IGCSE CP9 – Measuring Density | Edexcel International GCSE Physics

Theory — Density

Density is the mass per unit volume of a substance. It tells us how tightly packed the matter is.

The Density Equation

ρ = m / V

ρ = density (kg/m³ or g/cm³) · m = mass (kg or g) · V = volume (m³ or cm³)

If mass is in grams and volume in cm³, density comes out in g/cm³. Multiply by 1000 to convert to kg/m³.

Volume of Regular Solids

Cube/cuboid: V = l × w × h (measure length, width, height with ruler or vernier calipers)
Cylinder: V = π r² h (measure diameter with calipers → r = d/2, measure height)
Sphere: V = (4/3) π r³ (measure diameter with calipers → r = d/2)

Measure each dimension at least 3 times → take the mean

This reduces random errors. Use vernier calipers (precision 0.1 mm) for small objects.

Volume of Irregular Solids — Displacement Method

For objects with complex shapes, volume cannot be calculated from measurements. Instead, use Archimedes' principle: a submerged object displaces a volume of water equal to its own volume.

Fill a displacement (eureka) can to the spout level with water
Submerge the object — water overflows into a measuring cylinder
Volume of water collected = volume of object

Alternatively, use a measuring cylinder alone: record water level before and after submerging the object. ΔV = volume of object.

Known Densities for Comparison

Aluminium: 2.70 g/cm³
Iron / steel: 7.87 g/cm³
Copper: 8.96 g/cm³
Brass: 8.50 g/cm³
Perspex / acrylic: 1.18 g/cm³
Water: 1.00 g/cm³ (by definition)

Procedure

Equipment

Regular solid objects (cube, cylinder, sphere) · Ruler (mm scale) · Vernier calipers · Electronic balance · Calculator

1

Measure mass on balance

Place the object on the electronic balance. Record the mass m in grams. Take a single reading — mass measurement is precise and does not need repeating.

2

Measure dimensions with calipers

For each relevant dimension (length, diameter, height), take three measurements at different positions along the object and calculate the mean. This accounts for any non-uniformity.

💡 Vernier calipers read to 0.1 mm — much more precise than a standard ruler. For a cylinder, measure diameter at top, middle, and bottom.

3

Calculate volume

Use the appropriate formula for the shape. Convert all measurements to cm if mass is in grams (so density comes out in g/cm³).

4

Calculate density and compare

Use ρ = m/V. Compare your result to the known density of the material to find the percentage error. Repeat for several different objects/materials.

⚗️ Choose Regular or Irregular method. Select material and shape, adjust dimensions, then press ▶ Measure to get readings.

No readings yet.

Method

Shape

Material

Readings

Mass m—

Volume V—

True density—

Calculated Density

— g/cm³

Data Table

#	Method	Material	Mass / g	Volume / cm³	Density / g·cm⁻³	True ρ	% error
No readings yet — use the Simulation tab.

Questions

Question 1

A student measures an aluminium cylinder: diameter = 2.40 cm, height = 5.00 cm, mass = 61.1 g. Calculate (a) the volume of the cylinder, (b) the density of the aluminium, and (c) the percentage error compared to the true density of 2.70 g/cm³.

(a) r = d/2 = 1.20 cm. V = πr²h = π × (1.20)² × 5.00 = π × 1.44 × 5.00 = 22.6 cm³. (b) ρ = m/V = 61.1/22.6 = 2.70 g/cm³. (c) % error = |2.70 − 2.70|/2.70 × 100 = 0.0% — perfect result! In practice, small errors in measuring diameter have a large effect because r is squared in the formula, so even a 1% error in diameter causes a 2% error in volume and therefore density.

Question 2

Explain why the displacement method (eureka can) is used for irregular solids, and give two sources of error in this method.

For irregular solids, there is no simple geometric formula to calculate volume from measurements — the shape is too complex. The displacement method exploits Archimedes' principle: a submerged object displaces a volume of water exactly equal to its own volume, regardless of shape. Two sources of error: (1) The displacement can may not have been filled exactly to the spout level before the object was inserted — if slightly overfull, too much water overflows; if underfull, too little is collected. (2) Some water may remain in the can or on the spout rather than flowing into the measuring cylinder, so the collected volume is slightly less than the true volume — density is slightly overestimated as a result. A third source is reading error on the measuring cylinder (parallax error when reading the meniscus).

Question 3

A student measures a metal sphere using vernier calipers. They take three diameter readings: 3.22 cm, 3.24 cm, 3.21 cm. The mass is 148.5 g. Calculate the density and identify the metal.

Mean diameter d = (3.22 + 3.24 + 3.21)/3 = 9.67/3 = 3.223 cm. r = 1.612 cm. V = (4/3)πr³ = (4/3) × π × (1.612)³ = (4/3) × π × 4.195 = 17.56 cm³. ρ = m/V = 148.5/17.56 = 8.46 g/cm³. This is closest to brass (8.50 g/cm³) — % error = |8.46 − 8.50|/8.50 × 100 = 0.47%, an excellent result. It is not iron (7.87) or copper (8.96).