Define density, use density = mass divided by volume, and describe how to measure it

What this dot point is asking

SEAB wants you to define density, to use the formula density $=$ mass $\div$ volume, to find the volume of both regular and irregular solids, and to explain floating and sinking in terms of density. The big idea is that density tells you how much mass is packed into a given volume, which is why a small lead ball can be heavier than a large block of foam.

The answer

What density means

Density is the mass per unit volume of a substance. It tells you how tightly the matter is packed. A dense material such as lead has a lot of mass in a small volume; a low-density material such as cork has little mass for its size.

The formula is:

In symbols, $\rho = \dfrac{m}{V}$. Common units are grams per cubic centimetre ($\text{g cm}^{-3}$) or kilograms per cubic metre ($\text{kg m}^{-3}$). Water has a density of $1.0\ \text{g cm}^{-3}$, which is the same as $1000\ \text{kg m}^{-3}$.

Finding the volume of a regular solid

For a regular shape, calculate the volume from its measurements. For a box (cuboid), volume $=$ length $\times$ width $\times$ height. Measure each side with a rule, then multiply.

Finding the volume of an irregular solid

For an irregular object such as a stone, use the displacement method:

1. Part-fill a measuring cylinder with water and read the level.
2. Lower the object in gently until it is fully covered.
3. Read the new level. The rise in the water level equals the volume of the object.

Finding the density of a liquid

To find the density of a liquid, measure the mass of an empty measuring cylinder, add a known volume of the liquid, and measure the mass again. The mass of the liquid is the difference, and you read its volume from the cylinder. Then divide.

Floating and sinking

An object floats on a liquid if it is less dense than the liquid, and sinks if it is more dense. This is why wood ($\approx 0.6\ \text{g cm}^{-3}$) floats on water ($1.0\ \text{g cm}^{-3}$) but a steel nail sinks. A steel ship floats because its overall density, counting the air inside the hull, is less than that of water.

Examples in context

Example 1. Choosing materials. Aircraft use aluminium and carbon fibre because they have a low density, so the plane is lighter for its size and uses less fuel. A bridge cable uses dense, strong steel where strength matters more than weight. Density helps engineers choose the right material.

Example 2. Hot air balloons. Heating the air inside a balloon makes it expand, so the same mass of air takes up more volume and its density falls. When the air inside becomes less dense than the surrounding cool air, the balloon rises. Letting the air cool increases its density again and the balloon descends.

Try this

• Cue. A block has mass $500\ \text{g}$ and volume $250\ \text{cm}^3$. Find its density. [2 marks] $\rho = \dfrac{500}{250} = 2.0\ \text{g cm}^{-3}$.

• Cue. Describe how to measure the volume of a small irregular pebble. [2 marks] Use water displacement: read the water level in a measuring cylinder, add the pebble until covered, and the rise in level is the volume.

• Cue. A liquid has density $0.8\ \text{g cm}^{-3}$. A volume of $50\ \text{cm}^3$ is poured out. Find its mass. [2 marks] Mass $= \rho \times V = 0.8 \times 50 = 40\ \text{g}$.