Final answer:
The total force on a rectangular dam can be calculated using the average pressure due to the water depth and the area in contact with the water, resulting in a quadratic relationship with the depth showing that force increases with the square of the depth.
Step-by-step explanation:
To calculate the total force exerted on a rectangular dam, we need to determine the average pressure exerted by the water at depth and multiply it by the area of the dam in contact with the water.
The pressure exerted by a fluid at depth is given by P = ρgh, where ρ is the density of the liquid, g is the acceleration due to gravity, and h is the depth of the liquid. Since pressure increases linearly with depth, the average pressure exerted on the dam can be calculated as P = ρg(h/2), because the average depth (from the surface to the bottom) will be h/2. Now that we have the average pressure, we multiply it by the area in contact with the water (height × length of the dam), which is hL, to get the force.
So the total force, F, on the face of the dam is F = (ρg(h/2)) × (hL) = ρgh²L/2. This force increases with the square of the depth (h²) because of the quadratic relationship in the formula. The thickness of the dam is often increased with depth to balance this increasing force due to increasing pressure.