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 `10kgm^(2)`, the number of rotations made by the pulley before its direction of motion is reversed, is :

Options

A. more than 3 but less than 6
B. more than 6 but less than 9
C. more than 9
D. less then 3

Answer

A

Explanation

Here direction of motion will be reversed when force `F=0` or `20t-5t^(2)=0` or `t=4s`. If `alpha` is the angular acceleration then torque
`tau=Ialpha=Fror10xxalpha=(20t-5t^(2))xx2`
or `alpha=4t-t^(2) and omega=(d theta)/(dt)`
Also `(d theta)/(dt)=alpha.t`
or `d theta=alpha.tdt`
`d theta(4t-t^(2)).tdt=(4t^(2)-t^(3))dt`
Integrating both sides `theta=(4t^(3))/(3)-(t^(4))/(4)`
If n rotations are completed in 4s, then putting `t=4`
`therefore theta=2pin=(4xx64)/(3)-64=(64)/(3)~=3.4`
But `3lt3.4lt6`