Density

Density is one of the most fundamental properties of matter. It tells you how much mass is packed into a given volume and is essential for understanding why some objects float while others sink, why materials are chosen for specific engineering purposes, and how to identify unknown substances.

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Defining Density

Density is defined as mass per unit volume:

$\rho = \frac{m}{V}$ρ=Vm​

where:

• ρ (Greek letter rho) is density, measured in kg m⁻³ (SI unit) or g cm⁻³
• m is mass in kg (or g)
• V is volume in m³ (or cm³)

To convert between the two common unit systems: 1000 kg m⁻³ = 1 g cm⁻³. Water has a density of approximately 1000 kg m⁻³ or 1.0 g cm⁻³.

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Measuring Density of Regular Solids

For objects with regular geometric shapes (cubes, cuboids, cylinders, spheres), you can calculate volume directly from measured dimensions.

Method:

1. Measure the mass using a digital balance (record in kg or g).
2. Measure the relevant dimensions using a ruler, vernier callipers, or micrometer.
3. Calculate volume using the appropriate formula:
   • Cuboid: V = l × w × h
   • Cylinder: V = πr²h
   • Sphere: V = (4/3)πr³
4. Calculate density: ρ = m/V.

Worked example: A metal cylinder has mass 0.350 kg, diameter 2.40 cm, and height 5.00 cm.

• Radius = 2.40/2 = 1.20 cm = 0.0120 m
• Volume = π × (0.0120)² × 0.0500 = π × 1.44 × 10⁻⁴ × 0.0500 = 2.26 × 10⁻⁵ m³
• Density = 0.350 / 2.26 × 10⁻⁵ = 15 500 kg m⁻³

This is close to the density of tungsten (19 300 kg m⁻³) or lead (11 300 kg m⁻³), suggesting it could be a lead alloy.

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Measuring Density of Irregular Solids

When the shape is irregular and you cannot calculate volume from dimensions, you use displacement.

Method:

1. Measure the mass of the object on a balance.
2. Fill a measuring cylinder with water and record the initial volume V₁.
3. Carefully lower the object into the water (ensuring it is fully submerged and no air bubbles are trapped).
4. Record the new volume V₂.
5. Volume of object = V₂ − V₁.
6. Calculate density: ρ = m / (V₂ − V₁).

For larger objects, a eureka can (displacement can) can be used. Water overflows into a measuring cylinder through a spout when the object is submerged, and the collected water volume equals the object's volume.

Key source of error: Surface tension can cause water to cling to the spout, so you should wait until dripping stops completely before reading the volume.

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Measuring Density of Liquids

To find the density of a liquid:

1. Place an empty measuring cylinder on a balance and record its mass m₁.
2. Pour a known volume V of liquid into the measuring cylinder.
3. Record the new mass m₂.
4. Mass of liquid = m₂ − m₁.
5. Density = (m₂ − m₁) / V.

Read the meniscus at eye level to avoid parallax error. For water and most liquids, read from the bottom of the meniscus. For mercury, read from the top.

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Floating and Sinking

An object floats if its average density is less than the density of the fluid it is placed in. An object sinks if its average density is greater than the fluid density.

This explains why:

• A steel ship floats — the hull encloses a large volume of air, making the ship's average density much less than water.
• Ice floats on water — ice has a density of about 917 kg m⁻³, which is less than water's 1000 kg m⁻³.
• A helium balloon rises — helium (0.164 kg m⁻³) is less dense than air (1.225 kg m⁻³).

When an object floats, it displaces a weight of fluid equal to its own weight. This is a direct consequence of Archimedes' principle, which you will study in the next lesson.

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Comparing Densities of Materials

Different materials have vastly different densities:

Material	Density / kg m⁻³
Air (at sea level)	1.2
Cork	120
Ice	917
Water	1000
Aluminium	2700
Glass	2500
Iron/Steel	7800
Copper	8900
Lead	11 300
Gold	19 300

These values explain material selection in engineering. Aircraft use aluminium alloys (low density, reasonable strength) rather than steel. Electrical wiring uses copper (excellent conductor) despite its relatively high density because conductivity matters more than weight in that application.

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Density and the Particle Model

At the microscopic level, density depends on:

• The mass of individual atoms or molecules — heavier atoms mean higher density.
• How closely packed the particles are — tightly packed structures (like metals) have higher density.
• The arrangement of particles — diamond (3500 kg m⁻³) and graphite (2200 kg m⁻³) are both pure carbon, but their different crystal structures give different densities.

In solids, particles are closely packed in fixed positions, giving high density. In gases, particles are widely spaced, giving very low density. Liquids fall in between, with densities typically similar to (but slightly less than) the corresponding solid — water being a notable exception where the solid (ice) is less dense than the liquid.