A solid cylinder of mass $m$, cross-sectional area $A$, and density $\sigma$ floats vertically in a

Question

A solid cylinder of mass $m$, cross-sectional area $A$, and density $\sigma$ floats vertically in a liquid of density $\rho$. If it is slightly pushed down and released, it executes SHM. What is its time period?

Professor PC Thomas Classes · Answer

The restoring force is the extra buoyant force $F = -(A y \rho)g = -k y \implies k = A \rho g$. The time period $T = 2\pi\sqrt{m/k} = 2\pi\sqrt{m / (A \rho g)}$.