The problem describes a dam in the shape of an isosceles trapezoid and asks to calculate the force on the dam due to hydrostatic pressure. Let’s break down the problem step by step:

Given:

• Shape of the dam: isosceles trapezoid
• Height of the dam $h = 40 \, \text{m}$
• Top width of the dam $W_{\text{top}} = 70 \, \text{m}$
• Bottom width of the dam $W_{\text{bottom}} = 60 \, \text{m}$
• Water surface is 8 m below the top of the dam, so the height of water column is $h_{\text{water}} = 40 - 8 = 32 \, \text{m}$.

We need to calculate the hydrostatic force exerted on the dam, which depends on the depth, the area of the submerged surface, and the water density.

Hydrostatic Pressure Equation:

The force due to hydrostatic pressure $F$ is given by:

$F = \rho g A_{\text{avg}} \cdot h_{\text{avg}}$

Where:

• $\rho$ is the density of water ($1000 \, \text{kg/m}^3$),
• $g$ is the acceleration due to gravity ($9.81 \, \text{m/s}^2$),
• $A_{\text{avg}}$ is the area of the trapezoid submerged under water,
• $h_{\text{avg}}$ is the depth at the centroid of the water column (center of pressure).

Let’s calculate this step-by-step.

Step 1: Area of the submerged trapezoid

The trapezoidal area submerged is:

$A_{\text{sub}} = \frac{1}{2} (W_{\text{top}} + W_{\text{bottom}}) h_{\text{water}}$ Substituting the given values:

$A_{\text{sub}} = \frac{1}{2} (70 + 60) \times 32$

Step 2: Depth of the centroid (center of pressure)

For a vertical surface, the depth of the centroid $h_{\text{avg}}$ is:

$h_{\text{avg}} = \frac{h_{\text{water}}}{2} = \frac{32}{2} = 16 \, \text{m}$

Now, let’s calculate the force $F$. I’ll proceed with the calculations.The hydrostatic force exerted on the dam is approximately:

$F = 3.26 \times 10^8 \, \text{N}$

Let me know if you would like further details or explanations! Here are five related questions that could expand on this topic:

1. How is the pressure distribution along the dam's height calculated?
2. What role does the shape of the dam play in determining the hydrostatic force?
3. How would the force change if the water height were different?
4. Can hydrostatic force be minimized by altering the dam's design?
5. How would the calculation differ if the dam were not vertical?

Tip: When calculating hydrostatic force, always ensure that you find the depth of the centroid (average depth) correctly, as it significantly impacts the result.