Question

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

Answer 1

Answer: (A)

$\sqrt{\frac{\mu}{\alpha}}$

Answer 2

Let the bead starts slipping after time t

For critical condition

Frictional force provides the centripetal force $m\omega^{2}L = \mu R = \mu m \times a_{t} = ⥂ \mu mL\alpha$

m (αt)²L = μmLα ⇒ $t = \sqrt{\frac{\mu}{\alpha}}$ (As ω = αt)