Torque and the Lever Arm

Push a door near its hinge and almost nothing happens, even with real effort. Push the same force at the outer edge and it swings open easily. The force is identical, but the turning effect is not, because turning effect depends on where and how a force is applied relative to the pivot, not just its size.

Torque is tau = F r sin(theta), where r is the distance from the pivot to the point where the force is applied, and theta is the angle between the force vector and the lever arm (the line from pivot to that point). The sin(theta) factor exists because only the component of force perpendicular to the lever arm causes rotation. A component pointed along the lever arm, straight toward or away from the pivot, just stretches or compresses that line and turns nothing. You can picture this as F sin(theta) being the effective turning force, multiplied by r, or equivalently as F multiplied by r sin(theta), the perpendicular distance from the pivot to the force's full line of action. Both groupings give the same number.

This is the scalar version of torque as a cross product, tau = r x F, which you meet right alongside it: the magnitude is F r sin(theta), and the direction comes from the right-hand rule. Lever arm thinking also sets up everything after it in the unit, since Newton's second law for rotation, tau = I alpha, needs torque computed this way before you can connect it to angular acceleration.