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

The

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

second part another impulsive force acts on each body in the same direction as before, and the magnitude of this second impulsive force bears to the magnitude of the former a proportion which is constant for a given pair of bodies. This proportion lies between the values 0 and 1, both inclusive. When the proportion is 0 the bodies are termed inelastic; when it is greater than 0 and less than 1 the bodies are called imperfectly elastic; and when it is 1 the bodies are called perfectly elastic. This proportion is called the coefficient of elasticity, or the index of elasticity.

273. There are three assumptions involved in the preceding Article.

We assume that there is an epoch at which the velocities of the two bodies are equal: this will probably be admitted as nearly self-evident.

We assume that during each of the two parts into which the whole duration of the impact is divided by this epoch, the action on the two bodies is equal and opposite: this is justified by the Third Law of Motion.

We assume that the action on each body after the epoch is in the same direction as before, and bears a constant ratio to it: this assumption may be taken on trial as a matter to be tested by observation.

274. The theory of the collision of bodies appears to be chiefly due to Newton, who made some experiments on the subject, and recorded the results in his Principia. In his experiments the two balls used together seem always to have been formed of the same substance. He found that the value of the index of elasticity was for balls of 5 worsted about, for balls of steel about the same, for

8

balls of cork a little less, for balls of ivory for balls 9'

275. We have still to explain why the words elastic and inelastic are used in Art. 272. It appears from experiment that bodies are compressible in various degrees,

and recover more or less their original forms after the compression has been withdrawn ; so likewise they may be bent or twisted to some extent, and will recover their original forms when the forces which bent or twisted them cease to act this property is called elasticity. When one body impinges on another we may naturally suppose that the surfaces near the point of contact are compressed during the first part of the impact, and that they recover more or less their original forms during the second part of the impact.

276. By the aid of the principles which we have now explained the change of motion in bodies produced by collision may be calculated; but it is not suitable to our plan to enter on this calculation, and so we will merely state the result for some special cases. We now consider only direct collision.

277. Suppose that there is a collision between two balls equal in mass and perfectly elastic; then the two balls interchange velocities. There are various particular examples included in this single statement. Thus let a ball A impinge on a ball B at rest; then A is brought to rest, and B moves on with the velocity which A originally had. Again, let A and B be both moving in the same direction, and let A overtake B; then after the collision they both move on in the same direction as before, A with the velocity which B originally had, and B with the velocity which A originally had. Finally let A and B be moving in opposite directions and meet; then after the collision A moves backwards with the velocity which B originally had, and B moves backwards with the velocity which A originally had. Thus in fact if we suppose the two balls exactly alike, so that one cannot be distinguished from the other, the result is the same in all these examples as if one body had gone through the other, or as if one had passed close by the side of the other.

278. It is easy to verify the statement made with respect to the three preceding examples by experiments. The first example especially is interesting. We may take a row of equal elastic balls, say B, C, D at rest in a straight line, either close together or separated. Then let