What are density and buoyancy?

Pick up a cork and an iron block of the same size. The iron is about 33 times heavier. That difference is density: how much mass is packed into each bit of space.

ρ = m ÷ V

ρ (the Greek letter rho) is density, m is mass and V is volume. The SI unit is kg/m³; in the kitchen we use g/cm³. Water is 1 g/cm³, which is 1,000 kg/m³: one litre has a mass of one kilogram. To measure a density, weigh the object, find its volume (with a ruler, or by how much water it pushes aside) and divide.

Now lower a block into water. The water it pushes aside wants its place back, so it pushes up. Archimedes worked out how hard, over 2,200 years ago: the buoyant force equals the weight of the fluid pushed aside.

F = ρ_fluid × V_submerged × g

The upward push comes from pressure: water pushes harder on the bottom of the block, which is deeper, than on its top (see PressureClear). So:

If the object is less dense than the fluid, it sinks only until it has pushed aside its own weight of fluid, then it floats. The fraction under the surface is ρ_object ÷ ρ_fluid. Ice is 0.917 g/cm³, so about 92% of an ice cube, or of an iceberg in fresh water, hides below the surface.

If it is denser, even a fully sunk block can't push aside enough, so it sinks. It still feels lighter: the spring balance reads less underwater. Weight itself is gravity's pull (see GravityClear), and Newton's laws say the block settles where the forces balance (NewtonClear).

Steel is almost 8 times as dense as water, so a steel bolt sinks. A steel ship floats because of its shape. The hull is a huge hollow bowl: it pushes aside a lot of water while the steel itself takes up very little space. What matters is the average density of the whole ship, steel plus the air inside, and that is well below 1 g/cm³.

As you load cargo, the ship sinks deeper until the water it pushes aside weighs as much as ship plus cargo. That is its displacement. How deep the bottom sits is the draft. Load too much and there is too little hull left above the water, the freeboard, to ride out waves.

In 1876 a British MP, Samuel Plimsoll, got a law passed that made every ship carry a load line mark: a circle with a line through it. Next to it, short lines show how deep it may go in tropical, summer and winter seas, and in fresh water. Sea water (about 1.025 g/cm³) holds a ship up a little more than fresh water (1.000), so the same ship floats about 2.5% deeper in a river port like Kolkata than out at sea.

You can try it with clay. A 50 g lump sinks. Press the same clay into a bowl and it floats, and can even carry coins. A folded lotus leaf works even better: it is light, and its waxy skin keeps water out.

Archimedes is said to have used the same idea to test a king's crown. Gold is 19.3 g/cm³; silver is 10.5. A crown with silver mixed in is bigger than the same mass of pure gold, so it pushes aside more water.

Air is a fluid too, so Archimedes works in air. Everything around you is pushed up by the weight of the air it displaces. For you that is only about 0.08 kg worth, so you don't notice. For a balloon it is everything.

Heat air and its molecules spread out, so the same volume holds fewer of them. Air at 100 °C is about 20% lighter than air at 20 °C. Fill a 2,800 m³ envelope with it and the lighter air inside floats on the heavier air outside, lifting 600 kg or more. That is how the Montgolfier brothers flew in 1783. Pilots like to fly in cool early mornings: the colder the air outside, the bigger the difference, the more lift.

Helium is lighter still, 0.17 kg/m³ against air's 1.2, so a party balloon floats even when cold.

Water does the same. Water at 60 °C is 1.5% lighter than at 20 °C, so in a geyser the hot water rises and collects at the top, right where the outlet pipe draws from (see WaterHeaterClear). The cold inlet comes in at the bottom.

And the air in your room: cold air sinks. A split AC is mounted high on the wall so its cold air falls through the whole room and stirs it (see ACClear). Mount it low and the cold air pools on the floor while warm air stays floating under the ceiling. Heaters work best low for the same reason. This rising of warm fluid is convection (HeatClear). Sunshine warms the ground, warm moist air rises in bubbles called thermals, cools as it climbs and makes puffy clouds. Kites and eagles ride those thermals.

Your body is very close to the density of water. Fat is about 0.90 g/cm³, muscle and bone together about 1.10, so most people average 0.98 to 1.05 g/cm³. The big switch is your lungs (see LungsClear). Breathe all the way in and you add about 6 litres of air: almost everyone floats. Breathe out and you sink slowly. That is why swimming teachers say "take a big breath and lie back".

Fish have a built-in balloon, the swim bladder. Fish flesh is about 1.07 g/cm³, so a bladder of about 7% of the body lets a river fish hover without swimming; sea fish need only about 5% because sea water is denser. Submarines copy the trick with ballast tanks. An Indian Navy Kalvari-class submarine floats at 1,615 tonnes. Let about 160 tonnes of sea water into its tanks and it weighs exactly as much as the sea it displaces, so it can hover. Blow the tanks with compressed air and it rises.

Try the egg test in your kitchen. A fresh egg sinks in plain water. Stir in salt and the water gets denser, until the egg floats. Old eggs lose water through the shell and grow a bigger air pocket, so they stand on end, then float even in plain water.

Density also separates mixtures. Oil floats on water, honey sinks under both, and cream (malai) rises to the top of boiled milk because milk fat is lighter than the rest. Grit settles to the bottom of a water tank. In an RO purifier's storage tank a float switch rides up on the water and turns the pump off when the tank is full (see ROClear).

Almost everything gets denser as it cools. Water does too, until 4 °C. Below that it starts to expand again, and when it freezes it expands by about 9%: ice is only 0.917 g/cm³. So a lake cooling in winter first mixes until it is 4 °C all through. After that, the colder water stays on top, ice forms at the surface and floats, and the bottom stays at about 4 °C. Fish survive the winter under the lid of ice. If ice sank, lakes would freeze solid from the bottom up.

The Dead Sea, between Jordan and Israel, is so salty that its water is about 1.24 g/cm³. People float with a fifth of their body out of the water and read a newspaper lying on their back. You can't dive in it; you bob back up.

Astronauts practise spacewalks underwater. NASA's Neutral Buoyancy Laboratory in Houston is a pool 62 m long and 12 m deep. Divers add small lead weights and foam to each suited astronaut until they neither rise nor sink. It feels like floating in space, but it isn't quite: water drags on every move, and inside the suit you still feel your weight pressing on the shoulders.

Myth: heavy things sink. No: dense things sink. A 390 kg pine log floats while a 5 g nail sinks, because the log is 0.5 g/cm³ and the iron is 7.9. Buoyancy compares densities, not weights.

Myth: a boat floats better in deep water. No: the push depends on how much water the hull pushes aside, not on how much water is underneath. The same boat sits at the same depth in a 2 m canal and in the open ocean, as long as its bottom doesn't touch.