Understanding Center of Pressure and Center of Gravity

When a surface is submerged in a fluid, the fluid exerts pressure on the surface. This pressure varies with the depth of the fluid.

• Center of Gravity (CG): This is the point where the entire weight of the object (or surface, in this case, considering its material if any, or just the geometric center if it's a plane) is considered to act. For a uniform, symmetrical vertical plane surface, the center of gravity is usually at its geometric centroid.
• Center of Pressure (CP): This is the point where the total resultant hydrostatic force acts on the submerged surface. This resultant force is the sum of all the small forces exerted by the fluid pressure over the entire surface area.

Pressure Distribution on a Vertical Surface

Fluid pressure increases linearly with depth. On a vertical submerged surface, the pressure at the top edge is lower than the pressure at the bottom edge. This means the force distribution over the surface is not uniform; it's triangular (for atmospheric pressure at the surface) or trapezoidal (if the top edge is also submerged).

Consider a vertical plane surface submerged in a liquid:

Because the pressure and thus the force are greater at deeper parts of the surface, the resultant force's point of application (the Center of Pressure) will be shifted downwards compared to the geometric center (Center of Gravity).

Locating the Center of Pressure

The Center of Pressure is always located below the Center of Gravity for any submerged plane surface that is not horizontal.

• For a horizontal submerged surface, the pressure is uniform across the surface (if at a constant depth), so the Center of Pressure coincides with the Center of Gravity.
• For inclined or vertical submerged surfaces, the pressure increases with depth. This non-uniform pressure distribution means the resultant force acts at a point lower than the geometric center.

For a vertical plane surface, the Center of Pressure is located below the Center of Gravity.

Mathematically, the depth of the center of pressure below the free surface ($h_{cp}$) is given by:

\(h_{cp} = h_{cg} + \frac{I_{cg}}{A \cdot h_{cg}}\)

Where:

• \(h_{cg}\) is the depth of the Center of Gravity below the free surface.
• \(I_{cg}\) is the moment of inertia of the surface area about an axis passing through the Center of Gravity and parallel to the free surface.
• \(A\) is the area of the surface.

Since \(I_{cg}\), \(A\), and \(h_{cg}\) are all positive values, the term \(\frac{I_{cg}}{A \cdot h_{cg}}\) is always positive. This confirms that \(h_{cp} > h_{cg}\), meaning the Center of Pressure is always deeper than (and thus below) the Center of Gravity for a vertical submerged surface.