GATE 2025 CE (CE2) – Question 60

Fluid Mechanics · Properties of fluids, fluid statics · 2 marks · Numerical answer

The shaft of a 6 m wide gate in the figure will fail at a moment of 3924 kN.m about the hinge P. The maximum value of water depth $h$ (in m) that the gate can hold is ________________ (round off to the nearest integer).
Note: Density of water = 1000 kg/m$^3$, Acceleration due to gravity = 9.81 m/s$^2$

an inclined gate hinged at P on the floor, rising 4 m vertically over a horizontal distance of 3 m (length 5 m). The water is on the left side with the depth h measured from the level of P up to the water surface.

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Correct answer: 8

Explanation

The gate is 5 m long and inclined, so a point at the distance $s$ along it from P is at the height $0.8s$ above P. The pressure there is $\rho g(h - 0.8s)$. The moment about P per unit width is $\int_0^5\rho g(h - 0.8s)s\,ds = \rho g\left[\frac{hs^2}{2} - \frac{0.8s^3}{3}\right]_0^5 = \rho g(12.5h - 33.33)$. For the 6 m wide gate, $M = 6 \times 9.81 \times (12.5h - 33.33) = 3924$ kN m, so $12.5h - 33.33 = 66.67$ and $h = 8$ m.