Q. A pulley of radius $2m$ is rotated about its axis by a force $F = (20t - 5t^{2})$ newton (where $t$ is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $10kg\ m^{2}$, the number of rotations made by the pulley before its direction of motion if reversed, is
A) Less than 3
B) More than 3 but less than 6
C) More than 6 but less than 9
D) More than 9
Explanation
(b) To reverse the direction $\int_{}^{}{\tau d\theta = 0}$ (work done is zero)
$\tau = \left( 20t - 5t^{2} \right)2 = 40t - 10t^{2}$
$\alpha = \frac{\tau}{I} = \frac{40t - 10t^{2}}{10} = 4t - t^{2}$
$\omega = \int_{0}^{t}{\alpha\ dt = 2t^{2} - \frac{t^{3}}{3}}$
$\omega$ is zero at
$2t^{2} - \frac{t^{3}}{3} = 0 \Rightarrow t^{3} = 6t^{2} \Rightarrow t = 6\ \sec$
$\theta = \int_{}^{}{\omega dt} = \int_{0}^{6}{\left( 2t^{2} - \frac{t^{3}}{3} \right)dt}$
$= \left\lbrack \frac{{2t}^{3}}{3} - \frac{t^{4}}{12} \right\rbrack_{0}^{6} = 216\left\lbrack \frac{2}{3} - \frac{1}{2} \right\rbrack = 36\ rad$
No. of revolution $\frac{36}{2\pi}$ is less than 6
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.