UC-NRLF SbM 707 REESE LIBRARY OF T1IK UNIVERSITY OF CALIFORNIA. Q Received ^ yr^^^tf^_ ., 188 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 (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 (^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