A horizontal plank having mass `m` lies on a smooth horizontal surface. A sphere of same mass and radius `r` is spined to angular frequency `omega_(0)` and gently placed on the plank as shown in the figure. If coefficient of friction between the plank and the sphere is `mu`. Find the distance moved by the plank till sphere starts pure rolling on the plank. the plank is long enough.

Explanation

Let,
`a_(1)=` linear acceleration of sphere (towards right)
`a_(2)=` linear acceleration of plank (towrads left)
and `alpha=` angular retardation of sphere
`a_(1)=a_(2)=(mumg)/(m)=mugimpliesalpha=(mumgr)/((2)/(5)mr^(2))=(5)/(2)(mug)/(r)`
let pure rolling starts after time `t`. Then ,brgt `omegar-v=v`
`thereforeomegar=2v`
`(omega_(0)-alphat)r=(a_(1)t)`
Substituting the values,
`t=(2)/(9)(omega_(0)r)/(mug)`
`therefores=(1)/(2)(a_(2))t^(2)=(2omega_(0)^(2)r^(2))/(81mug)`