Hydrostatics on curved surfaces — the horizontal component of hydrostatic force on a curved surface equals which equivalent force?

Introduction / Context:
Curved submerged surfaces (e.g., gates, spillway profiles, elbows) experience hydrostatic forces with distinct horizontal and vertical components. Engineers resolve these components to design supports and anchors correctly.

• Static fluid; density rho and gravity g are constant.
• Curved surface submerged with a well-defined vertical projection area A_v.
• Hydrostatic pressure varies linearly with depth.

Concept / Approach:

The horizontal component on a curved surface equals the hydrostatic force on its vertical projection, acting through the center of pressure of that projection. The vertical component equals the weight of the imaginary fluid above the surface up to the free surface (upthrust concept).

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Integrating pressure components over the curved surface and comparing with the vertical projection yields identical F_h; this is a classic hydrostatics result.

Why Other Options Are Wrong:

“Weight of liquid vertically below …” pertains to vertical component, not horizontal; “product of pressure at centroid and area” ignores depth variation; “weight of liquid retained …” again is vertical component logic; “none” is incorrect.

Common Pitfalls (misconceptions, mistakes):

Confusing horizontal and vertical component formulae; using pressure at centroid instead of integrating for plane/curved differences.

Final Answer:

force on a vertical projection of the curved surface