Question

A horizontal plank having mass m lies on a smooth horizontal surfuce. A sphere of same mass and radius r is spined to an 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 the sphere starts pure rolling on the plank. The plank is long enough

Answer 1

Let, $a_1=$ linear aceeleration of sphere (towards right) $a_2=$ linear acceleration of plank (towards left)
and $\alpha =$ angular retardation of sphere
$a_1=a_2=\frac{\mu mg}{m}=\mu g$
$\alpha =\frac{\mu mgr}{\frac{2}{5}m{r}^2}=\frac{5}{2}\frac{\mu g}{r}$
Let pure rolling starts after lime ${}^{\prime }{t}^{\prime }$ Then
$\omega r-v=v$ $\therefore \omega r=2v$
$({\omega }_0-\alpha t)r=2\left(a_{1}t\right)$
Substituting the values.
$t=\frac{2}{9}\frac{{\omega }_0}{\mu g}$
$\therefore s=\frac{1}{2}\left(a_2\right)t^2=\frac{2{\omega }_0^2{r}^2}{81\mu g}$