Question

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 is 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 $9$

Answer 1

Answer: (B)

$\tau =\left(20t-5t^2\right)2=40t-10t^2$
$\alpha =\frac{\tau }{I}=\frac{40t-10t^2}{10}=4t-t^2$
$\omega =\underset{0}{\overset{t}{\int }}\alpha dt=2t^2-\frac{t^3}{3}$
When direction is reversed , $\omega$ is zero. So
$2t^2-\frac{t^3}{3}=0$
$\Rightarrow t^3=6t^2$
$\Rightarrow t=6s$
$\theta =\int \omega dt$
$=\underset{0}{\overset{6}{\int }}\left(2t^2-\frac{t^3}{3}\right)dt$
$={\left[\frac{2t^3}{3}-\frac{t^4}{12}\right]}_0^6=36rad$
Number of revolution $=\frac{36}{2\pi }$ = Less than $6$