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Question

A particle of mass m is attached to one end of a mass-less spring of force constant k, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time t=0 with an initial velocity u0. When the speed of the particle is 0.5 u0, it collides elastically with a rigid wall. After this collision

A
The time at which the maximum compression of the spring occurs is t=4π3mk
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B
The speed of the particle when it returns to its equilibrium position is u0
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C
The time at which the particle passes through the equilibrium position for the second time is t=5π3mk
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D
The time at which the particle passes through the equilibrium position for the first time is t=πmk
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Solution

The correct option is C The time at which the particle passes through the equilibrium position for the second time is t=5π3mk

(A) is correct due to conservation of energy.
(B) x=A sin ωt
dxdt=Aω cos ωt=u02=u0 cos ωt
cos ωt=12
ωt=π3
t=π3ω
Δt=2π3ω=2π3km, its wrong
(C) Maximum compression will occur at time
=2π3km+π2km=7π6km, hence it's wrong.
(D) Second time at equilibrium at
=2π3km+πkm=5π3km, so correct.

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