Question:
A solid sphere is set into motion on a rough horizontal surface with a linear speed $v$ in the forward direction and an angular speed $v / R$ in the anticlockwise direction as shown in figure. Find the linear speed of the sphere (a) when it stops rotating and (b) when slipping finally ceases and pure rolling starts.
Solution:
Conserving angular momentum about bottom point $L_{i}=L_{f}$
$m v R-I \omega=m v^{\prime} R$
$m v R-\left(\frac{2}{5} m R^{2}\right)\left(\frac{v}{R}\right)=m v^{\prime} R$
$\frac{3}{5} m v R=m v^{\prime} R$
$v^{\prime}=\frac{3 v}{5}$
Conserving angular momentum
$m v^{\prime} R=m v^{\prime \prime} R+I \omega^{n}$
$m\left(\frac{3 v}{5}\right) R=m v^{n} R+\left(\frac{2}{5} m R^{2}\right)\left(\frac{v !}{R}\right)$
$v^{m}=\frac{3 w}{7}$