question_answer

A uniform sphere of radius R is placed on a rough horizontal surface and given a linear velocity \[{{v}_{0}}\]and angular velocity \[{{\omega }_{0}}\] as shown. The sphere comes to rest after moving some distance to the right. It follows that:

A)  \[{{v}_{0}}=\,{{\omega }_{0}}R\]

B) \[2{{v}_{0}}=5{{\omega }_{0}}R\]

C) \[5{{v}_{0}}=2{{\omega }_{0}}R\]

D) \[2{{v}_{0}}={{\omega }_{0}}R\]

Correct Answer: C

Solution :

\[f=\mu mg,\,\,\,a=\mu g\]

\[\alpha =\frac{\mu mg}{\frac{2}{5}m{{R}^{2}}}=\frac{5}{2}\frac{g\mu }{R}\]

Now, \[t=\frac{{{v}_{0}}}{a}=\frac{{{\omega }_{0}}}{\alpha }\] or \[\frac{a}{a}=\frac{{{v}_{0}}}{{{\omega }_{0}}}\]

\[\therefore \frac{2R}{5}=\frac{{{v}_{0}}}{{{\omega }_{0}}};\] \[\therefore \,\,\,\,\,5{{v}_{0}}=2{{\omega }_{0}}R\]