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)

$\frac{5}{7}v_{0}$

Answer 2

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
$m v_{0} r =m v r+\frac{2}{5} m r^{2} \times \omega\,\,\, \left(\because \omega=\frac{v}{r}\right)$
$m v_{0} r =m v r+\frac{2}{5} m r^{2}\left(\frac{v}{r}\right)$
$v_{0} =v+\frac{2}{5} v$
$v_{0} =\frac{7}{5} v$
$\Rightarrow v=\frac{5}{7} v_{0}$