Power Delivered by Torque

Power delivered is simply the rate at which work is done. Differentiating the expression for work done by a torque, $dW=\tau\,d\theta$, with respect to time:

Since $\dfrac{d\theta}{dt}=\omega$, the angular velocity, this gives the instantaneous power delivered by torque:

This is the exact rotational analogue of the corresponding expression for instantaneous power delivered in translational motion,

$P=\vec F\cdot\vec v,$ …