Question

Water stands upto a height $h$ behind the vertical wall of a dam. What is the net horizontal force pushing the dam down the stream, if width of the dam is $\sigma$? ($\rho =$ density of water)

Answer 1

Let, an an infinitesimal $dy$ change in $h$

And the gauge pressure at a dept $y$ from the surface of the dam is

$p=pgy$

Here, density of water is $p$ and the acceleration due to gravity is $g$

The resultant force on the wall is

$F=\iint pdxy$ $.........\left(1\right)$

$F=\int pdy\int dx$

We already know that $\int dx=\sigma$

Now substituting $pgy$ for $p$ and $\sigma$ for $\int dx$ in equation $\left(1\right)$

$\begin{array}{c}F=\int pdy\int dx\\ =\int ydy\int \sigma \\ =pg\sigma {\int }_0^{h}ydy\\ =pg\sigma {{\left|\frac{y^2}{2}\right|}_0}^h\\ F=\frac{pg\sigma h^2}{2}\end{array}$