Two identical cylindrical vessels are kept on the ground and each contain the same liquid of density ρ. The area of the base of both vessels is S but the height of liquid in one vessel is x₁ and in the other is x₂. When both cylinders are connected through a pipe of negligible volume very close to the bottom, the liquid flows from one vessel to the other until it comes to equilibrium at a new height. The change in energy of the system in the process is :
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Question

Two identical cylindrical vessels are kept on the ground and each contain the same liquid of density ρ. The area of the base of both vessels is S but the height of liquid in one vessel is x₁ and in the other is x₂. When both cylinders are connected through a pipe of negligible volume very close to the bottom, the liquid flows from one vessel to the other until it comes to equilibrium at a new height. The change in energy of the system in the process is :

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G1 Physics · Accepted Answer

Step 1: Initially the gravitational potential energy of the two liquid columns is Uᵢ = (1/2)ρgS(x₁²+x₂²). At equilibrium both heights become (x₁+x₂)/2. Step 2: Final energy is U𝒇 = ρgS[(x₁+x₂)/2]². The decrease is Uᵢ−U𝒇 = (1/4)ρgS(x₂−x₁)². Correct answer: C.