A disc of radius  r is rotating about its center with an angular speed  ${\omega }_0$. It is gently placed on a rough horizontal surface. After what time it will start pure rolling?

Given radius of disc= r
Angular speed= ${\omega }_0$
Applying law of conservation of angular momentum about a point on the horizontal surface,

$\begin{array}{l}I{\omega }_0=mvr+I\omega \\ \text{}\Rightarrow \frac{m{r}^2}{2}{\omega }_0=mvr+\frac{m{r}^2}{2}\left(\frac{v}{r}\right)\\ \text{}\Rightarrow v=\frac{{\omega }_{0}r}{3}\end{array}$

Velocity of center of disc when it starts pure rolling is$\left(\frac{{\omega }_0\text{r}}{3}\right)$
 From  $v=u+at$

$\begin{array}{l}\Rightarrow v=at\\ \Rightarrow t=\frac{v}{a}(\because a=\mu g)\\ \Rightarrow t=\frac{{\omega }_{0}r}{3\mu g}\end{array}$