The work done by gravity equals the change in the potential energy of the system after the vessels are inter connected. We may regard the liquid in each vessel as equivalent to a point mass kept at their respective centers of mass. Remembering that the mass of the liquid is given by (Ahρ) and that the PE of a mass at a height h in earth's gravity is mgh, we have Total PE at start = (Ah1ρ)g
h
2
+(Ah2ρ)g
h2
2
=
Aρg
2
(h12+h22) After the vessels are connected, the height of liquid in each vessel is
(h1+h2)
2
Hence PE after connection = {Aρ(
h1+h2
2
)g(
h1+h2
2
)} =
Aρg
4
(h1+h2)2 Change in PE =
Aρg
4
{(h1+h2)2−2(h1+h2)2} = −
Aρg
4
(h1−h2)2 = - Aρg (
h1−h2
2
)2 This must be equal to the work done by gravity on the liquid. Thus the work done by gravity is Aρg(