Question

A solid sphere is rolling purely on a rough horizontal surface (coefficient of kinetic friction $=\mu )$ with speed of centre $=u$. It collides inelastically with a smooth vertical wall at a certain moment, the coefficient of restitution being $\frac{1}{2}$The sphere will begin pure rolling after a time

• A. $\frac{3u}{7\mu g}$
• B. $\frac{2u}{7\mu g}$
• C. $\frac{3u}{5\mu g}$
• D. $\frac{2u}{5\mu g}$

Answer 1

The correct option is A $\frac{3u}{7\mu g}$

$a=\frac{\mu mg}{m}=\mu g$
$\omega =\frac{u}{R}$
$\alpha =\frac{\left(\mu mg\right)R}{2/5m{R}^2}=\frac{5\mu g}{2R}$
Pure rolling will start when
$v=R\omega$
$(\frac{u}{2}-at)=R(\alpha t-\omega )$
or $(\frac{u}{2}-\mu gt)=R(\frac{5\mu g}{2R}t-\frac{u}{R})$
solving we get, $t=\frac{3u}{7\mu g}$