Buoyancy Calculator: Buoyant Force, Float or Sink

Buoyancy is the upward push a fluid exerts on anything placed in it, and Archimedes worked out the rule more than two thousand years ago: the force equals the weight of the fluid the object shoves out of the way.

Buoyant force F_b = fluid density × gravity × displaced volume

Where ρ_fluid is in kg/m³, g is 9.81 m/s², and V is the volume of fluid displaced, in m³. Notice what is missing. Nothing about the object’s own material appears anywhere in that formula, only how much fluid it pushes aside.

That is why a steel ship floats while a steel bolt sinks. The ship is shaped to displace an enormous volume of water before it is fully under, so the buoyant force grows large enough to match its weight. The bolt displaces only its own small volume, and the water can never push up hard enough.

Worked example: a wooden block

A block of wood at 600 kg/m³ measuring 0.2 m × 0.3 m × 0.1 m has a volume of 0.006 m³.

Its weight is 600 × 9.81 × 0.006 = 35.32 N. Fully submerged in fresh water it would feel 1,000 × 9.81 × 0.006 = 58.86 N pushing up. Buoyancy wins, so it floats, and it settles at the depth where the two match:

Fraction submerged = ρ_object ÷ ρ_fluid = 600 ÷ 1,000 = 0.6, so 60% of the block sits below the waterline.

There is a shortcut hiding in that last line. Divide mass by volume to get the object’s average density, then compare it with the fluid. Under 1,000 kg/m³ and it floats in fresh water; over it and it sinks. No force arithmetic needed. The force version is what you want when you need the actual number, for sizing a lift bag or a mooring float.

Common fluid densities

Seawater is only about 2.5 percent denser than fresh water, and that small edge is the whole reason you float a little higher in the ocean than in a lake. The Dead Sea takes it much further, which is why the photographs of people reading newspapers out there are not staged.

Buoyancy works in gases too

A 0.01 m³ helium balloon in sea-level air at 20°C gets 1.204 × 0.01 × 9.81 = 0.118 N of lift. Subtract the weight of the helium itself, 0.1786 kg/m³ × 0.01 m³ × 9.81 = 0.018 N, and about 0.100 N of net upward force is left. That works out to roughly 10 grams of payload per 10 litres of helium, which is why party balloons lift a ribbon and not much else.

Air density falls with both temperature and altitude, though. At 2,000 m it is nearer 1.0 kg/m³, so a balloon that only just lifts at sea level will not lift at all in Denver.

Where people get this wrong

Use the object’s average density, counting any trapped air, not the density of the material it is made from. Steel is 7,850 kg/m³ as a material and perhaps 200 kg/m³ as a whole ship. Get that distinction wrong and every hull you ever calculate will sink.

The other common slip is entering the total volume when only part of the object is under. For a fully submerged object the two are the same, but a floating hull displaces only what is below the waterline.

Submarines sit in the middle of all this on purpose. Their ballast tanks change the boat’s average density by a percent or two, which is enough to move between floating, hovering, and sinking without changing the hull at all.