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# Find the force $P$ needed to hold the $3 -\text{m}$-wide rectangular gate as shown if: (a) $l=2 \mathrm{~m}$ (b) $l=4 \mathrm{~m}$ (c) $l=5 \mathrm{~m}$

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We are asked to determine the required magnitude of the force $P$ to hold the gate in the given position.

First, we are going to formulate the equations for force $F$ and its location using the following equations. Then, we will get the sum of the moments about the hinge. The sum of moments should be in equilibrium to keep the gate in position.

\begin{aligned} &F=\gamma\cdot\overline{h}\cdot A\hspace{4.4cm}\tag{1} \end{aligned}

\begin{aligned} &y_P=\overline{y}+\dfrac{\overline{I}}{A\cdot\overline{y}}\hspace{4.3cm}\tag{2} \end{aligned}

where:

\begin{aligned} F&=\text{force exerted by water}\\ \gamma&=\text{specific weight of water}\\ \overline{h}&=\text{vertical height from the centroid}\\ &\phantom{=}\text{of the gate to the water surface}\\ A&=\text{area of the figure}\\ y_P&=\text{distance from the center of pressure}\\ &\phantom{=}\text{on the gate to the water surface}\\ \overline{y}&=\text{distance from the centroid to the}\\ &\phantom{=}\text{water surface}\\ \overline{I}&=\text{second moment of area of the gate} \end{aligned}

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