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Q.

In the figure shown rod AB is light and rigid while uniform rod CD has mass but is also rigid and hinged at midpoint O. Rod CD can rotate in the plane of diagram without friction about O. The time period of small oscillation of rod CD is T=2πmnk , where m is mass of rod CD and k is spring constant of each spring. Ignoring gravity, find the value of n ? (Assume the rod AB only translates)

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answer is 5.

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Detailed Solution

Effective k on Left side =2k3Let say rod is rotated by small angle θ in clockwise direction∴Extension in spring on left=Compression in spring on right=ℓ2θRestoring torque =FLrL+FRrR=(2k3ℓ2θ+kℓ2θ)(ℓ2)τ=Iα⇒−(2kθ12+kθ4)ℓ2=mℓ212α⇒α=−(5km)θ∴T=2πm5k
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