Object

Object material

Typical densities at room temperature (ice at 0 °C). Sets ρ₀.

Volume

Total volume of the object

01000

Object height

Height of the vertical prism; sets the draft and freeboard

0100

Floats

ρ₀ < ρ_f: buoyancy equals weight and the object floats partially submerged.

Results

Buoyant force E

4.903

Weight W

4.903

m = 0.5 kg

Net force

In equilibrium (E = W)

Submerged fraction

50% above the surface

Submerged volume

0.5 L

Draft T

10 cm

Freeboard: 10 cm

Max extra load

0.5 kg

ΔW = 4.903 N

Hydrostatics (Archimedes' Principle)

Fluid

Densities assumed at 20 °C and 1 atm. The fluid is at rest (hydrostatics).

Fundamentals & Explanation

The origin of buoyancy: pressure grows with depth

In a fluid at rest, hydrostatic pressure increases linearly with depth, P = P₀ + ρ_f·g·h. For a submerged body, the bottom face sits deeper than the top face and receives more pressure: the resultant of all pressure forces is a net vertical, upwardforce. That is Archimedes' buoyant force, and its magnitude is exactly the weight of the displaced fluid:

E = \rho_f \, g \, V_{\text{displaced}}

Buoyancy does not depend on the body's material or weight: only on the volume of fluid it displaces and the fluid's density. It acts at the center of buoyancy (the centroid of the submerged volume), a detail that becomes crucial when studying ship stability.

The flotation condition

Comparing the weight W = ρ_o·V·g with the maximum buoyancy E_max = ρ_f·V·g reduces to comparing densities: if ρ_o < ρ_f the body floats; if ρ_o > ρ_f it sinks; if they are equal it rests in neutral equilibrium at any depth. When it floats, the body sinks just enough for the displaced fluid to weigh as much as the body itself:

f = \frac{V_{\text{submerged}}}{V} = \frac{\rho_o}{\rho_f}

The iceberg is the canonical example: with 917 kg/m³ ice in 1025 kg/m³ seawater, f = 0.895 — nearly 90% of the volume sits underwater. The difference E_max − W is the reserve of buoyancy: the maximum extra load a floating body can take before submerging completely, Δm_max = (ρ_f − ρ_o)·V.

Apparent weight and densimetry: Archimedes' crown

A body denser than the fluid, hung from a scale inside it, registers an apparent weight W_app = W − Esmaller than its true weight. That difference can be measured precisely, and therein lies the method tradition attributes to Archimedes for unmasking King Hiero's crown: weighing in air and weighing submerged determines the density without damaging the piece,

\rho_o = \rho_f \, \frac{W_{\text{air}}}{W_{\text{air}} - W_{\text{app}}}

The same principle drives the hydrometer: a ballasted float sinks deeper in less dense liquids, and the graduated scale on its stem reads density directly. If the body is released while submerged, the net force W − E produces an initial acceleration a = g(1 − ρ_f/ρ_o), downward if it sinks and upward if it floats.

Buoyancy in gases: balloons

The principle makes no distinction between liquids and gases: a helium balloon floats in air for the same reason a cork floats on water. Because the density of air (1.225 kg/m³ at sea level) is tiny, large volumes are needed to generate useful lift. The load capacity of a balloon of volume V, gas density ρ_g and envelope mass m_env is:

L = (\rho_{\text{air}} - \rho_g)\, V g - m_{\text{env}}\, g

One cubic meter of helium at sea level lifts just over 1 kg; that is why weather balloons and airships are so voluminous. The hot-air balloon works the same way: heating the inside air lowers its density ρ_g below that of the ambient air.

Scope and limits of the model

Fluid at rest:this is hydrostatics. Once the fluid moves, dynamic forces appear — the bridge to Bernoulli's principle.
Rigid, incompressible body: the volume does not change with depth (a real gas balloon does compress).
Uniform fluid density: values at 20 °C and 1 atm; in gases the density drops with altitude.
Final equilibrium: the computed initial acceleration ignores hydrodynamic drag, which slows the real motion.

Quantities (SI)

Symbol	Quantity	Unit
E	Buoyant force	N
ρ_o, ρ_f	Object / fluid density	kg/m³
V, V_sub	Total / submerged volume	m³
T	Draft (submerged depth)	m
g	Standard gravity (9.80665)	m/s²

References

Çengel, Y. & Cimbala, J. Fluid Mechanics: Fundamentals and Applications (McGraw-Hill), ch. 3 “Pressure and Fluid Statics”.
White, F. M. Fluid Mechanics (McGraw-Hill), ch. 2.
Munson, Young & Okiishi. Fundamentals of Fluid Mechanics (Wiley), ch. 2.

Archimedes' Principle & Buoyancy

Object

Results

Hydrostatics (Archimedes' Principle)

Fundamentals & Explanation

The origin of buoyancy: pressure grows with depth

The flotation condition

Apparent weight and densimetry: Archimedes' crown

Buoyancy in gases: balloons

Scope and limits of the model

Quantities (SI)

References

The origin of buoyancy: pressure grows with depth

The flotation condition

Apparent weight and densimetry: Archimedes' crown

Buoyancy in gases: balloons

Scope and limits of the model

Quantities (SI)

References