question_answer

A solid cylinder is rolling down on an inclined plane of angle \[\theta ,\] The coefficient of static friction between the plane and cylinder is \[{{\mu }_{s}}\]. Then condition for the cylinder not to slip is

A)  \[\tan \,\,\theta \ge \,3{{\mu }_{s}}\]   

B)  \[\tan \,\,\theta >3{{\mu }_{s}}\]

C)  \[\tan \,\,\theta \le 3{{\mu }_{s}}\]   

D)  \[\tan \,\,\theta <3{{\mu }_{s}}\]

Correct Answer: C

Solution :

For the rolling of a solid cylinder acceleration \[a=\frac{g}{3}\sin \,\theta \] \[\therefore \] The condition for the cylinder to remain in equilibrium \[ma\le {{\mu }_{s}}R\] \[\Rightarrow \] \[\frac{1}{3}Mg\,\sin \,\theta \le \,Mg\,\cos \theta .{{\mu }_{s}}\] or \[{{\mu }_{s}}\ge \frac{1}{3}\,\tan \theta \] or \[\tan \theta \le 3{{\mu }_{s}}\]