Write 0.000\,045\ m in standard form and in micrometres. [2 marks]

OCR OCR 2018-style practice: The pressure p of an ideal gas is given by p = (1)/(3) c^2, where is density and c^2 is the mean square speed. Show that this equation is homogeneous with respect to units.

Work out the base units of each side. Pressure is force over area, so [p] = kg m s^-2m^2 = kg m^-1s^-2. The right-hand side: density [] = kg m^-3 and mean square speed [ c^2 ] = (m s^-1)^2 = m^2s^-2. Multiplying: kg m^-3 × m^2s^-2 = kg m^-1s^-2. Both sides reduce to kg m^-1s^-2, so the equation is homogeneous. The factor (1)/(3) is dimensionless and does not affect the units. Markers reward base units for p, base units for the right side, and the statement that they match.

EnglandPhysicsSyllabus dot point

How do physical quantities, units and dimensional checks underpin every calculation in physics?

Physical quantities and SI units: the seven base units, derived units in base-unit form, prefixes and standard form, estimation, and checking the homogeneity of equations by units.

A focused answer to the OCR H556 foundations content on physical quantities and units, covering the seven SI base units, derived units expressed in base units, prefixes and standard form, order-of-magnitude estimation, and checking the homogeneity of equations by their units.

Generated by Claude Opus 4.811 min answerUpdated 2026-06-02

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

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What this dot point is asking

OCR wants you to use the seven SI base units, express any derived unit in terms of those base units, work fluently with prefixes and standard form, make order-of-magnitude estimates, and check whether an equation is homogeneous (dimensionally consistent) using units.

The answer

The seven SI base units

A base unit is defined independently; a derived unit is built from base units by the defining equation of the quantity. Because the whole of physics is internally consistent, expressing a derived unit in base units is just a matter of following the definition.

Derived units in base-unit form

Other common results worth knowing: the pascal is $\text{kg m}^{-1}\text{s}^{-2}$ (force over area), the coulomb is $\text{A s}$ (current times time), and the volt is energy per unit charge, $\frac{\text{kg m}^2\text{s}^{-2}}{\text{A s}} = \text{kg m}^2\text{s}^{-3}\text{A}^{-1}$ . Deriving these from definitions is more reliable than memorising them.

Prefixes and standard form

When substituting into an equation, convert every prefixed quantity into base SI units first (so $4.7\ \mu\text{C}$ becomes $4.7 \times 10^{-6}\ \text{C}$ ). Mixing prefixed and base units is one of the most common causes of wrong answers by a factor of a thousand.

Homogeneity of equations

A physically valid equation must be homogeneous: every additive term has the same base units. To check, reduce each term to base units and compare. If they differ, the equation is certainly wrong. If they match, the equation is dimensionally possible, but homogeneity cannot detect a missing dimensionless factor such as $\frac{1}{2}$ or $2\pi$ .

Reducing power to base units two ways

Show that $P = Fv$ and $P = \frac{1}{2}\rho A v^3$ (drag power) have the same base units.

step 1: Base units of $Fv$

$[F] = \text{kg m s}^{-2}$ and $[v] = \text{m s}^{-1}$ , so $[Fv] = \text{kg m}^2\text{s}^{-3}$ , which is the watt.

step 2: Base units of $\rho A v^3$

$[\rho] = \text{kg m}^{-3}$ , $[A] = \text{m}^2$ , $[v^3] = \text{m}^3\text{s}^{-3}$ . Multiply: $\text{kg m}^{-3} \times \text{m}^2 \times \text{m}^3\text{s}^{-3} = \text{kg m}^2\text{s}^{-3}$ .

step 3: Compare

Both reduce to $\text{kg m}^2\text{s}^{-3}$ (the watt), so the two power expressions are dimensionally consistent.

Examples in context

Order-of-magnitude estimates check whether an answer is sensible: the mass of an adult is about $10^2\ \text{kg}$ , a human stride about $1\ \text{m}$ , and atmospheric pressure about $10^5\ \text{Pa}$ . Engineers reduce proposed formulae to base units to catch errors before building anything. Astronomers juggle prefixes constantly, from the femtometre scale of nuclei to the gigaparsec scale of the observable universe, so disciplined use of standard form is essential.

Try this

Q1. Express the joule in SI base units. [1 mark]

Cue. $1\ \text{J} = \text{kg m}^2\text{s}^{-2}$ (force times distance).

Q2. Write $0.000\,045\ \text{m}$ in standard form and in micrometres. [2 marks]

Cue. $4.5 \times 10^{-5}\ \text{m} = 45\ \mu\text{m}$ .

Q3. State what is meant by a homogeneous equation and one limitation of a homogeneity check. [2 marks]

Cue. Every term has the same base units; the check cannot reveal a missing dimensionless constant.

Exam-style practice questions

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

OCR 20183 marksThe pressure

p

of an ideal gas is given by

p = \frac{1}{3}\rho \langle c^2 \rangle

, where

\rho

is density and

\langle c^2 \rangle

is the mean square speed. Show that this equation is homogeneous with respect to units.

Show worked answer →

Work out the base units of each side. Pressure is force over area, so $[p] = \frac{\text{kg m s}^{-2}}{\text{m}^2} = \text{kg m}^{-1}\text{s}^{-2}$ .

The right-hand side: density $[\rho] = \text{kg m}^{-3}$ and mean square speed $[\langle c^2 \rangle] = (\text{m s}^{-1})^2 = \text{m}^2\text{s}^{-2}$ .

Multiplying: $\text{kg m}^{-3} \times \text{m}^2\text{s}^{-2} = \text{kg m}^{-1}\text{s}^{-2}$ .

Both sides reduce to $\text{kg m}^{-1}\text{s}^{-2}$ , so the equation is homogeneous. The factor $\frac{1}{3}$ is dimensionless and does not affect the units. Markers reward base units for $p$ , base units for the right side, and the statement that they match.

OCR 20202 marksA capacitor stores a charge of

4.7\ \mu\text{C}

at a potential difference of

9.0\ \text{V}

. Calculate the capacitance in nanofarads.

Show worked answer →

Capacitance is $C = \frac{Q}{V} = \frac{4.7 \times 10^{-6}}{9.0} = 5.22 \times 10^{-7}$ F.

Convert to nanofarads: $1\ \text{nF} = 10^{-9}$ F, so $5.22 \times 10^{-7}\ \text{F} = 522\ \text{nF}$ .

Markers reward the correct substitution with the micro prefix converted to $10^{-6}$ , and the final value converted to nanofarads (about $520\ \text{nF}$ ).

Related dot points

Sources & how we know this

OCR AS and A Level Physics A (H156, H556) specification — OCR (2015)