Define density as mass per unit volume; recall and use the equation ρ = m / V

1.4 Density

Learning Objective

Define density as mass per unit volume and use the equation \$\rho = \frac{m}{V}\$ to solve problems.

Definition

Density (\$\rho\$) is the amount of mass (\$m\$) contained in a given volume (\$V\$). It is a measure of how tightly matter is packed.

Formula

The relationship between density, mass and volume is expressed by:

\$\rho = \frac{m}{V}\$

where

• \$\rho\$ = density (kilograms per cubic metre, kg·m⁻³ or grams per cubic centimetre, g·cm⁻³)
• \$m\$ = mass (kg or g)
• \$V\$ = volume (m³ or cm³)

Rearranging the Equation

1. To find mass: \$m = \rho \times V\$
2. To find volume: \$V = \frac{m}{\rho}\$

Units and Conversions

Common units and their relationships:

Typical Densities of Common Materials

Worked Example

Problem: A solid block of metal has a mass of 540 g and a volume of 80 cm³. Determine its density.

1. Write down the formula: \$\rho = \dfrac{m}{V}\$
2. Substitute the given values: \$\rho = \dfrac{540\ \text{g}}{80\ \text{cm}^3}\$
3. Calculate: \$\rho = 6.75\ \text{g·cm}^{-3}\$
4. If required, convert to kg·m⁻³: \$6.75\ \text{g·cm}^{-3} \times 1000 = 6750\ \text{kg·m}^{-3}\$

Practice Questions

1. A wooden sphere has a mass of 0.45 kg and a volume of 0.0005 m³. Calculate its density in kg·m⁻³.
2. What volume will 250 g of a substance occupy if its density is 2.5 g·cm⁻³?
3. Identify which of the following materials is the most dense: aluminium (2.70 g·cm⁻³), water (1.00 g·cm⁻³), or air (0.0012 g·cm⁻³).

Common Mistakes to Avoid

• Mixing units – always ensure mass and volume are in compatible units before using the formula.
• Forgetting to convert density when required (e.g., from g·cm⁻³ to kg·m⁻³).
• Using the wrong symbol – \$\rho\$ denotes density, not \$d\$ or \$D\$.

Summary

Density is a fundamental property that links mass and volume. Mastery of the equation \$\rho = m/V\$ and the ability to rearrange it are essential for solving a wide range of physics problems.