Question

Write down an expression for the work done in rotating a body

Answer 1

To find:

Expression for work done in rotational motion of a body.

Calculation:

Let us consider the fact that angular acceleration causes change in angular velocity from $\omega_{1}$ to $\omega_{2}$

So , we can say that :

$\therefore \: angular \: acc. = \alpha = \dfrac{d( \omega)}{dt}$

Now , let dW amount of work be done to rotate the body by angle of $d\theta$ and let torque be $\tau$

$\therefore \: dW = \tau \times d \theta$

Torque can be represented in form of Moment Of Inertia and angular acceleration :

$= > \: dW = ( I \times \alpha ) \times d \theta$

Putting value of angular acceleration:

$= > \: dW = \{ I \times \dfrac{d( \omega)}{dt} \} \times d \theta$

$= > \: dW = I \times \dfrac{d( \theta)}{dt} \times d \omega$

$= > \: dW = I \times \omega \times d \omega$

Integrating on both sides:

$\displaystyle = > \: \int dW = I \times \int_{\omega_{1}}^{ \omega_{2}} \omega \times d \omega$

$\displaystyle = > \: W = I \times \int_{\omega_{1}}^{ \omega_{2}} \omega \times d \omega$

$\displaystyle = > \: W = I \times \bigg \{ \bigg(\dfrac{ {\omega_{2}}^{2} }{2} \bigg) - \bigg(\dfrac{ {\omega_{1}}^{2} }{2} \bigg) \bigg \}$

$\displaystyle = > \: W = \frac{1}{2} \times I \times \{ {(\omega_{2})}^{2} - {(\omega_{1})}^{2} \}$

So final answer is :

$\boxed{ \red{ \bold{ \: W = \frac{1}{2} \times I \times \{ {(\omega_{2})}^{2} - {(\omega_{1})}^{2} \}}}}$