Understanding the Centre of Pressure on Submerged Surfaces

When a plane surface is submerged in a fluid, it experiences a hydrostatic force due to the pressure exerted by the fluid. This pressure varies with depth, being higher at greater depths. The total hydrostatic force acting on the surface is the resultant of these varying pressure forces. The point at which this resultant force is considered to act is called the centre of pressure.

Another important point related to a submerged area is the centroid of the area. The centroid is the geometric center of the shape, often referred to as the centre of gravity if the surface were a uniform thin plate.

Locating the Centre of Pressure

The position of the centre of pressure is crucial for analyzing the stability and structural integrity of submerged objects like gates, dams, and tank walls. For a plane surface submerged in a liquid, the depth of the centre of pressure ($h_p$) from the free surface is related to the depth of the centroid ($h_c$) by the following formula:

$$\text{h}_p = \text{h}_c + \frac{\text{I}_{xx}}{\text{A h}_c}$$

Where:

• $\text{h}_p$ = Depth of the centre of pressure from the free surface.
• $\text{h}_c$ = Depth of the centroid of the area from the free surface.
• $\text{I}_{xx}$ = Moment of inertia of the submerged area about an axis passing through its centroid and parallel to the free surface.
• $\text{A}$ = Area of the submerged surface.

In this formula, $\text{I}_{xx}$ and $\text{A}$ are always positive values. The depth of the centroid $\text{h}_c$ is also measured from the free surface and is generally positive for a submerged surface.

From the formula, we can see that the term $\frac{\text{I}_{xx}}{\text{A h}_c}$ is always positive (assuming $\text{h}_c > 0$). Therefore, $\text{h}_p$ will always be greater than $\text{h}_c$. This means that the centre of pressure is always located deeper than the centroid of the submerged area.

Analyzing the Options

Let's examine the given options based on our understanding of the centre of pressure:

• Option 1: Is a point above the submerged area at which the resultant hydrostatic force is supposed to act.

  This statement correctly identifies the centre of pressure as the point where the resultant force acts. However, stating it is "above the submerged area" is generally incorrect. The centre of pressure is located within the fluid body exerting pressure on the surface, and its depth is typically important, not just whether it's "above" the entire area.

• Option 2: Should always coincided within the centre of submerged area.

  The "centre of submerged area" likely refers to the centroid. As established by the formula $\text{h}_p = \text{h}_c + \frac{\text{I}_{xx}}{\text{A h}_c}$, the centre of pressure only coincides with the centroid when $\frac{\text{I}_{xx}}{\text{A h}_c} = 0$. This happens only for a horizontally submerged plane surface, where the pressure is uniform across the surface, and $\text{h}_c$ is constant. For inclined or vertical surfaces, the pressure varies, and the centre of pressure is deeper than the centroid. Therefore, they do not always coincide.

• Option 3: Should be at the centre of gravity of the plane surface.

  The centre of gravity of the plane surface (assuming uniform material) is the same as its centroid. As discussed in Option 2, the centre of pressure is generally not at the centroid, except for horizontally submerged surfaces.

• Option 4: Is always below the centroid of area.

  Our analysis of the formula $\text{h}_p = \text{h}_c + \frac{\text{I}_{xx}}{\text{A h}_c}$ showed that $\text{h}_p > \text{h}_c$ for any non-horizontal submerged surface where $\text{I}_{xx} > 0$. Even for a horizontal surface, $\text{h}_p = \text{h}_c$, which can be considered "not above" the centroid's depth. For any varying pressure distribution (vertical or inclined surface), the centre of pressure is strictly below the centroid because pressure increases with depth, shifting the resultant force application point towards the deeper parts of the surface. Thus, this statement is correct.

Based on the formula and the principle that pressure increases with depth, the resultant hydrostatic force acts at a point lower than the geometric center (centroid) of the submerged area for any surface that is not horizontal.

Revision Table: Key Concepts

Additional Information: Factors Affecting Centre of Pressure

The location of the centre of pressure for a plane submerged surface is affected by several factors:

• Depth of Submergence: As the depth of submergence increases, the pressure distribution becomes more uniform relative to the average pressure, and the centre of pressure moves closer to the centroid. However, it always remains below the centroid.
• Orientation of the Surface: For a horizontal surface, pressure is uniform, and the centre of pressure coincides with the centroid. For vertical or inclined surfaces, pressure varies with depth, causing the centre of pressure to be below the centroid. The more vertical the surface, the further the centre of pressure is from the centroid (relative to the height of the surface).
• Shape of the Surface: The shape influences the moment of inertia ($I_{xx}$), which affects the distance between the centroid and the centre of pressure. Different shapes will have different $I_{xx}$ values for the same area, resulting in different centre of pressure locations relative to their centroids at the same depth and orientation.

Understanding the centre of pressure is vital in engineering design, particularly in fluid mechanics and structural engineering, to ensure the stability and safety of structures interacting with fluids.