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Question

# Determine the force $P$ needed to hold the $4$-$\mathrm{m}-$ wide gate in the position shown.

Solution

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Step 1
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We are asked to determine the required magnitude of the force acting on the top edge of the gate to keep it shut.

We will determine first the force exerted by water on the gate and its location. Then, we will get the sum of moments about the hinge to determine the magnitude of the force acting on the top edge. The following equations will be used:

\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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