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Question

A body of mass m hangs by as inextensible light string that passes over a smooth massless pulley that is fitted with a light spring of stiffness k as shown in the figure. If the body is released from rest, calculate the maximum elongation of the spring. Take g=10 m/s2.


A
mgk
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B
2mgk
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C
4mgk
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D
3mgk
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Solution

The correct option is C 4mgk

Let the maximum extension in the spring be x, then pulley will also fall through distance x
Applying string constraint:
l1+l2=constant
string l1 is fixed at one end, displacement at that end =0
Let mass m travels by x downwards
l1+l2=(l1x)+((l2x)+x)
x=2x (i)

Maximum elongation in spring will result when mass m stops, i.e its velocity vf=0
Applying energy conservation for the system:
Loss in gravitational PE of mass = Gain in elastic PE of spring
i.e mgx=12kx2
mg(2x)=12kx2 (from (i))
x=4mgk

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