Learning Objectives
4 objectivesBy the end of this note, you should be able to:
- Recall the SI base quantities and their units: mass, length, time, current, temperature.
- Express derived units as products or quotients of SI base units.
- Use SI base units to check the homogeneity of physical equations.
- Recall and use prefixes from pico (p) to tera (T) for base and derived units.
SI Base Quantities and Units
The SI base units form the foundation of all measurements in physics, with every other unit built from these fundamental five (for A-Level purposes). Each base quantity is measured using one defined unit, and no base unit can be expressed in terms of the others.
The five SI base units required for this syllabus are summarised below. Each is independent and serves as a building block for derived units.
| Base Quantity | Symbol | SI Unit | Unit Symbol |
|---|---|---|---|
| Mass | m | kilogram | kg |
| Length | l | metre | m |
| Time | t | second | s |
| Electric current | I | ampere | A |
| Thermodynamic temperature | T | kelvin | K |
The kilogram is unusual because it is the only base unit containing a prefix (kilo) in its name. Despite this, the kilogram is itself the base unit, not the gram.
MisconceptionMany students assume newton (N), joule (J), or pascal (Pa) are SI base units. They are not. Only mass, length, time, current, and temperature have base units in this syllabus. All other units are derived from these. Exam cue: When asked for a base unit, never write N, J, Pa, or W.
Derived Units from Base Units
A derived unit is formed by combining SI base units through multiplication or division, expressing physical quantities such as force, energy, or pressure. Every quantity in physics outside the five base quantities has a unit built from kg, m, s, A, and K.
To find a derived unit, start from the defining equation of the quantity, then substitute the base units of each variable. The table below shows common derived units expressed in base units.
| Quantity | Defining Equation | Derived Unit | In SI Base Units |
|---|---|---|---|
| Force | $F=ma$ | newton (N) | kg m s⁻² |
| Energy / Work | $W=Fs$ | joule (J) | kg m² s⁻² |
| Power | $P=\frac{W}{t}$ | watt (W) | kg m² s⁻³ |
| Pressure | $P=\frac{F}{A}$ | pascal (Pa) | kg m⁻¹ s⁻² |
| Charge | $Q=It$ | coulomb (C) | A s |
| Potential difference | $V=\frac{W}{Q}$ | volt (V) | kg m² s⁻³ A⁻¹ |
| Resistance | $R=\frac{V}{I}$ | ohm (Ω) | kg m² s⁻³ A⁻² |
| Frequency | $f=\frac{1}{T}$ | hertz (Hz) | s⁻¹ |
For example, the newton is found by substituting kg for mass and m s⁻² for acceleration in $F=ma$. This gives the base form kg m s⁻². The same logic applies to every derived unit.
Examiner InsightWhen asked to express a unit in base units, always write the unit using the symbols kg, m, s, A, K with negative powers for division. Never leave fractions or named units in your answer. Exam cue: Write Pa as kg m⁻¹ s⁻², not as N m⁻² or kg/(m s²).
Homogeneity of Physical Equations
A physical equation is homogeneous when both sides have identical SI base units, which is a necessary condition for the equation to be physically valid. If the base units do not match, the equation is definitely incorrect.
To test homogeneity, reduce every term on each side of the equation to its SI base units. If every term shares the same combination of base units, the equation is homogeneous. If any term differs, the equation cannot be correct.
Worked Example: Checking Homogeneity of an Equation of Motion
Use SI base units to check whether the equation ${v}^{2}={u}^{2}+2as$ is homogeneous, where v and u are velocities, a is acceleration, and s is displacement.
Step-by-step solution:
Left-hand side: v²
$$[{v}^{2}]=(m {s}^{-1}{)}^{2}$$
$$[{v}^{2}]={m}^{2} {s}^{-2}$$
Right-hand side, first term: u²
$$[{u}^{2}]=(m {s}^{-1}{)}^{2}$$
$$[{u}^{2}]={m}^{2} {s}^{-2}$$
Right-hand side, second term: 2as The number 2 is dimensionless, so it does not contribute base units.
$$[as]=(m {s}^{-2})(m)$$
$$[as]={m}^{2} {s}^{-2}$$
Comparison All three terms have base units of m² s⁻². Therefore the equation is homogeneous.
The equation passes the homogeneity test, so it is dimensionally consistent. This is a necessary but not sufficient check — a homogeneous equation may still contain an incorrect numerical factor.
MisconceptionHomogeneity proves an equation is dimensionally correct, not that it is physically correct. An equation like $s=ut+a{t}^{2}$ is homogeneous but missing the factor of ½. Homogeneity cannot detect missing or wrong constants. Exam cue: Never write “the equation is correct” — write “the equation is homogeneous” or “dimensionally consistent”.
Prefixes for Submultiples and Multiples
SI prefixes allow physical quantities to be written in compact form by indicating decimal multiples or submultiples of any base or derived unit. Each prefix is a power of ten, and the prefix attaches directly to the unit symbol.
| Prefix | Symbol | Multiplier |
|---|---|---|
| tera | T | 10¹² |
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| deci | d | 10⁻¹ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | μ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
When converting between prefixed and base units, replace the prefix with its multiplier. For example, 5 km = 5 × 10³ m, and 2 μF = 2 × 10⁻⁶ F. Always convert prefixed values to 
MisconceptionThe symbol M (mega, 10⁶) and m (milli, 10⁻³) differ by case only but represent factors that differ by 10⁹. Writing 5 mW when meaning 5 MW is a major error. Exam cue: Always check the case of prefix symbols carefully — capital letters denote multiples ≥ 10⁶ except kilo (k).
QUICK RECAP
Key Points
- Five SI base quantities: mass, length, time, current, temperature.
- Their units: kg, m, s, A, K.
- The kilogram is a base unit despite its prefix.
- Derived units are products or quotients of base units.
- Newton: kg m s⁻². Joule: kg m² s⁻². Watt: kg m² s⁻³.
- Pascal: kg m⁻¹ s⁻². Volt: kg m² s⁻³ A⁻¹. Ohm: kg m² s⁻³ A⁻².
- Coulomb: A s. Hertz: s⁻¹.
- Homogeneous equation: all terms share the same SI base units.
- Homogeneity is necessary but not sufficient for correctness.
- Homogeneity cannot detect missing constants like ½ or π.
- Prefixes: pico (10⁻¹²) to tera (10¹²) in powers of ten.
- Common prefixes: n (10⁻⁹), μ (10⁻⁶), m (10⁻³), k (10³), M (10⁶), G (10⁹).
- Convert prefixed values to base SI form before substituting into equations.
- M (mega) and m (milli) differ by 10⁹ — case matters.
CAN I…? PROGRESS CHECK
Self-Assessment
- Recall the five SI base quantities and their unit symbols.
- Express the newton, joule, pascal, watt, volt, and ohm in SI base units.
- Derive the base units of any quantity from its defining equation.
- Test whether a given equation is homogeneous using base units.
- Explain why homogeneity does not prove an equation is physically correct.
- Recall every prefix from pico to tera and its power of ten.
- Convert any prefixed value into standard SI form before calculation.
- Distinguish the symbols M (mega) and m (milli) in calculations.