Oscillation frequency with fluid added mass (${\omega }^2=k_{\mathrm{restoring}}/(m+m_a)$)

Fluid added mass increases inertia but need not alter the same restoring force. A submerged spring's constant buoyancy shifts equilibrium, whereas a pendulum's buoyancy reduces its restoring torque. Thus an unbounded-fluid sphere has spring frequency factor $\sqrt{\rho /(\rho +{\rho }_f/2)}$ and downward-pendulum factor $\sqrt{(\rho -{\rho }_f)/(\rho +{\rho }_f/2)}$. The latter stable configuration requires $\rho >{\rho }_f$.