Q. 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
Explanation
(a) Let the bead starts slipping after time $t$
For critical condition Frictional force provides the centripetal force
$m\omega^{2}L = \mu R = \mu m \times a_{t} = \mu Lm\alpha$
$\Rightarrow m(\alpha t)^{2}L = \mu mL\alpha$
$\Rightarrow t = \sqrt{\frac{\mu}{\alpha}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lbrack As\ \omega = \alpha t\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.