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 \[{{\mu}_{s}}\]. 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           \[\tan \theta \le 3{{\mu }_{s}}\]