Question

A solid sphere and a hollow sphere of equal mass and radius are placed over a rough horizontal surface after rotating it about its masss centre with same angular velocity ${\omega }_0$. Once the pure rolling starts let $v_1$ and $v_2$ be the linear speeds of their centres of mass. Let $t_1$ and $t_2$ be the times when pure rolling of solid sphere and of hollow sphere is started. Then

• A. $t_1=t_2$
• B. $t_1<t_2$
• C. $t_1>t_2$
• D. none of these

Answer 1

The correct option is B $t_1<t_2$

$ma=f⟹a=\frac{f}{m}$

Also, $fr=I\alpha ⟹a=\frac{I\alpha }{mr}$

Now, $v=0+at=\frac{I\alpha }{mr}t$ .................(1)

$w=w_o-\alpha t$

$\frac{v}{r}=w_o-\frac{mvr}{I}$ (using (1) and $v=rw$)

$⟹v=\frac{w_{o}r}{1+\frac{m{r}^2}{I}}$

As $I_{hollowsphere}>I_{solidsphere}⟹v_2>v_1$

From (1), $v\propto t$ as $a=\frac{f}{m}$ is same for both.

Thus $t_2>t_1$