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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 $u_0$. When the speed of the particle is $0.5 u_0$, it collies elastically with a rigid wall. After this collision :
$(A)$ the speed of the particle when it returns to its equilibrium position is $u_0$.
$(B)$ the time at which the particle passes through the equilibrium position for the first time is $t=\pi \sqrt{\frac{ m }{ k }}$.
$(C)$ the time at which the maximum compression of the spring occurs is $t =\frac{4 \pi}{3} \sqrt{\frac{ m }{ k }}$.
$(D)$ the time at which the particle passes througout the equilibrium position for the second time is $t=\frac{5 \pi}{3} \sqrt{\frac{ m }{ k }}$.
$(B,D)$
$(B,C)$
$(A,C)$
$(A,D)$
Solution
Displacement $x = A \sin \omega t$
Velocity $v = A \omega \cos \omega t =\frac{\omega A }{2}$
At the time of collision
$\cos \omega t=\frac{1}{2} $
$\omega t=\frac{\pi}{3} \Rightarrow t=\frac{2 \pi}{3}=\frac{\pi}{3} \sqrt{\frac{m}{k}}$
$Image$
for $(C)$ $ \quad time$$ =\frac{2 \pi}{3} \sqrt{\frac{m}{k}}+\frac{\pi}{2} \sqrt{\frac{m}{k}} $
$ =\frac{5 \pi}{6} \sqrt{\frac{m}{k}}$(So it is incorrect)
for $(D)$ $\quad time \quad=\frac{2 \pi}{3} \sqrt{\frac{m}{k}}+\pi \sqrt{\frac{m}{k}} $
$ =\frac{5 \pi}{3} \sqrt{\frac{m}{k}} \text { (So it is correct). }$
