Fluid Mechanics – Centre of pressure on a vertical plane surface in a static liquid Why does the centre of pressure always lie below the centroid of the surface area?

Introduction / Context:
For plane surfaces submerged in a static fluid, two key points are defined: the centroid (geometric center) and the centre of pressure (point of action of the resultant hydrostatic force). The centre of pressure is always found below the centroid for vertical (or inclined) surfaces. This question probes the physical reason behind that consistent observation.

Given Data / Assumptions:

• Static fluid (no motion), hence no shear stresses in the fluid.
• Plane surface is vertical and fully submerged.
• Fluid is incompressible with constant specific weight.
• Coordinate origin often taken at the free surface with depth measured downward.

Concept / Approach:

Hydrostatic pressure varies with depth according to p = rho * g * h. Because pressure increases linearly with depth, the lower parts of the surface experience greater pressure than the upper parts. The resultant hydrostatic force is therefore “weighted” toward the deeper region, shifting the line of action below the centroid where a uniform pressure would act. The centre of pressure location y_cp is given by y_cp = I_G / (A * y_c) + y_c for a vertical plane, where y_c is the depth of the centroid, A the area and I_G the second moment of area about the free surface (or about a horizontal axis through the centroid with transfer as needed). Since I_G / (A * y_c) is positive, y_cp exceeds y_c (i.e., lies deeper).

Step-by-Step Solution:

Verification / Alternative check:

For a rectangular plate with top at the free surface, the pressure diagram is triangular; the centroid of this triangle lies at 2/3 of depth, which is below the rectangle’s centroid at 1/2 depth—confirming the same principle graphically.

Why Other Options Are Wrong:

Option A confuses isotropy of pressure at a point with variation across depth. Option B is true for static fluids but does not explain the location shift. Option C is false; pressure is not constant with depth. Option E is not applicable.

Common Pitfalls:

Assuming constant pressure over a depth, or mixing local pressure isotropy with vertical hydrostatic variation. Also, forgetting that the centroid relates to geometry while the centre of pressure relates to force distribution.

Final Answer:

The liquid pressure increases linearly with depth