A uniform solid cylinder of length 0.20 m and density 500 kg m^-3 floats upright in water (density 1000 kg m^-3) in a very wide tank. Take g=10 m…

Let the cross-section be $A$. Restoring force for displacement $x$ is $\rho_w g A x$ and mass is $\rho_c A h$, so $\omega^2=\dfrac{\rho_w g}{\rho_c h}=\dfrac{1000\times10}{500\times0.2}=100\ \mathrm{s^{-2}}$ and $T=2\pi/10=\pi/5\ \mathrm{s}$: (A) is correct. Equilibrium submerged depth is $0.10\ \mathrm{m}$, so SHM holds only while the top stays above the surface and the bottom stays submerged, i.e. amplitude at most $0.10\ \mathrm{m}$; for $0.12\ \mathrm{m}$ the cylinder either becomes fully submerged or leaves the water, so (B) is false. In a lift with upward acceleration $a$, the effective gravity is $g+a=12$, and buoyancy and weight both scale, so $\omega^2=\rho_w g_{\rm eff}/(\rho_c h)=120$ and $T=2\pi/\sqrt{120}$: (C) is correct. For downward acceleration $2$, $g_{\rm eff}=8$, $\omega^2=80$, $T=2\pi/\sqrt{80}\neq\pi/5$, so (D) is false.