Hydrostatic force on plane areas: The resultant pressure force equals area times the pressure intensity at the centroid under which orientation(s) of the plane surface?

Introduction / Context:

Hydrostatic pressure varies linearly with depth in a fluid of constant specific weight. For any plane surface submerged in such a fluid, the resultant force is equivalent to the integral of pressure over area. A powerful result simplifies its magnitude to the product of area and pressure at the area centroid, irrespective of orientation.

Given Data / Assumptions:

• Incompressible fluid at rest with uniform gamma = rho * g.
• Plane surface fully submerged or exposed to hydrostatic head.
• Neglect surface tension and compressibility effects.

Concept / Approach:

The hydrostatic force on a plane area is F = ∫ p dA = ∫ (gamma * h) dA = gamma * (∫ h dA). By definition of centroid depth h_c, we have ∫ h dA = A * h_c. Thus F = gamma * A * h_c = p_c * A, where p_c is the pressure at the centroid. This holds for horizontal, vertical, or inclined planes; only the center of pressure location changes with orientation.

Step-by-Step Solution:

Verification / Alternative check:

The line of action (center of pressure) lies below the centroid except for horizontal surfaces; its depth is given by y_cp = I_xc / (A * h_c) + h_c. This confirms orientation affects only where the force acts, not its magnitude formula.

Why Other Options Are Wrong:

• Claiming the formula holds only for a specific orientation contradicts the general derivation.
• “None of these” is false because the relation is universally valid for any plane orientation.

Common Pitfalls:

• Confusing magnitude (F = p_c A) with the location of the resultant (center of pressure), which depends on second moments.

Final Answer:

All of the above