Density

How to measure and calculate density, and what density tells us about materials.

What is Density?

Density tells us how tightly packed a material is. A small but heavy object has high density.

\rho = \frac{m}{V}

$\rho$ = density (g/cm³ or kg/m³)
$m$ = mass (g or kg)
$V$ = volume (cm³ or m³)

Measuring Density

1
Measure the mass
Use a balance to find the mass in grams.
2
Find the volume
For regular shapes: use length, width, height formulas. For irregular shapes: use the displacement method — submerge in a measuring cylinder and read the volume of water displaced.
3
Calculate density
Divide mass by volume.

A pebble has mass 45 g. It displaces 18 cm³ of water. Find its density.

$\rho = \dfrac{45}{18} = 2.5 \text{ g/cm}^3$

Denser than water (1 g/cm³) → it sinks.

Density of Common Materials

Material	Density (g/cm³)
Air	0.0013
Ice	0.92
Water	1.00
Aluminium	2.70
Iron	7.87
Lead	11.34
Gold	19.32

Note

Why does ice float? Ice is less dense than liquid water (0.92 < 1.00), so it floats. This unusual property of water is what keeps lakes from freezing solid.

Real world: testing whether a ring is real gold

A jeweller suspects a gold ring ( $\rho_{\text{gold}} = 19.32\text{ g/cm}^3$ ) might be fake. The ring's mass is 18.2 g and water displacement test gives 1.9 cm³. Calculated density: $\rho = \frac{18.2}{1.9} \approx 9.6\text{ g/cm}^3$ — much less than gold. The ring is likely brass or another alloy. Density tests are used routinely in materials and forensic science.

Real world: hot-air balloon — comparing densities

Hot air (e.g. 100°C) is less dense than cool surrounding air (20°C). Heating 1 m³ of air reduces its density from about 1.20 kg/m³ to about 0.95 kg/m³. A large enough balloon volume creates enough buoyancy (low average density) to lift passengers. Density calculations determine the minimum required envelope volume.

Real world: submarine ballast tanks

A submarine adjusts its average density by filling or emptying ballast tanks with seawater ( $\rho = 1025\text{ kg/m}^3$ ). When the tank is full, average density exceeds seawater → submarine sinks. When tanks are blown with compressed air, average density falls below seawater → submarine rises. Density control governs every dive and ascent.

Tip When checking whether an object floats, compare densities. If

\rho_{\text{object}} < \rho_{\text{fluid}}

it floats; if greater, it sinks. No need to calculate forces unless the question asks for buoyant force.

Practice Problems

A wooden block has mass 200 g and volume 250 cm³. Find its density. Will it float in water?

Show answer

ρ = 200 ÷ 250 = 0.8 g/cm³. Since 0.8 < 1.0, it floats.

A liquid has density 1.2 g/cm³. You have 500 cm³ of it. What is its mass?

Show answer

m = ρ × V = 1.2 × 500 = 600 g.