Syllabus coverage
- Define density as mass per unit volume, and recall and use the equation density = mass / volume
- Describe how to determine the density of a liquid, including the appropriate calculations
- Describe how to determine the density of a regularly shaped solid, including the appropriate calculations
- Describe how to determine the density of an irregularly shaped solid which sinks in a liquid, using volume by displacement, including the appropriate calculations
- Determine whether an object floats based on density data
- Determine whether one liquid will float on another liquid based on density data, given that the liquids do not mix
Part of the IGCSE Physics syllabus 0625.
Density: mass per unit volume
Density is the mass per unit volume of a substance. It tells you how much matter is packed into each cubic centimetre or each cubic metre. A kilogram of feathers and a kilogram of lead have exactly the same mass, but the lead takes up far less space, so lead has a much greater density. Density is a property of the material rather than of the particular lump in front of you: a small nail and a heavy girder made of the same steel have the same density.
The equation is density = mass / volume, which is also written as rho = m / V. Take the mass in grams or kilograms, divide by the volume in cubic centimetres or cubic metres, and the answer is the density. Because a mass is divided by a volume, the unit of density is always a mass unit divided by a volume unit.
The two units you will meet are g/cm3, used for small laboratory objects, and kg/m3, used for larger objects and for the densities of liquids and gases. They are linked by a factor of 1000: 1 g/cm3 = 1000 kg/m3. Water has a density of about 1.0 g/cm3, which is 1000 kg/m3, and that value is worth remembering because it decides what floats.
Rearranging the equation and using it with numbers
The equation contains three quantities, so it rearranges in two ways. To find a mass, multiply: mass = density x volume. To find a volume, divide: volume = mass / density. Questions often give you the density and one other quantity and expect you to pick the right form for yourself.
Here is the calculation the right way round. A block of metal has a mass of 540 g and a volume of 200 cm3. Write the equation, then substitute: density = mass / volume = 540 / 200 = 2.7 g/cm3. The unit follows from the units you put in, because grams divided by cubic centimetres gives grams per cubic centimetre.
Now the other way round. A block of wood has a volume of 500 cm3 and a density of 0.60 g/cm3. The unknown is the mass, so use mass = density x volume = 0.60 x 500 = 300 g. The unit works too: g/cm3 multiplied by cm3 leaves grams, because the cm3 cancel.
You will sometimes be given a density in kg/m3 and a volume in cm3. Convert one of them before you calculate, or the answer will be out by a factor of 1000. To change g/cm3 into kg/m3, multiply by 1000; to change kg/m3 into g/cm3, divide by 1000. Aluminium, at 2.7 g/cm3, is 2700 kg/m3.
Measuring the density of a regularly shaped solid
A solid with a regular shape has a volume you can calculate from its dimensions, so you need only a balance for the mass and a ruler or vernier calipers for the lengths. Weigh the solid first, then work out its volume from the shape, then divide.
The formulas are: a cube, volume = side x side x side; a cuboid, volume = length x width x height; and a cylinder, volume = pi x (radius)2 x height. Measure each length more than once, in different places, and take an average, because the faces of a real block are never perfectly flat.
Worked example. A metal cuboid measures 5.0 cm by 4.0 cm by 2.0 cm and has a mass of 312 g. Its volume is 5.0 x 4.0 x 2.0 = 40 cm3, so its density is 312 / 40 = 7.8 g/cm3, which is the density of steel.
Measuring the density of a liquid
A liquid has no fixed shape, so you cannot measure its volume with a ruler. Pour it into a measuring cylinder and find its mass by weighing the container before and after.
The method: weigh the empty measuring cylinder and record its mass. Pour in the liquid and read the volume from the scale. Weigh the cylinder and the liquid together, then subtract the mass of the empty cylinder to find the mass of the liquid alone. Finally density = mass of liquid / volume of liquid.
Read the volume with your eye level with the bottom of the meniscus, the curved surface of the liquid. Looking from above or below the mark gives a parallax error and the reading will be wrong. Worked example: an empty cylinder has a mass of 45.0 g; with 50 cm3 of liquid in it the mass is 90.0 g. The mass of the liquid is 90.0 - 45.0 = 45.0 g, so its density is 45.0 / 50 = 0.90 g/cm3, about the density of cooking oil.
| Step | What to do | Why |
|---|---|---|
| 1 | Weigh the empty measuring cylinder | Its mass must be subtracted to give the mass of the liquid alone |
| 2 | Pour in the liquid and read the volume at eye level, at the bottom of the meniscus | Avoids a parallax error and gives the true volume of the liquid |
| 3 | Weigh the cylinder and the liquid together, then subtract the empty mass | Gives the mass of the liquid alone |
| 4 | Divide the mass of the liquid by its volume | Gives the density of the liquid, in g/cm3 |
Volume by displacement for irregular solids
A stone or a screw has no usable formula for its volume, so you find the volume of water it displaces instead. This is the displacement method, and it only works if the solid sinks, so that it can be pushed completely under the surface.
With a measuring cylinder: pour in enough water to cover the solid, read the starting volume V1, then lower the solid in on a thread until it is fully under the water and read the new volume V2. The volume of the solid is V2 - V1.
With a eureka can: fill the can until water just runs from the spout, wait until it stops dripping, then lower the solid in. Water equal in volume to the solid overflows into a measuring cylinder, and the volume collected is the volume of the solid. This is the better method for a large or awkwardly shaped object.
Worked example. A stone of mass 60 g makes the water level in a measuring cylinder rise from 50 cm3 to 74 cm3. The volume of the stone is 74 - 50 = 24 cm3, so its density is 60 / 24 = 2.5 g/cm3.
Floating and sinking: comparing densities
Whether an object floats depends on how its density compares with the density of the liquid around it. An object floats if its density is less than the density of the liquid, and sinks if its density is greater. If the two densities are equal, the object stays where it is put, suspended in the liquid.
Ice has a density of about 0.92 g/cm3 and water about 1.0 g/cm3, so ice floats. The two values are close, which is why an iceberg floats with only a small part above the surface: roughly a tenth of the ice is in the air and about nine tenths is below the waterline. Wood floats for the same reason, and a stone sinks because its density of about 2.5 g/cm3 is greater than that of water.
The same rule explains a ship. Steel has a density of about 7.8 g/cm3 and a solid block of it sinks, but a ship is hollow, so it contains a large volume of air. The average density of the whole ship, its total mass divided by its total volume including the air spaces, is less than the density of water, so the ship floats. If the hull is breached and floods, the average density rises above that of water and the ship sinks.
Gases have densities too, and the rule still applies. Warm air is less dense than the cold air around it, so a hot-air balloon rises: the balloon floats in air exactly as a block of wood floats in water.
Liquids that do not mix, and the particle picture
The same comparison works for two liquids that do not mix. The liquid with the lower density floats on top of the liquid with the higher density. Oil has a density of about 0.90 g/cm3 and water about 1.0 g/cm3, so a layer of oil always settles on top of a layer of water, which is why an oil-and-vinegar dressing separates. Mercury, at 13.6 g/cm3, sinks to the bottom of either of them.
You can predict the order of the layers from the numbers alone. List the liquids from the greatest density to the smallest; the first liquid on your list will be at the bottom of the container and the last will be at the top. This is exactly how a density column is built in the laboratory.
Why does density vary from material to material? In the particle model, a denser material has more particles packed into the same volume. The particles of a metal are heavy and closely packed, while the particles of a gas are far apart with mostly empty space between them, so a gas is hundreds of times less dense than a solid. Heating a substance usually makes it expand, so the same mass fills a larger volume and the density falls.
| Substance | Density in g/cm3 | Density in kg/m3 |
|---|---|---|
| Air | 0.0012 | 1.2 |
| Polystyrene foam | 0.02 | 20 |
| Oak wood | 0.70 | 700 |
| Ice | 0.92 | 920 |
| Water | 1.0 | 1000 |
| Aluminium | 2.7 | 2700 |
| Steel | 7.8 | 7800 |
| Lead | 11.3 | 11300 |
| Mercury | 13.6 | 13600 |
| Gold | 19.3 | 19300 |
Key terms you must be able to define
- Density
- The mass per unit volume of a substance, given by density = mass / volume and measured in kg/m3 or g/cm3.
- Displacement method
- Finding the volume of an irregular solid by measuring the volume of water it pushes aside when it is fully submerged.
- Eureka can
- A container with a spout, filled to the spout with water; an object lowered into it displaces its own volume of water, which overflows and is collected and measured.
- Meniscus
- The curved surface of a liquid in a measuring cylinder; the volume is read at eye level at the bottom of the meniscus.
- Parallax error
- An error in a reading caused by viewing the scale from the wrong angle, avoided by reading at eye level.
- Flotation
- An object floats in a liquid when its density is less than the density of the liquid, and sinks when its density is greater.
- Average density
- The total mass of an object divided by its total volume; a hollow steel ship floats because its average density, including the air inside, is less than the density of water.
- Density units
- g/cm3 is used for small laboratory objects and kg/m3 for larger ones; 1 g/cm3 = 1000 kg/m3.
Applied to the real world
Cambridge sets physics in unfamiliar contexts - a car braking, a power station, a hospital scanner, a satellite. Use these as models for how to apply the physics, then write your own.
You are asked to find the density of a small metal block. You weigh it on a balance to get 312 g, measure its sides with vernier calipers as 5.0 cm, 4.0 cm and 2.0 cm, and calculate the volume as 40 cm3. Dividing gives 7.8 g/cm3, which is the density of steel, so the block is identified as steel rather than aluminium at 2.7 g/cm3.
Density is used to test whether a piece of jewellery is really gold. A crown with a mass of 965 g and a volume of 50 cm3 has a density of 965 / 50 = 19.3 g/cm3, which matches gold. A crown of the same mass but a larger volume would have a lower density and could not be solid gold. This is the method Archimedes is said to have used.
An iceberg floats because the density of ice, about 0.92 g/cm3, is less than the density of sea water, about 1.0 g/cm3. Because the two densities are close, only about a tenth of the iceberg shows above the surface and the rest is hidden below, which is why icebergs are a danger to ships. Ships themselves float because their hollow hulls give them an average density less than that of water.
Oil and water do not mix, and oil has the lower density, so it floats as a separate layer on top of the water. The same thing happens in a bottle of oil-and-vinegar dressing, which is why it has to be shaken before use. If you pour oil into a pan of water, the oil spreads across the surface instead of sinking.
Polystyrene foam has a density of only about 20 kg/m3, while concrete is about 2400 kg/m3. A block of foam the same size as a concrete block is roughly a hundred times lighter, so foam is used for insulation and lightweight packing, while concrete is used where strength and mass are wanted.
The air inside a hot-air balloon is heated, so it expands and becomes less dense than the cooler air outside. The balloon therefore floats in the air in the same way that a block of wood floats in water. When the air inside cools, its density rises and the balloon descends.
Flashcards
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Exam-style questions
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Exam technique for this topic
- Write the equation, the substitution and the answer on separate lines, and give the unit every time. A density of 2.7 with no unit is a lost mark, because g/cm3 and kg/m3 differ by a factor of 1000.
- For an irregular solid the volume is the rise in the water level, V2 - V1. Using the final reading alone is the single most common error in this topic.
- Check the units before you substitute. If the mass is in grams and the volume in cm3 the density is in g/cm3; to turn that into kg/m3, multiply by 1000.
- In a floating question, compare the density of the object with the density of the liquid and say which is smaller. 'It is light' or 'it is heavy' scores nothing without that comparison.
- When you describe measuring the density of a liquid, say that the empty container is weighed first and that the volume is read at eye level at the bottom of the meniscus. Both are standard marks.
- For a hollow object such as a ship, use its average density, which is the total mass divided by the total volume including the air inside.
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Where this fits in IGCSE Physics
This page is one topic of Cambridge IGCSE Physics 0625. IGCSE Physics is the course home: every area and every topic in Cambridge's own order, the papers you sit and what each one tests, and the exam-practice tools. The full topic list, with each topic linked to its revision page, is on the IGCSE Physics syllabus page.