Work Done by Torque

Consider a rigid body rotating about a fixed axis perpendicular to the page, and a point $P$ on the body acted on by a tangential force $F$. As the body rotates, $P$ undergoes a small displacement $ds$ along its circular arc, and the work done by the force over this small displacement is, as in ordinary translational motion,

The small arc length $ds$, the corresponding small angle of rotation $d\theta$, and the radius $r$ (the distance of $P$ from the axis) are related by the standard geometric relation $ds=r\,d\theta$. Substituting:

The quantity $Fr$ is exactly the torque $\tau$ produced by the force about the axis, so

$dW=\tau\,d\theta.$ …