Density

Cambridge IGCSE Physics 0625 · Core and Extended

Syllabus points

• Define density as mass per unit volume and recall the equation.
• Determine the density of a regularly shaped solid, an irregularly shaped solid and a liquid.
• Predict whether an object will float or sink by comparing densities.
• Explain why a liquid or gas rises when it is heated, in terms of density.

The equation

density = mass / volume

Rearranged: mass = density x volume, and volume = mass / density.

Units are g/cm³ or kg/m³. The two are not interchangeable and the conversion is a factor of 1000:

1 g/cm³ = 1000 kg/m³

Water is 1.0 g/cm³, which is 1000 kg/m³. Anything you calculate can be checked against that: 8 g/cm³ is about right for iron, 2.7 for aluminium, 1.2 kg/m³ for air. A liquid coming out at 120 000 kg/m³ is denser than any material on Earth and the working is wrong.

Use the units to check the operation. Multiplying g/cm³ by cm³ leaves grams, a mass. Dividing would leave cm⁶/g, which is not a unit of anything.

Regular solids

Measure the sides with a ruler, calculate the volume, weigh it on a balance.

• Cuboid: volume = length x width x height
• Prism or cylinder: volume = cross-sectional area x length

Worked example. A block has a cross-sectional area of 25 cm² and a length of 20 cm, and a mass of 4000 g.

Volume = 25 x 20 = 500 cm³. Density = 4000 / 500 = 8.0 g/cm³, which is about right for steel.

Where dimensions are read from a scale rather than given, each one is the difference between two readings. Rules in these questions are deliberately not started at zero, and every wrong option is usually an endpoint used instead of a difference.

Irregular solids

Use displacement, as in 1.1.

1. Read the level in a measuring cylinder.
2. Lower the object in until fully submerged.
3. Read the new level.
4. Volume of the object = the rise, not the final reading.

Weigh the object separately, then divide.

Worked example. A cylinder holds 50 cm³ of water and sits on a balance reading 150 g. An object is lowered in; the level rises to 75 cm³ and the balance reads 210 g.

Volume = 75 − 50 = 25 cm³. Mass = 210 − 150 = 60 g. Density = 60 / 25 = 2.4 g/cm³.

Two subtractions, and skipping either gives one of the offered answers. Both readings are totals, because the cylinder and the water were already there.

Cork and other floating solids need a sinker to hold them under.

Liquids

Weigh an empty container, add the liquid, weigh again, and subtract to get the mass of the liquid alone. Read the volume off the cylinder.

Worked example. An empty flask has a mass of 34 g. With 20 cm³ of liquid added, the total is 50 g.

Mass of liquid = 50 − 34 = 16 g. Density = 16 / 20 = 0.80 g/cm³.

Any question quoting a "total mass" is telling you to subtract the container first.

Comparing densities

Density is a ratio, so neither number decides it alone.

• The heaviest object on a page is not necessarily the densest. A block of 68 g in 40 cm³ is 1.7 g/cm³, and one of 54 g in 20 cm³ is 2.7 g/cm³.
• Two objects with the same mass can have different densities, and two with the same volume can too.

Whenever a question shows several objects, work out every density rather than looking for the biggest mass. The row with the largest number in the mass column is usually there to catch anyone who does not.

Floating and sinking

An object floats in a fluid if its density is less than the fluid's, and sinks if it is greater.

• Ice at about 0.92 g/cm³ floats on water at 1.0.
• Steel at 8 g/cm³ sinks, yet a steel ship floats, because the ship is mostly air and its average density is well below 1.0.

The same rule explains why oil floats on water and why a balloon filled with helium rises through air.

Density and heating

Heating a fluid makes it expand. The same mass now occupies a larger volume, and since density = mass / volume, the density falls.

Less dense fluid surrounded by denser fluid is pushed upwards, so it rises. Cooler, denser fluid sinks to take its place, and the loop that results is convection. Air warmed by a hot kettle becomes less dense and rises; this is the standard question and the reasoning is always the same two steps.

Note what changes and what does not. Squeezing a piece of foam reduces its volume, so its density rises, but its mass and weight are unchanged, because squeezing adds no matter.

Common mistakes

• Using the final level in a measuring cylinder as the volume instead of the rise.
• Forgetting to subtract the mass of the container from a stated total.
• Choosing the object with the greatest mass as the densest.
• Mixing g/cm³ and kg/m³, or missing the factor of 1000 between them.
• Dividing volume by mass instead of mass by volume; check with the units.
• Saying heating increases the density of a gas.
• Saying that compressing something increases its mass.
• Reading dimensions off a scale as endpoints rather than differences.