3.1.1MeasurementsFoundation

Units and Measurement

Every quantity in physics is a number $and$ a unit — say "9.8" on its own and it means nothing; say "9.8 m s⁻²" and everyone knows exactly what you mean. This lesson covers the small set of $base units$ everything else is built from, how to check an equation is even dimensionally possible, and the prefixes that let you write $0.0000000000012$ m as $1.2$ pm instead.

30 min Video by Physics Online 3.1 Measurements and their errors

Unit Prefixes — A Level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

State the six SI base units used throughout A-Level physics and their quantities
Derive the unit of a quantity from its defining equation
Use homogeneity of units to check whether an equation could be correct
Convert between a value written with an SI prefix and standard form
Make a sensible order-of-magnitude estimate of a physical quantity

Base units

The SI system defines a small set of $base units$ , each an independent, precisely-defined standard, from which every other unit is built. At A-Level you need six: the kilogram (kg, mass), metre (m, length), second (s, time), ampere (A, electric current), kelvin (K, thermodynamic temperature) and mole (mol, amount of substance).

Every other unit in mechanics, electricity or thermal physics — newtons, joules, volts, ohms — is a combination of these six, and is therefore called a $derived unit$ .

Tip — If a question asks you to give an answer "in SI units" or "in base units", it wants kg, m, s (and A, K, mol where relevant) — not N, J or any other named derived unit.

Derived units and homogeneity

A derived unit comes directly from the defining equation of a quantity. Force is defined by $F = ma$ , so its unit is (unit of mass) × (unit of acceleration) $= kg m s^{- 2}$ — given the special name the newton (N). Energy, from $E = F d$ , has unit $kg m^{2} s^{- 2}$ , given the special name the joule (J).

Because every physically valid equation must balance dimensionally, checking that both sides of an equation reduce to the same combination of base units — called checking $homogeneity$ — is a genuinely useful way to test whether a formula you’ve derived (or half-remembered) could possibly be correct. It can never prove an equation is right (a missing factor of 2, or $π$ , changes nothing about the units), but it will immediately catch an equation that is dimensionally impossible.

1Left-hand side: energy, unit

kg m^{2} s^{- 2}

(the joule).

2Right-hand side:

[m] [v]^{2} = kg \times (m s^{- 1})^{2} = kg m^{2} s^{- 2}

3Both sides reduce to

kg m^{2} s^{- 2}

— the equation is homogeneous (the constant

\frac{1}{2}

has no units, so it doesn’t affect this check).

AnswerBoth sides give kg m² s⁻² — the equation is dimensionally consistent.

Tip — Homogeneity checks catch impossible equations, but can never confirm a numerical factor (like ½ or 2π) is correct — it’s a necessary test, not a sufficient one.

Prefixes

SI prefixes let you write very large or very small numbers compactly by attaching a multiplying power of ten to a unit, without changing the underlying quantity. A prefixed value must be converted back into the base unit (or standard form) before being used in any calculation.

p = 1 0^{- 12}, n = 1 0^{- 9}, μ = 1 0^{- 6}, m = 1 0^{- 3}, c = 1 0^{- 2}, k = 1 0^{3}, M = 1 0^{6}, G = 1 0^{9}, T = 1 0^{12}

Common SI prefixes: pico, nano, micro, milli, centi, kilo, mega, giga, tera.

1nano means

\times 1 0^{- 9}

650 nm = 650 \times 1 0^{- 9} m

3In standard form:

6.5 \times 1 0^{- 7}

Answer

6.5 \times 1 0^{- 7}

Tip — Always convert every prefixed quantity back to plain SI units before substituting into an equation — mixing nm and m (or mA and A) in the same calculation is one of the most common silly-mistake marks lost at A-Level.

Estimating physical quantities

A-Level physics regularly expects an $order-of-magnitude estimate$ : a rough calculation, using sensible everyday assumptions and approximate known values, that gets an answer to roughly the correct power of ten rather than an exact figure. The skill is choosing reasonable estimates for unknown quantities and combining them with a simple physical relationship.

1Mass of water:

ρ V = 1000 \times 250 \times 1 0^{- 6} = 0.25

kg.

2Number of moles:

n = \frac{mass}{molar mass} = \frac{0.25}{0.018} \approx 14

mol.

3Number of molecules:

N = n \times N_{A} = 14 \times 6.0 \times 1 0^{23} \approx 8 \times 1 0^{24}

Answer

\approx 8 \times 1 0^{24}

molecules — the exact number matters far less than getting the order of magnitude right.

Tip — For an estimation question, state your assumed values clearly and show the arithmetic — you are marked on sound reasoning and a sensible order of magnitude, not on matching one exact number.

Equation recap

F = ma \Rightarrow N = kg m s^{- 2}

The newton, derived from mass × acceleration.

E = F d \Rightarrow J = kg m^{2} s^{- 2}

The joule, derived from force × distance.

Common mistakes to avoid

Substituting a value with its prefix still attached (e.g. using "5" for 5 mm) directly into an equation.

Always convert to the base SI unit first (5 mm = 5×10⁻³ m) before using a value in a calculation.

Thinking that if an equation is homogeneous, it must be correct.

Homogeneity only rules out equations that are dimensionally impossible — it says nothing about missing numerical factors like ½ or π.

Giving an order-of-magnitude estimate to many significant figures.

An estimate should be quoted to at most 1 significant figure (or as a power of ten) — false precision defeats the point of an estimate.

Key takeaways

The six SI base units needed at A-Level are kg, m, s, A, K and mol.
Derived units (like N and J) come from the defining equation of a quantity, built from the base units.
Checking homogeneity (that both sides of an equation reduce to the same base units) can disprove but never prove an equation.
SI prefixes (p, n, μ, m, c, k, M, G, T) must be converted to plain SI units before use in a calculation.
Order-of-magnitude estimates use sensible assumed values to get an answer to roughly the right power of ten.

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Units and Measurement

Every quantity in physics is a number

and

a unit — say "9.8" on its own and it means nothing; say "9.8 m s⁻²" and everyone knows exactly what you mean. This lesson covers the small set of

base units

everything else is built from, how to check an equation is even dimensionally possible, and the prefixes that let you write

0.0000000000012

m as

1.2

pm instead.

30 min Video by Physics Online 3.1 Measurements and their errors