A long horizontal rod has a bead which can slide along its length and is initially placed at a distance L from one end A of the rod. The rod is set in angular motion about A with a constant angular acceleration, $\alpha$ . If the coefficient of friction between the rod and bead is $\mu$ , and gravity is neglected, then the time after which the bead starts slipping may not be:

Tangential Force ($F_t$ ) of the bead will be given by Normal reaction ($N$) while centripetal force ($F_C$ ) is provided by friction ($f$). The bead starts sliding when the centripetal force is just equal to limiting friction

$F_t$ is inwards

${\mathrm{F}}_{\mathrm{t}}=\mathrm{ma}=\mathrm{m}\alpha \mathrm{L}=\mathrm{N}$

Limiting value of friction  ${\left(f_r\right)}_{max}=\mu \mathrm{N}=\mu \mathrm{m}\alpha \mathrm{L}$  

Angular velocity at time ‘$t$’ is $\omega =\alpha t$  

Centripetal Force at time ‘$t$’ will be ${\mathrm{F}}_{\mathrm{c}}=\mathrm{mL}{\omega }^2=\mathrm{mL}{\alpha }^2{\mathrm{t}}^2 \therefore \mathrm{t}=\sqrt{\frac{\mu }{\alpha }}$

For $t>\sqrt{\frac{\mu }{\alpha }},F_C>{\left(f_r\right)}_{max}$  the bead starts sliding