Let force $F$ acts on a gate at $B$.

GATE CE 2025 Set 2 Fluid Mechanics and Hydraulic Machines - Hydrostatic Forces Question 1 English Explanation

$$ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\begin{aligned} F & =\rho g \bar{h} A \\ & =\rho g(h-4+2)(5 \times 6) \end{aligned} $$

$$ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\begin{aligned} & =1000 \times 9.81 \times(h-2) \times 30 \\ & =294.3 \times(h-2) \mathrm{kN} \end{aligned} $$

$$ \text { From figure, } \quad \sin \theta=\frac{h-\bar{h}_{c p}}{P B} $$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \begin{aligned} \frac{4}{5} & =\frac{h-\left[\bar{h}+\frac{I_{C G} \sin ^2 \theta}{A \bar{h}}\right]}{P B} \\ & =\frac{h-\left[h-2+\frac{\left.\frac{6 \times 5^3}{12} \times\left(\frac{4}{5}\right)^2\right]}{(h-2) \times 5 \times 6}\right]}{P B} \end{aligned} $$

$$ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\begin{aligned} &\Rightarrow \quad P B=\frac{5}{4}\left[2-\frac{4}{3(h-2)}\right]\\ &\text { Now, taking moment about } P \text {, }\\ &\begin{aligned} & F \times P B=3924 \\ & 294.3 \times(h-2) \times \frac{5}{4}\left[2-\frac{4}{3(h-2)}\right]=3924 \\ &(h-2) \times\left(2-\frac{4}{3(h-2)}\right)=\frac{32}{3} \\ & 2 h-4-\frac{4}{3}=\frac{32}{3} \\ & h=8 \mathrm{~m} \end{aligned} \end{aligned} $$

Hydrostatic Forces · Fluid Mechanics and Hydraulic Machines · GATE CE

Marks 2