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How do SI base units, derived units and prefixes give every physical quantity a consistent description?

SI base units, units derived from them, the use of standard prefixes, and checking equations for homogeneity using base units.

A focused answer to AQA A-Level Physics 3.1.1, covering the SI base units, how derived units are built from them, the standard prefixes from pico to tera, and how to test an equation for homogeneity using base units.

Generated by Claude Opus 4.89 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

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  1. What this dot point is asking
  2. SI base units
  3. Derived units
  4. Standard prefixes
  5. Checking homogeneity
  6. Try this

What this dot point is asking

AQA specification point 3.1.1 wants you to know the SI base units, how every other unit is derived from them, the standard prefixes and their powers of ten, and how to check that an equation is homogeneous (balanced) by reducing both sides to base units.

SI base units

The base units you need are listed below. Every other unit in physics is built from these.

These base units are defined independently, and all physical quantities are expressed as combinations of them. The kilogram, not the gram, is the base unit of mass, which is a frequent source of error.

Derived units

A derived unit is any combination of base units. You should be able to reduce a named unit to base units by starting from a defining equation.

The same approach handles any derived unit: the volt is kg m2 s−3 A−1\text{kg m}^2 \text{ s}^{-3} \text{ A}^{-1} (from V=PIV = \dfrac{P}{I}), and the pascal is kg m−1 s−2\text{kg m}^{-1} \text{ s}^{-2} (from p=FAp = \dfrac{F}{A}). The coulomb is A s\text{A s} (from Q=ItQ = It), and the ohm is kg m2 s−3 A−2\text{kg m}^2 \text{ s}^{-3} \text{ A}^{-2} (from R=VIR = \dfrac{V}{I}). Working a unit back to base units this way is a reliable method whenever you meet an unfamiliar combination, and it is the basis of the homogeneity check below.

Standard prefixes

Prefixes scale a unit by a power of ten. Learn these for the exam:

  • tera (T) =1012= 10^{12}
  • giga (G) =109= 10^{9}
  • mega (M) =106= 10^{6}
  • kilo (k) =103= 10^{3}
  • centi (c) =10−2= 10^{-2}
  • milli (m) =10−3= 10^{-3}
  • micro (μ\mu) =10−6= 10^{-6}
  • nano (n) =10−9= 10^{-9}
  • pico (p) =10−12= 10^{-12}

When substituting into equations, convert every prefixed quantity to its base unit value first, otherwise the powers of ten will be wrong.

Checking homogeneity

An equation is homogeneous if every term reduces to the same combination of base units. This is a fast sanity check, though it cannot confirm a dimensionless constant.

Try this

Q1. Express the pascal (Pa=N m−2\text{Pa} = \text{N m}^{-2}) in SI base units. [2 marks]

  • Cue. Start from N=kg m s−2\text{N} = \text{kg m s}^{-2}, then divide by m2\text{m}^2 to get kg m−1 s−2\text{kg m}^{-1} \text{ s}^{-2}.

Q2. Show that the equation P=12ρv3AP = \tfrac{1}{2}\rho v^3 A is homogeneous, where ρ\rho is density, vv is speed and AA is area. [3 marks]

  • Cue. Reduce both sides to base units; the left side (power) is kg m2 s−3\text{kg m}^2 \text{ s}^{-3}.

Q3. State the value of the prefix nano as a power of ten. [1 mark]

  • Cue. 10−910^{-9}.

Exam-style practice questions

Practice questions written in the style of AQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AQA 20183 marksShow that the equation E=12mv2E = \tfrac{1}{2}mv^2 is homogeneous with respect to its units, where EE is energy, mm is mass and vv is speed.
Show worked answer →

Reduce both sides to SI base units. Energy has units of the joule, which in base units is kg m2 s−2\text{kg m}^2 \text{ s}^{-2} (from E=FdE = Fd with N=kg m s−2\text{N} = \text{kg m s}^{-2}).

On the right, mv2mv^2 has units kg×(m s−1)2=kg m2 s−2\text{kg} \times (\text{m s}^{-1})^2 = \text{kg m}^2 \text{ s}^{-2}. The factor 12\tfrac{1}{2} is dimensionless.

Both sides reduce to kg m2 s−2\text{kg m}^2 \text{ s}^{-2}, so the equation is homogeneous.

Markers reward reducing energy to base units, reducing mv2mv^2 to base units, and stating the two match (with the dimensionless constant noted).

AQA 20213 marksExpress the pascal, the SI unit of pressure (Pa=N m−2\text{Pa} = \text{N m}^{-2}), in terms of SI base units, and explain why checking homogeneity cannot prove an equation is correct.
Show worked answer →

Start from the newton: N=kg m s−2\text{N} = \text{kg m s}^{-2}. Dividing by m2\text{m}^2 gives the pascal: Pa=kg m s−2÷m2=kg m−1 s−2\text{Pa} = \text{kg m s}^{-2} \div \text{m}^2 = \text{kg m}^{-1} \text{ s}^{-2}.

Checking homogeneity only confirms that every term has the same combination of base units. It cannot detect a wrong dimensionless numerical constant (for example writing 14\tfrac{1}{4} instead of 12\tfrac{1}{2}), because a pure number has no units, so a homogeneous equation can still be numerically wrong.

Markers reward the correct base-unit expression and explaining that dimensionless constants are not tested by homogeneity.

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