A simple pendulum of bob mass m is suspended vertically from O by a massless rigid rod of length L.

Question

A simple pendulum is suspended from a peg on a vertical wall. The pendulum is pulled away from the wall to a horizontal position (Fig. 1.19) and released. The ball hits the wall, the coefficient of restitution being 2/√5 . What is the minimum number of collisions after which the amplitude of oscillation becomes less than 60°? 

Answer 1

Let v₀ be the velocity of the bob just before the first collision.

The velocity of the bob just after the 1st collision is

v₁ will be the velocity of the bob just before 2nd collision. The velocity of the bob just after the second collision is

The velocity just after the nth collision is

We assume that after n collisions the amplitude of oscillation becomes 60°. 

n is slightly greater than 3. In fact

Thus the minimum number of collisions after which the amplitude becomes less than 60° is 4.