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Question

Two identical rods A and B are connected by a wire of length l in the middle as shown in figure. The wire is bent into an arc of a circle of radius R and released. If Y is Young's modulus of wire. Find time period of small oscillations. MOI of each rod is Ib :

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A
2lYr4
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B
2lIbYr4
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C
2IbYr4
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D
2lIbYπr4
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Solution

The correct option is D 2lIbYπr4
YiR=Ibard2θdt2
lR=2θ,i=πr44 and is called geometric moment of inertia.
Yπr44l(2θ)=Ibard2θdt2
or d2θdt2=Yπr42lIbθ
comparing with a=ω2θ
ω=Yπr42lIb
w=2πf=2πT
so T=2lIbYπr4

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