 
 
 Machines, definition of. 128 
 
 n different classes of. 131 
 
 Magnitude and direction of forces 40 
 
 Malleability 19 
 
 Mass, meaning of. 35 
 
 it and Weight, distinction between. 35 
 
 Matter, definition of 7 
 
 u divisibility of n 
 
 n indestructibility of 13 
 
 ti porosity of 14 
 
 n properties of 7 
 
 if the three states of 9 
 
 ' Mechanical advantage,' general 
 
 method of calculating 130 
 
 ' Mechanical advantage,' meaning of. 130 
 
 Metals, porosity of 14 
 
 Metric System, the 158 
 
 it ii rules for reduction in. 161 
 
 it ii units of force and 
 
 work in 163 
 
 Molecules, account of. 12 
 
 Moment of a force 143 
 
 Momentum, meaning of term 41 
 
 n method of expressing 41 
 
 ' n problems on 43 
 
 Motion, laws of. 44 
 
 n meaning of term 38 
 
 Movable pulleys 132 
 
 Natural Science, branches of 6 
 
 Nature, laws of 6 
 
 Neutral equilibrium 98 
 
 Newton, Sir Isaac, his laws of motion.44 
 n n investigation of 
 
 gravitation by 30 
 
 Non-elastic bodies, collision of. 81 
 
 Oar, principle of action of an 145 
 
 Osmose, meaning of 28 
 
 PAGE 
 Paper, method of splitting a sheet of. . 22 
 
 Parallelogram of forces 71 
 
 Pendulum, account of the 122-127 
 
 it length of the seconds .... 125 
 ' Perpetual Motion,' impossibility of..n6 
 
 Physics, definition of. 7 
 
 'Pile-driver,' description of a 120 
 
 Pisa, leaning tower of 102 
 
 Plumb-line 30 
 
 Porosity of Matter 14 
 
 Potential ( statical ) Energy 113 
 
 Poundal 66 
 
 ' Power ' in machines, meaning of.. . . 128 
 'Preserving one's balance,' meaning 
 
 of term 102 
 
 Pulleys, fixed and movable 131 
 
 tt systems of. 133-135 
 
 Questions and Exercises 164-176 
 
 Rack and Pinion, description of. 137 
 
 Reaction of a force, meaning of. 77 
 
 Recoil of a gun, explanation of. . . .^.779 
 Reflexion, angle of 84 
 
 if laws of 83 
 
 'Resistance' in machines, meaning of. 128 
 
 Resolution of Forces 73 
 
 Resultant force, meaning of 68 
 
 Rifle, recoil of a 79 
 
 Rocket, ascent of a, explained 80 
 
 Rod, method of finding the centre of 
 
 gravity of a 106 
 
 Rowing, use of the stretcher in 80 
 
 Rudder, action of a 76 
 
 Running before a jump, principle of. .48 
 
 Screws 152-154 
 
 Second Law of Motion . . 65 
 
 Skating, theory of 102 
 
 Slinging a stone, principle of. 53 
 
 Softness, meaning of. 18 
 
 Solid state, definition of. 10 
 
 Solids, table of densities of 37 
 
 Specific gravities, table of 37 
 
 Spring-balance, principle of 32 
 
 Stable equilibrium 98 
 
 States of Aggregation 9 
 
 Statical Energy 113 
 
 Statics, methods employed in 42 
 
 Steel, hardening and tempering of 21 
 
 Steelyard, description of 144 
 
180 
 
 ELEMENTARY DYNAMICS. 
 
 PAGE 
 
 Stretcher in rowing, use of a" 80 
 
 Sun, the, as a source of energy 114 
 
 Surface tension of liquids 23 
 
 Swing, theory of setting in motion a.. 105 
 
 Tenacity, meaning of 19 
 
 Third Law of Motion 77 
 
 Throwing a ball, principle of 49 
 
 ti the hammer, principle of.. . . 53 
 
 ' Top-heavy, ' meaning of term 98 
 
 Tossing oars, risk of upsetting in 104 
 
 Train of wheels and pinions 139 
 
 Tug of War, the game of 69 
 
 Unit of work 117 
 
 Unstable equilibrium 98 
 
 Velocity, meaning of term. 
 
 39 
 
 Viscosity. 17 
 
 Water, falling, as a source of energy.. 114 
 
 Wedges 151 
 
 'Weighing' things, meaning of 144 
 
 ii it methods of 31 
 
 Weight, meaning of. 29 
 
 n reasons for difference of 34 
 
 n and Mass, distinction between 35 
 
 Wheel and Axle 135-140 
 
 Whirling-table, description of 51 
 
 Windlass, description of 137 
 
 Wire-drawing 20 
 
 Work, meaning of in 
 
 11 metric unit of. 163 
 
 it relation of, to time. 120 
 
 IF table of different kinds of 121 
 
 I n unit'of 117 
 
 THE END. 
 
 Edinburgh : 
 Printed by W. & R. Chambers. 
 
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