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 displacement of the plank till the sphere starts pure rolling.

Options

A. `(omega_(0)^(2)r^(2))/(81mug)`
B. `(2omega_(0)^(2)r^(2))/(27mug)`
C. `(4omega_(0)^(2)r^(2))/(81mug)`
D. `(2omega_(0)^(2)r^(2))/(81mug)`

Answer

D

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)`