We know that when we apply force on any object in direction of the displacement of the object ,work is said to be done
Similarly force applied to the rotational body does work on it and this work done can be expressed in terms of moment of force (torque) and angular displacement θ
Consider the figure given below where a force F acts on the wheel of radius R pivoted at point O .so that it can rotate through point O
This force F rotates the wheel through an angle dθ and dθ is small enough so that we can regard force to be constant during corresponding time interval dt
Work done by this force is
dW=Fds
but ds=Rdθ
So
dW=FRdθ
Now FR is the torque Τ due to force F.so we have
dW=Τdθ ----(19)
if the torque is constant while angle changes from θ_{1} to θ_{2} then
W=Τ(θ_{2}-θ_{1})=ΤΔθ ---(20)
Thus work done by the constant torque equals the product of the torque and angular displacement
we know that rate of doing work is the power input of torque so
P=dW/dt=Τ(dθ/dt)=Τω