Question

Water stands at a depth D behind the vertical upstream face of a dam as shown in the figure. The force exerted by water per unit width of the dam is (Density of water is $\rho$):

• A. $\frac{1}{3}\rho g{D}^2$
• B. $\frac{1}{2}\rho g{D}^2$
• C. $\frac{1}{3}\rho gD$
• D. $\frac{1}{4}\rho g{D}^2$

Answer 1

The correct option is B $\frac{1}{2}\rho g{D}^2$

Pressure at a depth x = $\rho gx$

Force$dF$ on a horizontal of strip cross section of area $1.dx$

$\Rightarrow dF=\rho gxdx$

Net force per unit width = ${\int }_0^{D}dF=={\int }_0^D\rho gxdx=\frac{\rho g{x}^2}{2}{|}_0^D=\frac{\rho g{D}^2}{2}$

Answer. B