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Density

5.3

3 guided lessons · 27 checks · 5. Solids, Liquids and Gases → Density & Pressure

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The density equation

Density prices a material: how much mass is packed into each cube of its volume. The equation divides mass by volume, the triangle rearranges it, and the units follow whatever units the two inputs arrive in.

What you know already

A feather bag and a metal bag can share a size yet differ wildly in heft. These lessons define density, build the density equation and its triangle, compare materials state by state, and unlock the volume formulae the calculation needs first.

Key points of understanding

Density is how much mass a material packs into each unit of its volume.
ρ = m ÷ V, with density in kilograms per metre cubed, mass in kilograms and volume in metres cubed.
A formula triangle covering ρ, m or V reproduces every rearrangement of the equation.
Grams with centimetres cubed give grams per centimetre cubed; kilograms with metres cubed give kilograms per metre cubed.

Density triangle (2), IGCSE & GCSE Physics revision notes — To use a formula triangle, simply cover up the quantity you wish calculate and the structure of the equation is revealed

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1.The density equation

5.3

Key points

Density is how much mass a material packs into each unit of its volume.
ρ = m ÷ V, with density in kilograms per metre cubed, mass in kilograms and volume in metres cubed.
A formula triangle covering ρ, m or V reproduces every rearrangement of the equation.
Grams with centimetres cubed give grams per centimetre cubed; kilograms with metres cubed give kilograms per metre cubed.

First pass

Weigh it, measure the space it fills, divide the first by the second. Hide the symbol you want on the triangle and the other two show you whether to multiply or divide.

Analogy — A crowd count: people (mass) spread through a hall (volume) — a cramped hall has a dense crowd, an empty one does not.

Exam pointer — State the unit with the answer and match it to the input units — mixing grams with metres cubed is the classic unit slip.

Going deeper

The density equation ρ = m/V divides the mass of a sample by the volume it occupies, so equal masses in smaller volumes are denser. The unit follows the inputs: grams and centimetres cubed yield grams per centimetre cubed, kilograms and metres cubed yield kilograms per metre cubed, and the triangle encodes all three rearrangements.

Analogy — A population density map: the same city spread over a larger county reads as a lower figure — area here plays volume's role.

Exam pointer — Convert lengths to metres and masses to kilograms before dividing whenever the question wants kg/m³ — quote the unit to bank the mark.

Checks

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1.Which equation defines density?
2.A material's mass is measured in kilograms and its volume in metres cubed. Its density unit is
3.Two identical bags hold the same volume, one filled with feathers and one with metal. What can be said about their masses?
4.Why does a gas have a lower density than the same substance as a solid?
5.On the density formula triangle, covering V leaves
6.State the density equation in symbols and name each quantity with its unit.
Written response · recall
Your answer
7.State what the density of a material tells you about its mass and volume.
Written response · recall
Your answer
8.Explain why a balloon can occupy more space than a small lead bar yet contain far less mass.
Written response · explain why
Your answer
9.A cube measures 2 m along each edge and has a mass of 1600 kg. Calculate its density.
Written response · application
Your answer

2.Density across states and materials

5.3

The same substance changes its density with its state: packed particles make dense solids and liquids, while spread-out particles thin gases out. Material choice matters just as much — equal sizes, wildly different hefts.

Density comparison, downloadable AS & A Level Physics revision notes — A gas is less dense than the same substance in liquid or solid form

Key points

Objects made from low-density materials carry low masses; equal-sized objects of high-density materials carry high masses.
A bag of feathers is far lighter than the same bag filled with metal.
Gases are generally less dense than solids because their particles sit far apart.
The same substance is denser in solid or liquid form than as a gas.

First pass

Squeeze the same atoms closer together and the density climbs: solid tight, liquid close, gas sparse. Swap the material instead and the atom's own heft sets the scale — feathers float, lead sinks.

Analogy — Three classrooms: packed full (solid), half full (liquid), one pupil per corridor (gas) — same school, very different crowding.

Exam pointer — Comparisons must fix everything but one variable — equal volumes for material comparisons, equal substance for state comparisons.

Going deeper

Density tracks particle spacing: a substance's solid and liquid states pack particles closely, so both out-dense the gas state where the same mass disperses through far more volume. Across materials at equal volume, the heavier-particle, tighter-packed material wins the density contest outright.

Analogy — Packing for three trips: folded tight (solid), loosely rolled (liquid), scattered across the whole suitcase (gas) — same belongings, three bulks.

Exam pointer — When a question compares states, answer with particle spacing and the mass-over-volume consequence, not with 'gases are light', which explains nothing.

Checks

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1.Equal-sized blocks of wood and lead: which has the greater mass?
2.Water becomes steam. What happens to the density of the water you are watching?
3.Which statement about the three states of one substance is correct?
4.A workshop sorts offcuts by mass alone to guess density. What must it also know?
5.Which comparison is a fair density comparison?
6.State why gases are generally less dense than solids.
Written response · recall
Your answer
7.State which fills a same-size bag to a greater mass: feathers or metal.
Written response · recall
Your answer
8.Explain why comparing the densities of two different materials is fairest when their volumes match.
Written response · explain why
Your answer
9.A State changes from liquid to solid and its volume shrinks slightly while its mass stays constant. State what happens to its density.
Written response · application
Your answer

3.Finding volume first

5.3

Density questions often hide the volume inside the object's shape. Out come the geometry formulae — cubes, cuboids, cylinders and spheres — and only after the volume is rebuilt can the density equation divide.

Volume of shapes, downloadable AS & A Level Physics revision notes — Volumes of common 3D shapes

Key points

An object's volume is not always stated and must be calculated from its shape first.
Each regular shape carries its own volume formula — cube, cuboid, cylinder and sphere among them.
Once the volume is known, the density equation divides mass by volume as usual.
Lengths must enter the volume formulae in metres so the density lands in kilograms per metre cubed.

First pass

Measure the edges, apply the shape's own formula, then divide the mass by what you get. The geometry is the servant: it exists to feed the density equation a volume.

Analogy — A recipe: the shape's formula is the measuring jug step — only after pouring the volume can you cook the density.

Exam pointer — Write the volume formula explicitly before substituting — examiners award the method mark even where arithmetic slips.

Going deeper

Volume reconstruction precedes density calculation whenever the volume is not given directly: the object's shape selects the formula, the lengths are multiplied through in metres, and the resulting volume in metres cubed feeds ρ = m/V. Errors in the geometry propagate straight into the density, which is why the formula must be stated.

Analogy — A two-stage rocket: the geometry stage lifts off first, and only its burnout hands the mass-over-volume stage its payload.

Exam pointer — State the shape formula, convert lengths to metres, then divide — and carry the units through the volume step, not just the final answer.

Checks

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1.Why might a density calculation need a volume formula first?
2.Which volume belongs to a cuboid of edges 2 m, 3 m and 4 m?
3.An object's edges were measured in centimetres. What must happen before the volume feeds a kg/m³ density?
4.A cylinder's radius and height are known. What is the shape-formula step for its volume?
5.A slab measures 0.02 m × 0.4 m × 0.9 m and has a mass of 60 kg. Which working finds its density?
6.State the volume formula for a cube of edge length L.
Written response · recall
Your answer
7.State what must be calculated before the density equation when the volume is not given.
Written response · recall
Your answer
8.Explain why an error in the volume step propagates straight into the final density.
Written response · explain why
Your answer
9.A sphere of radius 0.1 m has a mass of 4.2 kg. Taking the sphere volume as 4 ÷ 3 × π × r³ and π ≈ 3, calculate its density.
Written response · application
Your answer

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1.The density equation

5.3