Question

one way to get a car unstuck is to tie one end of a strong rope to the car and the other end to a tree, then push the rope at it's midpoint at right angles to the rope. Draw a free body diagram and explain why even a small force on the rope can exert a large force on the car.

Answer 1

The tension in the rope at the midpoint can be decomposed into a force toward the tree, and a force toward the car. For the most part, these forces balance, what's "left over" is a force perpendicular to the rope, equal to 2 times the tension in the rope, times the sine of the displacement from a straight line between the tree and car. If the angle is small, the restoring force is approximately:

F = 2 * T * sin(theta) = 4 * T * dx / L

where L is the rope's length.

Rearrange to T = (L/4 dx) F. That is, the force on the car is the force of the sideways push, multiplied by L/(4 dx).

For a long rope that's reasonably taut when straight (large L, small dx), the amplification can be substantial.