Calculate the water pressure force on a rectangular dam

Calculate the water pressure force on a rectangular dam gate 20 m wide and 16 m high, whose top edge lies at the water surface.

Step 1: Pressure and force on a thin strip

With $y$ measured downward from the surface, a strip at depth $y$ of thickness $dy$ and width $w=20$ m has $dF=\rho gy\cdot w\,dy$.

Step 2: Integrate over the gate

\[F=\rho gw\int_0^h y\,dy=\rho gw\cdot\frac{h^2}{2},\qquad h=16\ \text{m}.\]

\[\boxed{F=\frac{\rho gwh^2}{2}}\]

Step 3: Numerical value

With $\rho=1000\ \text{kg/m}^3$, $g=9.8\ \text{m/s}^2$:

\[F=\frac{1000\times9.8\times20\times256}{2}=25{,}088{,}000\ \text{N}\approx25.1\ \text{MN}.\]

Step 4: Center of pressure

\[y_{\text{cp}}=\frac{\int_0^h y\cdot\rho gyw\,dy}{\int_0^h\rho gyw\,dy}=\frac{\int_0^hy^2\,dy}{\int_0^hy\,dy}=\frac{h^3/3}{h^2/2}=\frac{2h}{3}.\]

\[\boxed{y_{\text{cp}}=\frac{2h}{3}\approx10.67\ \text{m}}\]

The center of pressure sits below the gate's geometric mid-height ($h/2=8$ m) because pressure grows with depth, weighting the lower part of the gate more heavily than the upper part.