Density and Relative Density: Formulas, Units and Examples

Why does a small gold coin feel surprisingly heavy while a big plastic bucket feels light? The answer is density: how much mass is packed into each unit of volume. Density is one of the most useful fingerprints of a material — it identifies substances, predicts floating, and exposes fakes.

Relative density goes one step further by comparing a substance's density with that of water. Because it is a pure ratio, it needs no units and works in any measurement system — a favourite of jewellers, engineers and scientists alike.

Key Concepts

• Density (ρ): mass per unit volume, ρ = m ÷ V, with the SI unit kg/m³.
• Relative density (RD): density of the substance divided by density of water (1000 kg/m³); also called specific gravity. It has no units.
• Fingerprint value: pure substances have characteristic densities — gold 19,300 kg/m³, iron 7870 kg/m³, ice 917 kg/m³.
• Float/sink rule: RD > 1 means denser than water (sinks); RD < 1 means lighter (floats).
• Temperature effect: most substances expand when heated, so density falls slightly with rising temperature — including water near boiling.

Formulas

ρ = m ÷ V, with mass m in kilograms and volume V in cubic metres, giving kg/m³. Relative density = ρ_substance ÷ ρ_water, where ρ_water = 1000 kg/m³. So RD = ρ ÷ 1000 numerically when ρ is in kg/m³. Because both numerator and denominator carry the same units, RD is dimensionless. Volume is often found by fluid displacement: submerge the object and measure the displaced water's volume.

Worked Example

A jeweller tests a "gold" bangle of mass 0.386 kg. Lowered into water, it displaces 2 × 10⁻⁵ m³ (20 cubic centimetres) of water. Its density is ρ = 0.386 ÷ (2 × 10⁻⁵) = 19,300 kg/m³, and its relative density is 19,300 ÷ 1000 = 19.3 — exactly matching pure gold, so the bangle is genuine. If the same bangle had displaced 25.7 cm³ instead, the density would be 0.386 ÷ (2.57 × 10⁻⁵) ≈ 15,000 kg/m³ (RD ≈ 15), well below gold's 19.3 — revealing a fake. This displacement test is the modern version of Archimedes' own "Eureka" experiment.

Common Misconceptions

• "Density and weight are the same thing." Density is mass per unit volume — a property of the material. Weight depends on gravity too: your density is the same on the Moon, but your weight is one-sixth.
• "Relative density has units." It is a ratio of two densities, so units cancel completely — RD = 19.3, not "19.3 kg/m³."
• "Denser objects always sink." Only relative to the fluid they are in: an iron nail (RD ≈ 7.9) sinks in water but floats in mercury (RD ≈ 13.6), because mercury is denser still.

Real-World Applications

• Gold and jewellery testing: the displacement method exposes counterfeit metals quickly and non-destructively.
• Battery maintenance: a hydrometer measures the electrolyte's relative density to judge a car battery's charge.
• Cooking: oil (RD ≈ 0.9) floats on water — which is why oil spills spread on the sea and water won't mix into oil.
• Construction: aggregates and concrete mixes are specified by density for strength and durability.
• Oceanography: layers of seawater stack by density, driving deep ocean currents that regulate climate.

Density packs mass into volume; relative density compares that packing with water as a clean, unit-free number. Memorise a few fingerprints — gold 19.3, iron 7.9, ice 0.92 — and you can identify materials, predict floating, and catch a fake with a glass of water.