A block of mass $$2M$$ is attached to a massless spring with $$spring-constant$$ k. This block is connected to two other blocks of masses M and $$2M$$ using two massless pulleys and strings. The acceleration of the blocks are $$a1$$, $$a2$$ and $$a3$$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $$x0$$. Which of the following $$option(s)$$ $$is/are$$ correct ?[g is the acceleration due to gravity. Neglect friction]

Question

A block of mass $$2M$$ is attached to a massless spring with $$spring-constant$$ k. This block is connected to two other blocks of masses M and $$2M$$ using two massless pulleys and strings. The acceleration of the blocks are $$a1$$, $$a2$$ and $$a3$$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $$x0$$. Which of the following $$option(s)$$ $$is/are$$ correct ?[g is the acceleration due to gravity. Neglect friction]

Infinity Learn · Accepted Answer

Using constraint $$relationa2$$−$$a1=a1$$−$$a3Also$$,$$2Mg$$−$$T=2Ma3Mg$$−$$T=Ma22T$$−$$kX=2Ma1$$ Solving, $$a1=4g7$$−$$3kx14M$$⇒$$a1=$$−$$3k14Mx$$−$$8Mg3kThis$$ equation represents SHM of block $$2M$$ attached with spring.Comparing with $$a=-$$ω$$2x-xmean$$⇒ω$$=3k14M$$ and xmean $$=8Mg3k$$ Since,  $$a1=0$$, at two points $$x=0$$ and $$x=x0$$,⇒ Amplitude $$=x02$$⇒$$x02=8Mg3k$$⇒$$x0=16Mg3k$$ Now, put $$x=x04=4Mg3k$$⇒$$a1=4g7$$−$$3k14M$$⋅$$4Mg3k$$⇒$$a1=2g7When$$ extension is $$x02$$ , the block has maximum velocity,$$vmax=$$ω$$A=3k14M.8Mg3k$$

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