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Question

In an ideal pulley particle system, mass m2 (>m1) is connected with a vertical spring of stiffness k. If mass m2 is released from rest, when the spring is undeformed, find the maximum compression of the spring.


A
x=2(m2m1)gk
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B
x=2(m2)gk
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C
x=2(m2+m1)gk
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D
x=(m2m1)gk
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Solution

The correct option is A x=2(m2m1)gk
Work done by tension and normal reaction at the pulley are zero (since they are internal constraint forces)
So, the total mechanical energy of the system will be conserved
ΔM.E=0
or ΔK+ΔU=0


Let x be the maximum compression of the spring.
Then, ΔUsp=k2x2......(i)

m2 will descend by the same distance x.
i.e ΔUgr2=m2gx.....(ii)
and m1 will ascend by x
i.e ΔUgr1=m1gx....(iii)

Since both m1 and m2 comes to rest simultaneously.
ΔK=0.....(iv)

From conservation of energy :
ΔK+ΔU=0
i.e [ΔUsp+ΔUgr1+ΔUgr2]+ΔK=0
(12kx2+m1gxm2gx)+0=0
x=2(m2m1)gk

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