11. A circular plate of 1 m diameter is immersed in water in such a way that its plane makes an angle of 40° with the horizontal and its top edge is 1.25 mm below the water surface. Find the total pressure on the plate and the point where at acts.

We can solve this problem in two parts: 

1. **Finding the total hydrostatic pressure on the plate.**
2. **Finding the point of action of the pressure (center of pressure).**

### Given:
- Diameter of the circular plate, \( d = 1 \, \text{m} \).
- Radius of the plate, \( r = \frac{d}{2} = 0.5 \, \text{m} \).
- The angle between the plate and the horizontal, \( \theta = 40^\circ \).
- The depth of the top edge of the plate below the water surface, \( h_1 = 1.25 \, \text{mm} = 0.00125 \, \text{m} \).
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \).
- Acceleration due to gravity, \( g = 9.81 \, \text{m/s}^2 \).

### 1. Total Hydrostatic Pressure on the Plate

The total hydrostatic pressure on a surface submerged in a fluid is given by:

\[
F = \rho g A \overline{h}
\]

Where:
- \( F \) is the total force (pressure),
- \( \rho \) is the density of the fluid (water),
- \( g \) is the acceleration due to gravity,
- \( A \) is the area of the plate,
- \( \overline{h} \) is the depth of the centroid of the plate from the water surface.

#### Area of the circular plate:
\[
A = \pi r^2 = \pi (0.5)^2 = 0.7854 \, \text{m}^2
\]

#### Depth of the centroid (\( \overline{h} \)):

The centroid of a circular plate is located at its center. Since the top edge of the plate is 1.25 mm below the water surface and the plate is inclined, the depth of the centroid from the surface is:

\[
\overline{h} = h_1 + r \sin(\theta) = 0.00125 + 0.5 \sin(40^\circ)
\]

We can now compute the total force \( F \).

### 2. Location of the Center of Pressure

The center of pressure is the point where the total hydrostatic force acts. For a plane surface inclined at an angle to the horizontal, the depth of the center of pressure \( h_{cp} \) is given by:

\[
h_{cp} = \overline{h} + \frac{I_G}{A \overline{h}}
\]

Where:
- \( I_G \) is the second moment of area (area moment of inertia) about the centroid of the plate, for a circular plate this is \( I_G = \frac{\pi r^4}{4} \).

Let's calculate both the total force \( F \) and the center of pressure location. I'll begin with the calculations.### Results:

1. **Total Hydrostatic Pressure on the Plate**:  
   The total force acting on the plate due to hydrostatic pressure is approximately \( 2485.89 \, \text{N} \).

2. **Location of the Center of Pressure**:  
   The center of pressure is located at a depth of approximately \( 0.516 \, \text{m} \) below the water surface.

Would you like more details or have any further questions?

Here are some related questions to expand your understanding:
1. How would the force change if the plate was horizontal?
2. What happens to the center of pressure if the diameter of the plate is doubled?
3. How does the angle of inclination affect the hydrostatic force?
4. What if the plate was submerged deeper in the water, how would the total force change?
5. How does the fluid's density affect the total pressure?

**Tip**: The center of pressure is always deeper than the centroid because the pressure increases with depth, creating a torque that shifts the pressure point downward.

A circular plate of 1 m diameter is immersed in water in such a way that its plane makes an angle of 40° with the horizontal and its top edge is 1.25 mm below the water surface. Find the total pressure on the plate and the point where it acts.

Learn how to calculate the total hydrostatic pressure on a circular plate immersed in water at a 40° angle and find the center of pressure. This problem involves hydrostatic principles, trigonometry, and geometry. Includes step-by-step calculations and key formulas.

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