A uniform cylinder of length L and mass M having cross-sectional area… | JEE Main 2013 (25 Apr, Shift 1)

Q19JEE Main 2013 · 25 APR · SHIFT 1Difficult

Other SHM Systems

A uniform cylinder of length L and mass M having cross-sectional area A is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. When the cylinder is given a downward push and released, it starts oscillating vertically with a small amplitude. The time period T of the oscillations of the cylinder will be:

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Answer: (d)

Extra downward displacement $x$ adds spring force $kx$ and extra buoyancy $A\sigma g x$ , both upward.
Net restoring constant $k + A\sigma g$ , so $T = 2\pi\sqrt{\frac{M}{k + A\sigma g}}$ , answer (d).

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