A long horizontal plank of mass `m` is lying on a smooth horizontal surface. A sphere of same mass `m` and radius `r` is spinned about its own axis with angular velocity we and gently placed on the plank. The coefficient of friction between the plank and the sphere is `mu`. After some time the pure rolling of the sphere on the plank will start. Answer the following questions.

Find the time `t` at which the pure roiling starts

Options

A. `(omega_(0)r)/(9mug)`
B. `(2omega_(0)r)/(9mug)`
C. `(omega_(0)r)/(3mug)`
D. `(2omega_(0)r)/(mug)`

Answer

B

Explanation

`a_(1)=(mumg)/m=mug, a_(2)=(mumg)/m=mug`
`v_(1)=a_(1)t=mu"gt", v_(2)=a_(2)t=mu"gt"`

`tau=mumgr`
`omega=omega_(0)`+`alphat,`
where `alpha=-tau/I=(mumgr)/(2/5mr^(2))=-5/2(mug)/r`
`implies omega=omega_(0)-5/2(mug)/rt`
When the pure rolling starts
`omega=(v_(1)+v_(2))/r`
`implies omega_(0)-5/2(mu"gt")/r=(2mu"gt")/r`
`implies t=(2omega_(0)r)/(9mug)`
Velocity of the sphere `v_(2)=mu"gt"=(2omega_(0)r)/9`
` s_(P)=1/2a_(1)t^(2)=1/2mug((2omega_(0)r)/(9mug))^(2)=2/81(omega_(0)^2r^(2))/(mug)`