Question

A pulley os radius 2m is rotated about its axis by a force `F= (20 t- 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 rotaitons 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

`tau=Fr=I alpha`
`(20t-5t)^(2)(2)=10alpha`
`alpha=4t-t^(2)`
`int_(0)^(theta)domega=int_(0)^(t)(4t-t^(2))dt`
`omega=2t^(2)-(t^(3))/(3)`
`int_(0)^(theta)d theta=int_(0)^(t)(2t^(2)-(t^(3))/(3))dt`
`theta=(2t^(3))/(3)-(t^(4))/(12)`
`omega=0=2t^(2)-(t^(3))/(3)impliest=6s`
At `t=6 s`
`theta=(2)/(3)(6)^(3)-((6)^(4))/(12)=36rad`
Number of rotations `n=(theta)/(2pi)=(18)/(pi)`
`3 lt n lt 6`