Question

A body starts rotating from rest under the influence of a variable torque $\tau =10{\cos }^2\theta$ where, $\tau$ is in $\text{J}$. Find the total work done by the torque to rotate the body from $0^{\circ }$ to ${180}^{\circ }$.

• A. $\text{Zero}$
• B. $5\pi \text{J}$
• C. $900\text{J}$
• D. $10\pi \text{J}$

Answer 1

The correct option is B $5\pi \text{J}$

Given, $\tau =10{\cos }^2\theta$
Since we know that,
Work done by variable torque
$W=\int \tau d\theta$
Now, work done to rotate the body from $0^{\circ }$ to ${180}^{\circ }$ i.e $\pi \text{rad}$ is given by
$W={\int }_0^{\pi }10{\cos }^2\theta d\theta$
$\Rightarrow W=10{\int }_0^{\pi }{\cos }^2\theta d\theta$
$\Rightarrow W=10{\int }_0^{\pi }\frac{(\cos 2\theta +1)}{2}d\theta$ $(\text{as}{\cos }^2\theta =\frac{\cos 2\theta +1}{2})$
$\Rightarrow W=5{\int }_0^{\pi }(\cos 2\theta +1)d\theta$
$\Rightarrow W=5({\left[\frac{\sin 2\theta }{2}\right]}_0^{\pi }+{\left[\theta \right]}_0^{\pi })$
$\Rightarrow W=5(0+\pi )=5\pi \text{J}$
So, option B is correct.