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 den

Question

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 ρ at the equilibrium position. When the cylinder is given a small downward push and released it starts oscillating vertically with a small amplitude. If the force constant of the spring is k , the frequency of oscillation of the cylinder is

TestHub · Accepted Answer

When cylinder is displaced by an amount x from its meanposition, spring force and upthrust both will increase. Hence,
Net restoring force = extra spring force + extra upthrust

Or $F = - (kx + Ax ρ g)$

Or $a = - (\frac{k + ρ Ag}{M}) x$

Now, $f = \frac{1}{2 π} \sqrt{|\frac{a}{x}|}$

$= \frac{1}{2 π} \sqrt{\frac{k + ρ Ag}{M}}$

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