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Question

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 starts from rest in angular motion about A with a constant angular acceleration a .If the coefficientof friction between the rod and the bead is u and gravity is neglected then what is the time after which the bead starts slipping

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Solution

Assuming the given motion takes place in Horizontal plane.

Steps:
1) The bead will start sliding when centripetal force is equal to limiting friction.
i. e.
mw2L=f(max)
=> mw2L=uN ----(1)

2) We know that,
Here. Normal Reaction is provided by Tangential force.
F(t) = mLα =N ---(2)

3 ) Substituting value of N in eq. (1),
Now,
At time t,.
w = 0 + αt
w = αt

Now,
mw2L=umLα
=>m(αt)2L=umLα
=>αt2=u
=> t=uα .

At time t , bead will start sliding.

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