Turning back to what was said about Galileo, it will be remembered that he showed that all bodies, big and small, light and heavy, fell to the earth at the same speeds. What is that speed? Let us denominate by G the number of feet per second of increase of motion produced in a body by the earth’s action during one second. Then the velocity at the end of that second will be V = GT. The space fallen through will be S = 1/2 GT².
What I want to know then is this: how far will a body under the action of gravity fall in a second of time?
This, of course, is a matter for measurement. If we can get a machine to measure seconds, we shall be able to do it; but inasmuch as falling bodies begin by falling sixteen feet in the first second and afterwards go on falling quicker and quicker, the measurements are difficult. Galileo wanted to see if he could make it easier to observe. He said to himself, “If I can only water down the force of gravity and make it weaker, so that the body will move very slowly under its action, then the time of falling will be easier to observe.” But how to do it? This is one of those things the discovery of which at once marks the inventor.
Fig. 25.
The idea of Galileo was, instead of letting the body drop vertically, to make it roll slowly down an incline, for a body put upon an incline is not urged down the incline with the same force which tends to make it fall vertically.
Can any law be discovered tending to show what the force is with which gravity tends to drag a mass down an incline?
There is a simple one, and before Galileo’s time it had been discovered by Stevinus, an engineer. Stevinus’ solution was as follows. Suppose that A B C is a wedge-shaped block of wood. Let a loop of heavy chain be hung over it, and suppose that there is a little pulley at C and no friction anywhere. Then the chain will hang at rest. But the lower part, from A to B, is symmetrical; that is to say, it is even in shape on both sides. Hence, so far as any pull it exerts is concerned, the half from A to D will balance the other half from B to D. Therefore, like weights in a scale, you may remove both, and then the force of gravity acting down the plane on the part A C will balance the force of gravity acting vertically on the part C B. Now the weight of any part of the chain, since it is uniform, is proportional to its length. Hence, then, the gravitational force down the plane of a piece whose weight equals C A is equal to the gravitational force vertically of a piece whose weight equals C B. In other words, the force of gravity acting down a plane is diminished in the ratio of C B to C A.
But when a body falls vertically, then, as we have seen, S = 1/2 GT², where S is the space it will fall through, G the number of feet per second of velocity that gravity, acting vertically on a body, will produce in it in a second, and T the number of seconds of time. If then, instead of falling vertically, the body is to fall obliquely down a plane, instead of G we must put as the accelerating force
G × (vertical height of the end of the plane)/(length of the plane).