Module 2 · Foundations of Physics

Units, Prefixes and Homogeneous Equations

Specification: OCR A H556 | Section: 2.1 Physical quantities and units | Focus: SI base units, derived units, prefixes, conversions, standard form, and checking equations by dimensional consistency

By the end of this topic you should be able to…

name the SI base quantities and their units
write common derived units in base-unit form
convert confidently between prefixes and units
use standard form sensibly in unit conversions
check whether an equation is homogeneous
explain what homogeneity can and cannot prove

Big idea: units are not just labels. They are a quick way to check whether a line of physics makes sense.

Part 1

Physical quantities, units and why they matter

A physical quantity has a number and a unit. For example, 3.2 m or 12 s.

Units matter because they tell you what the number means. Saying “the distance is 5” is incomplete. Saying “the distance is 5 m” is a usable physics statement.

OCR habit

Always include units in calculations, graph labels and final answers unless the quantity is genuinely unitless.

Part 2

SI base units

OCR A lists these SI base quantities and units:

Base quantity	Unit name	Unit symbol
Mass	kilogram	kg
Length	metre	m
Time	second	s
Current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol

Common misconception

Students often try to treat newton, joule or watt as base units. They are derived units.

Part 3

Derived units

Derived units are built from base units.

Quantity	Unit	Base-unit form
Velocity	m s⁻¹	m s⁻¹
Acceleration	m s⁻²	m s⁻²
Momentum	kg m s⁻¹	kg m s⁻¹
Density	kg m⁻³	kg m⁻³
Force	N	kg m s⁻²
Energy	J	kg m² s⁻²
Power	W	kg m² s⁻³
Pressure	Pa	kg m⁻¹ s⁻²

Being able to switch between named units and base-unit form is very useful for checking equations.

Part 4

Prefixes and powers of ten

Prefixes are shortcuts for powers of ten. They are closely linked to standard form.

Prefix	Symbol	Meaning
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	10³
mega	M	10⁶
giga	G	10⁹
tera	T	10¹²

Standard form link

For example, 4.7 ms = 4.7 × 10⁻³ s and 3.2 kV = 3.2 × 10³ V.

Easy trap

m can mean metre or the prefix milli. Context matters. For example, m s⁻¹ means metres per second, but ms means millisecond.

Part 5

How to convert between units and prefixes

A reliable method is:

convert the starting value into base units using powers of ten
convert from base units into the target unit

Example: 7.5 ms to s

7.5 ms = 7.5 × 10⁻³ s = 0.0075 s

Example: 0.0042 kV to V

0.0042 kV = 0.0042 × 10³ V = 4.2 V

Exam trap

When moving from a small unit to a larger unit, the number usually gets smaller. When moving from a large unit to a smaller unit, the number usually gets bigger.

Part 6

What does homogeneous mean?

An equation is homogeneous if both sides have the same base units.

Example: F = ma

Left side: F = N = kg m s⁻²
Right side: ma = kg × m s⁻² = kg m s⁻²

The units match, so the equation is homogeneous.

Why this is useful

it helps catch algebra mistakes
it helps check whether an equation could be right
it builds confidence in rearranging equations

Part 7

What homogeneity cannot prove

Passing a homogeneity check does not prove that an equation is physically correct.

Correct units, wrong physics

E = mv² has the same units as energy, but the correct kinetic energy equation is E = 1/2 mv².

Missing constants

An equation can have the right units but still be missing a constant or numerical factor.

Direction and meaning

Units do not tell you everything about vectors, signs, or whether the model makes physical sense.

Best exam wording

If the units match, say the equation is dimensionally consistent or homogeneous. Do not say this proves the equation is definitely correct.

Interactive

Unit converter and homogeneity checker

Mode

Value From To

3.2 mV = 0.0032 V
Step 1: 3.2 mV = 3.2 × 10⁻³ V. Step 2: so the answer is 0.0032 V.

Worked Examples

Worked examples

Worked example 1

Convert 6.4 ms into seconds.

milli means 10⁻³

6.4 ms = 6.4 × 10⁻³ s

Answer = 0.0064 s

Worked example 2

Convert 3.5 kV into volts.

kilo means 10³

3.5 kV = 3.5 × 10³ V

Answer = 3500 V

Worked example 3

Convert 8.2 × 10⁻⁶ m into micrometres.

micro means 10⁻⁶

8.2 × 10⁻⁶ m = 8.2 μm

Worked example 4

Write the unit of force in SI base units.

Use F = ma

mass has unit kg

acceleration has unit m s⁻²

So force has unit kg m s⁻²

Worked example 5

Show that P = E / t is homogeneous.

Energy has base units kg m² s⁻²

Divide by time s

kg m² s⁻² / s = kg m² s⁻³

This is the unit of power, so the equation is homogeneous

Worked example 6

A student suggests F = mv for force. Show that this is not homogeneous.

Force has units kg m s⁻²

mv has units kg × m s⁻¹ = kg m s⁻¹

The powers of s do not match

So the equation is not homogeneous

Worked example 7

Convert 0.45 mm into metres.

milli means 10⁻³

0.45 mm = 0.45 × 10⁻³ m

Answer = 4.5 × 10⁻⁴ m

Worked example 8

The equation for density is ρ = m / V. Show that the unit is kg m⁻³.

Mass has unit kg

Volume has unit m³

So density has unit kg / m³ = kg m⁻³

Knowledge

Knowledge Check

Name two SI base quantities and their units.

2 marks

Any two from mass kg, length m, time s, current A, temperature K, amount of substance mol

What prefix symbol means 10⁻⁶?

1 mark

What does it mean for an equation to be homogeneous?

1 mark

Both sides have the same base units / dimensions

State one limitation of a homogeneity check.

1 mark

Matching units do not prove an equation is physically correct

Exam Practice

Exam-Style Questions

State what is meant by a physical quantity.

1 mark

A measurable property with a numerical value and a unit

A voltage is written as 2.6 kV.

a) Write this in volts. [1 mark]
b) Write this in millivolts. [2 marks]

3 marks

a) 2.6 kV = 2600 V
b) 2600 V = 2.6 × 10⁶ mV

Write the following in SI base units:

a) force [1 mark]
b) energy [1 mark]
c) power [1 mark]

3 marks

a) kg m s⁻²
b) kg m² s⁻²
c) kg m² s⁻³

A student writes that 7.2 ms = 7200 s. Explain the mistake and give the correct answer.

2 marks

milli means 10⁻³, not 10³
7.2 ms = 0.0072 s

Show that the equation p = mv for momentum is homogeneous.

3 marks

Momentum has units kg m s⁻¹
mv has units kg × m s⁻¹ = kg m s⁻¹
So the equation is homogeneous

A student suggests the equation P = Ft for power.

a) Show that this equation is not homogeneous. [3 marks]
b) State the correct base units of power. [1 mark]

4 marks

a) F has units kg m s⁻²
Ft has units kg m s⁻² × s = kg m s⁻¹
Power has units kg m² s⁻³, so units do not match
b) kg m² s⁻³

Convert the following:

a) 4.5 μs into s [1 mark]
b) 0.008 m into mm [1 mark]
c) 3.6 × 10⁹ Hz into GHz [1 mark]

3 marks

a) 4.5 × 10⁻⁶ s
b) 8 mm
c) 3.6 GHz

Explain why a successful homogeneity check is useful, but does not prove an equation is correct.

3 marks

It shows that both sides have consistent units
This means the equation could be correct
But an equation can have the right units and still be missing a constant or have the wrong physical relationship

Topic Summary

Base units

Learn the SI base quantities and units exactly.

Prefixes

Think in powers of ten, and use standard form when it keeps working clear.

Derived units

Break named units into base-unit form when checking equations.

Homogeneity

Matching units is necessary for a correct equation, but not sufficient.

N = kg m s⁻²

J = kg m² s⁻²

W = kg m² s⁻³

μ = 10⁻⁶

k = 10³