Question 14.4: The Force on a Dam Water is filled to a height H behind a da...
The Force on a Dam
Water is filled to a height H behind a dam of width w (Fig. 14.5, page 364). Determine the resultant force exerted by the water on the dam.
Conceptualize Because pressure varies with depth, we cannot calculate the force simply by multiplying the area by the pressure. As the pressure in the water increases with depth, the force on the adjacent portion of the dam also increases.
Categorize Because of the variation of pressure with depth, we must use integration to solve this example, so we categorize it as an analysis problem.
Analyze Let’s imagine a vertical y axis, with y = 0 at the bottom of the dam. We divide the face of the dam into narrow horizontal strips at a distance y above the bottom, such as the red strip in Figure 14.5. The pressure on each such strip is due only to the water; atmospheric pressure acts on both sides of the dam.
Use Equation 14.4 to calculate the pressure due to the water at the depth h:
P=P_0+\rho g h (14.4)
P = ρgh = ρg(H – y)
Use Equation 14.2 to find the force exerted on the shaded strip of area dA = w dy :
dF = P dA = ρg(H – y)w dy
Integrate to find the total force on the dam:
F=\int P d A=\int_0^H \rho g(H-y) w d y=\frac{1}{2} \rho g w H^2Finalize Notice that the thickness of the dam shown in Figure 14.5 increases with depth. This design accounts for the greater force the water exerts on the dam at greater depths.