UPSC NDA I 2023 General Ability Solved Paper

Concept:Potential energy of a liquid column is proportional to the height of its center of mass above the reference level.Explanation:Let the total mass of liquid be $m$.Initially, the liquid occupies container X up to height $h$.The center of mass of a uniform column of height $h$ is at $\frac{h}{2}$ from the base.Thus, initial potential energy: $P_1 = m g \cdot \frac{h}{2} = \frac{mgh}{2}$.When the valve is opened, liquid flows until both containers have equal height and mass.Final height in each container is $\frac{h}{2}$, and mass in each is $\frac{m}{2}$.The center of mass of each column is at $\frac{1}{2} \cdot \frac{h}{2} = \frac{h}{4}$ from the base.Final total potential energy: $P_2 = 2 \times \left( \frac{m}{2} g \cdot \frac{h}{4} \right) = \frac{mgh}{4}$.Comparing: $\frac{P_1}{P_2} = \frac{\frac{mgh}{2}}{\frac{mgh}{4}} = 2$, so $P_1 = 2P_2$.Answer:C. $P_1 = 2P_2$