The depth of the center of pressure on a vertical rectangular gate (4 m wide and 3 m high) with water up to top surface is
2.0 m
Calculating Center of Pressure on a Vertical Rectangular Gate
The question asks for the depth of the center of pressure on a vertical rectangular gate. We are given the dimensions of the gate and told that the water level is up to the top surface of the gate. The center of pressure is the point where the total hydrostatic force acts on the submerged surface.
Understanding the Center of Pressure
When a surface is submerged in a fluid, the pressure exerted by the fluid increases with depth. This varying pressure creates a total force, and the point where this resultant force acts is called the center of pressure. For a vertical plane surface, the center of pressure is always below the centroid of the area because the pressure is higher at greater depths.
Formula for Depth of Center of Pressure
For a plane surface submerged in a fluid, the depth of the center of pressure (\(h_p\)) from the free surface is given by the formula:
\[h_p = \bar{h} + \frac{I_G}{\bar{h} \cdot A}\]
Where:
- \(h_p\) is the depth of the center of pressure from the free surface.
- \(\bar{h}\) is the depth of the centroid of the submerged area from the free surface.
- \(I_G\) is the moment of inertia of the submerged area about an axis passing through its centroid and parallel to the free surface.
- \(A\) is the area of the submerged surface.
Applying the Formula to the Vertical Rectangular Gate
The gate is a vertical rectangle with:
- Width (b) = 4 m
- Height (d) = 3 m
The water is up to the top surface, meaning the top edge of the rectangle is at the free surface of the water.
1. Calculate the depth of the Centroid (\(\bar{h}\))
For a rectangle submerged vertically with its top edge at the free surface, the centroid is at the middle of the height. The depth of the centroid from the free surface is:
\[\bar{h} = \frac{\text{Height}}{2} = \frac{d}{2}\]
Substituting the given height:
\[\bar{h} = \frac{3 \text{ m}}{2} = 1.5 \text{ m}\]
2. Calculate the Area (A)
The area of the rectangular gate is:
\[A = \text{Width} \times \text{Height} = b \times d\]
Substituting the dimensions:
\[A = 4 \text{ m} \times 3 \text{ m} = 12 \text{ m}^2\]
3. Calculate the Moment of Inertia (\(I_G\))
The moment of inertia of a rectangle about an axis passing through its centroid and parallel to its base (which is parallel to the free surface in this case, considering the width 'b' as the base parallel to the surface) is:
\[I_G = \frac{b \cdot d^3}{12}\]
Substituting the dimensions:
\[I_G = \frac{4 \text{ m} \times (3 \text{ m})^3}{12} = \frac{4 \times 27}{12} \text{ m}^4\]
\[I_G = \frac{108}{12} \text{ m}^4 = 9 \text{ m}^4\]
4. Calculate the Depth of the Center of Pressure (\(h_p\))
Now substitute the calculated values of \(\bar{h}\), \(A\), and \(I_G\) into the formula for \(h_p\):
\[h_p = \bar{h} + \frac{I_G}{\bar{h} \cdot A}\]
\[h_p = 1.5 \text{ m} + \frac{9 \text{ m}^4}{1.5 \text{ m} \times 12 \text{ m}^2}\]
\[h_p = 1.5 \text{ m} + \frac{9}{18} \text{ m}\]
\[h_p = 1.5 \text{ m} + 0.5 \text{ m}\]
\[h_p = 2.0 \text{ m}\]
The depth of the center of pressure on the vertical rectangular gate is 2.0 m from the free surface.
| Parameter | Value |
|---|---|
| Gate Width (b) | 4 m |
| Gate Height (d) | 3 m |
| Depth of Centroid (\(\bar{h}\)) | 1.5 m |
| Area (A) | 12 m\(^2\) |
| Moment of Inertia (\(I_G\)) | 9 m\(^4\) |
| Depth of Center of Pressure (\(h_p\)) | 2.0 m |
Revision Table: Center of Pressure Calculation
Let's quickly review the key steps and formulas used in calculating the depth of the center of pressure for this specific case.
| Concept | Formula/Description | Value for this gate |
|---|---|---|
| Depth of Centroid (\(\bar{h}\)) | For vertical rectangle with top at surface, \(\bar{h} = d/2\) | 1.5 m |
| Area (A) | Area of rectangle, \(A = b \times d\) | 12 m\(^2\) |
| Moment of Inertia (\(I_G\)) | For rectangle about centroidal axis parallel to width, \(I_G = b \cdot d^3 / 12\) | 9 m\(^4\) |
| Depth of Center of Pressure (\(h_p\)) | General formula: \(h_p = \bar{h} + I_G / (\bar{h} \cdot A)\) | 2.0 m |
Additional Information: Hydrostatic Force and Pressure
The hydrostatic force on a submerged plane surface is calculated as \(F = \rho \cdot g \cdot \bar{h} \cdot A\), where \(\rho\) is the fluid density and \(g\) is the acceleration due to gravity. For this gate, the hydrostatic force would be \(F = \rho \cdot g \cdot 1.5 \text{ m} \cdot 12 \text{ m}^2 = 18 \rho g\) Newtons (or kN, depending on units). The center of pressure is the point where this total force effectively acts. Its location depends on the shape of the submerged area and its orientation (vertical, inclined, horizontal) relative to the free surface.
For a vertical plane surface, the center of pressure is always below the centroid. The distance between the centroid and the center of pressure is given by \(I_G / (\bar{h} \cdot A)\). In this case, this distance is 0.5 m (2.0 m - 1.5 m).
Important Questions from Hydrostatic Force
The centre of pressure of a plane submerged surface
In the context of hydrostatics, the resultant hydrostatic force acting on a submerged plane surface passes through which of the following points?
If a planar surface is immersed in a liquid, the resultant liquid pressure acts at a point called ___________.
The resultant of all normal pressure acts
Which is the law that states that the intensity of pressure at a point in a fluid at rest is the same in all directions?
