Question

A long horizontal plank of mass 'm' is lying on a smooth horizontal surface. A solid sphere of same mass ${}^{\prime }{m}^{\prime }$ and radius ${}^{\prime }{r}^{\prime }$ is spinned about its own axis with angular velocity ${\omega }_0$ 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. Find the time $t$ at which the pure rolling starts.

Answer 1

Let velocity of centre of mass $=v$

Frictional force $f=\mu mg$

i.e, $ma=\mu mg$ $a=$ retardation of $CM$

now torque $\tau fR=I\alpha$

$\alpha =\frac{fR}{I}=\frac{5\mu g}{2R}(\because I_{sphere}=\frac{2\mu R^2}{5})$

$\therefore w=\alpha t=\frac{5\mu g}{2R}t$

Now if pure rolling has started then, $v=wR$

So $v-\mu gt=5\mu gt/2t=\frac{2v}{7\mu g}$