Hence, then, the space _b c_, by which the body has been pulled in, is represented by half F, but _a b_, the space which the body would have travelled forwards, will be represented by V, the velocity of the body in a second; but if the motion be such that the distance _b c_ travelled in a second is very small, then the triangles _a b d_ and _a b c_ are approximately similar, and the smaller _a b_ is the more nearly similar they are. Whence then (a b)/(b c) = (a d)/(a b), that is to say (a b)² = a d × b c. But _a b_ represents the space which would have been traversed by the body in one second at the rate it was going, and hence is equal to V; _a d_ is the diameter of the circle, and hence equals 2 R; _b c_ is the space through which the body has been drawn in the second by the attractive force F, and therefore equals half F. Whence then V² = 2 R × half F = R F. We took a second as the limit of time during which the motion was to be considered. Of course any other time could have been taken. Now what is true of the motion of a body during a very short time is also true of the body during the whole of its path, assuming that the path is a circle, and that F remains constant, as it obviously will if the path is a circle, and the velocity is uniform. Whence then we may generally say that if a body is being whirled round at the end of a string the strain F on the string is directly proportional to the square of the velocity, and is inversely proportional to the length of the string. The time of rotation, is of course = length of the path ÷ velocity = (2πR)/V = (2πR)/√(R F) = 2π√(R/F). Whence then we see that for motion in a circle of a mass under the attraction of a centripetal force, or pull of a string, the time of rotation will be uniform, provided that the centripetal force always varies as the radius of the path. From this it is evident that a body fixed on to an elastic thread where the pull varies as the extension would make its rotations always in equal times. If your sling consists of elastic, whirl as you will, you can only whirl the body round so many times in a second, and no more. Any increase in your efforts only makes the string stretch, and the circle get bigger. The velocity of the body in its path of course increases, but the time it takes to go once round is invariable. It also follows that if a body hung by a string of length _l_, under the action of gravity, be travelling in a circle round and round, then, _if the circle is a small one compared with the length of the string_, the inward acceleration _f_ towards the centre will be approximately proportional to the radius _r_ of the circle, and the time of rotation will be t = 2π√(r/f). But in this case _f_, the inward acceleration, is to _g_ the acceleration downwards of gravity as A B:A P or f/g = (A B)/(A P) = (A P)/(O P) = r/l. [Illustration: FIG. 61.] [Illustration: FIG. 62.]