Correct option is (a) \(\sqrt {\frac \mu \alpha}\)
Tangential force (F1) of the bead will be given by the normal reaction (N), while centripetal force (Fc) is provided by friction (fr). The bead starts sliding when the centripetal force is just equal to the limiting friction.
Therefore, Ft = ma = mαL = N
∴ Limiting value of friction
(fr)max = µN = µmαL …(i)
Angular velocity at time t is ω = at
∴ Centripetal force at time t will be
Fc = mLw2 = mLa2 t2 …(ii)
Equating equation (i) and (ii), we get t = \(\sqrt {\frac \mu \alpha}\)
For t > \(\sqrt {\frac \mu \alpha}\), Fc > (fr)max i.e. , the bead starts sliding.
In the figure Ft is perpendicular to the paper inwards.