Question

A pulley of radius $2$ $m$ 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 $10$ $kg{m}^2$, 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)

more than 3 but less than 6

$\alpha =\frac{T}{I}=4t-t^2$
$\frac{d\omega }{dt}=4t-t^2$
$\omega ={2t}^2-\frac{t^3}{3}$
When $\omega =0$, $t=6s$
$\omega =\frac{d\theta }{dt}=2t^2-\frac{t^3}{3}$
$d\theta =(2t^2-\frac{t^3}{3})dt$
$\theta =\frac{2t^3}{3}-\frac{t^4}{12}$
$\theta \left(in6s\right)=36rad$
$2\pi n=36$
$n=\frac{36}{2\pi }<6$.