UPSEE Solved Model Paper 1

The work done by gravity equals the change in the potential energy of the system after the vessels are interconnected. We may regard the liquid in each vessel as equivalent to a point mass kept at their respective centres of gravity. Remembering that the mass of the liquid is given by (Ahp) and the PE of a mass at a height h in earth's gravity is mgh, we have
Total PE at start = (Ah1p)g

h1

2

+(Ah2p)g

h2

2

Apg

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

1

)g(

h1+h2

2

)}
=

Aρg

4

(h1+h2)2
Change in PE

Aρg

4

{(h1+h2)2−2(h12+h22)}
= −

4ρg

4

(h1−h2)2
= −Aρg(

h1−h2

1

)2
This must be equal to the work done by gravity on the liquid. Thus the work done by gravity is
Aρg(

h1+h2

1

)2
Hence the correct choice is (c).