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Question

Find the natural frequency of the system shown in the figure. The pulleys are smooth, massless and the spring is unstretched in the equilibrium position.


A
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B
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C
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D
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Solution

The correct option is A
Let x be the displacement of the block from the equilibrium position.
Since the length of the string remains the same, the spring extends by x = (2x).
The energy of the system is
E=12kx2+12mv2=12k(2x)2+12mv2
Differentiate with respect to time
dEdt4kxdxdt+mvdvdt=0 (energy conservation)
4kxv+mvd2xdt2=0
4kg+md2xdt2=0d2xdt2+4kxm=0
T=2πm4k=πmk

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