question_answer

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 \[{{\upsilon }_{0}}.\]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)  \[\frac{6}{7}{{v}_{0}}\]

Correct Answer: C

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 \[I\omega =\text{cosntant}\] \[m{{v}_{0}}r=mvr+\frac{2}{5}m{{r}^{2}}\times \omega \]  \[\left( \because \,\omega =\frac{v}{r} \right)\] \[m{{v}_{0}}r=mvr+\frac{2}{5}m{{r}^{2}}\left( \frac{v}{r} \right)\] \[{{v}_{0}}=v+\frac{2}{5}v\] \[{{v}_{0}}=\frac{7}{5}v\Rightarrow v=\frac{5}{7}{{v}_{0}}\]