Question

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

$\frac{5}{7}{v}_0$

Let the velocity of the sphere when it starts pure rolling be $v$ and angular velocity be $w$.

As the net torque about the line $O$ is zero, thus angular momentum is conserved about $O$

$\therefore$ $L_i=L_f$

$m{v}_{o}r=mvr+I\omega$

$m{v}_{o}r=mvr+\frac{2}{5}m{r}^2\times \frac{v}{r}$ $(\because v=r\omega )$

$\therefore$ $v_o=v+\frac{2}{5}v$ $⟹v=\frac{5}{7}{v}_o$