Torque from a Force at an Angle

Push straight into a door or straight out along a wrench handle, and nothing rotates no matter how hard you push. That force is pointed along the line to the pivot, so it has nothing to turn. Pull at a slant and part of your force pulls along the handle, wasted, while the rest pushes sideways on it, which is the part that actually creates rotation. The sideways part is F sin(theta), so torque is tau = F r sin(theta), with theta measured between the force and the handle itself.

There are two equivalent ways to get there. One: find the component of the force perpendicular to the handle, F sin(theta), and multiply by the full handle length r. Two: find the lever arm, the perpendicular distance from the pivot to the line along which the force acts, which works out to r sin(theta), and multiply by the full force F. Same answer, same sine, just grouped differently — shrink the force or shrink the distance, the product is identical.

This is the angled cousin of 5.2's tau = F r for perpendicular forces: set theta = 90 degrees there and sin(theta) = 1, so the formulas agree. It also sets up torque as a vector product (r x F) later, where the sine is already built into the cross product definition. Keep this sine straight and the rest of rotational dynamics, net torque, angular acceleration, rolls out the same way net force did in Unit 1.