S.I. units

What you'll learn

What S.I. base quantities and base units are, and which ones OCR expects you to know.
How derived units such as newtons, joules and volts are built from base units.
How to check whether an equation is homogeneous using S.I. base units.
How to convert prefixes and label graph axes/table columns correctly.

Why units matter

Physics is quantitative: you usually describe a physical quantity using a number and a unit. For example, saying a time is “2.4” is incomplete; saying it is 2.4 s tells someone what has been measured.

The Système International d’Unités, usually written SI, is the internationally agreed system of units. OCR may write this as S.I. units. Using the same units worldwide makes measurements comparable, repeatable and useful in science and engineering.

Definition

Physical quantity and unit

A physical quantity is a measurable property, such as mass, length, time or current. A unit is the agreed standard used to measure that quantity, such as kilogram, metre, second or ampere.

The whole topic is a toolkit: start from base units, build derived units, handle prefixes, then use units to check equations and present data clearly.

Concept map showing SI base quantities, derived units, prefixes and graph label convention

S.I. base quantities and base units

A base quantity is treated as independent: it is not defined by combining other quantities. A base unit is the SI unit for a base quantity.

For OCR A Physics, the key base quantities in this section are:

Base quantity	SI base unit	Unit symbol
mass	kilogram	kg
length	metre	m
time	second	s
electric current	ampere	A
thermodynamic temperature	kelvin	K
amount of substance	mole	mol

Officially, SI also includes luminous intensity, measured in candela, symbol cd. It is not a major calculation unit in H556, but it is part of the full SI base-unit set.

Tip

Writing unit symbols

Unit symbols are case-sensitive and are not pluralised: write 5 kg, not 5 kgs; 300 K, not 300 °K. A capital letter can completely change the meaning, such as m for milli/metre but M for mega.

Derived units

A derived unit is made by combining base units according to the equation defining the quantity.

For example:

speed has units metre per second, written m s^-1
acceleration has units metre per second squared, written m s^-2
momentum has units kg m s^-1
density has units kg m^-3

Key Idea

Unit algebra

Treat unit symbols like algebraic factors: multiply them, divide them and cancel them using the physical equation.

Example

Deriving units for momentum and density

Momentum is given by $p = m v$ , where mass has unit kg and velocity has unit $s−1\mathrm{m\,s^{-1}}$ .
Multiply the units in the equation:

\mathrm{kg} \times \mathrm{m\,s^{-1}} = \mathrm{kg\,m\,s^{-1}}

so momentum has units $s−1\mathrm{kg\,m\,s^{-1}}$ .

Density is given by $ρ=mV\rho = \frac{m}{V}$ , where mass has unit kg and volume has unit $m3\mathrm{m^3}$ .
Divide the units:

\frac{\mathrm{kg}}{\mathrm{m^3}} = \mathrm{kg\,m^{-3}}

so density has units $m−3\mathrm{kg\,m^{-3}}$ .

Named units you should recognise

Many derived units have special names. You still need to know that they are built from base units.

Unit name	Symbol	Common quantity	Base-unit form
hertz	Hz	frequency	s^-1
newton	N	force	kg m s^-2
joule	J	energy, work done	kg m^2 s^-2
watt	W	power	kg m^2 s^-3
pascal	Pa	pressure	kg m^-1 s^-2
coulomb	C	charge	A s
volt	V	potential difference	kg m^2 s^-3 A^-1
ohm	Ω	resistance	kg m^2 s^-3 A^-2
farad	F	capacitance	kg^-1 m^-2 s^4 A^2
tesla	T	magnetic flux density	kg s^-2 A^-1
weber	Wb	magnetic flux	kg m^2 s^-2 A^-1

The same letter can mean different things depending on context. For example, italic $W$ is often work done in equations such as $Fx\cos\theta$ , but upright W is the unit watt.

Common Mistake

Tesla or tera?

T on its own as a unit symbol means tesla. T before a unit as a prefix means tera, or $10^{12}$ , such as TW for terawatt.

Checking homogeneity of equations

An equation is homogeneous if every term has the same base units. This is one of the quickest ways to spot impossible equations.

Definition

Homogeneous equation

A physical equation is homogeneous when the units on the left-hand side match the units on the right-hand side, and all terms being added or subtracted have the same units.

Constants such as $12\frac{1}{2}$ and functions such as $sin⁡θ\sin\theta$ or $cos⁡θ\cos\theta$ have no units. They are dimensionless.

Example

Checking work done equation units

Check whether $Fx\cos\theta$ is homogeneous.

Force has units newtons, and from $F = ma$ :

\mathrm{N} = \mathrm{kg\,m\,s^{-2}}

Displacement $x$ has unit m, and $cos⁡θ\cos\theta$ is dimensionless.

Multiply the right-hand side units:

\mathrm{kg\,m\,s^{-2}} \times \mathrm{m} = \mathrm{kg\,m^2\,s^{-2}}

Work done is energy, measured in joules, and:

\mathrm{J} = \mathrm{kg\,m^2\,s^{-2}}

Both sides match, so $Fx\cos\theta$ is homogeneous.

Common Mistake

What homogeneity cannot prove

A homogeneous equation is not necessarily correct. For example, a missing dimensionless constant could still leave the units unchanged. But a non-homogeneous equation is definitely wrong.

Prefixes

A prefix is added before a unit to show a power-of-ten multiplier. Prefixes let you write very large or very small values neatly.

Prefix	Symbol	Multiplier
pico	p	10^-12
nano	n	10^-9
micro	µ	10^-6
milli	m	10^-3
centi	c	10^-2
deci	d	10^-1
kilo	k	10^3
mega	M	10^6
giga	G	10^9
tera	T	10^12

In calculations, it is usually safest to convert everything into SI base units before substituting into equations.

Example

Using prefixes in a calculation

A resistor of resistance 4.7 kΩ carries a current of 3.2 mA. Find the potential difference across it.

Convert the prefixed quantities into SI units:

\times 10^3\,\Omega

\times 10^{-3}\,\mathrm{A}

Use the equation $V = I R$ :

\times 10^{-3})(4.7 \times 10^3)

Combine the numbers and powers of ten:

15.04\,\mathrm{V}

To two significant figures, the potential difference is 15 V.

Common Mistake

Cubed prefixes

Be careful when a prefixed length unit is squared or cubed. Since 1 cm = $10^{-2}$ m, then 1 cm^3 = $10^{-6}$ m^3, not $10^{-2}$ m^3.

Labelling graph axes and table columns

OCR expects graph axes and table headings to follow the convention:

quantity / unit

For example:

time / s
speed / m s^-1
acceleration / m s^-2
force / N
potential difference / V

This means the numerical values in the column or on the axis are the quantity divided by that unit. The unit belongs in the heading, not repeated in every data cell.

Example

Labelling a velocity-time graph

You are plotting velocity against time and want the units of the gradient.

Label the horizontal axis as time / s and the vertical axis as velocity / m s^-1.
A gradient is change in vertical quantity divided by change in horizontal quantity, so the gradient units are:

\frac{\mathrm{m\,s^{-1}}}{\mathrm{s}}

Simplify the units:

\mathrm{m\,s^{-2}}

This is the unit of acceleration, as expected for a velocity-time graph.

Tip

A quick axis-label check

If the axis label has only a unit, such as “m s^-1”, it is incomplete. Ask: what quantity is being measured? Write “speed / m s^-1” or “velocity / m s^-1”.

Exam technique

In the exam

Convert prefixed quantities to SI units before using formulae, especially mA, kΩ, cm and mm.
For unit questions, start from the equation and replace each quantity with its base units.
For graphs and tables, use the format quantity / unit, and remember gradient units are vertical-axis units divided by horizontal-axis units.

Self review

Check yourself

What are the SI base units for mass, length, time, current, temperature and amount of substance?
Show that density has units kg m^-3.
How should you label the y-axis for a graph of force measured in newtons?

What you'll learn

What S.I. base quantities and base units are, and which ones OCR expects you to know.
How derived units such as newtons, joules and volts are built from base units.
How to check whether an equation is homogeneous using S.I. base units.
How to convert prefixes and label graph axes/table columns correctly.

Why units matter

Definition

Physical quantity and unit

A physical quantity is a measurable property, such as mass, length, time or current. A unit is the agreed standard used to measure that quantity, such as kilogram, metre, second or ampere.

The whole topic is a toolkit: start from base units, build derived units, handle prefixes, then use units to check equations and present data clearly.

Concept map showing SI base quantities, derived units, prefixes and graph label convention

S.I. base quantities and base units

A base quantity is treated as independent: it is not defined by combining other quantities. A base unit is the SI unit for a base quantity.

For OCR A Physics, the key base quantities in this section are:

Base quantity	SI base unit	Unit symbol
mass	kilogram	kg
length	metre	m
time	second	s
electric current	ampere	A
thermodynamic temperature	kelvin	K
amount of substance	mole	mol

Officially, SI also includes luminous intensity, measured in candela, symbol cd. It is not a major calculation unit in H556, but it is part of the full SI base-unit set.

Tip

Writing unit symbols

Unit symbols are case-sensitive and are not pluralised: write 5 kg, not 5 kgs; 300 K, not 300 °K. A capital letter can completely change the meaning, such as m for milli/metre but M for mega.

Derived units

A derived unit is made by combining base units according to the equation defining the quantity.

For example:

speed has units metre per second, written m s^-1
acceleration has units metre per second squared, written m s^-2
momentum has units kg m s^-1
density has units kg m^-3

Key Idea

Unit algebra

Treat unit symbols like algebraic factors: multiply them, divide them and cancel them using the physical equation.

Example

Deriving units for momentum and density

Momentum is given by $p = m v$ , where mass has unit kg and velocity has unit $s−1\mathrm{m\,s^{-1}}$ .
Multiply the units in the equation:

\mathrm{kg} \times \mathrm{m\,s^{-1}} = \mathrm{kg\,m\,s^{-1}}

so momentum has units $s−1\mathrm{kg\,m\,s^{-1}}$ .

Density is given by $ρ=mV\rho = \frac{m}{V}$ , where mass has unit kg and volume has unit $m3\mathrm{m^3}$ .
Divide the units:

\frac{\mathrm{kg}}{\mathrm{m^3}} = \mathrm{kg\,m^{-3}}

so density has units $m−3\mathrm{kg\,m^{-3}}$ .

Named units you should recognise

Many derived units have special names. You still need to know that they are built from base units.

Unit name	Symbol	Common quantity	Base-unit form
hertz	Hz	frequency	s^-1
newton	N	force	kg m s^-2
joule	J	energy, work done	kg m^2 s^-2
watt	W	power	kg m^2 s^-3
pascal	Pa	pressure	kg m^-1 s^-2
coulomb	C	charge	A s
volt	V	potential difference	kg m^2 s^-3 A^-1
ohm	Ω	resistance	kg m^2 s^-3 A^-2
farad	F	capacitance	kg^-1 m^-2 s^4 A^2
tesla	T	magnetic flux density	kg s^-2 A^-1
weber	Wb	magnetic flux	kg m^2 s^-2 A^-1

The same letter can mean different things depending on context. For example, italic $W$ is often work done in equations such as $Fx\cos\theta$ , but upright W is the unit watt.

Common Mistake

Tesla or tera?

T on its own as a unit symbol means tesla. T before a unit as a prefix means tera, or $10^{12}$ , such as TW for terawatt.

Checking homogeneity of equations

An equation is homogeneous if every term has the same base units. This is one of the quickest ways to spot impossible equations.

Definition

Homogeneous equation

A physical equation is homogeneous when the units on the left-hand side match the units on the right-hand side, and all terms being added or subtracted have the same units.

Constants such as $12\frac{1}{2}$ and functions such as $sin⁡θ\sin\theta$ or $cos⁡θ\cos\theta$ have no units. They are dimensionless.

Example

Checking work done equation units

Check whether $Fx\cos\theta$ is homogeneous.

Force has units newtons, and from $F = ma$ :

\mathrm{N} = \mathrm{kg\,m\,s^{-2}}

Displacement $x$ has unit m, and $cos⁡θ\cos\theta$ is dimensionless.

Multiply the right-hand side units:

\mathrm{kg\,m\,s^{-2}} \times \mathrm{m} = \mathrm{kg\,m^2\,s^{-2}}

Work done is energy, measured in joules, and:

\mathrm{J} = \mathrm{kg\,m^2\,s^{-2}}

Both sides match, so $Fx\cos\theta$ is homogeneous.

Common Mistake

What homogeneity cannot prove

A homogeneous equation is not necessarily correct. For example, a missing dimensionless constant could still leave the units unchanged. But a non-homogeneous equation is definitely wrong.

Prefixes

A prefix is added before a unit to show a power-of-ten multiplier. Prefixes let you write very large or very small values neatly.

Prefix	Symbol	Multiplier
pico	p	10^-12
nano	n	10^-9
micro	µ	10^-6
milli	m	10^-3
centi	c	10^-2
deci	d	10^-1
kilo	k	10^3
mega	M	10^6
giga	G	10^9
tera	T	10^12

In calculations, it is usually safest to convert everything into SI base units before substituting into equations.

Example

Using prefixes in a calculation

A resistor of resistance 4.7 kΩ carries a current of 3.2 mA. Find the potential difference across it.

Convert the prefixed quantities into SI units:

\times 10^3\,\Omega

\times 10^{-3}\,\mathrm{A}

Use the equation $V = I R$ :

\times 10^{-3})(4.7 \times 10^3)

Combine the numbers and powers of ten:

15.04\,\mathrm{V}

To two significant figures, the potential difference is 15 V.

Common Mistake

Cubed prefixes

Be careful when a prefixed length unit is squared or cubed. Since 1 cm = $10^{-2}$ m, then 1 cm^3 = $10^{-6}$ m^3, not $10^{-2}$ m^3.

Labelling graph axes and table columns

OCR expects graph axes and table headings to follow the convention:

quantity / unit

For example:

time / s
speed / m s^-1
acceleration / m s^-2
force / N
potential difference / V

This means the numerical values in the column or on the axis are the quantity divided by that unit. The unit belongs in the heading, not repeated in every data cell.

Example

Labelling a velocity-time graph

You are plotting velocity against time and want the units of the gradient.

Label the horizontal axis as time / s and the vertical axis as velocity / m s^-1.
A gradient is change in vertical quantity divided by change in horizontal quantity, so the gradient units are:

\frac{\mathrm{m\,s^{-1}}}{\mathrm{s}}

Simplify the units:

\mathrm{m\,s^{-2}}

This is the unit of acceleration, as expected for a velocity-time graph.

Tip

A quick axis-label check

If the axis label has only a unit, such as “m s^-1”, it is incomplete. Ask: what quantity is being measured? Write “speed / m s^-1” or “velocity / m s^-1”.

Exam technique

In the exam

Convert prefixed quantities to SI units before using formulae, especially mA, kΩ, cm and mm.
For unit questions, start from the equation and replace each quantity with its base units.
For graphs and tables, use the format quantity / unit, and remember gradient units are vertical-axis units divided by horizontal-axis units.

Self review

Check yourself

What are the SI base units for mass, length, time, current, temperature and amount of substance?
Show that density has units kg m^-3.
How should you label the y-axis for a graph of force measured in newtons?

What you'll learn

Why units matter

S.I. base quantities and base units

Derived units

Named units you should recognise

Checking homogeneity of equations

Prefixes

Labelling graph axes and table columns

Review S.I. units by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

S.I. units

What you'll learn

Why units matter

S.I. base quantities and base units

Derived units

Named units you should recognise

Checking homogeneity of equations

Prefixes

Labelling graph axes and table columns

Review S.I. units by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

S.I. units

What you'll learn

Why units matter

S.I. base quantities and base units

Derived units

Named units you should recognise

Checking homogeneity of equations

Prefixes

Labelling graph axes and table columns

Review S.I. units by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

S.I. units Revision Guide

S.I. units

What you'll learn

Why units matter

S.I. base quantities and base units

Derived units

Named units you should recognise

Checking homogeneity of equations

Prefixes

Labelling graph axes and table columns

Review S.I. units by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

S.I. units Revision Guide