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Question

A ball suspended by a thread of length l at the point O on an inclined was as shown. The inclination of the wall with the vertical is α. The thread is displaced through a small angle β(>α) away from the vertical and the ball is released. If the collision between the wall and the ball is elastic, find the period of oscillation of pendulum.
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Solution

Obviously for small β the ball execute part of S.H.M. Due to the perfectly elastic collision the velocity of ball simply reversed. As the ball in S.H.M. (|θ|<α on the left) its motion law is differential from can be written as
¨θ=glθ=ω20θ (1)
If we assume that ball is released from the extreme position, θ=β at t=0, the solution of differential equation would be taken in the formθ=βω0t=βcosglt (2)
If t be the time taken by the ball to go from the extreme position θ=β to the wall .i.e.
θ=α, then Eqn (2) can be rewritten as
α=βcosglt
or t=lgcos1(αβ)=lg(πcos1αβ)
Thus the sought time T=2t=2lg(πcos1αβ)
=2lg(π2+sin1αβ), [because sin1+cos1=π/2]

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