Vertical oscillations of a floating cylinder (${\omega }^2={\rho }_{f}gA/m=g/h_0$)

A vertical cylinder of constant cross-sectional area $A$, partly immersed to depth $h_0$ in a fluid of mass density ${\rho }_f$, satisfies $mg={\rho }_{f}gA{h}_0$ by Archimedes' principle. A small extra downward displacement $z$ increases the buoyant force by ${\rho }_{f}gAz$, so $m\ddot{z}=-{\rho }_{f}gAz$. This is simple harmonic motion with the displayed angular frequency and oscillation period $2\pi \sqrt{h_0/g}$. The ideal model neglects fluid added inertia, drag and surface effects; it requires partial immersion throughout the motion.