MHTCET 22 Apr 2025 Shift 1 Paper

Concept:The net horizontal force on the tank equals the rate of change of momentum of the liquid streams escaping through the two holes.This is governed by Torricelli's theorem for efflux velocity.Explanation:Let the area of each hole be $A$ and the density of the liquid be $\rho$.Take the upper hole at depth $x$ and the lower hole at depth $x+h$ from the free surface.By Torricelli's theorem, the efflux velocities are:$$v_1 = \sqrt{2g(x+h)} \quad \text{(lower hole)}, \qquad v_2 = \sqrt{2gx} \quad \text{(upper hole)}.$$Mass of liquid discharged per second from a hole is:$$\frac{dm}{dt} = \rho A v.$$Therefore, momentum discharged per second is:$$\frac{dp}{dt} = \rho A v^2.$$Since the two holes face opposite directions, the momentum fluxes oppose each other.Thus, the net horizontal force on the tank is:$$F = \rho A v_1^2 - \rho A v_2^2 = \rho A (v_1^2 - v_2^2).$$Substituting the velocities:$$F = \rho A \left[2g(x+h) - 2gx\right] = 2\rho A g h.$$Since $2\rho A g$ is constant, we get $F \propto h$.Answer:$$F \propto h$$Hence, the correct option is D. $h$.