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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=5π3√mk (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=2π3√km, its wrong (C) Maximum compression will occur at time =2π3√km+π2√km=7π6√km, hence it's wrong. (D) Second time at equilibrium at =2π3√km+π√km=5π3√km, so correct.

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