Centre of pressure for a vertical plane surface: Select the correct formula for the depth h of the centre of pressure from the free surface (IG = second moment of area about the horizontal centroidal axis, A = area of immersed surface, x = depth of centroid).

Introduction / Context:
The centre of pressure is the point of application of the resultant hydrostatic force on a plane surface. For a vertical surface, pressure increases linearly with depth, shifting the resultant below the centroid. The exact depth ensures moment equivalence between the distributed pressure and a single resultant force.

Given Data / Assumptions:

• Vertical plane surface fully or partially submerged.
• Hydrostatic condition (no motion).
• Free surface exposed to uniform atmospheric pressure.

Concept / Approach:
Resultant hydrostatic force F = ρ * g * A * x. Taking moments about the free surface and equating to the moment of the resultant at depth h yields h = x + IG / (A * x). Since IG/(A * x) is positive, h > x, meaning the centre of pressure lies below the centroid for a vertical surface.

Step-by-Step Solution:

Verification / Alternative check:
For a thin vertical rectangle with centroid at x, the addition IG/(A*x) is nonzero, so h is always below the centroid, matching intuition from the linear pressure distribution.

Why Other Options Are Wrong:

• x − IG/(A * x): Would place h above the centroid, contradicting physics.
• IG/(A * x) alone: Omits the centroidal depth term.
• (A * x)/IG: Inverts the relationship and has wrong dimensions.

Common Pitfalls:
Forgetting to include the centroidal term x; confusing IG about the centroid with second moment about the free surface.

Final Answer:
h = x + (IG / (A * x))