Solve the following :
Question:

A uniform wheel of radius $R$ is set into rotation about its axis at an angular speed $\omega$. This rotating wheel is now placed on a rough horizontal surface with its axis horizontal. Because of friction at the contact, the wheel accelerates forward and its rotation decelerates till the wheel starts pure rolling on the surface. Find the linear speed of the wheel after it starts pure rolling.

Solution:

By angular momentum conservation at bottom point since ${ }^{\tau_{e x t}}=0$

$L_{i}=L_{f}$

$I \omega=m v^{\prime} R+I \omega^{\prime}$

$\left(\frac{m R^{2}}{2}\right) \omega=m v^{\prime} R+\left(\frac{m R^{2}}{2}\right)\left(\frac{v y}{R}\right)$

$v^{p}=\frac{\omega R}{3}$