Question

A solid sphere of mass $M$ and radius $r$ slips on a rough horizontal plane. At some instant it has translational velocity $v_o$ and rotational velocity about the center $v_0/2r$. The translational velocity after the sphere starts pure rolling is

• A. $\frac{6v_o}{7}$in backward direction
• B. $\frac{7v_o}{6}$ in backward direction
• C. $\frac{6v_o}{7}$in forward direction
• D. $\frac{7v_o}{6}$ in forward direction

Answer 1

The correct option is C $\frac{6v_o}{7}$in forward direction

Draw a diagram of given problem.

Calculate translational velocity after sphere starts pure rolling.
As we know, angular momentum is conserved at lowest point.
Therefore,
$m{v}_{0}r+\frac{2}{5}m{r}^2\frac{v_0}{2r}=m{v}^{\prime }r+\frac{2}{5}m{r}^2\frac{v^{\prime }}{r}$
$\frac{10m{v}_{0}r+2m{v}_{0}r}{10}=\frac{7m{v}^{\prime }r}{5}$
Hence, translational velocity is
$v^{\prime }=\frac{6v_0}{7}$