Question

A solid uniform sphere resting on a rough horizontal plane is given a horizontal impulse directed through its center so that it starts sliding with an initial velocity $v_0$. When it finally starts rolling without slipping the speed of its center is

• A. $\frac{2}{7}{v}_0$
• B. $\frac{3}{7}{v}_0$
• C. $\frac{5}{7}{v}_0$
• D. $\frac{6}{7}{v}_0$

Answer 1

Answer: (C)

Let, the final velocity be $v$

So, angular momentum will remain conserved along point of contact
By conservation of angular momentum
Angular momentum will remain conserved along point of contact
$I\omega =$ constant
$m{v}_{0}r=mvr+\frac{2}{5}m{r}^2\times \omega \left(\because \omega =\frac{v}{r}\right)$
$m{v}_{0}r=mvr+\frac{2}{5}m{r}^2\left(\frac{v}{r}\right)$
$v_0=v+\frac{2}{5}v$
$v_0=\frac{7}{5}v$
$\Rightarrow v=\frac{5}{7}{v}_0$