# Work done and power in rotational motion

## work done and power in rotational motion

- 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

- In vector notation

P=**Τ**.**ω**

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