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Q.

Two identical cylindrical vessels are kept on the ground and each contain the same liquid of density d. The area of the base of both vessels is S but the height of liquid in one vessel is $x_{1}$ and in the other, $x_{2}$ . 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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a

$\frac{1}{4} g d S {(x_{2} - x_{1})}^{2}$

b

$\frac{3}{4} g d S {(x_{2} - x_{1})}^{2}$

c

$g d S {(x_{2} + x_{1})}^{2}$

d

$g d S (x_{2}^{2} + x_{1}^{2})$

answer is A.

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$\begin{array}{l} Δ U = U_{i} - U_{f} \\ = (d s x_{1}) g (\frac{x_{1}}{2}) + (d s x_{2}) g (\frac{x_{2}}{2}) - 2 \times {d s \frac{(x_{1} + x_{2})}{2} g \frac{(x_{1} + x_{2})}{4}} \\ = \frac{1}{2} d g s (x_{2} - x_{1}) \times \frac{(x_{2} - x_{1})}{2} \\ = \frac{1}{4} d g s {(x_{2} - x_{1})}^{2} \end{array}$

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Two identical cylindrical vessels are kept on the ground and each contain the same liquid of density d. The area of the base of both vessels is S but the height of liquid in one vessel is x1 and in the other, 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