Two identical cylindrical vessels with their bases at same level each contains a liquid of density ρ. The height of the liquid in one vessel is h1 and that in the other vessel is h2. The area of either base is A. The work done by gravity in equalizing the levels when the two vessels are connected, is

If h is the common height when they are connected, by conservation of massρA1h1+ρA2h2=ρhA1+A2h = (h1+h2)2                    as A1=A2=A given As (h12) and (h22) are heights of initial centre of gravity of liquid in two vessels., the initial potential energy of the system  Ui=h1Aρgh12+h2Aρh22=ρgAh12+h222--------------(i)When vessels are connected the height of centre of gravity of liquid in each vessel will be h2, i.e. h1+h24 [as h=h1+h22Final potential energy of the systemUf=h1+h22Aρgh1+h24=Aρgh1+h224--------(ii)Work done by gravityW=Ui−Uf=14ρgA2h12+h22−h1+h22=14ρgAh1-h22