question_answer

A long horizontal rod has a bead which can slide along its length, and initially placed at a distance \[L\] from one end \[A\] of the rod. The rod is set in angular motion about \[A\] with constant angular acceleration\[\alpha \]. If the coefficient of friction between the rod and the bead is \[\mu \], and gravity is neglected, then the time after which the bead starts slipping is

A) \[\sqrt{\frac{\mu }{\alpha }}\]               

B) \[\frac{\mu }{\sqrt{\alpha }}\]

C) \[\frac{1}{\sqrt{\mu \alpha }}\] 

D) Infinitesimal

Correct Answer: A

Solution :

[a] Let the bead starts slipping after time t. for critical condition frictional force provides the centripetal force \[m{{\omega }^{2}}L=\mu R=\mu m\times {{a}_{t}}=\mu Lm\alpha \]  \\[\Rightarrow m{{\omega }^{2}}L=\mu R=\mu m\times {{a}_{t}}=\mu Lm\alpha \]\[\Rightarrow m{{(\alpha t)}^{2}}L=\mu mL\alpha t=\sqrt{\frac{\mu }{\alpha }}\]