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To solve the problem in the image, where you are tasked with calculating the fluid force on one side of a semicircular plate, we will use principles from fluid mechanics. Here's a step-by-step approach:

### Information provided:
- **Radius of the plate, \( r = 8 \, \text{ft} \)** (semicircular plate).
- **Water depth, \( h = 12 \, \text{ft} \)**.
- **Weight-density of water, \( \rho = 62.4 \, \text{lb/ft}^3 \)**.

The fluid force on the plate is determined by integrating the pressure at different depths. The pressure \( P \) at a depth \( y \) below the surface of the water is:

\[
P(y) = \rho \cdot y
\]

### Steps:

1. **Find the depth \( y \) for each horizontal strip** of the semicircular plate and calculate the pressure acting at that depth. The plate is located vertically, so the pressure varies along its height.
  
2. **Break the semicircular plate into differential horizontal strips** at different depths below the water surface and calculate the force on each strip.

   The fluid force on a small strip of area \( dA \) is given by:
   \[
   dF = P(y) \cdot dA = \rho \cdot y \cdot dA
   \]
   
   For a semicircular plate of radius \( r \), the differential area \( dA \) for a horizontal strip is the width of the strip times the thickness \( dy \), which is:
   \[
   dA = 2 \cdot \sqrt{r^2 - x^2} \cdot dy
   \]
   where \( x \) is the distance from the center of the plate.

3. **Set up the integral** over the height of the plate. The total force \( F \) is:
   \[
   F = \int_{0}^{r} \rho \cdot y \cdot 2 \cdot \sqrt{r^2 - y^2} \, dy
   \]

4. **Integrate** to find the total force.

I will now compute the total fluid force using these steps.The total fluid force on one side of the semicircular plate is approximately **21,299.2 lb**.

Would you like further details or have any questions about the process?

Here are 5 follow-up questions to explore:

1. How would the fluid force change if the water depth was increased?
2. What happens to the fluid force if the plate is fully submerged?
3. How would you calculate the fluid force if the plate was rectangular instead of semicircular?
4. How does the shape of the object affect the distribution of fluid pressure?
5. How would the result differ if the plate was oriented horizontally instead of vertically?

**Tip**: When working with fluid mechanics problems, always ensure that the units of density and area are consistent for proper force calculation.

Calculate the fluid force on one side of a semicircular plate of radius 8 ft that rests vertically on its diameter at the bottom of a pool filled with water to a depth of 12 ft. Assume the weight-density of water is 62.4 lb/ft^3.

Learn how to calculate the fluid force on one side of a semicircular plate submerged vertically in water. This problem uses principles of fluid mechanics, including pressure-depth relationships and integration techniques. Suitable for college-level fluid mechanics students.

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