Buoyancy Force

Overview

Buoyancy force is the upward force exerted on an object immersed in a fluid, equal to the weight of fluid displaced by the object (Archimedes' principle).

Variables

Symbol	Description	Unit
$F_b$	Buoyancy force	N
$\rho_f$	Fluid density	kg/m³
$V$	Displaced volume	m³
$g$	Gravitational acceleration	m/s²

Formula

$F_b = \rho_f \cdot V \cdot g$

Calculator

Notes

• Results are approximate and should be verified for critical applications
• Input values should be within reasonable engineering ranges

Archimedes' Principle

Archimedes' principle states that the upward buoyant force exerted on a body fully or partially submerged in a fluid equals the weight of the fluid that the body displaces.

This leads to two important conditions:

• If the body weighs more than the fluid → it sinks
• If the body weighs less than the fluid → it floats

Worked Example: Buoyant Force on a Floating Box

A plastic box with length 0.3 m and width 0.4 m is submerged 0.1 m into water ($\rho = 1000 \text{ kg/m}^3$).

The submerged volume is:

$V = (0.3 \text{ m})(0.4 \text{ m})(0.1 \text{ m}) = 0.012 \text{ m}^3$

Applying the buoyant force formula:

$F = V \rho g = (0.012 \text{ m}^3)(1000 \text{ kg/m}^3)(9.81 \text{ m/s}^2) = 119 \text{ N}$

This 119 N upward force supports the box at the water surface, balancing the weight of the displaced water column.

Physical Mechanism of Buoyancy

The upward buoyant force is not a fundamental force but the result of a pressure differential within the fluid. Fluid pressure increases with depth. For a submerged object, the pressure on its lower surface is greater than the pressure on its upper surface. This net pressure difference creates an upward force, which is the buoyant force.

The buoyant force acts vertically upward through the center of buoyancy, which is the centroid of the volume of fluid displaced by the object. This is the point where the force can be considered to act for stability analysis.

Common Fluid Densities

Equilibrium Conditions for Floating Bodies

For a body floating at equilibrium, the buoyant force equals the object's weight:

$F_b = W_{body}$

This leads to the draft equation — the submerged depth $h$ of a floating prism with uniform cross-section area $A$:

$h = \frac{m_{body}}{\rho_{fluid} \cdot A}$

The fraction submerged for a floating object is determined by the density ratio:

$\frac{V_{submerged}}{V_{total}} = \frac{\rho_{body}}{\rho_{fluid}}$

Design Criteria Summary

Condition	Mathematical Criteria	Result
$\rho_{body} < \rho_{fluid}$	$W_{body} < F_b$ (when fully submerged)	Floats
$\rho_{body} = \rho_{fluid}$	$W_{body} = F_b$	Neutral buoyancy
$\rho_{body} > \rho_{fluid}$	$W_{body} > F_b$	Sinks

Metacentric Height (Stability)

For floating vessels, stability is governed by the metacentric height $GM$:

$GM = BM - BG$

where:

• $BM = \frac{I}{V_{submerged}}$ = distance between center of buoyancy and metacenter
• $I$ = second moment of waterplane area (m⁴)
• $BG$ = distance between center of buoyancy (B) and center of gravity (G)

Stability criterion:

• $GM > 0$: Stable (self-righting)
• $GM < 0$: Unstable (capsizes)
• $GM = 0$: Neutral stability

Buoyancy Applications

Application	Key Principle
Ship design	Hull volume displaces enough water to support cargo weight
Submarines	Ballast tanks control average density relative to seawater
Hydrometers	Instrument floats at depth proportional to liquid density
Hot air balloons	Heated air has lower density than surrounding atmosphere
Offshore platforms	Large displacement volume provides upward buoyant force

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References

• Original Source