Calculating the Average Pressure and Force Exerted: What Force Must a Dam Withstand?

The average pressure $\overline{P}$ due to the weight of the water is the pressure at the average depth $\overline{h}$ of 40.0 m, since pressure increases linearly with depth.

The average pressure due to the weight of a fluid is

Entering the density of water from Table 11.1 and taking $\overline{h}$ to be the average depth of 40.0 m, we obtain

The force exerted on the dam by the water is the average pressure times the area of contact:

We have already found the value for $\overline{P}$. The area of the dam is $A=\text{80.0 m}\times \text{500 m}=4.00\times {\text{10}}^4{\text{m}}^2$, so that

Although this force seems large, it is small compared with the $1.96\times {\text{10}}^{\text{13}}\text{N}$ weight of the water in the reservoir—in fact, it is only $\mathrm{0.0800\%}$ of the weight. Note that the pressure found in part (a) is completely independent of the width and length of the lake—it depends only on its average depth at the dam. Thus the force depends only on the water’s average depth and the dimensions of the dam, not on the horizontal extent of the reservoir. In the diagram, the thickness of the dam increases with depth to balance the increasing force due to the increasing pressure.epth to balance the increasing force due to the increasing pressure.