Fig. 60.
The strain on the body, due to the force tending to pull it inwards, we shall designate by F, meaning by F the number of feet of velocity that would in one second be imparted to the body by the attractive force.
Suppose that at some given instant of time the body is at a point a. At that instant its direction will be along a b, tangential to the circle at a, and that is the path it would take if the centripetal or attractive force ceased to act just as the body got to a. In that case the body would be whirled off like a stone from a sling along the line a b, and would at the end of a given time, let us suppose a second, arrive at b. But it is not so whirled off; it is attracted towards O and pulled inwards, and comes to c. Hence, then, the attractive force acting during one second must have been sufficient to pull the mass in from b to c. But we know that if an accelerating force (F) acts on a body for a second it produces a final velocity equal to F at the end of the second, and an average velocity half F during the second.
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).