Density

1. Definition

A. Density

Density, ρ, is the mass per unit volume of a substance.

ρ = m/V

• ρ = density (kg m⁻³)
• m = mass (kg)
• V = volume (m³)
• Density is a scalar quantity.

Data table

Example material (ρ = 2.7 g cm⁻³)
Volume (cm³)	Mass (g)
0	0
2	5
4	11
6	16
8	22
10	27
12	32

2. Key Ideas

• Rearrangements:
  • m = ρ V
  • V = m/ρ
• Use consistent units:
  • kg with m³ → kg m⁻³
  • g with cm³ → g cm⁻³
• Useful conversions:
  • 1 g = 1 × 10⁻³ kg
  • 1 cm³ = 1 × 10⁻⁶ m³
  • 1 g cm⁻³ = 1000 kg m⁻³
• Density of the same material is the same anywhere at the same temperature, because mass and volume do not depend on gravitational field strength.

3. Detailed Explanations

A. What density tells you

Density compares how “packed” the matter is:

• higher density → more mass in the same volume
• lower density → less mass in the same volume

B. Volume (how you measure it)

Volume, V, is the amount of space occupied by a three-dimensional object.

• SI unit of volume is m³.
• Other common units: cm³ and mL (where 1 cm³ = 1 mL).

How volume is found in exams:

• regular solid: use dimensions (e.g. V = lwh)
• irregular solid: use water displacement
• liquid: read the volume in a measuring cylinder

C. Using the density equation (and rearranging)

Density is given by:

ρ = m/V

Rearranging gives:

• m = ρ V (use this when density and volume are known)
• V = m/ρ (use this when density and mass are known)

D. Unit conversions you should memorise

1. Converting between cm³ and m³:

2. Converting between g cm⁻³ and kg m⁻³:

1 g cm⁻³ = 10⁻³ kg/10⁻⁶ m³ = 10³ kg m⁻³

E. Densities of common substances (typical values)

F. Link to pressure in liquids (next lesson)

Density appears directly in hydrostatic pressure:

p = hρ g

See: Hydrostatic Pressure.

4. Common Mistakes

• Using weight (N) instead of mass (kg) in ρ = m/V.
• Mixing units (e.g. mass in g but volume in m³) without converting.
• Forgetting that volume conversions are cubed:
  • 1 cm³ = 10⁻⁶ m³ (not 10⁻²).
• For displacement questions: using the final reading as the volume instead of (final - initial).

5. Exam Tips

• Start with the formula you need (ρ = m/V, m = ρ V, or V = m/ρ).
• Write units beside your values before you calculate.
• State the final density with the correct unit (kg m⁻³ or g cm⁻³).
• In practical-style questions, describe measurements clearly: “measure mass with a balance” and “find volume by displacement”.

6. Worked Examples

Example 1: Convert unitsCore

Aluminium has density 2.7 g cm⁻³. Convert this to kg m⁻³.

Show Answer

Use 1 g cm⁻³ = 1000 kg m⁻³:

2.7 g cm⁻³ = 2.7 × 1000 = 2.7 × 10³ kg m⁻³

Example 2: Mass from density and dimensions (SI units)Core

A block of concrete has dimensions 0.40 m × 0.30 m × 0.10 m and density 2500 kg m⁻³. Find its mass.

Show Answer

Volume: V = (0.40)(0.30)(0.10) = 0.012 m³

Mass: m = ρ V = (2500)(0.012) = 30 kg

Example 3: Density of an irregular solid (displacement)Core

An irregular stone has mass 120 g. Water in a measuring cylinder rises from 50 cm³ to 95 cm³ when the stone is fully immersed. Find the density of the stone.

Show Answer

Volume of stone (displaced water): V = 95 - 50 = 45 cm³

Density: ρ = m/V = 120/45 = 2.67 g cm⁻³

In SI units: 2.67 g cm⁻³ = 2.67 × 10³ kg m⁻³

Example 4: Mass and volume from density (copper)Core

Copper has density 9.0 g cm⁻³. Determine:

1. the mass of 5.0 cm³
2. the volume of 63 g

Show Answer

1. Mass: m = ρ V = (9.0)(5.0) = 45 g

2. Volume: V = m/ρ = 63/9.0 = 7.0 cm³

Example 5: Density from mass and volume (aluminium sheet)Core

An aluminium sheet has mass 200 g and volume 73 cm³. Find its density.

Show Answer

ρ = m/V = 200/73 = 2.74 g cm⁻³

Example 6: Mass and volume from density (lead)Core

Lead has density 11 g cm⁻³. Determine:

1. the mass of 4.0 cm³
2. the volume of 55 g

Show Answer

1. Mass: m = ρ V = (11)(4.0) = 44 g

2. Volume: V = m/ρ = 55/11 = 5.0 cm³

7. Mind Stretchers

Mind stretcher 1: Does density change when you cut an object?Extension

A cube has density 8.0 g cm⁻³. It is cut into 8 identical smaller cubes. What is the density of each smaller cube? Explain.

Show Answer

Density stays the same.

When you cut the cube, both the mass and the volume of each piece become 1/8 of the original, so the ratio ρ = m/V is unchanged.

Mind stretcher 2: Mixed units (volume in cm³)Extension

A piece of wood has mass 0.60 kg and density 800 kg m⁻³. Find its volume in cm³.

Show Answer

Volume in m³: V = m/ρ = 0.60/800 = 7.5 × 10⁻⁴ m³

Convert to cm³ (since 1 m³ = 10⁶ cm³): V = (7.5 × 10⁻⁴)(10⁶) = 7.5 × 10² = 750 cm³

8. Practice and next step

Complete the Pressure quiz, then continue to What is pressure? to use force and area before studying liquid columns.