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Question

A small ball is suspended from a fixed point O by means of an ideal string of length l. The ball is first taken aside such that the string becomes horizontal and then released from rest. At the bottom, it collides with a fixed obstacle. The coefficient of restitution is e. The maximum angular deflection of the string after nth collision is


A
sin1(e2n)
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B
cos1(e2n)
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C
sin1(1e2n)
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D
cos1(1e2n)
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Solution

The correct option is D cos1(1e2n)
Let, v1,v2,v3,,vn be the velocities acquired by the ball immediately after 1st,2nd,3rd,nth collision respectively.

So, the velocities after collisions be

v1=evo (after 1st collision)
where, vo=(2gl)12

v2=e2v0 (after 2nd collision)


vn=env0 (after n collision)

After nth collision, let θ be the maximum angular deflection.


From the law of conservation of energy,
12mv2n=mgl(1cosθ)

12e2nv20=gl(1cosθ)

12e2n2gl=gl(1cosθ)

e2n=1cosθ

cosθ=1e2n

θ=cos1(1e2n)

Hence, (C) is the correct answer.

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