Cambridge IGCSE0972

Density

Physics 0972 Chapter Notes

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Density
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In this chapter

  1. 1Understanding Density
  2. 2The Density Formula Triangle
  3. 3Measuring Density in the Lab
  4. 4Floating and Sinking

1. Understanding Density

Density is a fundamental property of a substance that tells us how 'compact' it is. It measures how much mass (or 'stuff') is packed into a certain amount of space (volume). For example, a block of lead and a block of wood of the same size have very different masses because lead is much denser. We calculate density using the formula: density = mass / volume. A substance with a low density, like a sponge, has less mass in a given volume compared to a high-density substance like a steel ball.

ρ = m / V

Density = Mass / Volume

Key term

Density: The mass per unit volume of a substance.

Examiner insight

Examiners expect you to be able to define density in terms of mass and volume and state the correct units.

Common pitfall

Confusing density with mass. A large object is not necessarily denser than a small one. For example, a huge log is less dense than a small pebble.

Fun fact

The densest known material in the universe is found in neutron stars, where a sugar-cube-sized amount of material would have a mass of over 100 million tonnes.

Worked example 12 marks

A block of granite has a mass of 5400 g and a volume of 2000 cm³. Calculate the density of the granite in g/cm³.

  1. 1

    Step 1: Write down the formula for density.

  2. 2

    Density = Mass / Volume

  3. 3

    Step 2: Substitute the given values into the formula.

  4. 4

    Density = 5400 g / 2000 cm³

  5. 5

    Step 3: Calculate the result.

  6. 6

    Density = 2.7 g/cm³

Recap

  • Density is a measure of how much mass is contained in a given volume.
  • The formula for density is ρ = m / V.
  • High density means a lot of mass is packed into a small volume.
  • Low density means a small amount of mass is spread over a large volume.
  • The standard unit for density is kilograms per cubic metre (kg/m³), but grams per cubic centimetre (g/cm³) is also commonly used.

Quick check

  1. If two objects have the same volume but object A has more mass, which object is denser?1 mark

2. The Density Formula Triangle

The relationship between density (ρ), mass (m), and volume (V) can be remembered using a formula triangle. This is a very useful tool for rearranging the equation to find any one of the three quantities if you know the other two. To use it, cover the quantity you want to find. The position of the other two quantities shows you the calculation to perform. If they are side-by-side, you multiply. If one is on top of the other, you divide.

ρ = m / V

m = ρ × V

V = m / ρ

Key term

Formula Triangle: A visual aid used to help rearrange the three-variable equations for density, mass, and volume.

Examiner insight

Examiners award marks for showing the correct rearrangement of the formula before substituting numbers. Don't just write down the answer from your calculator.

Common pitfall

Using inconsistent units is the most common mistake. For example, trying to calculate density using a mass in grams and a volume in cubic metres without converting first.

Worked example 13 marks

The density of air is approximately 1.3 kg/m³. What is the mass of air in a room measuring 5 m × 4 m × 3 m?

  1. 1

    Step 1: Calculate the volume of the room.

  2. 2

    Volume (V) = length × width × height = 5 m × 4 m × 3 m = 60 m³

  3. 3

    Step 2: State the formula for mass, rearranged from the density equation.

  4. 4

    Mass(m) = Density (ρ) × Volume (V)

  5. 5

    Step 3: Substitute the values for density and volume.

  6. 6

    m = 1.3 kg/m³ × 60 m³

  7. 7

    Step 4: Calculate the mass.

  8. 8

    m = 78 kg

Worked example 22 marks

Gold has a density of 19.3 g/cm³. What is the volume of a gold bar with a mass of 965 g?

  1. 1

    Step 1: State the formula for volume, rearranged from the density equation.

  2. 2

    Volume (V) = Mass(m) / Density (ρ)

  3. 3

    Step 2: Substitute the values for mass and density.

  4. 4

    V = 965 g / 19.3 g/cm³

  5. 5

    Step 3: Calculate the volume.

  6. 6

    V = 50 cm³

Recap

  • Use the formula triangle to easily find the correct rearrangement.
  • To find mass, calculate: m = ρ × V.
  • To find volume, calculate: V = m / ρ.
  • Always ensure your units are consistent before you calculate.
  • If density is in kg/m³, volume must be in m³ and mass will be in kg.
  • If density is in g/cm³, volume must be in cm³ and mass will be in g.

Quick check

  1. A substance has a density of 2 g/cm³. What is the mass of 10 cm³ of it?1 mark
  2. An object has a mass of 100 g and a density of 5 g/cm³. What is its volume?1 mark

3. Measuring Density in the Lab

To find the density of any object, you always need to measure two things: its mass and its volume. Mass is always measured using a top-pan balance. The method for finding the volume depends on the object's shape.

For a regular solid (like a cube or cuboid), measure its length, width, and height with a ruler, then calculate the volume (V = l × w × h).

For an irregular solid (like a stone), you must use the displacement method. Submerge the object in a measuring cylinder of water and measure the change in volume. The volume of the object is equal to the volume of water it displaces (Final volume - Initial volume). Alternatively, use a Eureka can (displacement can); the volume of water that overflows into a measuring cylinder when the object is submerged is the object's volume.

For a liquid, find its mass by measuring the mass of an empty measuring cylinder, then the mass of the cylinder with the liquid in it. The difference is the liquid's mass. The volume is read directly from the scale on the measuring cylinder.

Volume of a cuboid = length × width × height

Volume of object = Final water level - Initial water level

Key term

Displacement: The technique of finding the volume of an object by measuring the volume of fluid it displaces when fully submerged.

Fun fact

The story of Archimedes and the king's crown is a classic example of using displacement. He realised a fake crown mixed with cheaper, less dense silver would have a larger volume for the same mass, and would therefore displace more water than a pure gold one.

Worked example 13 marks

A student wants to find the density of a small, irregular stone. They measure its mass as 84 g. They then put 50 cm³ of water into a measuring cylinder. When the stone is fully submerged in the water, the water level rises to 80 cm³. Calculate the density of the stone.

  1. 1

    Step 1: Find the volume of the stone using the displacement method.

  2. 2

    Volume of stone = Final water level - Initial water level

  3. 3

    Volume = 80 cm³ - 50 cm³ = 30 cm³

  4. 4

    Step 2: State the formula for density.

  5. 5

    Density (ρ) = Mass(m) / Volume (V)

  6. 6

    Step 3: Substitute the known mass and the calculated volume.

  7. 7

    ρ = 84 g / 30 cm³

  8. 8

    Step 4: Calculate the density.

  9. 9

    ρ = 2.8 g/cm³

Recap

  • To find density, you must measure mass and volume.
  • Use a top-pan balance to measure mass.
  • For regular solids, calculate volume from measured dimensions.
  • For irregular solids, find volume by water displacement.
  • The volume of an irregular object is the amount the water level rises when it is submerged.
  • For liquids, measure mass by difference and read volume from the measuring cylinder.

Quick check

  1. What two pieces of apparatus are essential for finding the density of an irregular stone?2 marks

4. Floating and Sinking

Whether an object floats or sinks in a fluid (a liquid or a gas) depends on a simple comparison of densities. An object will float if its average density is less than the density of the fluid it is placed in. An object will sink if its average density is greater than the density of the fluid. If the densities are exactly equal, the object will be suspended, neither sinking nor rising. This is why a log (density ~0.75 g/cm³) floats on water (density 1.0 g/cm³), but a rock (density ~2.7 g/cm³) sinks. It's also why a steel ship can float: although steel is very dense, the ship's hull contains a huge volume of air, making its *average* density (total mass divided by total volume) less than that of water.

Key term

Average Density: The total mass of an object divided by its total volume, which is the property that determines whether it will float or sink.

Examiner insight

Examiners look for a direct comparison of densities in your explanation. Clearly state which density is greater and link this to your conclusion about floating or sinking.

Common pitfall

Incorrectly stating that 'heavy things sink' and 'light things float'. A massive aircraft carrier floats, while a tiny pebble sinks. It is density, not mass, that is the deciding factor.

Worked example 14 marks

The density of water is 1.0 g/cm³ and the density of petrol is 0.80 g/cm³. A block of polythene has a density of 0.95 g/cm³. Will the block float or sink in(a) water, and(b) petrol? Explain your answers.

  1. 1

    (a) In water:

  2. 2

    Step 1: Compare the density of polythene to the density of water.

  3. 3

    Density of polythene (0.95 g/cm³) < Density of water (1.0 g/cm³).

  4. 4

    Step 2: State the conclusion based on the comparison.

  5. 5

    Since the polythene is less dense than water, it will float.

  6. 6

    (b) In petrol:

  7. 7

    Step 1: Compare the density of polythene to the density of petrol.

  8. 8

    Density of polythene (0.95 g/cm³) > Density of petrol (0.80 g/cm³).

  9. 9

    Step 2: State the conclusion based on the comparison.

  10. 10

    Since the polythene is denser than petrol, it will sink.

Recap

  • An object floats if its density is less than the fluid's density.
  • An object sinks if its density is greater than the fluid's density.
  • This principle applies to both liquids and gases.
  • The average density of an object determines if it floats, which is why steel ships can float.
  • Ice floats because it is one of the few substances that is less dense in its solid form than its liquid form.

Quick check

  1. The density of mercury is 13.6 g/cm³. Will a block of solid gold (density 19.3 g/cm³) float or sink in liquid mercury?1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A block of aluminium has a mass of 540 g and a volume of 200 cm³. A block of steel has a mass of 780 g and a volume of 100 cm³. Which material is denser? Show your calculations.4 marks
  2. The density of petrol is 800 kg/m³. A car's fuel tank has a volume of 0.05 m³. Calculate the mass of petrol required to fill the tank.2 marks
  3. A student measures the mass of a rock to be 120 g. They place it in a Eureka can filled with water, and 40 cm³ of water is collected in a measuring cylinder. Calculate the density of the rock.2 marks
  4. A plastic bag filled with air has a volume of 0.008 m³. When the air in the bag is squeezed into a rigid container, the mass of the container (with air) increases from 0.020 kg to 0.0304 kg. Calculate the density of the air.3 marks
  5. A block of wood has dimensions 10 cm × 5 cm × 4 cm and a mass of 150 g. (a) Calculate its density. (b) Will it float in water (density 1.0 g/cm³)? Explain your answer.4 marks
  6. An object made of copper (density = 8.9 g/cm³) has a mass of 445 g. What is its volume?2 marks
  7. Explain, in terms of particles, why a solid is typically denser than a gas.2 marks
  8. A measuring cylinder contains 40 ml of water. A bunch of keys with a mass of 96 g is lowered into the cylinder, and the water level rises to 52 ml. Given that 1 ml = 1 cm³, calculate the density of the keys.3 marks
  9. The density of lead is 11,400 kg/m³ and the density of aluminium is 2,700 kg/m³. What mass of lead has the same volume as 540 kg of aluminium?4 marks
  10. A hollow gold sphere has an outer radius that defines a volume of 30 cm³. Its mass is 386 g. The density of solid gold is 19.3 g/cm³. Is the sphere made of pure gold, or is it a fake? Justify your answer with a calculation.3 marks

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