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

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, 1 2 π ( k - Aρg M ) 1 / 2, 1 2 π k + A ρ g M 1 / 2, 1 2 π ( k + ρgL 2 M ) 1 / 2, 1 2 π ( k + Aρg Aρg ) 1 / 2

MARKS App · 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 = - k + ρ Ag M x Now, f = 1 2 π a x = 1 2 π k + ρ Ag M

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