Hydrostatic Forces, Buoyancy & Floatation Notes

Last reviewed 16 Sept 2026 · 6 min read

Total pressure on plane surfaces

Total pressure $F$ is the resultant force of fluid pressure on a surface; the centre of pressure is the point where it acts. Because pressure increases with depth, the centre of pressure lies below the centroid of a submerged vertical or inclined surface.

FormulaPlane surface (any inclination)

F = ρ g A \overset{ˉ}{h}

$\overset{ˉ}{h}$ = vertical depth of the centroid of the area below the free surface.

Depth of centre of pressure:

h^{*} = \overset{ˉ}{h} + \frac{I _{G} sin ^{2} θ}{A h ˉ}

$I_{G}$ = second moment of area about the centroidal axis parallel to the free surface; $θ$ = inclination of the surface to the horizontal ( $θ$ = 90° for vertical: $h^{*} = \overset{ˉ}{h} + I_{G} / (A \overset{ˉ}{h})$ ).

Horizontal surface: pressure uniform; $F = ρ g A h$ acting at the centroid.
The centre of pressure approaches the centroid as depth increases.
For a vertical rectangle with top edge at the surface: $h^{*} = \frac{2}{3} d$ . For a vertical triangle with base at the surface (apex down): $h^{*} = d /2$ ; apex at the surface (base down): $h^{*} = \frac{3}{4} d$ .

Pressure diagram method

For rectangular surfaces, the total force equals the volume of the pressure prism (area of the pressure diagram × width) acting through its centroid — convenient for gates and dam faces, and when water acts on both sides.

Curved surfaces

The force on a curved surface is found from components:

FormulaCurved surfaces

Horizontal component $F_{H}$ = force on the vertical projection of the curved surface: $F_{H} = ρ g A_{v} \overset{ˉ}{h}_{v}$ , acting at the centre of pressure of that projection.
Vertical component $F_{V}$ = weight of the liquid vertically above the curved surface up to the free surface (real or imaginary), acting through the centroid of that volume.
Resultant $F = F_{H}^{2} + F_{V}^{2}$ , inclined at $tan^{- 1} (F_{V} / F_{H})$ to the horizontal; for a circular surface it passes through the centre.

Applications: radial (Tainter) gates, curved dam faces, pipe bends, domes and spherical tanks.

Buoyancy

Archimedes' principle: a body wholly or partly immersed in a fluid experiences an upward buoyant force equal to the weight of fluid displaced, acting through the centre of buoyancy (centroid of the displaced volume).

A body floats when its weight equals the buoyant force of the immersed part.
Fraction submerged of a floating body = (specific gravity of body)/(specific gravity of liquid).

Metacentre and stability

When a floating body tilts slightly, the centre of buoyancy shifts. The metacentre M is the point where the vertical through the new centre of buoyancy meets the original vertical axis.

FormulaMetacentric height

B M = \frac{I}{V}, GM = B M - B G = \frac{I}{V} - B G

$I$ = second moment of the waterline plane area about the axis of tilting (the longitudinal axis for rolling — use the smaller I); $V$ = volume displaced; $B$ = centre of buoyancy; $G$ = centre of gravity ( $B G$ positive when G is above B).

Condition	Floating body	Submerged body
Stable	M above G ( $GM > 0$ )	B above G
Neutral	M coincides with G	B coincides with G
Unstable	M below G ( $GM < 0$ )	B below G

Typical metacentric heights: merchant ships about 0.3–1.2 m; warships higher. A large GM means a stiff ship with quick, uncomfortable rolling; small GM gives slow comfortable rolling but less reserve against capsizing.

Experimental metacentric height

Move a known weight $w$ across the deck by distance $x$ and measure the angle of heel $θ$ :

GM = \frac{w x}{W tan θ}

( $W$ = total weight including $w$ .)

Period of rolling

T = 2 π \frac{k ^{2}}{g GM}

$k$ = radius of gyration about the rolling axis.

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Hydrostatic Forces, Buoyancy & Floatation