Buoyancy

Archimedes' principle

The buoyant force on an object immersed in a fluid equals the weight of the fluid displaced by the object:

F_buoyant = ρ_fluid · V_submerged · g

where ρ_fluid is the fluid's density, V_submerged is the volume of fluid displaced (= submerged volume of object), g is gravitational acceleration.

The buoyant force is always UPWARD, opposing gravity.

Floating, sinking, neutral

For an object placed in a fluid, three outcomes:

Object density vs fluid	Behavior	Example
ρ_object < ρ_fluid	Floats — partially submerged	Wood in water; helium balloon in air; ship
ρ_object = ρ_fluid	Neutral — stays where placed	Fish with adjusted swim bladder
ρ_object > ρ_fluid	Sinks — falls through fluid	Rock in water; lead in air

For floating objects, the fraction submerged equals ρ_object / ρ_fluid:

fraction_submerged = ρ_object / ρ_fluid

Iceberg in water (ρ_ice = 0.92, ρ_seawater = 1.025): 0.92/1.025 = 0.90 — about 90% underwater, 10% above. The "tip of the iceberg."

Common densities (kg/m³)

Substance	Density (kg/m³)
Air (sea level, 20°C)	1.20
Helium	0.18
Hydrogen	0.09
Cork	240
Wood (oak)	700-900
Ice	917
Liquid water	1,000
Seawater	1,025
Aluminum	2,700
Steel	7,850
Lead	11,340
Mercury	13,534
Gold	19,300
Osmium (densest natural)	22,590

Real-world applications

System	How it uses buoyancy
Ships	Hollow hull → low average density → floats
Hot-air balloons	Heat air inside, reducing density < surrounding air
Submarines	Variable ballast tanks adjust average density
Fish swim bladders	Gas-filled organ for neutral buoyancy
Hydrometers	Floating cylinder; depth indicates fluid density
Lava lamp	Wax expands/cools changing density relative to oil
SCUBA buoyancy compensators	Inflatable vest adjusts diver buoyancy
Hot air vs cold	Convection currents driven by density differences

JavaScript — buoyancy calculations

// Buoyant force
function buoyantForce(fluidDensity, submergedVolume, g = 9.81) {
  return fluidDensity * submergedVolume * g;
}

// Object weight in fluid (apparent weight)
function apparentWeight(mass, objectDensity, fluidDensity, g = 9.81) {
  const volume = mass / objectDensity;
  const F_buoy = buoyantForce(fluidDensity, volume, g);
  return mass * g - F_buoy;
}

// Floating: how much of an object is submerged?
function fractionSubmerged(objectDensity, fluidDensity) {
  return objectDensity / fluidDensity;
}

// Iceberg
console.log(`Iceberg in seawater: ${(fractionSubmerged(917, 1025) * 100).toFixed(1)}% submerged`);  // 89.5%

// Ship floats: hollow hull average density
function shipFloats(massShip, hullVolume, fluidDensity = 1025) {
  // Average density of ship-as-volume = mass / hull volume
  const avgDensity = massShip / hullVolume;
  return avgDensity < fluidDensity;
}

// 50,000-ton ship with 100,000 m³ hull
console.log(`50000-ton ship floats? ${shipFloats(50e6, 1e5)}`); // density 500 < 1025, yes

// Helium balloon lift
function balloonLift(volume_m3, gasMass, skinMass, gasDensity, airDensity = 1.20) {
  // Net upward force = (air_displaced_weight) - (gas_weight + skin_weight)
  const buoy = airDensity * volume_m3 * 9.81;
  const balloon_weight = (gasMass + skinMass) * 9.81;
  return buoy - balloon_weight;
}

// 1 m³ helium (0.18 kg/m³ → 0.18 kg of He, ~0.05 kg latex skin)
const lift = balloonLift(1, 0.18, 0.05, 0.18);
console.log(`1 m³ helium balloon lift: ${lift.toFixed(2)} N (~ ${(lift/9.81).toFixed(2)} kg)`);
// ~10 N = ~1 kg lift

// Hot-air balloon: heat air inside to reduce its density
function hotAirBuoyancy(volume_m3, T_inside_K, T_outside_K, P_atm = 101325) {
  // Ideal gas: density ∝ 1/T at constant P
  const M_air = 0.029;  // kg/mol
  const R = 8.314;
  const rho_inside = P_atm * M_air / (R * T_inside_K);
  const rho_outside = P_atm * M_air / (R * T_outside_K);
  const lift = (rho_outside - rho_inside) * volume_m3 * 9.81;
  return { rho_inside, rho_outside, lift };
}

// 2000 m³ balloon, hot air at 100°C, outside 20°C
console.log(hotAirBuoyancy(2000, 373, 293));
// Outside ρ ~ 1.20, inside ρ ~ 0.95 → density diff 0.25 × 2000 = 500 kg
// Lift = 500 × 9.81 ≈ 4900 N = 500 kg lift

Where buoyancy shows up

• Naval engineering. Ship design, dock-loading, displacement calculations, stability analysis.
• Aviation. Hot-air balloons, helium dirigibles, weather balloons.
• Submarine/diving. Submarine ballast control, SCUBA buoyancy compensators, surface vs depth control.
• Geology. Continental drift (continents float on denser mantle); isostatic equilibrium.
• Atmosphere. Convection (warm air rises due to lower density), thermal updrafts used by birds and gliders.
• Astronomy. Density layering in stars; lighter elements rise, heavier sink (stratified composition).
• Engineering. Hot-water heating systems use natural convection; chimneys exploit buoyancy of hot smoke.

Common mistakes

• Confusing displaced weight with object weight. Buoyancy = weight of FLUID displaced, not weight of OBJECT. Floating object — buoyancy = object weight (forces balance). Submerged object — buoyancy = displaced fluid weight (might be less than object weight).
• Forgetting buoyancy depends on submerged volume, not total volume. Partially submerged object only displaces what's underwater. Floating ship displaces water volume up to the waterline, not full hull.
• Using density of object instead of fluid. Buoyant force depends on fluid's density. Same object in different fluids → different buoyancy.
• Ignoring atmospheric buoyancy. Air density (1.2 kg/m³) is small but nonzero. For very precise weighing, "weight in air" is slightly less than "true weight" by air buoyancy. Standard physics ignores it.
• Confusing apparent weight with real weight. Apparent (in fluid) = real - buoyancy. Submerged objects can feel weightless even though their real weight is unchanged.
• Treating non-uniform fluid as uniform. Compressible fluids (air at altitude, ocean depths) have variable density. Buoyant force at one depth differs from another.