Pushing a block up a ramp

g / The applied force FA pushes the block up the frictionless ramp.

Figure (g) shows a block being pushed up a frictionless ramp at constant speed by an applied force F_A. How much force is required, in terms of the block's mass, m, and the angle of the ramp, θ?

h / Three forces act on the block. Their vector sum is zero.

i / If the block is to move at constant velocity, Newton's first law says that the three force vectors acting on it must add up to zero. To perform vector addition, we put the vectors tip to tail, and in this case we are adding three vectors, so each one's tail goes against the tip of the previous one. Since they are supposed to add up to zero, the third vector's tip must come back to touch the tail of the first vector. They form a triangle, and since the applied force is perpendicular to the normal force, it is a right triangle.

Figure (h) shows the other two forces acting on the block: a normal force, F_N, created by the ramp, and the weight force, F_W, created by the earth's gravity. Because the block is being pushed up at constant speed, it has zero acceleration, and the total force on it must be zero. From figure (i), we find

|F_A| = |F_W| sin θ
= mg sin θ .

Since the sine is always less than one, the applied force is always less than mg, i.e., pushing the block up the ramp is easier than lifting it straight up. This is presumably the principle on which the pyramids were constructed: the ancient Egyptians would have had a hard time applying the forces of enough slaves to equal the full weight of the huge blocks of stone.

Essentially the same analysis applies to several other simple machines, such as the wedge and the screw.