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# A plane gate of uniform thickness holds back a depth of water as shown. Find the minimum weight needed to keep the gate closed.

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The hydrostatic force acting on the gate by

$\begin{equation} F=\rho g A \bar{x} \end{equation}$

where $\rho$ is the water density, $g$ is the gravity acceleration, $A$ is the gate area, $\bar{x}$ is the vertical distance between centroid of the gate from the free surface. The gate area can be calculated as

$\begin{equation} A=L \times w \end{equation}$

where $L$ is the length of the plate and $w$ is the plate width. We substitute $3 \mathrm{~m}$ for $L$ and $2 \mathrm{~m}$ for $w$, so

\begin{equation} \begin{aligned} A &=3 \times 2 \\ &=6 \mathrm{~m}^{2} \end{aligned} \end{equation}

The centroid of the plate from the free surface can be calculated with

$\begin{equation} \bar{x}=\frac{L}{2} \sin \theta \end{equation}$

and by replacing $3 \mathrm{~m}$ for $L$ and $30^{\circ}$ for $\theta$, we have

\begin{equation} \begin{aligned} \bar{x} &=\frac{3}{2} \times \sin 30^{\circ} \\ &=0.75 \mathrm{~m} \end{aligned} \end{equation}

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