Introduction to SI Units Revision Notes | A Level AQA Physics

1. Why a Universal Language of Measurement Matters

Imagine two physicists — one in Tokyo, one in Manchester — measuring the same electric current but recording it in different units. Their results would be meaningless to each other. The International System of Units (SI), from the French Système International d'Unités, solves this by giving every scientist on Earth the same measurement language.

At A Level, you will encounter quantities ranging from the mass of a proton ($1.67 \times 10^{-27}$ kg) to the distance to a nearby star ($4 \times 10^{16}$ m). Without a coherent unit system and a set of agreed prefixes, comparing or combining these numbers would be impossible. Every equation you use this year ultimately traces back to a small set of base units.

2. The SI Base Units You Need to Know

There are seven SI base units in total. For AQA A Level Physics, you are required to know six of them — the candela (the unit of luminous intensity) is explicitly excluded from your specification. You do not need to memorise the formal definitions of these quantities, but you must know which unit belongs to which quantity.

Notice that temperature uses the kelvin, not the degree Celsius. The kelvin scale starts at absolute zero (0 K = −273.15 °C), which makes it the physically meaningful choice for equations such as the ideal gas law. When a question gives temperature in °C, convert it: $T(\text{K}) = T(°\text{C}) + 273$.

The mole deserves a special mention. It is the SI unit for amount of substance — a count of particles (atoms, molecules, ions, electrons) expressed in units of Avogadro's number ($N_A = 6.02 \times 10^{23}$ mol$^{-1}$). You will use it whenever you connect macroscopic measurements to particle-level behaviour.

That retrieval attempt is worth doing seriously — examiners regularly award a straightforward mark for correctly stating a base unit, and students frequently lose it by writing °C instead of K, or by confusing the symbol for ampere (A) with the symbol for area.

3. Building Derived Units from Base Units

Every other unit in physics is a derived unit — built by multiplying or dividing base units according to the defining equation for that quantity. This is not just bookkeeping; it is a powerful consistency check. If the units on both sides of your equation do not match, something in your working is wrong.

Start with something familiar: speed. Speed is defined as distance divided by time, so its unit is metres divided by seconds — written as m s$^{-1}$ (or equivalently m/s). The unit follows directly from the formula, with no extra information needed.

Now consider a more complex case: the newton (N), the unit of force. Newton's second law states $F = ma$, so force has units of kg × m s$^{-2}$. The newton is therefore shorthand for kg m s$^{-2}$. Whenever you see a named derived unit — newton, joule, pascal, watt — you can always unpack it into base units.

You do not need to memorise every row of that table. What you need is the method: start from the defining equation, substitute the units of each quantity, and simplify. The step-by-step walkthrough below shows exactly how to do this under exam conditions.

Worked example: Deriving the Base Units of the Joule

Show that the joule (J), the unit of energy, is equivalent to kg m$^{2}$ s$^{-2}$ in base units.

1. $W = F \times d$ — Work done equals force multiplied by displacement. This is the equation that defines the joule. [M1]
2. Unit of $F$ = N = kg m s$^{-2}$ — The newton is itself a derived unit. Unpack it using $F = ma$: unit of $a$ is m s$^{-2}$, so unit of $F$ is kg × m s$^{-2}$.
3. Unit of $d$ = m — Displacement is a length — a base unit, so no unpacking needed.
4. Unit of $W$ = kg m s$^{-2}$ × m = kg m$^{2}$ s$^{-2}$ — Combine by adding exponents for matching base units: m$^{1}$ × m$^{1}$ = m$^{2}$. [A1]
5. 1 J = 1 kg m$^{2}$ s$^{-2}$ — This is the accepted base-unit expression for the joule. Any quantity with these base units is an energy.

Answer: 1 J = 1 kg m$^{2}$ s$^{-2}$

4. SI Prefixes: Scaling Units for the Real World

Physics spans an enormous range of scales. Writing $0.000000001$ m every time you refer to a nanometre is impractical and error-prone. SI prefixes attach to any unit to shift its scale by a defined power of ten, keeping numbers manageable without changing the underlying measurement.

The prefix replaces the power of ten in your number. So 5 km means $5 \times 10^{3}$ m, and 300 nm means $300 \times 10^{-9}$ m = $3.00 \times 10^{-7}$ m. Before substituting into any equation, always convert prefixed units back to their SI base form — this is where a large proportion of unit-conversion errors originate in exam scripts.

Worked example: Converting Prefixed Units Before Calculation

A resistor dissipates a power of 2.0 mW when a current of 4.0 mA flows through it. Calculate the resistance of the resistor in ohms.

1. $P = 2.0 \text{ mW} = 2.0 \times 10^{-3} \text{ W}$ — The prefix milli means $10^{-3}$. Replace mW with $\times 10^{-3}$ W before proceeding. [M1]
2. $I = 4.0 \text{ mA} = 4.0 \times 10^{-3} \text{ A}$ — Again, milli = $10^{-3}$. Both quantities are now in SI base units.
3. $P = I^{2} R \quad \Rightarrow \quad R = \frac{P}{I^{2}}$ — Rearrange for resistance. [M1]
4. $I^{2} = (4.0 \times 10^{-3})^{2} = 4.0 \times 10^{-3} \times 4.0 \times 10^{-3} = 16 \times 10^{-6} = 1.6 \times 10^{-5} \text{ A}^{2}$ — Expand the square explicitly: $4.0 \times 4.0 = 16$ and $10^{-3} \times 10^{-3} = 10^{-6}$.
5. $R = \frac{2.0 \times 10^{-3}}{1.6 \times 10^{-5}} = \frac{2.0}{1.6} \times 10^{-3-(-5)} = 1.25 \times 10^{2} = 125 \text{ } \Omega$ — Divide the coefficients ($2.0 \div 1.6 = 1.25$) and subtract exponents ($-3 - (-5) = +2$). [A1]
6. Unit of $R$: $\frac{\text{W}}{\text{A}^{2}} = \frac{\text{kg m}^{2} \text{s}^{-3}}{\text{A}^{2}} = \text{kg m}^{2} \text{s}^{-3} \text{A}^{-2} = \Omega$ ✓ — The unit of the answer matches the ohm expressed in base units, confirming the formula was applied correctly.

Answer: $R = 125 \ \Omega$

5. Using Base Units to Check Equations

Earlier you saw how to derive the base units of a quantity from its defining equation. That same process works in reverse: you can check whether an unfamiliar equation is homogeneous — meaning the base units on the left-hand side match those on the right-hand side. If they do not match, the equation is definitely wrong. If they do match, the equation is at least dimensionally consistent (though it could still have a missing numerical constant).

This technique is sometimes called a unit check or unit analysis. AQA does not require you to perform formal dimensional analysis, but checking units is an expected part of good exam technique and is often rewarded implicitly when you show your working clearly.

Worked example: Checking Homogeneity: Kinetic Energy Formula

Verify that the equation $E_k = \frac{1}{2}mv^{2}$ is homogeneous in SI base units.

1. Unit of $E_k$ = J = kg m$^{2}$ s$^{-2}$ — Energy is measured in joules. From the earlier derivation, 1 J = 1 kg m$^{2}$ s$^{-2}$.
2. Unit of $m$ = kg — Mass is a base quantity — no derivation needed.
3. Unit of $v$ = m s$^{-1}$, so unit of $v^{2}$ = (m s$^{-1}$)$^{2}$ = m$^{2}$ s$^{-2}$ — Square both the unit of distance and the unit of time: m$^{1 \times 2}$ = m$^{2}$ and s$^{-1 \times 2}$ = s$^{-2}$.
4. Unit of $\frac{1}{2}mv^{2}$ = kg × m$^{2}$ s$^{-2}$ = kg m$^{2}$ s$^{-2}$ — The numerical factor $\frac{1}{2}$ is dimensionless and does not affect the unit check.
5. LHS: kg m$^{2}$ s$^{-2}$ = RHS: kg m$^{2}$ s$^{-2}$ ✓ — The units match, so the equation is homogeneous. This confirms the formula is dimensionally consistent.

Answer: The equation $E_k = \frac{1}{2}mv^{2}$ is homogeneous; both sides have units of kg m$^{2}$ s$^{-2}$.

6. Exam Technique: Getting Every Mark on Units Questions

Unit questions appear in two main forms at A Level: state the unit (one mark, no working required) and show that the unit of [quantity] is [base units] (two or three marks, working required). The second type is where students most often drop marks — not because they lack knowledge, but because they skip steps that the mark scheme rewards explicitly.

For a show that question, write out the defining equation, substitute the unit of each variable, and simplify step by step. Never go straight from the equation to the answer in one line — the method marks are in the intermediate steps. For a calculate question involving prefixed units, show the conversion explicitly as a separate line before substituting.

When a question asks you to state a base unit, give both the full name and the symbol if the question is ambiguous about which it wants. Writing 'kilogram (kg)' takes two seconds and removes any doubt. Examiners accept either, but clarity never costs marks.

That question targets the two most common confusions: treating named derived units (newton, joule) as base units, and including the candela because it sounds like it should be on the list. Both errors reflect a misunderstanding of what 'base unit' means — a unit that cannot be broken down further into other SI units.

7. Bringing It Together: A Unit-Consistency Checklist

Every calculation you carry out this year should pass through the same mental checklist. This is not extra work — it is the habit that separates students who consistently score full marks on calculation questions from those who drop one or two marks through avoidable unit errors.

• Identify every quantity in the equation and note its unit.
• Convert any prefixed units to SI base units before substituting (show this conversion as a separate line).
• Substitute values with units attached — never substitute bare numbers.
• After calculating, check that the unit of your answer matches the expected unit for that quantity.
• If the question asks you to express a unit in base form, start from the defining equation and work through each variable systematically.
• For temperature: confirm whether the equation requires kelvin or accepts Celsius, and convert if necessary.

The six base units — kilogram, metre, second, ampere, kelvin, and mole — are the foundation on which every other unit in your A Level course is built. Mastering their symbols, their associated quantities, and the method for deriving units from equations gives you a reliable self-checking tool that works across every topic, from mechanics to electromagnetism to thermal physics.