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Applications of Integrals · Work & Force

Hydrostatic Force

F=∫abρg⋅d(y)⋅w(y) dyF = \int_a^b \rho g \cdot d(y) \cdot w(y)\, dy

The force of fluid pressure on a submerged vertical surface, where d(y) is the depth and w(y) is the width at depth y.

  • 2 worked examples
  • 14 in Applications of Integrals

Worked examples

Find the hydrostatic force on a vertical square plate (2m × 2m) with top edge at water surface (ρ = 1000 kg/m³).
  1. Width w(y) = 2 at all depths. Depth d(y) = y, y from 0 to 2.
  2. F = ∫₀² 1000(9.8)(y)(2) dy = 19600 ∫₀² y dy = 19600[y²/2]₀² = 19600(2) = 39200 N

Answer: 39,200 N = 39.2 kN

1 more worked examplePremium

A vertical rectangular plate is 3 m wide and 2 m tall, with its top edge 1 m below the surface of the water (ρ = 1000 kg/m³).

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Common questions

What is the Hydrostatic Force?

The force of fluid pressure on a submerged vertical surface, where d(y) is the depth and w(y) is the width at depth y. It is one of the applications of integrals formulas in the CalcRef reference.

How do you use the Hydrostatic Force?

Worked example. Find the hydrostatic force on a vertical square plate (2m × 2m) with top edge at water surface (ρ = 1000 kg/m³). Width w(y) = 2 at all depths. Depth d(y) = y, y from 0 to 2. F = ∫₀² 1000(9.8)(y)(2) dy = 19600 ∫₀² y dy = 19600[y²/2]₀² = 19600(2) = 39200 N. Answer: 39,200 N = 39.2 kN.

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