Physical quantities and the SI units

Five what?

Your friend says the road to the market is "five". Five what? Five metres is a few steps. Five kilometres is an hour's walk. The number alone tells you nothing.

So every measurement has two parts: a number and a unit. "$5\ \text{km}$" has both, and now you can picture the walk.

What a measurement really says

A physical quantity is anything we can measure: a length, a mass, a time, a speed, a force, a temperature. To measure it, we compare it with a fixed amount of the same kind, called the unit.

When we say a rope is $2.4\ \text{m}$ long, we mean: it is $2.4$ times as long as one metre. In short,

The rope does not care which unit we choose. Measure it in feet or in centimetres and the number changes, but the length stays the same. Try it.

You should find that a unit twice as big gives a number half as big. The number and the unit always multiply to the same length.

What makes a good unit

Long ago people used the length of a foot, or the width of a thumb. Your foot and mine differ, so we would never agree. A good unit is:

• well defined, so everyone means the same thing;
• the same everywhere and always;
• easy to reproduce in any laboratory;
• of a sensible size for what we measure.

Base and derived quantities

We do not need a separate unit for every quantity. We choose a few base quantities, give each a unit, and build everything else from them.

Speed is a length divided by a time, so its unit is metre per second, $\text{m s}^{-1}$. We call speed a derived quantity, and $\text{m s}^{-1}$ a derived unit.

The SI system

In 1960 scientists agreed on one system for the whole world: the Système International, or SI. It has seven base quantities.

Two angle units also appear often: the radian for an angle, and the steradian for a solid angle. An angle in radians is an arc divided by a radius, a length over a length, so it has no dimensions.

Watch out. The base unit of mass is the kilogram, not the gram. It is the only base unit with a prefix in its name.

Units from constants of nature

Until 2019, the kilogram was the mass of one metal cylinder kept near Paris. If it got dusty, the kilogram changed! Now every base unit is fixed by a constant of nature, which is the same everywhere in the universe.

• The second comes from the light of the caesium-133 atom: $9\,192\,631\,770$ of its cycles make one second.
• The metre comes from the speed of light, fixed at exactly $299\,792\,458\ \text{m/s}$.
• The kilogram comes from Planck's constant, $h$.
• The ampere, the kelvin and the mole come from the charge of the electron, the Boltzmann constant and Avogadro's number.

You do not need the digits. Remember the idea: any laboratory can rebuild the units from nature.

Derived units, from formulas

To find a derived unit, put the units into the formula. Force is mass times acceleration:

This unit gets its own name, the newton (N). In the same way:

A worked example

A school bag of mass $4\ \text{kg}$ is lifted $1.5\ \text{m}$ onto a shelf, against a weight of $40\ \text{N}$. How much work is done, and what is that in base units?

1. Work. $W = Fs = 40\ \text{N} \times 1.5\ \text{m} = 60\ \text{N m} = 60\ \text{J}$.
2. Base units. $\text{N m} = (\text{kg m s}^{-2})(\text{m}) = \text{kg m}^2\,\text{s}^{-2}$.

So $60\ \text{J} = 60\ \text{kg m}^2\,\text{s}^{-2}$. The joule is just a short name.

Writing units the right way

• A unit named after a person has a capital symbol (N, J, Pa) but a small name (newton, joule).
• Symbols take no plural s: $5\ \text{kg}$, never $5\ \text{kgs}$.
• Leave a space between the number and the symbol, and put no full stop after the symbol.
• Do not mix styles: write $\text{m s}^{-1}$ or $\text{m/s}$, not "metre/s".

Practise with new numbers

Try it in Python

The program changes a cricket ball's mass and speed to SI units, and then finds its kinetic energy.

Remember these

• A measurement is a number times a unit. The unit is part of the answer.
• SI has seven base units: m, kg, s, A, K, mol, cd. Every other unit is built from them.
• Find a derived unit by putting units into the formula: $\text{N} = \text{kg m s}^{-2}$, $\text{J} = \text{kg m}^2\,\text{s}^{-2}$.
• The base units are fixed by constants of nature, so any laboratory can rebuild them.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Is density a base quantity or a derived quantity? What is its SI unit?

2. Power is work done per second. Write the watt in SI base units.

3. A label says "Weight: 2 Kgs." Write it the SI way.

4. Why was the old kilogram, a metal cylinder kept in France, replaced?