STEM Calculators

Hydrostatic Force Tool

Math Calculus • Applications of Integrals

Written by STEM Calculators Team Published Updated
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8. 8. Hydrostatic Force Tool

Computes hydrostatic force on a vertical submerged surface: \(F=\int P\,dA=\rho g\int_{a}^{b} h(y)\,w(y)\,dy\), where \(w(y)\) is plate width and \(h(y)\) is depth below the fluid surface.

Inputs
Variable: y. Constants: pi, e. Functions: sin cos tan ln log sqrt abs exp. Implicit multiplication allowed: 2y, (y+1)(y-1).
Pressure: \(P(y)=\rho g\,h(y)\).
Must be even.
Click a preset to load and calculate.
Ready
Pressure diagram
Drag to pan • wheel to zoom • arrows length ∝ \(P(y)=\rho g h(y)\).
plate pressure curve surface \(h\approx0\)
x: 0, y: 0, zoom: 60 px/unit
Results
Enter \(w(y)\), \(h(y)\), \([a,b]\), \(\rho\), and \(g\), then click Calculate.

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Frequently Asked Questions

What is the formula for hydrostatic force on a vertical plate?

For a vertical plate, pressure at depth h is P(y) = ρ g h(y). The total force is F = ∫ P dA = ρ g ∫[a,b] h(y) w(y) dy, where dA = w(y) dy.

How do I choose w(y) and h(y) for common plate shapes?

For a rectangle, w(y) is a constant width W. If the fluid surface is at y = H, a common depth function is h(y) = H - y (deeper points have larger h).

Why would I clamp h(y) < 0 to 0?

Negative depth means a point is above the fluid surface, so it should contribute no pressure and no force. Clamping prevents above-surface regions from producing unphysical negative pressure.

How many Simpson segments N should I use?

N must be even, and larger N usually increases accuracy for curved functions. Start with a moderate even value and increase it if the integral changes noticeably.

What units should I use to get force in newtons?

Use ρ in kg/m^3, g in m/s^2, and w(y) and h(y) in meters. With these units, the computed force is in newtons (N).

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