Hydrostatics — Centre of Pressure for a Vertically Immersed Plane For a vertically immersed plane surface with area A, centroid depth y_c below the free surface, and second moment of area about its centroid I_G, the distance of the centre of pressure from the free surface is:

Introduction / Context: The centre of pressure is the point of action of the resultant hydrostatic force on a submerged surface. Locating it is vital for gate design, retaining structures, and tank walls to compute moments and stability.

Given Data / Assumptions:

• Plane surface of area A; centroid located at depth y_c from the free surface.
• Hydrostatic condition with fluid of constant density.
• I_G is the second moment of area about an axis through the centroid parallel to the free surface.

Concept / Approach: The pressure varies linearly with depth: p = ρ g y. The resultant force acts at the centre of pressure located below the centroid for vertical planes. Applying the standard hydrostatics result gives y_cp = y_c + (I_G / (A y_c)).

Step-by-Step Solution:

Verification / Alternative check: Since I_G > 0, y_cp > y_c for a vertical plane, agreeing with physical intuition: pressure is greater at deeper points, pulling the resultant below the centroid.

Why Other Options Are Wrong: Omitting y_c or using a minus sign contradicts the derivation; taking a square root ignores the linear depth relation.

Common Pitfalls: Confusing I_G about the centroid with I_0 about the free surface; using the formula for an inclined plane without adjusting depths.

Final Answer: y_cp = y_c + I_G / (A y_c).