Question

A sphere of mass $m$ is placed on a rough surface of coefficient of friction $\mu$. Initially the sphere is given velocity $v_0$ and angular velocity ${\omega }_0$ as shown in the figure. Find out the value of ${\omega }_0$ so that the sphere will come to rest after a certain time, sphere has radius $r$.

Answer 1

Let at time $t$ it stops.
Then at time $t$
$v_0-\mu gt=0$ ($a=\mu g$ as sphere skids)
$t=v_0/\mu g$
Also, $-\mu mgr=\frac{2}{5}{mr}^2\alpha$
$\alpha =-\frac{5}{2}\frac{\mu g}{r}$
$\omega ={\omega }_0-\frac{5}{2}\frac{\mu g}{r}t$
$\Rightarrow {\omega }_0=\frac{5}{2}\frac{v_0}{r}$