Article — Buoyancy Calculator (Archimedes Principle)
Buoyancy Calculator (Archimedes Principle)
Buoyant force equals the weight of fluid displaced: F_b = ρ × V × g. A 1 m³ object in fresh water gets 9810 N of lift. The same object in sea water (1025 kg/m³) gets 10,050 N. Whether it floats depends on whether F_b can balance the object's own weight.
What buoyancy is
Buoyancy is the upward push a fluid exerts on anything immersed in it. The push exists because fluid pressure increases with depth — the bottom of a submerged object sits in higher-pressure fluid than the top, and the difference is a net upward force. Archimedes proved around 250 BC that this net force exactly equals the weight of the fluid that the object pushes aside. Two and a half millennia later, no experiment has found a violation.
The same principle holds in liquids and gases. Air buoyancy is normally negligible because air is light, but for a hot-air balloon or helium balloon it dominates. In water it dominates for any object less dense than water — wood, ice, boats. In mercury, even cast iron floats.
The Greek king Hiero II reportedly asked Archimedes to detect whether a goldsmith had cheated on a crown. Archimedes used buoyancy: a pure-gold crown of known weight would displace exactly the same volume as a pure-gold lump of the same weight. If the crown displaced more, it was alloyed with silver. The story is the original recorded use of nondestructive density testing.
The buoyancy formula
F_b = ρ × V × g. Three inputs, one output. ρ is the density of the fluid in kg/m³. V is the displaced volume — for a fully submerged object, that equals the object's volume; for a partially submerged floater, it equals just the underwater portion. g is gravitational acceleration, 9.81 m/s² on Earth.
The equation is dimensionally clean: kg/m³ times m³ times m/s² gives kg·m/s² = N. There is no fitted constant, no correction term, no temperature factor. Real-world buoyancy calculations only get messy when ρ varies (warm water versus cold) or when V is hard to measure (irregular shapes need water-displacement measurement).
Will it float or sink?
The float-or-sink rule reduces to a density comparison. If the object's average density is less than the surrounding fluid, it floats. If greater, it sinks. If equal, it hovers — neutral buoyancy.
Cork (240) Floats — 76% aboveIce (917) Floats — 8% aboveWater (1000) NeutralConcrete (2400) SinksIron (7870) SinksGold (19300) Sinks fastFor floating objects, the submerged fraction equals the density ratio. Ice with 917 kg/m³ sits 91.7% underwater in fresh water — the visible part of an iceberg is the famous one-eighth, which is roughly correct (in sea water, ice is 10.5% above the surface).
Buoyancy in different fluids
Switch fluids and the buoyant force on the same object scales linearly with fluid density. Sea water gives 2.5% more lift than fresh water, which is why swimmers find ocean swimming slightly easier. The Dead Sea (1240 kg/m³) gives 24% more lift; people float so high there that drowning is nearly impossible by accident.
Mercury at 13,546 kg/m³ lifts iron blocks. A solid iron cube placed on mercury floats with about 42% of its volume above the surface, because iron (7870) is only 58% as dense as mercury. Demonstrations use this to startle students — heavy metal lifting heavier metal seems wrong until the density numbers settle the question.
Buoyancy in air versus water
Air buoyancy is real but usually small. Sea-level air has density 1.225 kg/m³, roughly 1/800 of water. A 1 m³ object in air gets about 12 N of lift — under a kilogram of equivalent weight. For people and most objects, that lift is buried in the noise of gravity, friction, and air drag.
When buoyancy in air does matter: precision weighing. A laboratory scale measuring sub-milligram differences must correct for air buoyancy because a mass of one density displaces more air than a mass of higher density. NIST publishes correction formulas; calibration of national standards uses them routinely.
Helium balloons rise because helium (0.166 kg/m³) is less dense than air. A 1 m³ helium balloon experiences 12 N of air buoyancy, but its own weight (helium + skin) is only about 1 N. Net upward force: roughly 11 N. That lifts a small payload — the basis of weather radiosondes and party balloon trains.
Apparent weight under water
Submerging an object reduces its felt weight by the buoyant force. A 10 kg iron block weighs 98.1 N in air. Its volume is 10 / 7870 = 1.27 × 10⁻³ m³. In water that volume gets ρVg = 1000 × 0.00127 × 9.81 = 12.5 N of buoyancy. Apparent weight in water: 98.1 − 12.5 = 85.6 N. The block still feels heavy and still sinks, but it's noticeably lighter under water.
Divers exploit this. Neutral-buoyancy training tanks let astronauts simulate space-walks: ballast brings the suit + diver to ρ_water, and net force is near zero. Movement in a neutral-buoyancy tank closely resembles weightlessness, which is why NASA built one.
Common buoyancy mistakes
The four classic errors.
The density in F_b = ρVg is the fluid's density, not the object's. Using the object's density gives an answer with the right units but the wrong physics. Always pull ρ from the surrounding medium.
Second, ignoring partial submersion. A floating boat displaces only the underwater volume, not its total hull volume. Treating a floater as fully submerged overestimates F_b by a factor of (total / submerged) volume.
Third, confusing buoyancy with weight. Buoyancy is the upward force; weight is the downward gravity force. They are independent — gravity does not change with submersion, only the apparent weight (net force) does.
Fourth, assuming buoyancy changes with depth in water. It doesn't. F_b depends on displaced volume and fluid density; both are constant for water at any reasonable depth.