A uniform cylinder of mass 0.50 kg and cross-sectional area 50 cm^2 floats upright in water of density 1000 kg m^-3, partly submerged, while being…

Let the cylinder be displaced downward by $x$ from equilibrium. The extra displaced volume is $Ax$, so the buoyant force rises by $\rho gAx$ and acts upwards. The spring, which is already stretched, also exerts an extra upward force $kx$. Both are restoring and add, giving $m\ddot{x}=-(k+\rho gA)x$ and an effective stiffness $k_{eff}=k+\rho gA$. Here $\rho gA=1000\times 9.8\times 5\times10^{-3}=49\ \mathrm{N\,m^{-1}}$, so $k_{eff}=100\ \mathrm{N\,m^{-1}}$. The period is $T=2\pi\sqrt{m/k_{eff}}=2\pi\sqrt{0.50/100}=0.444\ \mathrm{s}$. Using the spring alone gives $0.62\ \mathrm{s}$, and doubling the buoyancy term gives $0.36\ \mathrm{s}$.