Density

CIE A-Level Physics· 25 min read

1. Definition and Units of Density★☆☆☆☆⏱ 10 min

📘 Definition

Density

\rho = \frac{m}{V}

An intensive bulk property of matter, calculated as the total mass of a substance divided by its total volume.

Example:

Pure water at standard temperature and pressure has a density of 1000 kg m^{-3}.

Density is an intensive property, which means it does not change with the amount of substance. A 1 kg block of iron and a 10 kg block of iron have the same density, unlike mass which is an extensive property that scales with amount of substance.

\rho = \frac{m}{V}

The SI unit of density is kg m^{-3}, but g cm^{-3} is also commonly used. The conversion factor between the two is: $1 g cm^{- 3} = 1000 kg m^{- 3}$ .

📐 Worked Example

A rectangular aluminum block has mass 135 g, with side lengths 2 cm × 3 cm × 5 cm. Calculate its density in SI units.

1
Calculate the volume of the block:
$V = 2 \times 3 \times 5 = 30 \text{ cm}^3$
2
Convert mass and volume to SI units:
$m = 135 \text{ g} = 0.135 \text{ kg} \\ V = 30 \times (10^{-2} \text{ m})^3 = 3 \times 10^{-5} \text{ m}^3$
3
Substitute into the density formula:
$\rho = \frac{0.135}{3 \times 10^{-5}} = 4500 \text{ kg m}^{-3}$

2. Measuring Density Experimentally★★☆☆☆⏱ 15 min

To find density experimentally, you measure mass (usually with a balance) and volume separately. The method for finding volume depends on the type of substance:

Regular solids: Calculate volume from measured side lengths.
Irregular sinking solids: Use water displacement in a measuring cylinder.
Irregular floating solids: Use a sinker to fully submerge the object, or use a eureka can.
Liquids: Subtract the mass of the empty container from the total mass to get liquid mass, read volume directly.

📐 Worked Example

An irregular stone has mass 56 g. When placed in a measuring cylinder with 20 cm³ of water, the level rises to 41 cm³. Calculate the density of the stone in g cm⁻³.

1
Find the volume of the stone from displacement:
$V = V_{\text{final}} - V_{\text{initial}} = 41 - 20 = 21 \text{ cm}^3$
2
Calculate density from the formula:
$\rho = \frac{m}{V} = \frac{56}{21} \approx 2.67 \text{ g cm}^{-3}$
3
Convert to SI units if required: $2.67 g cm^{- 3} = 2670 kg m^{- 3}$

3. Applications of Density★★☆☆☆⏱ 15 min

Density is used to predict whether an object will float or sink in a fluid. An object will float if its average density is less than or equal to the density of the fluid, and sink if its average density is higher. This principle is used in many CIE exam problems involving buoyancy and upthrust.

📐 Worked Example

A hollow plastic sphere has a total mass of 25 g and a total volume of 100 cm³. Will it float in pure water? Justify your answer.

1
Calculate the average density of the sphere:
$\rho_{\text{sphere}} = \frac{25 \text{ g}}{100 \text{ cm}^3} = 0.25 \text{ g cm}^{-3}$
2
Density of pure water = $1 g cm^{- 3}$
3
Conclusion: Since $0.25 g cm^{- 3} < 1 g cm^{- 3}$ , the sphere will float.

4. Common Pitfalls

Wrong move:

Forgetting to convert units between g cm⁻³ and kg m⁻³

Why:

Most students incorrectly use 1 g cm⁻³ = 1 kg m⁻³, leading to answers off by a factor of 1000

Correct move:

Memorize that 1 g cm⁻³ = 1000 kg m⁻³, check the required unit before finalizing your answer

Wrong move:

Including the mass of the container when calculating liquid density

Why:

This adds extra mass to the calculation, leading to a density value that is too high

Correct move:

Always subtract the mass of the empty container from the total mass to get the mass of the liquid only

Wrong move:

Only measuring displacement of the part above water for floating objects

Why:

Volume is the total space occupied by the object, so partial displacement gives an incorrectly low volume and high density

Correct move:

Fully submerge the floating object with a sinker, then subtract the displacement of the sinker to get the correct total volume

Wrong move:

Assuming a heavier object is always denser than a lighter one

Why:

Density depends on both mass and volume, a large light object can be less dense than a small heavy one

Correct move:

Always calculate density as mass divided by volume, never compare densities only by comparing masses

5. Quick Reference Cheatsheet

Concept	Formula / Rule	Unit / Value
Density	$ρ = m / V$	SI: kg m⁻³
Unit conversion	$1 g cm^{- 3} = 1000 kg m^{- 3}$
Density of pure water		$1000 kg m^{- 3} = 1 g cm^{- 3}$
Object floats	$ρ_{o bj ec t} \leq ρ_{f l u i d}$
Object sinks	$ρ_{o bj ec t} > ρ_{f l u i d}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 13
Density unit conversion multiple choice
2022 · 21
Experimental density calculation
2021 · 31
Practical measurement of irregular density

Going deeper

syllabusCIE 9702 A-Level Physics SyllabusUnit 4 content reference

What's Next

Density is a foundational core concept for fluid statics, pressure, and buoyancy, all key topics in Unit 4 of the CIE A-Level 9702 Physics syllabus. A solid understanding of density, unit conversions, and experimental measurement is critical to solving problems involving pressure in liquids, upthrust, and Archimedes' principle, which regularly appear in multiple-choice, structured, and practical paper questions. Density also comes up in topics involving materials properties, including stress and strain. Experimental questions on measuring density of irregular solids are common in Paper 3 practical assessments, so mastering the methods covered here will help you score highly in that component.