Buoyancy Calculator — Calculate Buoyant Force Using Archimedes's Principle

Archimedes's Principle

Archimedes's principle states: Any object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.

The buoyant force formula:

F_b = ρ_fluid × V_displaced × g

Where F_b is the buoyant force (N), ρ_fluid is the density of the fluid (kg/m³), V_displaced is the volume of fluid displaced by the submerged object (m³), and g is gravitational acceleration (9.81 m/s² on Earth).

The principle is attributed to Archimedes of Syracuse (287–212 BC), who reportedly discovered it while stepping into a bath and observing the water level rise. The insight — that the displaced water volume equals the submerged object volume — led to a method for determining the volume and density of irregularly shaped objects.

How to Use the Buoyancy Calculator

1. Open the buoyancy calculator.
2. Enter the fluid density (kg/m³). Select a preset (water, seawater, mercury) or enter a custom value.
3. Enter the volume of the object (m³) — or the submerged volume if only partly submerged.
4. Enter the object mass (kg) or density (kg/m³) to determine whether it floats or sinks.
5. Click Calculate. The tool returns the buoyant force (N), the weight of fluid displaced, and whether the object floats or sinks.

Common Fluid and Material Densities

Float, Sink, or Neutral? The Density Rule

The simplest rule for determining whether an object floats or sinks:

• If object density < fluid density: the object floats (partially submerged)
• If object density = fluid density: the object is neutrally buoyant (can remain at any depth)
• If object density > fluid density: the object sinks

This rule applies to solid objects of uniform density. Hollow objects (like a ship or a submarine) have an effective density lower than the material they are made of, because they include air. The effective density is total mass divided by total volume (including hollow spaces).

Real-World Buoyancy Examples

Fraction Submerged for Floating Objects

For a floating object, only part of it is submerged — the submerged fraction equals the ratio of object density to fluid density:

Fraction submerged = ρ_object / ρ_fluid

For ice (917 kg/m³) in water (1,000 kg/m³): fraction submerged = 917/1,000 = 0.917 = 91.7%. This is why 90% of an iceberg is below the waterline — the density ratio of ice to water directly gives the submerged fraction.

For a ship in seawater: if the ship has an effective density of 500 kg/m³ (half of seawater's 1,025 kg/m³), it floats with approximately 49% submerged. Adding cargo increases effective density and causes the ship to sit lower in the water.

Ships: The Hollow Hull Principle

Steel has a density of ~7,850 kg/m³ — much greater than seawater at 1,025 kg/m³. Yet steel ships float because the hull is hollow and filled with air. The critical factor is not the density of the steel, but the average density of the entire ship (steel + air + contents divided by total volume including the hollow space).

The Plimsoll line (load line) painted on the hull of cargo ships marks the maximum safe depth of submersion under different conditions (freshwater vs. seawater, tropical vs. North Atlantic). Loading beyond the Plimsoll line increases effective density beyond safe limits for the conditions.

Submarines and Neutral Buoyancy

A submarine adjusts its buoyancy by filling or emptying ballast tanks with seawater. With tanks empty (filled with air), the submarine floats. As tanks fill with water, effective density increases and the submarine descends. When tanks are filled to achieve an effective density matching seawater, the submarine is neutrally buoyant — able to maintain depth without constant engine power.

To surface, compressed air is blown into the ballast tanks, expelling water and reducing effective density below that of seawater. This is a direct practical application of Archimedes's principle controlled by adjusting the displaced volume of water.

Buoyancy in Aerostatics (Balloons)

The same principle applies to objects in air. A hot-air balloon floats because heated air inside the balloon is less dense than the surrounding cooler air. The buoyant force is the weight of the air displaced by the balloon's volume. For a balloon to lift off, the buoyant force must exceed the weight of the balloon, basket, passengers, and fuel.

Helium balloons float because helium (density 0.164 kg/m³) is much less dense than air (1.2 kg/m³). The buoyant force per unit volume = (1.2 − 0.164) × 9.81 ≈ 10.2 N/m³. A 1 m³ helium balloon generates about 10.2 N of lift — sufficient to lift about 1 kg.

Apparent Weight in Fluid

An object submerged in fluid appears lighter than it is in air. The apparent weight is:

Apparent weight = Actual weight − Buoyant force = mg − ρ_fluid × V × g

This is why objects feel lighter in water — your body, for example, displaces about 70 L of water (70 kg equivalent), reducing your apparent weight by approximately 700 N. An 80 kg person weighs about 784 N in air but has an apparent weight of only ~84 N when fully submerged in water (if density is ~1,143 kg/m³) or even less if closer to water density.

Apparent weight measurement (weighing in and out of water) was Archimedes's original method for determining the density of irregularly shaped objects — the same technique used in hydrostatic weighing for body fat percentage measurement.

Common Questions

Why does salt water feel more buoyant than fresh water?

Seawater (density ~1,025 kg/m³) is denser than fresh water (1,000 kg/m³) because of dissolved salts. A denser fluid exerts a greater buoyant force for the same displaced volume: F_b = ρ_fluid × V × g. The Dead Sea (density ~1,240 kg/m³) is so dense that humans float effortlessly without any swimming motion — the high salt concentration makes it impossible to sink.

Does buoyancy depend on the depth of submersion?

For incompressible fluids (liquids), buoyancy depends only on the volume displaced and the fluid density — not on the depth. A 1 m³ object submerged 1 m or 100 m experiences the same buoyant force. For gases (compressible fluids), density varies with pressure, so buoyancy changes with depth — this is why deep-sea fish with gas bladders can have problems if brought to the surface too quickly.

Can something be buoyant in a gas?

Yes — anything less dense than the surrounding gas is buoyant in that gas. Hydrogen and helium balloons are buoyant in air. Hot-air balloons use heated, lower-density air. The International Space Station is "buoyant" in an extremely thin upper atmosphere — though at those altitudes, orbital mechanics (centripetal acceleration) rather than buoyancy determines its trajectory.