Calculating Total Hydrostatic Pressure on a Vertical Submerged Surface

The question asks us to find the total pressure force exerted by water on a vertical rectangular plane surface. We are given the dimensions of the rectangle and its position relative to the free water surface.

Understanding Total Pressure Force

The total pressure force on a submerged plane surface is the resultant force due to the varying pressure of the fluid over the surface area. For a plane surface, this total force (F) is given by the formula:

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

Where:

• \(\rho\) is the density of the fluid.
• \(g\) is the acceleration due to gravity.
• \(\bar{h}\) is the depth of the centroid of the submerged area from the free surface.
• \(A\) is the total area of the submerged surface.

Given Parameters

• Width of the rectangular plane = \(2 \, m\)
• Depth of the rectangular plane = \(3 \, m\)
• The plane is vertical.
• Upper edge is horizontal and \(2 \, m\) below the free water surface.
• Fluid is water (\(\rho \approx 1000 \, kg/m^3\)).
• Let's use \(g \approx 9.81 \, m/s^2\) for acceleration due to gravity.

Step-by-Step Calculation

1. Calculate the Area (A)

The area of the rectangular surface is the product of its width and depth.

\[A = \text{Width} \times \text{Depth} = 2 \, m \times 3 \, m = 6 \, m^2\]

2. Determine the Depth of the Centroid (\(\bar{h}\))

For a rectangle, the centroid is located at the geometric center. The distance of the centroid from the upper edge is half of the depth.

Distance of centroid from upper edge = \(\frac{\text{Depth}}{2} = \frac{3 \, m}{2} = 1.5 \, m\)

The upper edge is located \(2 \, m\) below the free surface. Therefore, the depth of the centroid from the free surface (\(\bar{h}\)) is the sum of the depth of the upper edge and the distance of the centroid from the upper edge.

\[\bar{h} = \text{Depth of upper edge} + \text{Distance of centroid from upper edge}\]

\[\bar{h} = 2 \, m + 1.5 \, m = 3.5 \, m\]

3. Calculate the Total Pressure Force (F)

Now, substitute the values of \(\rho\), \(g\), \(\bar{h}\), and \(A\) into the formula \(F = \rho g \bar{h} A\).

\[F = (1000 \, kg/m^3) \times (9.81 \, m/s^2) \times (3.5 \, m) \times (6 \, m^2)\]

\[F = 1000 \times 9.81 \times 3.5 \times 6 \, N\]

\[F = 9810 \times 21 \, N\]

\[F = 206010 \, N\]

To express the force in kiloNewtons (kN), divide by 1000.

\[F = \frac{206010}{1000} \, kN = 206.01 \, kN\]

Conclusion

The total pressure force acting on the vertical rectangular plane surface is \(206.01 \, kN\).