# SI base and derived units

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u1-si-base-and-derived-units/

This sub-topic covers the foundation of all physical measurement: the SI system of units. You will learn the seven base units, how to derive units for other quantities, and how to use units to check consistency of physical equations.

**Prerequisites:** [Introduction to physical quantities and measurement](https://www.owlsprep.com/study/cie-9702-u1-introduction-to-physical-quantities/)

## Learning objectives

- Identify the seven SI base quantities and their corresponding units
- Distinguish clearly between base units and derived units
- Derive SI units for derived physical quantities from defining equations
- Check the homogeneity of physical equations using unit analysis

## SI Base Quantities and Units

**SI Base Quantities and Units** — The International System (SI) defines seven fundamental base quantities, each with a standardised base unit. All other physical quantities are built from combinations of these base quantities.

*Example:* Length is a base quantity, with base unit the metre (m).

| Base Quantity | SI Base Unit | Unit Symbol |
| --- | --- | --- |
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |

> **tip**
>
> CIE exams only require you to memorize the first six base units listed above. Luminous intensity is almost never tested.

**Worked example:** Which of the following is an SI base unit: joule, newton, kilogram, watt?

1. Recall the definition of an SI base unit: base units are the seven fundamental units defined in the SI system.
2. Joule (energy), newton (force), and watt (power) are all combinations of base units, so they are derived units.
3. Kilogram is the base unit of mass, so it is the correct answer.

## Derived Quantities and Derived Units

**Derived Units** — A derived unit is the unit of a derived physical quantity, obtained by combining base units according to the algebraic relation that defines the quantity.

Any physical quantity that is not a base quantity is a derived quantity, so its unit must be a derived unit. To find the derived unit of a quantity, start by writing the quantity in terms of base quantities using its definition, then replace each quantity with its corresponding base unit.

**Worked example:** Derive the SI derived unit for pressure, given that pressure $P = \frac{F}{A}$ where $F$ is force and $A$ is area.

1. First, express force in terms of base quantities: force $F = ma$, where $m$ is mass and $a$ is acceleration.
2. Acceleration $a = \frac{\Delta v}{t}$, and velocity $v = \frac{d}{t}$, so acceleration has units:
3. $$[a] = \frac{\text{m}}{\text{s}^2}$$
4. Therefore, force has units:
5. $$[F] = \text{kg} \cdot \frac{\text{m}}{\text{s}^2} = \text{kg m s}^{-2} = \text{N}$$
6. Area $A$ has units of length squared, so $[A] = \text{m}^2$.
7. Substitute into the pressure definition:
8. $$[P] = \frac{[F]}{[A]} = \frac{\text{kg m s}^{-2}}{\text{m}^2} = \text{kg m}^{-1} \text{s}^{-2}$$
9. This is the SI derived unit for pressure, which is also given the special name pascal (Pa).

## Homogeneity of Physical Equations

**Homogeneous Equation** — A physical equation is homogeneous (dimensionally consistent) if every term in the equation has the same overall units.

If an equation is not homogeneous, it cannot be physically correct. However, a homogeneous equation is not guaranteed to be correct, because it can still have wrong numerical constants or missing terms. Checking homogeneity is a useful tool to catch errors in your working and verify unfamiliar equations.

**Worked example:** Check if the equation $v^2 = u^2 + 2as$ is homogeneous, where $v$ and $u$ are velocities, $a$ is acceleration, and $s$ is distance.

1. Find the units of the left-hand side (LHS):
2. $$[v^2] = (\text{m s}^{-1})^2 = \text{m}^2 \text{s}^{-2}$$
3. Find units of each term on the right-hand side (RHS): the first term $u^2$ has the same units as $v^2$, so $[u^2] = \text{m}^2 \text{s}^{-2}$.
4. The constant $2$ is dimensionless, so we only calculate units for $as$:
5. $$[2as] = [a][s] = (\text{m s}^{-2})(\text{m}) = \text{m}^2 \text{s}^{-2}$$
6. All terms on both sides have matching units, so the equation is homogeneous.

> **tip**
>
> Numerical constants (like the 2 in this example) have no units, so you can always ignore them when checking homogeneity.

## Common pitfalls

- **Wrong:** Confusing base quantities with base units in exam answers
  - Why it fails: Students often mix up the name of the quantity and the name of its unit when answering multiple choice or written questions
  - Correct: Always read the question carefully: if asked for a base unit, give the unit (e.g. kilogram, not mass); if asked for a base quantity, give the quantity name.
- **Wrong:** Listing named derived units as base units
  - Why it fails: Many students incorrectly assume that units with special names (like newton or joule) are base units
  - Correct: Remember only the seven core SI base units are base units; any unit named after a scientist is almost always a derived unit.
- **Wrong:** Claiming a homogeneous equation is definitely physically correct
  - Why it fails: Homogeneity only confirms dimensional consistency, it does not check for wrong constant factors or missing terms
  - Correct: Only ever state that an inhomogeneous equation is definitely wrong; a homogeneous equation is only possibly correct, not proven correct by unit checking.
- **Wrong:** Including units of dimensionless constants when deriving units
  - Why it fails: Students often incorrectly try to account for constants like 2, $\frac{1}{2}$ or $\pi$ when calculating derived units, which leads to unnecessary mistakes
  - Correct: Ignore all dimensionless numerical constants when deriving or checking units, only account for the units of physical quantities.

## Cheatsheet

| Category | Key Information |
| --- | --- |
| SI base units | m (length), kg (mass), s (time), A (current), K (temperature), mol (amount of substance) |
| Common derived units | Force: N = kg m s⁻², Pressure: Pa = kg m⁻¹ s⁻², Energy: J = kg m² s⁻², Power: W = kg m² s⁻³ |
| Homogeneity rule | Inhomogeneous equation = definitely wrong; Homogeneous = possibly correct |
| Derivation rule | Ignore dimensionless constants when finding/checking units |

## What's next

Mastering SI units is the foundational first step for all of A-Level Physics, as every topic from kinematics to electromagnetism relies on consistent unit analysis. Next, you will build on this knowledge to learn about SI prefixes and orders of magnitude, which allow you to handle the very large and very small quantities common across all areas of physics. Unit checking is also a critical skill for any calculation: a quick check of homogeneity can catch algebraic mistakes in your working before you lose exam marks, and this skill will be especially useful when you start solving more complex problems in mechanics later in the course.

- [Unit prefixes](https://www.owlsprep.com/study/cie-9702-u1-unit-prefixes/)
- [Estimation of physical quantities](https://www.owlsprep.com/study/cie-9702-u1-estimation-of-physical-quantities/)
- [Measurement uncertainties](https://www.owlsprep.com/study/cie-9702-u1-measurement-uncertainties/)

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