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upwards, and if this upward pressure is sufficient W will be supported. It is found by theory and by trial that the advantage of this combination of Levers is expressed by the product of the numbers which express the advantages of the separate Levers. For example, suppose that AK is 3 times KB, that BL is 4 times LC, and that CM is 5 times MD; then the advantages of the Levers separately are expressed by 3, 4, and 5 respectively, and the advantage of the combination is expressed by 3×4 × 5, that is by 60. Hence any Power at A will support a Weight of 60 times that amount at D. If we suppose the Power to be a little greater than is necessary for equilibrium the Weight will be moved, but in order to raise the Weight through any space the Power must descend through 60 times that space.

263. Combinations of Wheels and Axles are often used. The Wheel of each of the pieces which form the combination is made to act on the Axle of the next by means of teeth or of a strap. It is found by theory and by trial that the advantage of this combination is expressed by the product of the numbers which express the advantages of the separate pieces.

264. The Differential Axle, or Chinese Wheel.

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This machine may be considered as a combination of the Wheel and Axle with a single moveable Pully. Two

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cylinders of different radii have a common axis with which they are firmly connected; the axis is supported in a horizontal position so that the two cylinders can turn as one body round the axis. A string has one end fastened to the larger cylinder, is coiled several times round the cylinder, then leaves it, passes under a moveable Pully and is coiled round the smaller cylinder to which the other end is fastened. The string is coiled in opposite ways round the two cylinders, so that as it winds off one it winds on the other. A Weight W is hung from the moveable Pully; and the equilibrium is maintained by a Power P applied at the end of a handle attached to the axis. It is found by theory and by trial that there is equilibrium on this machine when the Power is to the Weight in the same proportion as half the difference of the radii of the two cylinders is to the length of the arm at which the Power acts. Thus by making the difference of the radii of the two cylinders sufficiently small we can secure any amount of mechanical advantage.

265. It is not difficult to shew that the preceding statement is consistent with the principle of Art. 208. For suppose the Power to be a little greater than is necessary for equilibrium; thus the Weight will be raised. Let the Power describe the circumference of the circle of which Then from the smaller the Power-arm is the radius. cylinder a piece of the string is unwound equal in length to the circumference of the cylinder; and on the larger cylinder a piece of the string is wound equal in length to the circumference of the cylinder. Thus the excess of the circumference of the larger cylinder over the circumference of the smaller is equal to the whole length of string which is removed from the hanging position; so that each of the two vertical portions is shortened by half this length, which is therefore the space through which the Weight is raised. Thus the same number which expresses the proportion of the Weight to the Power when there is equilibrium, expresses also the proportion of the space passed over by the Power to the space passed over by the Weight when there

is motion.

266. Hunter's Screw, or the Differential Screw.

AB is a right circular cylinder, having a Screw traced on its surface; this fits into a corresponding groove cut in the block CE, which forms part of the rigid framework CDFE. The cylinder AB is hollow, and has a thread cut in its inner surface, so that a second Screw GH can work in it. second Screw does not turn round, for it has a crossbar KL the ends of which are constrained by smooth grooves, so that the piece

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GHLK can only move up and down. The machine is used to produce a great pressure on any substance placed between KL and the fixed base on which the framework CDFE stands; this pressure we will call the Weight, and denote by W the Power P is applied by a handle at the top of the outer screw. It is found by theory and by trial that there is equilibrium in this machine when the Power is to the Weight in the same proportion as the difference of the distances between two consecutive threads in the two Screws is to the circumference of the circle having the Power-arm for radius.

267. It is not difficult to shew that the preceding statement is consistent with the principle of Art. 208. For suppose the Power to be a little greater than is necessary for equilibrium, then the Weight will be moved. Let the outer Screw be turned round once. The whole space passed over by the end of the Power-arm, estimated in the direction of the Power-arm, is equal to the circumference of the circle having the Power-arm for radius, as in Art. 258. By turning round the outer Screw the piece KL descends through a space equal to the distance between two consecutive threads; at the same time some of the lower Screw enters into the other, namely a length equal to the distance

between two consecutive threads. Therefore, on the whole, the piece KL descends through a space equal to the difference of the distances between two consecutive threads in the two Screws. Thus the same number which expresses the proportion of the Weight to the Power, when there is equilibrium, expresses also the proportion of the space passed over by the Power estimated in the direction of the Power to the space passed over by the Weight, when there is motion.

XVIII. COLLISION OF BODIES.

268. In the last nine Chapters we have been concerned mainly with questions relating to equilibrium; we now return to some which relate to motion.

When the application of force results in motion we measure the force by the momentum which is produced in a definite time, as for instance, one second; and as long as we keep to the action of force on the same body we may measure the force by the velocity which is produced. One of the forces with which we are familiar is gravity, which takes an appreciable time to produce a moderate velocity. There are however other forces which seem to produce a large velocity almost instantaneously. For example, when a cricket-ball is driven back by a blow from a bat the original velocity of the ball is taken away and a new one is given to it in a contrary direction; the velocity taken away, and also that given, are very large, while the whole operation takes place in an extremely brief time. Similarly, when a bullet is discharged from a gun a very large velocity is given to the bullet in an extremely brief time. Forces which produce such effects as these are called impulsive forces; and the following is the usual definition: An impulsive force is a force which produces a large change of motion in an extremely brief time.

269. Thus impulsive forces do not differ in kind from other forces but only in degree; and an impulsive force is merely a force which acts with very great intensity during a very brief time. As the laws of motion may be taken to

be true whatever be the intensity of the forces which produce the motion, we can apply these laws to the action of impulsive forces. But since the duration of the action of an impulsive force is too brief to be appreciated, we cannot measure such a force by the momentum produced in a definite time; it is usual to measure an impulsive force by the whole momentum which it produces.

270. We shall not have to consider the result of the simultaneous action of impulsive forces and ordinary forces for the following reason: the impulsive forces are so much more intense than the ordinary forces that during the brief period of simultaneous action the latter do not produce an effect of any importance in comparison with that produced by the former. Thus, to make a supposition which is not extravagant, an impulsive force might produce a velocity of 1000 feet per second in less than one tenth of a second, while the earth's attraction in one tenth of a second would produce a velocity of about 3 feet per second.

The words impact and impulse are often used as abbreviations for the action of an impulsive force, or for impulsive action.

271. We are now about to consider some questions relating to the collision of two bodies; the bodies may be considered to be small spheres of uniform substance. We shall not take account of any possible rotation of these spheres; that is to say, the motion we are about to consider is that which all the particles of the body have in common, leaving out such as may be different for different particles. The collision of spheres is called direct when at the instant of contact the centres of the spheres are moving in the straight line in which the impulse takes place, that is, in the straight line which joins the centres of the two spheres; the collision is called oblique when this condition is not fulfilled.

272. When one body impinges directly on another, the following is considered to be the nature of the mutual action. The whole duration of the impact is divided into two parts. During the first part a certain impulsive force acts in opposite directions on the two bodies, of such an amount as to render their velocities equal. During the

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