Cambridge IGCSE0625

Density

Physics 0625 Chapter Notes

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Density
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In this chapter

  1. 1Density: The 'Packedness' of Matter
  2. 2Mastering Density Calculations and Units
  3. 3How to Measure the Density of Solids
  4. 4How to Measure the Density of a Liquid
  5. 5Why Things Sink or Float

1. Density: The 'Packedness' of Matter

Density is a fundamental property of matter that describes how much 'stuff' (mass) is packed into a given amount of space (volume). A small, heavy object like a lead fishing weight is very dense, while a large, light object like a party balloon is not. The formal definition is mass per unit volume. We calculate it using the formula: density = mass / volume. This relationship means that for the same volume, a denser material will have more mass.

ρ = m / V

m = ρ × V

V = m / ρ

Key term

Density (ρ): Density is the mass of a substance per unit volume, a measure of how tightly matter is packed together.

Examiner insight

Examiners award marks for correctly stating the density formula and for using the correct symbols (ρ for density, m for mass, V for volume).

Common pitfall

Confusing mass with weight. Density is calculated using mass (the amount of matter, in kg or g), not weight (the force of gravity on that matter, in N).

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 540 g and a volume of 200 cm³. Calculate its density in g/cm³.

  1. 1

    Step 1: Write down the formula for density.

  2. 2

    ρ = m / V

  3. 3

    Step 2: Substitute the given values for mass and volume.

  4. 4

    ρ = 540 g / 200 cm³

  5. 5

    Step 3: Calculate the result.

  6. 6

    ρ = 2.7 g/cm³

Recap

  • Density is the mass per unit volume of a substance.
  • The symbol for density is the Greek letter rho (ρ).
  • The formula is ρ = m / V.
  • This formula can be rearranged to find mass (m = ρ × V) or volume (V = m / ρ).
  • A denser object has more mass in the same volume compared to a less dense object.

Quick check

  1. What are the three quantities related by the density formula?1 mark
  2. If two objects have the same volume, which one has more mass: the denser one or the less dense one?1 mark

2. Mastering Density Calculations and Units

In physics, using the correct units is crucial for getting the right answer. The official SI unit for density is kilograms per cubic metre (kg/m³). However, in a lab setting, it's often more convenient to measure mass in grams(g) and volume in cubic centimetres (cm³), giving a density in g/cm³. You must be able to convert between these two units. The key relationship to remember is: 1 g/cm³ = 1000 kg/m³. Always check that your units are consistent before you substitute values into the density formula.

1 g/cm³ = 1000 kg/m³

1 kg/m³ = 0.001 g/cm³

Key term

SI Unit: The internationally agreed system of units for scientific measurement; for density, the SI unit is kilograms per cubic metre (kg/m³).

Examiner insight

Marks are frequently lost due to incorrect unit conversions. Always write down your conversion step to make your working clear and to help avoid errors.

Common pitfall

Mixing units in a calculation, for example, using a mass in grams with a volume in cubic metres without first converting one of them.

Worked example 13 marks

Calculate the mass of a steel bar that has the same volume as a 5400 kg block of aluminium. The density of aluminium is 2700 kg/m³ and the density of steel is 7800 kg/m³.

  1. 1

    Step 1: Find the volume of the aluminium block. Rearrange the density formula to V = m / ρ.

  2. 2

    V_aluminium = 5400 kg / 2700 kg/m³ = 2 m³

  3. 3

    Step 2: The steel bar has the same volume.

  4. 4

    V_steel = 2 m³

  5. 5

    Step 3: Calculate the mass of the steel bar using m = ρ × V.

  6. 6

    m_steel = 7800 kg/m³ × 2 m³

  7. 7

    m_steel = 15600 kg

Worked example 21 mark

The density of gold is 19.3 g/cm³. What is its density in kg/m³?

  1. 1

    Step 1: Recall the conversion factor.

  2. 2

    1 g/cm³ = 1000 kg/m³

  3. 3

    Step 2: Multiply the density in g/cm³ by 1000.

  4. 4

    Density in kg/m³ = 19.3 × 1000

  5. 5

    Density in kg/m³ = 19300 kg/m³

Recap

  • The SI unit for density is kg/m³.
  • A common alternative unit is g/cm³.
  • To convert from g/cm³ to kg/m³, multiply by 1000.
  • To convert from kg/m³ to g/cm³, divide by 1000.
  • Ensure all quantities are in consistent units before calculating.

Quick check

  1. The density of water is 1 g/cm³. What is its density in kg/m³?1 mark

3. How to Measure the Density of Solids

To find the density of any object, you need to measure two things: its mass and its volume. The method for finding the volume depends on whether the object has a regular or irregular shape.

Regular Solids (e.g., a cube, cuboid):

  1. Measure its mass using a top-pan balance.
  2. Measure its length, width, and height using a ruler.
  3. Calculate its volume using the formula V = length × width × height.
  4. Calculate density using ρ = m / V.

Irregular Solids (e.g., a stone, a key):

  1. Measure its mass using a top-pan balance.
  2. Find its volume using the displacement method. Submerge the object in water in a measuring cylinder. The volume of the object is equal to the rise in the water level (Final volume - Initial volume). Alternatively, use a displacement can (Eureka can) and collect the overflowed water in a measuring cylinder.
  3. Calculate density using ρ = m / V.

Volume of a cuboid = length × width × height

Volume of object = Final water level - Initial water level

Key term

Displacement Method: A technique to measure the volume of an irregular object by measuring the volume of the fluid it displaces when fully submerged.

Examiner insight

When describing an experiment, clearly list the apparatus used (e.g., 'top-pan balance', 'ruler'), the specific measurements taken, and the final calculation performed.

Fun fact

The story of Archimedes shouting 'Eureka!' after discovering the principle of displacement in his bath is one of the most famous anecdotes in the history of science.

Worked example 13 marks

A student wants to find the density of a small metal cube. The cube has a mass of 48 g and each side has a length of 2.0 cm. Calculate its density.

  1. 1

    Step 1: Calculate the volume of the cube.

  2. 2

    V = length × width × height = 2.0 cm × 2.0 cm × 2.0 cm = 8.0 cm³

  3. 3

    Step 2: Use the density formula ρ = m / V.

  4. 4

    ρ = 48 g / 8.0 cm³

  5. 5

    ρ = 6.0 g/cm³

Worked example 23 marks

A rock of mass 60 g is placed in a measuring cylinder containing 50 cm³ of water. The water level rises to 75 cm³. Calculate the density of the rock.

  1. 1

    Step 1: Find the volume of the rock by displacement.

  2. 2

    Volume of rock = Final water level - Initial water level

  3. 3

    V = 75 cm³ - 50 cm³ = 25 cm³

  4. 4

    Step 2: Use the density formula ρ = m / V.

  5. 5

    ρ = 60 g / 25 cm³

  6. 6

    ρ = 2.4 g/cm³

Recap

  • To find the density of any solid, measure its mass and volume.
  • Mass is measured with a top-pan balance.
  • The volume of a regular solid is found by measuring its dimensions and calculating.
  • The volume of an irregular solid is found by water displacement.
  • Density is then calculated using ρ = m / V.

Quick check

  1. Which two instruments are needed to find the density of a rectangular wooden block?2 marks
  2. How do you find the volume of a small, irregularly shaped stone?1 mark

4. How to Measure the Density of a Liquid

Finding the density of a liquid follows the same principle: measure its mass and volume. The procedure is straightforward:

  1. Place an empty, dry measuring cylinder on a top-pan balance and record its mass (e.g., m₁).
  2. Pour the liquid into the measuring cylinder. Record the volume of the liquid (V) by reading the scale at eye level from the bottom of the meniscus.
  3. Place the measuring cylinder containing the liquid back on the balance and record the new, total mass (e.g., m₂).
  4. Calculate the mass of the liquid by subtracting the mass of the empty cylinder from the total mass (m = m₂ - m₁).
  5. Finally, calculate the density of the liquid using the formula ρ = m / V.

Mass of liquid = (Mass of cylinder + liquid) - (Mass of empty cylinder)

Key term

Meniscus: The curved upper surface of a liquid in a tube, which should be read from the bottom of the curve at eye level to get an accurate volume measurement for water.

Common pitfall

Forgetting to subtract the mass of the container (the measuring cylinder) and using the total mass in the density calculation. This gives an incorrectly high density value.

Worked example 13 marks

An empty measuring cylinder has a mass of 120 g. When 100 cm³ of oil is poured into it, the total mass becomes 205 g. Calculate the density of the oil.

  1. 1

    Step 1: Find the mass of the oil alone.

  2. 2

    Mass of oil = Total mass - Mass of empty cylinder

  3. 3

    m = 205 g - 120 g = 85 g

  4. 4

    Step 2: The volume of the oil is given as 100 cm³.

  5. 5

    V = 100 cm³

  6. 6

    Step 3: Calculate the density using ρ = m / V.

  7. 7

    ρ = 85 g / 100 cm³

  8. 8

    ρ = 0.85 g/cm³

Recap

  • To find a liquid's density, you need the mass and volume of the liquid itself.
  • Measure the mass of an empty cylinder first.
  • Measure the volume of the liquid in the cylinder.
  • Measure the combined mass of the cylinder and liquid.
  • Subtract the cylinder's mass to find the liquid's mass.
  • Calculate density using ρ = m / V.

Quick check

  1. A student measures the mass of a beaker of water as 250 g. The empty beaker has a mass of 150 g. What is the mass of the water?1 mark

5. Why Things Sink or Float

Whether an object sinks or floats in a fluid (a liquid or a gas) is determined by a simple comparison of densities. The rule is:

  • If an object is denser than the fluid it is placed in, it will sink.
  • If an object is less dense than the fluid, it will float.
  • If an object has the same density as the fluid, it will remain suspended within it (it will neither sink nor float to the top).

For example, the density of pure water is about 1.0 g/cm³. Wood has a density of about 0.75 g/cm³. Since 0.75 is less than 1.0, wood floats in water. Steel has a density of about 7.8 g/cm³. Since 7.8 is greater than 1.0, a solid piece of steel sinks in water.

Key term

Floating: An object floats when it is supported by a fluid because its average density is less than the density of the fluid.

Examiner insight

To earn full marks when explaining why something floats or sinks, you must explicitly state and compare the density of the object with the density of the fluid.

Fun fact

An iceberg floats with about 90% of its volume submerged because ice is about 10% less dense than seawater. The phrase 'tip of the iceberg' comes from this fact.

Worked example 12 marks

The table shows the density of various substances. Will a solid block of polythene float or sink in petrol? Explain your answer. (Density of polythene = 0.95 g/cm³, Density of petrol = 0.80 g/cm³).

  1. 1

    Step 1: Compare the density of the object (polythene) with the density of the fluid (petrol).

  2. 2

    Density of polythene = 0.95 g/cm³

  3. 3

    Density of petrol = 0.80 g/cm³

  4. 4

    Step 2: State the comparison.

  5. 5

    0.95 g/cm³ > 0.80 g/cm³. The density of polythene is greater than the density of petrol.

  6. 6

    Step 3: State the conclusion based on the rule.

  7. 7

    Because polythene is denser than petrol, it will sink.

Recap

  • An object sinks if its density is greater than the fluid's density.
  • An object floats if its density is less than the fluid's density.
  • The density of pure water is a key reference: 1.0 g/cm³ or 1000 kg/m³.
  • This principle applies to both liquids and gases.
  • A ship made of steel floats because its average density (including the air in its hull) is less than water's density.

Quick check

  1. An object has a density of 1200 kg/m³. Will it float or sink in pure water (density 1000 kg/m³)?1 mark

End-of-chapter exercise

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

  1. A piece of copper has a mass of 178 g and a volume of 20 cm³. Calculate the density of copper.2 marks
  2. A block of wood measures 5 cm × 10 cm × 4 cm and has a mass of 150 g. Calculate its density in g/cm³. Will it float in water (density 1.0 g/cm³)?4 marks
  3. The density of mercury is 13600 kg/m³. What is the volume of 680 kg of mercury?2 marks
  4. Describe, step-by-step, how you would accurately measure the density of a small, irregularly shaped stone in a school laboratory.5 marks
  5. A storage tank measures 4 m × 2 m × 1.5 m. Calculate the mass of petrol it can hold if the density of petrol is 800 kg/m³.3 marks
  6. An empty bottle has a mass of 30 g. When filled with water (density 1.0 g/cm³), its mass is 80 g. When filled with a different liquid, its mass is 74 g. Calculate the density of the other liquid.4 marks
  7. A block of material has a mass of 390 g and a volume of 50 cm³. Use the table to identify the material. | Substance | Density (g/cm³) | | --- | --- | | Steel | 7.8 | | Aluminium | 2.7 | | Glass | 2.5 |3 marks
  8. A crown has a mass of 1.8 kg. When it is submerged in water, it displaces 100 cm³ of water. Is the crown pure gold? Explain your answer. (Density of gold = 19.3 g/cm³, Density of water = 1.0 g/cm³).4 marks
  9. A hollow plastic cube has external side lengths of 10 cm. Its mass is 90 g. Calculate the average density of the cube. Explain why this value is much lower than the density of solid plastic (approx 0.9 g/cm³).3 marks
  10. Calculate the weight of water in a swimming pool that is 25 m long, 10 m wide and has a water depth of 2 m. (Density of water = 1000 kg/m³, g = 10 N/kg).4 marks

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