Question

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 h_{1} 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

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Solution

The correct option is **C**

If h is the common height when they are connected, by conservation of mass

ρA1h1+ρA2h2 = ρh(A1+A2)

h = (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 = (h1Aρ)g h12+(h2Aρ)h22 = ρgA(h21+h22)2 ....(i)

When vessels are connected the height of centre of gravity of liquid in each vessel will be h2,

i.e. (h1+h2)4[as h = (h1+h2)2]

Final potential energy of the system

Uf = [(h1+h2)Aρ2]g(h1+h24) = Aρg[(h1+h2)24] ....(ii)

Work done by gravity

W = Ui−Uf = 14ρgA[2(h21+h22)−(h1+h2)2]

= 14ρgA(h1∼h2)2

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