Question

A hollow sphere of radius $2\text{m}$ is projected horizontally along a rough surface with velocity $v_0$ and angular velocity ${\omega }_0$. Find out their ratio so that the sphere stops moving after some time.

• A. $\frac{2}{3}$
• B. $\frac{4}{3}$
• C. $1$
• D. $\frac{3}{2}$

Answer 1

The correct option is B $\frac{4}{3}$


The hollow sphere is moving rightward. Hence, to oppose the rotational motion of the sphere, friction will act leftward at the contact point. Hence, about point P , ${\tau }_p=0$

Therefore, applying conservation of angular momentum about point ($P$),
${\overrightarrow{L}}_i={\overrightarrow{L}}_f$
When the sphere stops moving finally, both velocity and angular velocity become zero.
$\Rightarrow I_{com}\times {\omega }_0-m{v}_{0}R=0+0$
$\Rightarrow \frac{2}{3}m{R}^2\times {\omega }_0-m{v}_{0}R=0$
$\Rightarrow \frac{v_0}{{\omega }_0}=\frac{2R}{3}$
Now putting $R=2\text{m}$,
$\frac{v_0}{{\omega }_0}=\frac{2\times 2}{3}=\frac{4}{3}$