Density and Pressure

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Define and use density
  2. Define and use pressure
  3. Derive, from the definitions of pressure and density, the equation for hydrostatic pressure ∆p = ρg∆h
  4. Use the equation ∆p = ρg∆h
  5. Understand that the upthrust acting on an object in a fluid is due to a difference in hydrostatic pressure
  6. Calculate the upthrust acting on an object in a fluid using the equation F = ρgV (Archimedes' principle)

1. Density

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What is density?

Density is the mass per unit volume of a substance. In simple terms, it tells us how tightly packed the matter (stuff) is in a given space. A dense material has a lot of mass squeezed into a small volume.

Formula:

ρ = m / V

Where:

  • ρ (the Greek letter rho) = density in kilograms per cubic metre (kg m⁻³)
  • m = mass in kilograms (kg)
  • V = volume in cubic metres (m³)

Understanding the formula:

  • If you have more mass in the same volume, the density increases
  • If you have the same mass spread over a larger volume, the density decreases

Units: The SI unit for density is kg m⁻³ (kilograms per cubic metre). Sometimes you might see g cm⁻³ (grams per cubic centimetre), but always convert to SI units in exams.

Key ideas about density:

  • Solids are usually more dense than liquids, which are more dense than gases
  • This is because particles in solids are packed tightly together, so there's more mass in a given volume
  • In liquids, particles are less tightly packed
  • In gases, particles are very spread out, so there's much less mass in the same volume

Calculating volume for different shapes:

Before you can calculate density, you often need to find the volume of an object. Here are the formulas for common shapes:

  • Cube: V = L³ (where L is the length of one side)
  • Cuboid (rectangular box): V = length × width × height
  • Cylinder: V = πr²h (where r is the radius and h is the height)
  • Sphere: V = (4/3)πr³ (where r is the radius)

Example calculation:

A metal block has a mass of 73 kg and dimensions 0.85 m × 0.50 m × 0.04 m. What is its density?

Step 1: Calculate the volume

  • V = length × width × height
  • V = 0.85 × 0.50 × 0.04
  • V = 0.017 m³

Step 2: Use the density formula

  • ρ = m / V
  • ρ = 73 / 0.017
  • ρ = 4294 kg m⁻³ (or 4300 kg m⁻³ to 2 significant figures)

Unit conversions - important tips:

When converting units, remember:

  • To convert from a larger unit to a smaller unit, you multiply
    • Example: 125 m = 125 × 100 = 12,500 cm
  • To convert from a smaller unit to a larger unit, you divide
    • Example: 5 g = 5 ÷ 1000 = 0.005 kg

For volume conversions, remember to cube the conversion factor:

  • 1 cm³ = 1 ÷ (100)³ = 1 × 10⁻⁶ m³
  • 1 mm³ = 1 ÷ (1000)³ = 1 × 10⁻⁹ m³

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