SI units
One language for science
- Long ago every place used its own units — feet, cubits, pounds.
- Science and trade were a mess of conversions.
- The SI system fixes this: one shared set of units for everyone.
The five base units
- Every unit in physics is built from just five base units.
- Learn these first, then build the rest.
Select all the SI base units.
The five SI base units are the kilogram, metre, second, ampere and kelvin. The newton and joule are derived units, built from the base units.
Fun fact: the kilogram
- Until 2019 the kilogram was a real metal cylinder kept in Paris.
- Now it is fixed by a constant of nature instead.
Build a derived unit from its equation
- A derived unit is base units multiplied or divided.
- Build it from the equation that defines the quantity.
- Simple ones: speed $= \dfrac{\text{m}}{\text{s}}$, acceleration $= \dfrac{\text{m}}{\text{s}^2}$.
Match each derived unit to its form in SI base units.
Each comes from its defining equation: force gives the newton, energy the joule, power the watt, and pressure the pascal.
Worked example: the watt in base units
- power = energy ÷ time, so $\text{W} = \dfrac{\text{J}}{\text{s}}$.
- Put in the joule: $\dfrac{1}{\text{s}} \times \dfrac{\text{kg}\cdot\text{m}^2}{\text{s}^2}$.
- Simplify: $\dfrac{\text{kg}\cdot\text{m}^2}{\text{s}^3}$.
Checking equations: homogeneity
- An equation is homogeneous when both sides have the same base units.
- In $v^{2} = u^{2} + 2as$, every term works out to $\dfrac{\text{m}^2}{\text{s}^2}$ — so it balances.
Is the equation $v = u + at$ homogeneous?
Yes — every term is a speed. $v$ and $u$ are $\dfrac{\text{m}}{\text{s}}$, and $at = \dfrac{\text{m}}{\text{s}^2} \times \text{s} = \dfrac{\text{m}}{\text{s}}$.
Matching units do not prove an equation is right — the number could still be wrong. But if the units don't match, it is wrong for sure.
Using base units, which equation is not homogeneous (so must be wrong)?
In $v = u + a$, $a$ is $\dfrac{\text{m}}{\text{s}^2}$ but $u$ is $\dfrac{\text{m}}{\text{s}}$ — the units do not match, so it cannot be right. The other three are homogeneous.
Prefixes
- A prefix scales a unit by a power of ten.
- You must know the ten in the syllabus, from pico to tera.
Put these prefixes in order, smallest factor first.
In order of size: nano, micro, milli, then kilo, mega. Each step up this list is a larger power of ten.
Converting a prefixed unit
- Replace each prefix with its power of ten, then simplify.
- kilo $=10^{3}$, so $2\ \text{kN} = 2 \times 10^{3}\ \text{N} = 2000\ \text{N}$.
A force is $2\ \text{kN}$. How many newtons is that?
kilo means $\times 10^{3}$, so $2\ \text{kN} = 2 \times 10^{3}\ \text{N} = 2000\ \text{N}$.
Watch the squares
- For a squared unit like $\text{mm}^2$, square the factor too.
$(10^{-3}\ \text{m})^{2} = 10^{-6}\ \text{m}^{2}$, not $10^{-3}\ \text{m}^{2}$.
What is $(10^{-3}\ \text{m})^{2}$, written in $\text{m}^2$?
Square the factor too: $(10^{-3})^{2} = 10^{-6}$, so $(10^{-3}\ \text{m})^{2} = 10^{-6}\ \text{m}^2$ — not $10^{-3}$.
You've got it
- every unit is built from the five base units: kg, m, s, A, K
- a derived unit comes from the equation that defines it
- check an equation with homogeneity — and when a prefix is squared, square its factor too