A vessel with square cross-section and height of 6 m is vertically partitioned. A small window of 100 ~cm^2 with hinged door is fitted at a depth of 3 m in the partition wall. One part of the vessel is filled completely with water and the other side is filled with the liquid having density 1.5 \times 10^3 ~kg / m^3 . What force one needs to apply on the hinged door so that it does not get opened ? (Acceleration due to gravity =10 ~m / s^2 )

Properties of Matter PYQ — JEE Main 2025 · 2 Apr · Shift 1 · Q73 with Solution

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Correct answer: 150

Area of window: $A = 100\,\text{cm}^2 = 100 \times 10^{-4} = 10^{-2}\,\text{m}^2$
Depth of window below liquid surface: $h = 3\,\text{m}$
Density of water: $\rho_w = 1000\,\text{kg/m}^3$
Density of other liquid: $\rho_l = 1.5 \times 10^3 = 1500\,\text{kg/m}^3$
Acceleration due to gravity: $g = 10\,\text{m/s}^2$

Since both compartments are filled completely and the window is at the same depth on both sides, hydrostatic pressure at the window is:

On water side: $P_w = \rho_w g h = 1000 \times 10 \times 3 = 30000\,\text{Pa}$
On denser liquid side: $P_l = \rho_l g h = 1500 \times 10 \times 3 = 45000\,\text{Pa}$

$\Delta P = P_l - P_w = 45000 - 30000 = 15000\,\text{Pa}$

$F = \Delta P \cdot A = 15000 \times 10^{-2} = 150\,\text{N}$

So, the external force required to keep the hinged door from opening is:

$\boxed{150\,\text{N}}$

Stored correct answer = 150

This matches our derived answer.