Calculate Total Hydrostatic Pressure Force on a Circular Plate

The total hydrostatic pressure force acting on a submerged plane surface is determined using the formula:

\(F = \rho g A \bar{h}\)

Where:

• \(\rho\) is the density of the fluid (water in this case).
• \(g\) is the acceleration due to gravity.
• \(A\) is the area of the submerged surface.
• \(\bar{h}\) is the vertical distance from the free surface of the fluid to the centroid of the submerged surface.

Given Parameters and Unit Conversion for Hydrostatic Force Calculation

We are provided with the following information:

• Radius of the circular plate, \(R = 750 \text{ mm}\).
• Area of the circular plate, \(A = 1.767 \times 10^6 \text{ mm}^2\).
• Distance between the free surface of water and the plate is 3000 mm. For a vertically submerged circular plate, the centroid is located at its geometric center. The question phrasing suggests this distance is the vertical depth to the plate. Based on the options, it is highly likely that 3000 mm represents the vertical distance from the free surface to the centroid of the plate. So, we take \(\bar{h} = 3000 \text{ mm}\).
• Density of water, \(\rho = 1000 \text{ kg/m}^3\) (assuming the unit \(\rm kg/m^2\) in the question is a typographical error and should be \(\rm kg/m^3\)).
• Acceleration due to gravity, \(g = 9.81 \text{ m/s}^2\).

For calculation, it's important to use consistent units (SI units). We convert millimeters to meters:

• \(\bar{h} = 3000 \text{ mm} = 3000 \times 10^{-3} \text{ m} = 3 \text{ m}\)
• \(A = 1.767 \times 10^6 \text{ mm}^2 = 1.767 \times 10^6 \times (10^{-3} \text{ m})^2 = 1.767 \times 10^6 \times 10^{-6} \text{ m}^2 = 1.767 \text{ m}^2\)

Step-by-Step Calculation of Total Hydrostatic Pressure Force

Now, substitute the converted values into the hydrostatic force formula \(F = \rho g A \bar{h}\):

\(F = (1000 \text{ kg/m}^3) \times (9.81 \text{ m/s}^2) \times (1.767 \text{ m}^2) \times (3 \text{ m})\)

Performing the multiplication:

\(F = (1000 \times 9.81) \times 1.767 \times 3 \text{ N}\)

\(F = 9810 \times 1.767 \times 3 \text{ N}\)

\(F = 17333.07 \times 3 \text{ N}\)

\(F = 51999.21 \text{ N}\)

Conclusion on Total Hydrostatic Pressure Force

The calculated total hydrostatic pressure force is approximately \(51999.21 \text{ N}\).

Comparing this result with the provided options, the closest value is \(52002.81 \text{ N}\).

Therefore, the total hydrostatic pressure force acting on the circular plate is \(52002.81 \text{ N}\).