Q. A pulley of radius 2 m is rotating about its axis by a force $F\ $=(20t - ${5t}^{2}$) N (where $t$ is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis rotation is 10 ${kgm}^{2}$, the number of rotations made by the pulley before its direction of motion if reserved, is
A) More than 3 but less than 6
B) More than 6 but less than 9
C) More than 9
D) Less than 3
Explanation
(a) $F = 20t - 5t^{2}$
$\alpha = \frac{FR}{I} = 4t - t^{2}$
$\Rightarrow$ $\frac{d\omega}{dt} = 4t - t^{2}$
$\Rightarrow$ $\int_{0}^{\omega}{d\omega} = \int_{0}^{t}{\left( 4t - t^{2} \right)dt}$
$\Rightarrow$ $\omega = 2t^{2} - \frac{t^{3}}{3}$
When direction is reversed,
$\omega = 0,\ ie,\ t = 0,\ 6$s
Now, $d\theta = \omega\ dt$
$\int_{0}^{\theta}{d\theta} = \int_{0}^{6}{\left( 2t^{2} - \frac{t^{3}}{3} \right)dt}$
$\Rightarrow$ $\theta = \left\lbrack \frac{2t^{3}}{3} - \frac{t^{4}}{12} \right\rbrack_{0}^{6}$
$\Rightarrow$ $\theta = 144 - 108 = 36\ rad$
$\therefore$ Number of rotations,
$n = \frac{\theta}{2\pi} = \frac{36}{2\pi} < 6$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.