Question

The height of water in a vessel is $h.$ The vessel wall of width $b$ is at an angle $\theta$ to the vertical. The net force exerted by the water on the wall is:

• A. $\frac{1}{3}\rho b{h}^{2}g\cos \theta$
• B. $\frac{1}{2}b{h}^2\rho g$
• C. $\frac{1}{2}\rho b{h}^{2}g\sec \theta$
• D. $zero$

Answer 1

The correct option is C $\frac{1}{2}\rho b{h}^{2}g\sec \theta$

$z=y\sec \theta$ $\Rightarrow$ $dz=dy\sec \theta$
$P=\left(\rho gy\right)$
$dA=\left(b\right)dz=\left(b\right)dy\sec \theta$
$dF=PdA=\left(\rho gb\sec \theta \right)ydy$
$\therefore$ $F={\int }_0^{h}dF=\frac{1}{2}\rho b{h}^{2}g\sec \theta$