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(m2−m1)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=(m2−m1)gk
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Solution
The correct option is Ax=2(m2−m1)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+m1gx−m2gx)+0=0 ⇒x=2(m2−m1)gk