Hydrostatic Force on a Plane Surface

F = p_{c} A = ρ g h_{c} A

Resultant fluid force on one side of a plane area equals the pressure at the centroid times the area. hc is the vertical depth of the centroid. The force is normal to the plane and acts at the center of pressure, which is below the centroid on an inclined or vertical surface.

$F$ (N): Resultant hydrostatic force
$p_{c}$ (Pa): Pressure at the centroid
$A$ (m²): Surface area
$h_{c}$ (m): Vertical depth of the centroid

Step-by-step solved example

A vertical 2 m × 3 m gate has its centroid 4.0 m below the free surface of water (ρ = 1000 kg/m³, g = 9.81). Find F.

Resultant F = p_c A on a plane surface; acts at the center of pressure.

1. Area and centroid pressure
A = 6.00 m². pc = ρghc = 1000×9.81×4.0 = 39.24 kPa.
2. Force
F = pc A = 39.24e3 × 6.00 = 235 kN.
$F = 235 kN$

Answer: 235 kN

10 practice questions

0/10 correct

1.F on 2.0 m² at hc = 5.0 m in water (γ = 9.81 kN/m³) is

2.pc is

3.The resultant acts through

4.On a horizontal plane the force is

5.A = 1.5 m², hc = 2.0 m, γ = 9.81 kN/m³. F =

6.F on a face is independent of

7.An inclined plane uses hc equal to

8.The resultant is

9.3 m × 4 m plane, hc = 6 m, water. F ≈

10.Doubling the depth of a submerged horizontal plate