Answer :
To solve this problem, we'll calculate two things: the total pressure (also known as the hydrostatic force) acting on the surface, and the position of the center of the pressure.
1. Total Pressure (Hydrostatic Force):
The hydrostatic force on a surface submerged in a fluid can be calculated using the following formula:
[tex]F = \rho \cdot g \cdot A \cdot h_c[/tex]
Where:
- [tex]F[/tex] is the hydrostatic force (total pressure) on the surface.
- [tex]\rho[/tex] is the density of the fluid (for water, [tex]\rho = 1000 \text{ kg/m}^3[/tex]).
- [tex]g[/tex] is the acceleration due to gravity [tex](9.81 \text{ m/s}^2)[/tex].
- [tex]A[/tex] is the area of the submerged surface.
- [tex]h_c[/tex] is the depth of the centroid of the submerged area below the free surface of the fluid.
For the given problem:
- The plane surface is [tex]2 \text{ m} \times 3 \text{ m}[/tex], so the area [tex]A = 6 \text{ m}^2[/tex].
- The centroidal depth [tex]h_c[/tex] can be calculated as the average depth of the plane from the water surface. Since the plane makes an angle of [tex]30°[/tex] and the upper edge is 1.5 m deep, the depth of the centroid is:
[tex]h_c = 1.5 \text{ m} + \frac{3 \text{ m}}{2} \sin(30°) = 1.5 + 1.5 \cdot 0.5 = 2.25 \text{ m}[/tex]
Now, substituting these values into the formula:
[tex]F = 1000 \cdot 9.81 \cdot 6 \cdot 2.25 = 132435 \text{ N}[/tex]
2. Position of the Center of Pressure:
The center of pressure is the point at which the total pressure force acts, and it is always below the centroid. The depth of the center of pressure, [tex]h_p[/tex], can be calculated using the formula:
[tex]h_p = h_c + \frac{I_g}{A \cdot h_c}[/tex]
Where:
- [tex]I_g[/tex] is the second moment of area (moment of inertia) about the horizontal axis passing through the centroid. For a rectangle, [tex]I_g = \frac{b \cdot d^3}{12}[/tex].
- [tex]b = 2 \text{ m}[/tex] is the width, and [tex]d = 3 \text{ m}[/tex] is the depth.
[tex]I_g = \frac{2 \cdot 3^3}{12} = \frac{54}{12} = 4.5 \text{ m}^4[/tex]
Substituting all known values:
[tex]h_p = 2.25 + \frac{4.5}{6 \cdot 2.25} = 2.25 + 0.333 = 2.583 \text{ m}[/tex]
Therefore, the hydrostatic force acting on the surface is approximately [tex]132,435 \text{ Newtons}[/tex], and the center of pressure is located at a depth of approximately [tex]2.583 \text{ meters}[/tex] below the water surface.
