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.
Fig. 61.
Fig. 62.
Whence then the time of rotation of this body would be if the circle of rotation was small