Question

A plank of mass $M$ rests on a smooth horizontal plane. A sphere of mass $m$ and radius $r$ is paced on the rough upper surface of the plank and the plank is suddenly given a velocity $v$ in the direction of its length. Find the time after which the sphere beings pure rolling, if the coefficient of friction between the plank and the sphere is $\mu$ and the plank is sufficiently long.

Answer 1

Let $t$ be the after which slipping between the sphere and plank disappears.
For the sphere,
$N=mg$ and $\mu N=m{a}_S$
$\Rightarrow a_S=\mu g$
And $\tau =I\alpha$
$\Rightarrow \mu mgr=\frac{2}{5}m{r}^2\alpha$
$\Rightarrow \alpha =\frac{5\mu g}{2r}$
For the plank,
After time $t$
Velocity of plank $V_P=V-\frac{\mu mg}{M}t$,
Velocity of sphere, $V_S=\mu gt$
And Velocity of sphere, $\omega =\frac{5\mu g}{2r}t$
For no slipping, the point of contact of sphere should have the same plank,
$\Rightarrow V_S+\omega r=V_P$
Substituting and solving,
$\mu gt+\left(\frac{5\mu g}{2r}t\right)r=v-\frac{\mu mg}{M}t$
$\Rightarrow t=\frac{2v}{g(2\mu +5\mu +2\mu m/M)}=\frac{2v}{\mu g[7+\left(2m/M\right)]}$