A uniform solid cylinder of height 0.20 m and density 600 kg m^-3 floats upright in water of density 1000 kg m^-3 in a tank fixed in a lift that…

In the frame of the lift the effective gravity is $g_{eff}=g+a=9.8+2.2=12.0\ \mathrm{m\,s^{-2}}$. For floating, the submerged depth is $h=(\rho_b/\rho_w)H=0.6\times0.20=0.12\ \mathrm{m}$ (independent of $g_{eff}$). If displaced by $y$ downward, the restoring force is the extra buoyancy $\rho_wAy\,g_{eff}$, and the mass is $\rho_bAH$. So $\rho_bAH\,\ddot y=-\rho_wA\,g_{eff}\,y$ which is simple harmonic with $\omega^2=\dfrac{\rho_wg_{eff}}{\rho_bH}=\dfrac{g_{eff}}{h}$. Hence $T=2\pi\sqrt{h/g_{eff}}=2\pi\sqrt{0.12/12}=2\pi(0.1)=0.628\ \mathrm{s}\approx0.63\ \mathrm{s}$. Using $g=9.8$ ignoring the lift gives $0.70\ \mathrm{s}$. The result also shows why a lift accelerating upward shortens the period: the heavier effective weight of the liquid increases the stiffness of the buoyant restoring force by the same factor $g_{eff}/g$ while the mass of the cylinder is unchanged.