A uniform solid cylinder of mass 0.80 kg and cross-sectional area 40 cm^2 floats upright in water (density 1000 kg m^-3) in a wide tank. A light…

The spring joins the cylinder to the floor and is stretched, so it pulls the cylinder down. Displace the cylinder downward by $x$: the spring is stretched less, so its downward pull falls by $kx$ (an upward change in the net force), and the extra submerged length raises the buoyant force by $\rho gAx$. Both changes are upward and proportional to $x$, so the net restoring constant is $k_{eff}=k+\rho gA=60+1000\times10\times4\times10^{-3}=100\ \mathrm{N\,m^{-1}}$. Hence $T=2\pi\sqrt{m/k_{eff}}=2\pi\sqrt{0.80/100}=0.56\ \mathrm{s}$. The wrong values come from omitting the spring ($0.89\ \mathrm{s}$, only buoyancy), omitting buoyancy ($0.73\ \mathrm{s}$, only the spring) or combining the two stiffnesses as if in series ($1.15\ \mathrm{s}$); the two restoring effects act on the same body in the same direction, so they add.