Buoyancy

What is buoyancy?

Take a block of wood and a block of brass with exactly the same mass and drop both into a tank of water: the wood floats and the brass sinks.^[2] The reason is density. Brass is denser than water, while the wood is less dense.^[2] The force that settles the matter is buoyancy, the upward force on any object in any fluid.^[1]

In physics, liquids and gases both count as fluids, because they yield to shearing (sliding) forces, whereas solids resist them.^[2] So buoyancy acts in air as well as in water: helium-filled balloons tug upward on their strings because of air’s buoyant effect.^[1]

The buoyant force is always present, whether an object floats, sinks or stays suspended.^[1] Even objects that sink, like an anchor, are partly supported by the water while they are submerged.^[1]

Why fluids push up

Buoyancy starts from one fact: pressure increases with depth in a fluid.^{[1, 4]} Pressure is the force pressing at right angles on a surface, divided by the area it presses on.^[2] Its SI unit is the pascal (Pa), named after the French mathematician and physicist Blaise Pascal (1623–1662).^[2]

The increase can be huge. MIT’s course notes work out that the pressure 4 km down in Earth’s ocean is 40 × 10⁶ Pa greater than at the surface.^[3] That is 400 times 100 kPa, which the same notes equate with 0.987 atm (standard atmospheres), so roughly 395 atmospheres (our own calculation).^[3]

Now picture any object under water. Its bottom is deeper than its top, so the pressure on its bottom is higher than the pressure on its top.^{[1, 4]} The fluid therefore pushes up on the bottom harder than it pushes down on the top.^[1] The difference between the two is a net upward force: the buoyant force, or simply buoyancy.^{[1, 4]}

Archimedes’ principle

Archimedes’ principle says that the buoyant force on an object equals the weight of the fluid it displaces, meaning the fluid it pushes out of the way.^{[1, 4, 13]} It applies whether the object is wholly or only partly under the surface.^{[1, 13]}

There is a neat way to see why. Imagine the object removed, with the space it took up filled by fluid instead.^[1] That fluid would simply stay put, its weight supported by the fluid around it, so the surrounding fluid must push up on it with a force equal to its weight.^{[1, 3]} Put the object back and the surroundings push on it in just the same way, so the buoyant force equals the weight of the fluid displaced by the object.^[1]

It follows that the buoyant force depends on three things: the density of the fluid, the strength of gravity and the volume of fluid displaced.^[3]

Zoom in far enough and the buoyant force results from a very large number of collisions of the fluid’s molecules; MIT’s notes call Archimedes’ principle the large-scale description of that force.^[3] The rule is named after the Greek mathematician and inventor Archimedes (ca. 287–212 BCE), who stated it long before the idea of force was well established.^[1] His story, and the legend attached to it, are in the history section below.

Float, sink or hover

An object in a fluid has two forces to weigh against each other: its own weight, pulling down, and the buoyant force, pushing up.^[1]

• If the buoyant force is greater than the object’s weight, the net force is upward, and the object rises when released.^[1]
• If the buoyant force is less than the weight, the object sinks when released.^[1]
• If the two are equal, the object can stay at its present depth, either fully under water, like a neutrally buoyant diver, or partly above the surface, like a floating object.^[1]

What ultimately decides the contest is the object’s average density: its mass divided by its volume.^{[2, 1]} If an object’s average density is less than that of the fluid around it, it floats; an object denser than the fluid sinks.^[1] The reason is that a denser fluid holds more mass, and so more weight, in the same volume, so the fluid the object displaces outweighs the object.^[1] A stone in water is the opposite case: it is denser than water, so the buoyant force on it is less than its weight and it accelerates downward.^[3]

Plenty of things float on denser fluids: oil on water, a hot-air balloon in the atmosphere, a bit of cork in wine, an iceberg in salt water and the hot wax in a lava lamp.^[1] A less obvious example, according to the OpenStax textbook, is mountain ranges, which float on the higher-density crust and mantle beneath them.^[1]

How much sits below the surface?

How deep a floating object rides depends on how its density compares with the fluid’s.^[1] The fraction of it that is submerged equals the object’s density divided by the fluid’s density.^[1]

• An iceberg. OpenStax gives the density of ice at 0 °C as 9.17 × 10² kg/m³ and of sea water at 0 °C as 1.030 × 10³ kg/m³.^[2] Dividing one by the other, about 89% of a floating iceberg is under water and about 11% shows above it (our own calculation: 917 ÷ 1030 ≈ 0.89).^{[2, 1]}
• A foam block. MIT’s density table lists Styrofoam at 75 kg/m³ and water at 1.00 × 10³ kg/m³, so a block of it floats with only about 7.5% of its volume under water (our own calculation: 75 ÷ 1000).^{[3, 1]}
• A person. In an OpenStax worked example, a 60.0 kg woman floats in fresh water with 97.0% of her volume submerged when her lungs are full of air, which gives her an average density of 970 kg/m³.^[1]
• A cargo ship. An unloaded ship has a lower average density than the same ship loaded, so less of it is submerged.^[1]

Why a steel ship floats

A lump of clay dropped in water sinks, but mould the same lump into the shape of a boat and it floats.^[1] Because of its shape, the clay boat displaces more water than the lump did and feels a greater buoyant force, even though its mass is the same.^[1] The same is true of steel ships.^[1]

MIT’s course notes explain it this way: the dense steel of a ship displaces some water, but a much larger volume of water is displaced by the air inside the hull.^[3] The average density of the whole object is what ultimately determines whether it floats.^[1]

As a ship is lowered into the water, it displaces more and more water, until the weight of the displaced water equals the weight of the ship; at that point, it floats.^[13]

Salt water and fresh water

Salt water is heavier because of the salt it contains, so a ship has to displace a smaller volume of it to equal its own weight.^[13] MIT’s table gives sea water as 1.03 × 10³ kg/m³ and water as 1.00 × 10³ kg/m³, so a ship moving from fresh water into the sea displaces about 3% less volume and rides slightly higher (our own calculation: 1.00 ÷ 1.03 ≈ 0.97).^{[3, 1]}

People feel the same effect. A McGill University article says it is almost impossible to sink in the Dead Sea, because of what it calls an astounding salt content, which it gives as about 34%.^[13] No second source we checked gives this figure, so it is that article’s estimate.

Apparent weight: why things feel lighter underwater

Objects appear to weigh less when they are submerged, and physicists call this reduced reading the object’s apparent weight.^[1] The apparent loss of weight equals the weight of the fluid displaced.^[1]

Apparent weight is also a practical tool. Weigh a coin in air, then again while it is submerged in a liquid, and you can calculate the coin’s density if the liquid’s density is known; that density is an indication of whether the coin is genuine.^[1]

The math (optional)

A few short formulas hold everything above together. Here ρ (the Greek letter rho) stands for density, m for mass, V for volume and g for the acceleration due to gravity.

• Density is mass per unit volume, ρ = m/V, and its SI unit is kg/m³.^[2] The metric system was originally devised so that water would have a density of 1 g/cm³, which is the same as 10³ kg/m³.^[2]
• Pressure is force per unit area, p = F/A.^[2] In a fluid of constant density, the pressure at a depth h is the atmosphere’s pressure plus the pressure due to the weight of the fluid above: p = p₀ + ρhg.^[2] This equation only holds for a fluid of constant density.^[2]
• Weight of a piece of material: w = mg = ρVg.^[2]
• Archimedes’ principle: F_B = w_fl, the weight of the fluid displaced.^[1] Combining this with the weight formula gives F_B = ρ_flVg, where V is the volume of fluid displaced, which is why the buoyant force depends on the fluid’s density, gravity and that volume.^{[3, 2]}
• Fraction submerged for a floating object = ρ_obj ÷ ρ_fl.^[1]

MIT’s course notes give these densities, at 0.00 °C and 100 kPa unless another condition is noted:^[3]

Material	Density (kg/m3)
Helium	0.179
Air (at sea level)	1.20
Styrofoam	75
Ethanol	0.81 × 103
Ice	0.92 × 103
Water	1.00 × 103
Seawater	1.03 × 103
Blood	1.06 × 103
Aluminium	2.70 × 103
Iron	7.87 × 103

How much lighter underwater? (our own calculation.) A solid object that is fully submerged displaces its own volume of fluid, so the share of its weight that seems to vanish is ρ_fl ÷ ρ_obj.^{[1, 2]} With the table values, a solid block of iron seems about 13% lighter in water (1.00 ÷ 7.87 ≈ 0.13), and a block of aluminium about 37% lighter (1.00 ÷ 2.70 ≈ 0.37).^{[3, 1]}

A helium balloon’s payload (our own calculation). A classroom buoyancy problem asks how much payload can a balloon weighing 80 kg, with a capacity of 1200 m³, support when it is filled with helium?^[4] Using MIT’s densities for helium and for air at sea level:^[3]

1. The balloon displaces 1200 m³ of air, with a mass of 1.20 × 1200 = 1,440 kg; by Archimedes’ principle, the weight of that air is the buoyant force.^{[3, 1]}
2. The helium inside has a mass of 0.179 × 1200 ≈ 215 kg.^[3]
3. That leaves about 1,440 − 215 − 80 ≈ 1,145 kg for the payload.^{[4, 3]}

This is only an illustration: it ignores the volume of the fabric and of the payload, and real air density varies, since NIST computes it from measurements of temperature, pressure and relative humidity.^[6]

A puzzle: the rock in the bowl. Put a rock in a salad bowl floating in a beaker of water, then take the rock out and let it sink to the bottom. Will the water level go up or down?^[3] MIT’s answer is that the water level drops.^[3] Here is why, in our own words. While the rock rides in the bowl, the floating bowl must displace extra water weighing as much as the rock.^{[13, 1]} On the bottom, the rock displaces only its own volume of water, and because the rock is denser than water, that water weighs less than the rock.^[1] Less water displaced means a lower water level.

Buoyancy in air

Hot-air balloons and other lighter-than-air craft are called aerostatic machines: they rely on differences in air density for lift.^[4] A hot-air balloon rises because the warmer air inside it is less dense than the cooler air outside the balloon.^[4] Aircraft wings make lift in a completely different way, described on the Lift page.

MIT’s notes also point out that water vapour, lighter than air, can cause convection currents that form clouds.^[3]

Air is so much less dense than water that its buoyant force on an object floating in water can usually be neglected.^[3] But it cannot always be ignored, as the next section shows.

Buoyancy on the scales

When an object is weighed in air, the force it produces has two parts: one proportional to the object’s mass, and one proportional to its volume, which is the buoyant force.^[6] Under some circumstances, that second part is large enough to need correcting.^[6]

The US National Institute of Standards and Technology (NIST) warns in its standard procedure that, if uncorrected, air buoyancy is frequently the largest source of error in mass measurement.^[6] It says an air buoyancy correction should be made in all high-accuracy mass determinations, and that the corrections are calculated directly from Archimedes’ principle.^[6] The correction may matter even at modest accuracy when the object being weighed has a density very different from that of the standard weights, and NIST gives weighing water as an example.^[6]

Weighing water (our own calculation). NIST’s conventions use a reference density of 8.0 g/cm³, a reference temperature of 20 °C and a normal air density of 0.0012 g/cm³.^[6] Take 1 kg of water, at MIT’s table value of 1.00 × 10³ kg/m³ (1.00 g/cm³), and balance it against weights of density 8.0 g/cm³.^{[3, 6]} The water takes up 1,000 cm³, so the air buoys it up by the weight of about 1.2 g of air, while the much smaller weights, at about 125 cm³, get a lift of only about 0.15 g.^{[6, 1]} Uncorrected, the balance would make the water look about 1 g (roughly 0.1%) lighter than it is: about 998.95 g instead of 1,000 g.^{[6, 1]}

Neutral buoyancy: hovering underwater

Neutral buoyancy is the equal tendency of an object to sink or float.^[5] An item made neutrally buoyant through a combination of weights and flotation devices seems to hover under water.^[5]

NASA uses this to train astronauts. The mission of its Neutral Buoyancy Laboratory (NBL) is to prepare for space missions involving spacewalks, and the pool measures 202 ft in length, 102 ft in width and 40 ft in depth.^[5] Neutral buoyancy is the best method available for astronauts to train for spacewalks.^[5]

It is not the same as weightlessness, though, and there are two differences.^[5] First, suited astronauts in the NBL are neutrally buoyant but not truly weightless: they still feel their weight inside their suits.^[5] Second, water drag hinders motion, which makes some tasks easier and others harder to perform than in zero gravity.^[5]

Changing buoyancy on purpose

Some machines and animals change their buoyancy so they can go up or down.

• Submarines have adjustable density, using ballast tanks, so that they may float or sink as desired.^[1]
• Argo floats are autonomous robots that alternately dive and rise through the top 2,000 metres of the ocean, collecting climate observations as they go.^[7] A hydraulic system controls a float’s buoyancy by adjusting the amount of oil in an external bladder.^[7] To descend, the float withdraws oil from the bladder into the instrument, which increases its density and lets it sink lower and lower.^[7] Once the float reaches its deepest point, the bladder is inflated with oil, causing it to rise to the surface; on the way up, it measures temperature and salinity at different depths.^[7]
• Crewed submersibles. NOAA explains that human-occupied vehicles (HOVs) carry buoyancy packs that help keep the vehicle light, so it can easily ascend at the end of a dive.^[8]
• Fish. Many bony fishes have a swim bladder, a gas-filled organ that forms as a pouch from the gut and helps control the fish’s buoyancy.^[9]

Staying upright: ship stability

Floating is only half the job: a ship also has to stay upright. The ship’s weight acts straight down at its centre of gravity.^[10] The buoyant force acts straight up at the centre of buoyancy, which lies at the centroid (the geometric centre) of the ship’s underwater volume.^[10]

When no outside forces act, the two forces are lined up vertically, so they produce no turning effect.^[10] When an outside turning effect (a moment) heels the ship over, the underwater shape changes and the centre of buoyancy moves.^[10] Where it moves relative to the centre of gravity defines how stable the ship is as it heels.^[10] The US Naval Academy’s notes call the sideways distance between the lines of the weight and the buoyant force the righting arm (GZ).^[10] If the ship’s internal moment balances the external one, the ship stays heeled at that angle; otherwise, it keeps heeling until it capsizes.^[10]

Common misconceptions

• “Heavy things sink and light things float.” Density matters, not weight: a block of wood and a block of brass with exactly the same mass behave differently, the wood floating and the brass sinking.^[2] Shape matters too, since a lump of clay sinks while the same clay made into a boat floats.^[1]
• “Things that sink feel no buoyant force.” The buoyant force is always present, whether an object floats, sinks or is suspended.^[1] Even a sinking anchor is partly supported by the water.^[1]
• “The deeper something goes, the stronger the buoyant force.” Pressure rises with depth, but the buoyant force is the weight of the fluid displaced, so it depends on the fluid’s density, gravity and the displaced volume, not on depth as such.^{[1, 3]} A neutrally buoyant diver, for instance, can remain suspended at their present depth.^[1]
• “Buoyancy only happens in water.” Helium-filled balloons tugging on their strings show air’s buoyant effect.^[1] If it isn’t corrected, air buoyancy is frequently the largest source of error in precise mass measurement.^[6]

A short history

Archimedes was a native of Syracuse, in Sicily.^[12] OpenStax gives his dates as about 287–212 BCE.^[1] He was killed in 212 BC, during the capture of Syracuse by the Romans in the Second Punic War; the Getty Museum says it was a Roman soldier who killed him.^{[12, 11]}

His treatise On Floating Bodies lays down the basic principles of hydrostatics, the physics of fluids at rest, and contains his most famous theorem, which gives the weight of a body immersed in a liquid and is now called Archimedes’ principle.^[12] In it he also studied the stability of floating bodies of different shapes and specific gravities (relative densities).^[12]

The crown and the bath. A famous story says that Hieron asked Archimedes to find out whether a crown was solid gold.^[11] In the Getty Museum’s telling, this led to his legendary discovery that a solid displaces a volume of liquid equal to its own volume, which supposedly made him leap from his bath and run naked through the streets crying “Eureka!” (I have found it).^[11] How far we can rely on this is questionable: a McGill University article notes that we first hear of the story from the Roman architect and engineer Vitruvius, in the 1st century BC, some two hundred years after the moment supposedly happened.^[13] The same article adds that, however fanciful the tale, there is no doubt that Archimedes really did formulate the principle of buoyancy, which explains, among other things, why ships float.^[13]

How the text survived. A book known as the Archimedes Palimpsest contains On Floating Bodies, which expands on the famous observation about a solid submerged in a liquid.^[11] The same book is the only surviving source for two works by Archimedes, now fully legible thanks to imaging technology.^[11] Latin and Arabic translations of Archimedes also survive, but the Getty Museum notes that the palimpsest’s Greek version brings us closer to his original words.^[11]

Buoyancy at work today. According to NOAA, until the first Argo floats were deployed in 2000, less than 1 percent of the ocean below the upper few hundred metres was being monitored routinely.^[7]

Try it yourself

• The clay boat. Drop a lump of modelling clay into a bowl of water and it sinks; shape the same clay into a boat and it floats, because it now displaces more water.^[1]
• Weigh something underwater. Hang a small, heavy object from a luggage scale, then lower it into a bucket of water: the reading drops, because the object’s apparent weight is less by the weight of the water it displaces.^[1]
• An ice cube. Float an ice cube in a glass of water and see how little of it sticks out: with MIT’s table values for ice and water, about 92% of it should be under the surface (our own calculation: 0.92 ÷ 1.00).^{[3, 1]}
• The rock and the bowl. Float a small bowl in a basin with a pebble in it, mark the water level on the side, then drop the pebble into the water; MIT’s worked answer to this puzzle predicts that the level will fall.^[3]

Going further

Read Fluid Mechanics for the bigger picture of pressure and flow, and Bernoulli’s Principle for what happens when fluids move. Lift and Drag explain how moving air holds aircraft up and slows them down, and Gravity covers the weight that buoyancy works against. The free OpenStax University Physics textbook and MIT’s Classical Mechanics course listed below go deeper, with worked problems.