Q.Derive an expression for the work done by torque.

Work done by torque. For a rigid body rotating about a fixed axis, a tangential force $F$ acting at a point that undergoes a small displacement $ds$ along the circular arc does work $dW=F\,ds$. Using the geometric relation between arc length and angle, $ds=r\,d\theta$, this becomes $dW=(Fr)\,d\theta$, and recognising $Fr$ as the torque $\tau$ produced by the force,

This is the direct rotational parallel of the translational $dW=F\,ds$, with torque replacing force and angular displacement replacing linear displacement. Integrated over a finite rotation, the total work done by a (constant) torque through an angle $\theta$ is $W=\tau\theta$.

Power delivered by torque. Power is the rate of doing work; differentiating $dW=\tau\,d\theta$ with respect to time,

the direct rotational parallel of the translational $P=\vec F\cdot\vec v$. A motor or engine delivering a fixed power $P$ can therefore trade off torque against angular speed — high torque at low $\omega$, or low torque at high $\omega$ — exactly as a translational engine trades off force against speed. …