CalcRef

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.

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

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