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

## The correct options are A the speed of the particle when it returns to its equilibrium position is u0 D the time at which the particle passes through the equilibrium position for the second time is t=5π3√mk.v=u0sinωt (Suppose t1 is the time of collision) u02=u0cosωt1=>t1=π3ωNow the particle returns to equilibrium position at time, t2=2t1 i.e. 2π3ω with the same mechanical energy i.e. its speed will u0.Let t3 is the time at which the particle passes through the equilibrium position for the second time.t3=T2+2t1=πω+2π3ω=5π3ω=5π3√mkEnergy of the particle and spring remains conserved.

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