Figure 26 represents an inclined plane supporting a ball A which is free to roll on an axle through its center. A cord attached to the yoke of the axle passes over a guide pulley B to a counterweight W. The weight W is then pulling against the ball in a direction parallel to the face of the plane and is preventing the ball from rolling down.
Now it is easy to see that the weight W does not need to be so heavy as the ball to keep the ball from rolling, since part of the weight of the ball is supported by the plane. In other words, the ball naturally tends to fall straight down in the direction of the dotted line XY, just as though it were dropped from the hand and fell to the floor.
By a diagram of similar triangles, it can be proved that the length and height of the inclined plane are proportional to the weights A and W. For example, if in Fig. 26 we make the height of the plane 1 ft. and its length 2 ft., we know that the weight W need only be one-half as heavy as the weight of the ball to keep it from rolling down the plane. Stated as a proportion this would be, •
Fig. 26. - Inclined Plane.
Weight A : Weight W = 2 ft. : 1 ft.
We will now study the relative movements of the weights if the height of the inclined plane is one-half its length. In Fig. 26 when the ball rolls from the top of the plane to the bottom it has traveled 2 ft. on the plane but has dropped only 1 ft. in a vertical direction. By this we know that the distance the ball travels on the plane is to the vertical distance it moves through as 2 is to 1, when the height of the plane is one-half its length.
It has now been proved that there is a definite ratio or relation between the height and length of the plane and the weight of the ball and counterweight, and also between the distances the ball moves along the plane and perpendicular to it. Whatever the height or length of the plane, these relations always hold true.
I. To find the counterweight or force, multiply the weight on the plane by the height of the plane and divide by the length of the plane.
II. To find the weight on the plane, multiply the force by the length of the plane and divide by the height of the plane.