An ice cube of size l = 50 cm is floating in a tank (base area $\mathrm{A}=2\mathrm{m}\times 2\mathrm{m}$) partially filled with water. water is Density of ${\mathrm{\rho }}_{\mathrm{w}}$$=1000 {\mathrm{kgm}}^{-3}$ and that of ice is ${\mathrm{\rho }}_{\mathrm{ice}}=900 {\mathrm{kg}}^{-3}$ Calculate change increase in gravitational potential energy (in J) when ice melts completely. (g = 10 ms^-2)

As the density of the ice $\left({\mathrm{\rho }}_{\text{ice }}\right)$ is less than the density of the water $\left({\mathrm{\rho }}_{\text{water }}\right)$, hence the ice cube float partially immersed.
In case of floatation
Buoyancy force = weight of cube

$\mathrm{B}=\mathrm{mg}\text{ or }{\mathrm{l}}^2{\mathrm{h}}_0{\mathrm{\rho }}_{\text{water }}\mathrm{g}={\mathrm{l}}^3{\mathrm{\rho }}_{\text{ice }}\mathrm{g}$

$\Rightarrow {\mathrm{h}}_0=\mathrm{l}\times \frac{{\mathrm{\rho }}_{\text{ice }}}{{\mathrm{\rho }}_{\text{water }}}=50\times \frac{900}{1000}=45\mathrm{cm}$

Centre of mass of ice cube is at a depth of 25 cm from its upper surface. But the upper surface is at a height of 5 cm from water surface.
Therefore, depth of the centre of the mass of ice cube from free water surface is h₁ = 25 - 5 = 20 cm from free water surface.
When ice melts, water is formed and that water occupies the space which was displaced by the cube. Therefore, depth of centre of mass of the water body formed after the melting of ice cube is

 ${\mathrm{h}}_2=\frac{45}{2}=22.5\mathrm{cm}$

it means that the centre of mass move down through distance

$\mathrm{\Delta h}={\mathrm{h}}_2-{\mathrm{h}}_1=22.5-20=2.5\mathrm{cm}$

lt means gravitational potential energy of the system will decrease. 

Decrease in gravitational potential energy of the system

$\mathrm{\Delta U}=\mathrm{mg\Delta h}=(0.5{)}^3\times 1000\times 10\times \frac{2.5}{1000}=31.25\mathrm{J}$

Since, level of free water surface does not rise or fall, hence size of tank is of no concern.