Edexcel IGCSE Physics · Spec 5.1-5.2
Density: Using the Equation
Worked examples using the density equation.
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Density: Using the Equation
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Explained
Using the density equation, and keeping the units straight
Density equals mass divided by volume. Rearranged, mass equals density multiplied by volume, and volume equals mass divided by density.
A formula triangle helps if you use one, with mass on top and density and volume side by side underneath. Cover the quantity you want and the arrangement of the other two tells you what to do.
Three worked examples, one for each rearrangement
A block has a mass of 300 g and a volume of 100 cubic centimetres. Density is 300 divided by 100, which is 3 g per cubic centimetre.
A piece of aluminium has a volume of 25 cubic centimetres and aluminium has a density of 2.7 g per cubic centimetre. Mass is 2.7 multiplied by 25, which is 67.5 g.
A 500 g sample of a liquid has a density of 0.8 g per cubic centimetre. Volume is 500 divided by 0.8, which is 625 cubic centimetres.
In each case the working is one line, and the difficulty in exam questions is almost never the arithmetic. It is the units.
The unit conversion
The SI unit is kilograms per cubic metre. Grams per cubic centimetre is the laboratory unit.
The factor between them is 1000, but not for the reason people expect. One metre is 100 centimetres, so one cubic metre is 100 cubed cubic centimetres, which is 1 000 000. Combined with 1000 grams in a kilogram, the two factors partly cancel and leave 1000.
So 1 g per cubic centimetre is 1000 kg per cubic metre. Water is the value to hold on to, being exactly that in both systems, and it doubles as a sanity check: anything denser than water sinks in it.
The safest habit is to convert before calculating rather than afterwards, and to work in one system throughout.
What the mark scheme accepts and rejects
An Edexcel International GCSE Physics mark scheme works through a density measurement in three parts, and the third is unusually generous in a way worth understanding.
Having found a mass of 9.2 g and a volume of 7 cubic centimetres by displacement, the calculation is 9.2 divided by 7. One mark for the substitution, one for the evaluation, and one for the unit. On the unit mark it says: any dimensionally correct unit that matches the candidate's evaluation.
It then prints four answers that all score full marks: 1300 kilograms per cubic metre, 1.3 grams per cubic centimetre, 1300 grams per litre, and 1.3 times ten to the minus three kilograms per cubic centimetre.
Read that list again. The last one is not a unit anybody teaches, and it is credited, because the number and the unit agree with each other. What is being marked is not whether you chose the expected units but whether your answer is internally consistent.
Which means the unit is a mark you can secure by being careful rather than by remembering a convention. Write down the units you actually used for the mass and the volume, divide one by the other, and quote that. It also means an answer of 1.3 labelled kilograms per cubic metre loses the mark, because the number belongs to a different unit.
Earlier in the same question the mark scheme covers the mass measurement, crediting zeroing or taring the balance and standing it on a flat and level surface, and then ignoring both repeat and average and any reference to significant figures or decimal places. Not every good laboratory habit is relevant to every question, and here the relevant ones are the ones that affect a single reading.
Finding the volume you need
For a regular solid, calculate it from measured dimensions: length times width times height for a cuboid.
For an irregular solid, use displacement. Lower it into a measuring cylinder of water and subtract the first reading from the second, which is what turned 30 and 37 into 7 cubic centimetres in the question above.
For a liquid, measure it directly in a measuring cylinder, and find its mass by weighing the cylinder empty and then full.
Where a question involves a mixture or a hollow object, work out the total mass and the total volume separately and divide at the end. Densities cannot be averaged directly unless the volumes happen to be equal.
Spec 5.1-5.2
What you need to know
- Use density equals mass over volume
- Rearrange it to find mass or volume
- Keep the units consistent
Active recall
Quick check
Answer each question before opening the answer.
What is the density of a 300 g block with a volume of 100 cm³?
ρ = m ÷ V = 300 ÷ 100 = 3 g/cm³.
How could you find the volume of an irregular solid?
By displacement — measure the volume of water it pushes out of the way.
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