Question

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

• A. $2h\sigma g$
• B. $\frac{h^2\sigma \rho g}{2}$
• C. $\frac{h^2\sigma \rho g}{4}$
• D. $\frac{h\sigma \rho g}{4}$

Answer 1

The correct option is B $\frac{h^2\sigma \rho g}{2}$

Let's consider,

$\sigma =$ width of the dam

$h=$ height of the dam

$\rho =$ density of water

The pressure at the small element of height $yy$,

$P_y=y\rho g$. . . . . .(1)

The force due to element $dy$ is

$dF=P_y\left(dA\right)=y\rho g\left(\sigma dy\right)=\rho \sigma gydy$

Net force, $F=\int dF=\rho \sigma g{\int }_0^{h}ydy$

$F=\rho \sigma g\frac{y^2}{2}{|}_0^h$

$F=\frac{\rho \sigma g{h}^2}{2}$

The correct option is B.