In the last topic we found the density of a substance by dividing its mass by its volume. That gives a number in kilograms per cubic metre or grams per cubic centimetre, and those numbers are awkward to hold in your head. There is a tidier way to say the same thing, which is to compare every substance with one particular substance that everybody has: water.
Relative density
The density of an object is compared with the density of water at 4 degrees Celsius, and from that comparison the relative density of the object is found. The density of water at 4 degrees Celsius is 1 gram per cubic centimetre.
The ratio of the density of any substance to the density of water at 4 degrees Celsius is called relative density. From the relative density we get information about how much less or how much more the mass of any substance is, compared with the same volume of water at 4 degrees Celsius.
| Relative density \( = \dfrac{\text{density of the object}}{\text{density of water at 4 °C}} \) |
Since relative density is a ratio between two densities, it has no unit at all. The units on the top and the bottom are the same, so they cancel out and only a bare number is left.
Density and relative density
| SN | Name of the substance | Density (kg/m³) | Density (g/cm³) | Relative density |
|---|---|---|---|---|
| 1 | Ice | 920 | 0.92 | 0.92 |
| 2 | Aluminium | 2700 | 2.7 | 2.7 |
| 3 | Iron | 7800 | 7.8 | 7.8 |
Look at that table again and notice something that looks like a printing error. The density in grams per cubic centimetre and the relative density are exactly the same number, every single time. That is not a mistake and it is not a coincidence either; it happens because of the unit we chose. Relative density is the density of the substance divided by the density of water, and the density of water at 4 degrees Celsius happens to be exactly 1 gram per cubic centimetre. Dividing any number by one leaves it unchanged. So if you already know a density in grams per cubic centimetre, you already know the relative density, and you do not have to calculate anything. This is enormously convenient, and it is worth understanding rather than just noticing, because the trick fails completely if the density is given in kilograms per cubic metre. Iron is 7800 kg/m³, but its relative density is 7.8 and not 7800, because water in those units is 1000 rather than 1.
Worked example 1: the relative density of gold
If the density of gold is given as 19 grams per cubic centimetre, what is its relative density? The density of water at 4 degrees Celsius is 1 gram per cubic centimetre.
| Step | Working |
|---|---|
| Given | Density of gold = 19 g/cm³, density of water at 4 °C = 1 g/cm³ |
| Formula | \( \text{RD} = \dfrac{19 \text{ g/cm}^3}{1 \text{ g/cm}^3} \) |
| Answer | RD = 19, with no unit, because the units cancel |
The relative density of gold is therefore 19. That single number tells you something you can picture at once: for the same volume, the mass of gold is 19 times greater than the mass of water. Fill a small bottle with water and it might hold 100 grams; fill the same bottle with gold and it would hold 1900 grams, which is nearly two kilograms. This is why a small gold ornament feels surprisingly heavy in the hand, and it is also the reason a fake made from a lighter metal gives itself away the moment somebody weighs it properly.
Floating and sinking
If the relative density of any object is more than 1, that object sinks in water, and if its relative density is less than 1, that object floats in water. In the same way, if the density of an object is greater than the density of a liquid, the object sinks in that liquid, and if the density of the object is less than that of the liquid, it floats.
Almost everybody arrives at this topic believing that heavy things sink and light things float, and it is worth destroying that idea straight away with two examples. A sewing needle has a mass of about one gram and it sinks the instant you drop it into water. A cargo ship has a mass of many thousands of tonnes and it floats. So being heavy cannot be what decides it. Notice what the rule above actually compares: not the mass of the object, and not its size, but its density set against the density of the liquid. Heaviness on its own is meaningless here, because a big enough piece of anything is heavy. The only question that matters is whether this substance is more crowded than water or less crowded than water, and that is a question about the material rather than about the amount of it. This is exactly why density had to be introduced before floating could be explained.
Worked example 2: an iron box that floats
If the mass of an iron box measuring 1 m by 0.5 m by 0.2 m is 20 kg, find the density of that iron box. On the basis of the figure you have calculated, will that iron box sink or float in water? Give the reason.
| Step | Working |
|---|---|
| Volume of the iron box | \( V = 1 \times 0.5 \times 0.2 = 0.1 \) m³ |
| Mass of the iron box | m = 20 kg |
| Formula | \( D = \dfrac{m}{V} \) |
| Substitute | \( D = \dfrac{20}{0.1} = 200 \) kg/m³ |
| Compare with water | 200 kg/m³ is less than the density of water, which is 1000 kg/m³ |
| Conclusion | The iron box floats in water |
Look at that answer again, because at first sight it seems impossible. Iron has a density of 7800 kg/m³, which is nearly eight times that of water, and a nail made of iron sinks without hesitation. Yet this box, also made of iron, comes out at 200 kg/m³ and floats. The two figures do not disagree, because they are densities of two different things. The 7800 is the density of solid iron. The 200 is the density of the box, and the box is not solid iron at all: it is a thin shell of iron with a great deal of air inside it, and the volume in the calculation is the volume of the whole box, air included. Air is very light, so the same twenty kilograms of iron has been spread across a much larger volume, and spreading mass over more volume is exactly what lowers density. That single idea is how every ship in the world floats. Steel is denser than water, but a ship is a steel shell wrapped around an enormous amount of air, and the ship as a whole is less dense than water. Punch a hole in it, let the water take the place of the air, and the average density rises past 1000 and the ship goes down.
The dividing line is 1, but only for water
Read the rule once more and notice that it comes in two versions. The first version says that a relative density above 1 sinks and below 1 floats, and that version works only for water, because relative density is measured against water in the first place. The second version is the general one: an object sinks in a liquid if the object is denser than that liquid, and floats if it is less dense. The number 1 is not magic; it simply happens to be where water sits on its own scale.
You can watch that dividing line move for yourself. Put a fresh egg into a glass of plain water and it sinks, because an egg is slightly denser than water. Now stir several spoonfuls of salt into the water and put the egg back. It floats. Nothing about the egg has changed at all: it has the same mass, the same volume and the same density it had a minute ago. What changed is the liquid. Dissolved salt adds mass to the water without adding much volume, so the salt water is denser than plain water, and once the liquid becomes denser than the egg, the egg rises. This is also why swimming is noticeably easier in the sea than in a river.
| Object | Relative density | In water it will |
|---|---|---|
| Cork | About 0.24 | Float high, with most of it above the surface |
| Wood | About 0.7 for most kinds | Float low, with much of it under the surface |
| Ice | 0.92 | Just barely float, which is why an iceberg is mostly hidden |
| Aluminium | 2.7 | Sink |
| Iron | 7.8 | Sink quickly |
Notice the second column doing more work than just saying float or sink. Cork at 0.24 floats with about a quarter of itself under the water, and wood at 0.7 floats with about seven tenths of itself submerged, which is why a log rides so much lower than a cork. Ice at 0.92 barely wins, so about nine tenths of an iceberg lies below the surface and only a tenth is visible. The relative density does not just decide whether a thing floats; it decides how much of it sticks out.
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