Density

Imagine holding a football-sized lump of expanded polystyrene in one hand and a golf-ball-sized lump of lead in the other. The polystyrene is enormous; the lead is tiny. Yet the little lead ball is far heavier. How can something so small out-weigh something so big?

The answer is density: how much mass is crammed into a given volume of a material. Lead packs a huge amount of stuff into every cubic centimetre — its tiny atoms are heavy and jammed close together. Polystyrene is mostly trapped air, so each cubic centimetre holds almost nothing. Density isn't about how big something is, or even how heavy it is — it is about how tightly the mass is packed.

The density formula

To turn "how tightly packed" into a number, we simply divide the mass by the volume it fills. Density is written with the Greek letter \rho ("rho"):

\rho = \dfrac{m}{V}

Here m is the mass (how much stuff there is) and V is the volume (how much space it takes up). Because it is mass divided by volume, density is measured in kilograms per cubic metre (\text{kg/m}^3) or, for smaller everyday objects, grams per cubic centimetre (\text{g/cm}^3).

The two units describe exactly the same idea at different scales, and they are easy to swap between:

1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3.

The density of a material is its mass per unit volume:

Density is a property of the material itself — a fixed number like 1\ \text{g/cm}^3 for water — not of the size of the lump you happen to have.

How dense is everyday stuff?

Every material has its own density. Learning a few benchmark values gives you a feel for the scale — and each one is just m/V measured for that material:

Notice the pattern from our previous lesson on the states of matter: for the same substance the solid is usually denser than the liquid, which is far denser than the gas — because the particles are packed closest in a solid and spread furthest apart in a gas.

Worked example 1 — finding the density

A pupil puts a small metal block on a balance and finds its mass is 240\ \text{g}. Measuring its sides, they work out its volume is 30\ \text{cm}^3. What is its density, and what might the metal be?

Step 1 — write the formula.

\rho = \dfrac{m}{V}

Step 2 — put the numbers in.

\rho = \dfrac{240\ \text{g}}{30\ \text{cm}^3} = 8\ \text{g/cm}^3.

Step 3 — interpret it. A density of 8\ \text{g/cm}^3 is very close to iron/steel (7.9\ \text{g/cm}^3), so the block is probably iron. It is 8 times denser than water, so it will sink like a stone.

Worked example 2 — finding the mass

A sculptor needs a solid aluminium part with volume 50\ \text{cm}^3. Aluminium has density 2.7\ \text{g/cm}^3. How much will the part weigh out at?

Here we know \rho and V and want m, so use the rearranged formula m = \rho\,V:

m = \rho\,V = 2.7\ \tfrac{\text{g}}{\text{cm}^3} \times 50\ \text{cm}^3 = 135\ \text{g}.

The \text{cm}^3 cancels, leaving grams — a good check that you used the right formula.

Worked example 3 — finding the volume

A gold ring has mass 38.6\ \text{g}. Gold has density 19.3\ \text{g/cm}^3. What volume of gold is in the ring?

Now we know m and \rho and want V, so use V = \dfrac{m}{\rho}:

V = \dfrac{m}{\rho} = \dfrac{38.6\ \text{g}}{19.3\ \text{g/cm}^3} = 2\ \text{cm}^3.

You could check this by dropping the ring into a measuring cylinder of water: the water level should rise by exactly 2\ \text{cm}^3. That trick — measuring volume by the water an object pushes aside — is called displacement, and it is the only easy way to measure the volume of an awkward shape.

Floating and sinking

Density decides one of the most useful things in physics: whether an object floats or sinks. The rule is beautifully simple.

An object floats if it is less dense than the fluid around it, and sinks if it is denser. A cork (\approx 0.2\ \text{g/cm}^3) bobs on water; an iron nail (\approx 7.9\ \text{g/cm}^3) plummets. It has nothing to do with size: a giant wooden log floats while a tiny steel screw sinks, because it is the density that matters, not the mass on its own.

This is also why oil floats on water (oil is less dense), and why a hot-air balloon rises: heating the air inside makes it spread out and become less dense than the cooler air outside.

See it for yourself

Below is a block dropped into a tank of water. Drag the mass and volume sliders and watch the live density readout \rho = m/V and the float-or-sink verdict. Try to make the very same block float and then sink: pile on mass (heavier, denser) to push it under, or swell the volume (bigger, but the same mass spread thinner) to lift it back up. The block settles lower in the water the closer its density gets to 1\ \text{g/cm}^3.

Almost everything gets denser when it freezes — the particles slow down and pack tighter. Water is the great rebel. When water freezes into ice, the molecules lock into an open, six-sided crystal pattern that actually holds them further apart than in liquid water. So ice is slightly less dense (0.92\ \text{g/cm}^3) than the water it came from (1\ \text{g/cm}^3) — and less-dense things float.

That one strange quirk is a very big deal. Because ice floats, a frozen pond freezes from the top down, leaving liquid water — and the fish in it — alive underneath the ice sheet. If ice sank, ponds and even oceans would freeze solid from the bottom up, and life as we know it might never have got going.

It's an old trick question, and the answer is: they weigh exactly the same — one kilogram is one kilogram, whatever it's made of. What people are really picturing is the difference in density. To gather a whole kilogram of feathers you would need a gigantic, room-filling sack, because feathers are so low-density. A kilogram of steel is a small, dense block you could hold in one hand. Same mass, wildly different volumes — and that difference in volume is density doing its work.