Density
Density — definition and calculations
Defining density
Density is the mass per unit volume of a substance.
\[ \rho = \frac{m}{V} \]
| Symbol | Quantity | Unit |
|---|---|---|
| \( \rho \) (rho) | Density | kg/m³ or g/cm³ |
| \( m \) | Mass | kg or g |
| \( V \) | Volume | m³ or cm³ |
Rearrangements of \( \rho = m / V \):
\[ m = \rho V \qquad V = \frac{m}{\rho} \]
A useful memory aid — cover the quantity you want in this triangle:
Cover \( m \) → \( \rho \times V \); cover \( \rho \) → \( m \div V \); cover \( V \) → \( m \div \rho \).
Units and conversion
SI unit: kg/m³ (used in calculations with SI quantities)
Common alternative: g/cm³ (convenient for lab work) \[ 1\,\text{g/cm}^3 = 1000\,\text{kg/m}^3 \] Examples: water = 1.0 g/cm³ = 1000 kg/m³; iron = 7.9 g/cm³ = 7900 kg/m³.
Common alternative: g/cm³ (convenient for lab work) \[ 1\,\text{g/cm}^3 = 1000\,\text{kg/m}^3 \] Examples: water = 1.0 g/cm³ = 1000 kg/m³; iron = 7.9 g/cm³ = 7900 kg/m³.
| Material | Density (kg/m³) | Density (g/cm³) |
|---|---|---|
| Air (at room temperature) | 1.2 | 0.0012 |
| Ice | 917 | 0.92 |
| Water (liquid) | 1000 | 1.00 |
| Sea water | 1025 | 1.03 |
| Wood (typical) | 600–900 | 0.6–0.9 |
| Aluminium | 2700 | 2.70 |
| Iron / steel | 7900 | 7.90 |
| Lead | 11 300 | 11.3 |
| Gold | 19 300 | 19.3 |
Worked calculations
Example 1 — find density
A block of aluminium has mass 540 g and volume 200 cm³. Calculate its density.
\( \rho = \dfrac{m}{V} = \dfrac{540}{200} = 2.7\,\text{g/cm}^3 \)
In SI units: \( 2.7\,\text{g/cm}^3 = 2700\,\text{kg/m}^3 \)
Example 2 — find mass
A piece of iron has volume 50 cm³. Iron has density 7.9 g/cm³. Find the mass.
\( m = \rho V = 7.9 \times 50 = 395\,\text{g} \)
Example 3 — find volume
A liquid has density 1.2 g/cm³ and mass 360 g. Find its volume.
\( V = \dfrac{m}{\rho} = \dfrac{360}{1.2} = 300\,\text{cm}^3 \)
Floating and sinking
An object placed in a fluid will:
- Float if its density is less than the density of the fluid (\( \rho_\text{object} < \rho_\text{fluid} \)).
- Sink if its density is greater than the density of the fluid (\( \rho_\text{object} > \rho_\text{fluid} \)).
- Be neutrally buoyant (hover at any depth) if \( \rho_\text{object} = \rho_\text{fluid} \).
In density calculations, always check that mass and volume units are consistent. If mass is in grams and volume is in cm³, the answer is in g/cm³. If mass is in kg and volume is in m³, the answer is in kg/m³. Mixing units gives a wrong answer.