https://file.aiquickdraw.com/mathbot/file/1727464389133153s0e7a.jpg

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.

A dam has the shape of an isosceles trapezoid with a height of 40 m. The width at the top is 70 m and the width at the bottom is 60 m. The surface of the water is 8 m below the top of the dam. Find the force on the dam due to hydrostatic pressure.

This problem calculates the force exerted on a trapezoidal dam due to hydrostatic pressure. The dam has a height of 40 m, with a water surface 8 m below the top. Learn to compute the force using geometry and physics principles.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation