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 atarta rolling without shpping the speed of its centre is

• A

  $\frac{2}{7} v_{0}$
• B

  $\frac{3}{7} v_{0}$
• Correct answer
  C

  $\frac{5}{7} v_{0}$
• D

  $\frac{6}{7} v_{0}$

Solution

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 $$ \begin{aligned} I_{\omega} &=\text { constant } \\ m v_{0} r &=m v r+\frac{2}{5} m r^{2} \times \omega \quad\left(\because \omega=\frac{v}{r}\right) \\ m v_{0} r &=m v r+\frac{2}{5} m r^{2}\left(\frac{v}{r}\right) \\ v_{0} &=v+\frac{2}{5} v \\ v_{0} &=\frac{7}{5} v \Rightarrow v=\frac{5}{7} v_{0} \end{aligned} $$

Hence, the correct answer is option: 'C'.