Work Done by a Torque

Think about cranking a winch. Push with the same steady force on the handle, but crank it through more turns, and you do more work — your arms feel it. The force at each instant hasn't changed, but you kept applying it over a longer path. That path, for rotation, is measured in angle, not meters. So rotational work piles up as torque acts over more and more radians, exactly like ordinary work piles up as a force acts over more and more meters.

The reason force times distance and torque times angle give the same answer comes straight from how torque is built. Torque equals force times the radius at which it acts. The point where the force is applied moves a distance equal to radius times angle (in radians) as the object turns. Multiply force by that distance: force times radius times angle. Force times radius is torque, so you get torque times angle. The radius cancels out of the bookkeeping — it's baked into both torque and the distance traveled, so the work comes out identical to F times d. A steady 5 N m torque turning a wheel through 3 radians does 5 x 3 = 15 joules of work, no matter the wheel's size. On a torque versus angle graph, that work is literally the area under the curve, just like force versus distance.

This is why a torque on a locked wheel, however strong, does zero work: the wheel sweeps out zero angle, so there's no path to do work over. That same W = torque x angle shows up next in the work-energy theorem for rotation, linking it to rotational kinetic energy and setting up power as torque times angular velocity.