Hydrostatics – Total Force on an Inclined Plane Surface For a plane area A immersed in a liquid and inclined at angle theta to the free surface, the total hydrostatic pressure equals w * A * x_cg, where w is specific weight and x_cg is the vertical depth of the area's centroid below the free surface.

Introduction:
Determining the resultant hydrostatic force on plane surfaces is basic to gate, dam, and tank design. The magnitude depends on the area, specific weight, and the centroidal depth, not directly on the surface orientation for the force magnitude itself.

Given Data / Assumptions:

• Liquid at rest with specific weight w.
• Plane surface area A inclined at angle theta to the free surface.
• Centroid located at vertical depth x below the free surface.

Concept / Approach:

Pressure varies linearly with depth: p = w * z. The resultant force equals the integral of pressure over area, which reduces to F = w * A * x_cg, where x_cg is the centroid depth measured vertically. Orientation affects center of pressure location but not the magnitude expression.

Step-by-Step Solution:

Verification / Alternative check:

For a vertical rectangular gate, x is the depth of its centroid; substituting gives the standard textbook expression and matches experimental measurements of total force.

Why Other Options Are Wrong:

wA: Ignores depth variation of pressure. wx: Misses area dimension. wAx/sin theta: Angle does not enter the force magnitude in this form; it affects the center of pressure. wAcos theta: Not dimensionally consistent for force without x.

Common Pitfalls:

Measuring x along the plate instead of vertically; confusing the force magnitude with the line of action location (which does depend on theta).

Final Answer:

wAx