SI units, prefixes and homogeneity

A number on its own says nothing in physics: 5 could be five metres, five seconds or five kilograms. Every measured quantity is a number multiplied by a unit, and the units used in this lesson can be built from the six SI base units listed below. Writing units in base units lets you convert data safely and, before doing any physics, spot an equation that cannot possibly be right.

SI base quantities and prefixes

H1 Physics uses these six SI base quantities, each with its own unit. SI has seven base quantities in total; the seventh is luminous intensity, measured in candela (cd).

A derived quantity is defined by an equation that links it to base quantities.

Prefixes stand for powers of ten:

Capital and small letters mean different things: m is milli, M is mega. The base unit of mass already carries a prefix: it is the kilogram, so 1 g = 10⁻³ kg.

When a prefixed unit is squared or cubed, the prefix is squared or cubed too, because it belongs to every length in the product:

Derived units in base units

To write a derived unit in base units, take the equation that defines the quantity and replace each quantity by its unit.

Each row uses units found above it. For the volt, J C⁻¹ = (kg m² s⁻²)/(A s) = kg m² s⁻³ A⁻¹. Rebuilding a unit from its equation is safer than memorising a string of symbols.

Guided practice 1

An area and a spring constant

About 4 min

Problem

Convert 3.6 cm² to m². A spring obeys F = kx, where x is its extension. Express the unit of k in SI base units.

Convert the area, then build the unit

Area

m²Unit: m²

Check answer

Hints

Hint 1

area

Square the conversion factor: 1 cm² = (10⁻² m)².

Hint 2

spring constant

Rearrange to k = F/x, then replace force and extension by their base units.

Show solution step by step

1. Convert the area

   Method

   Square the factor 10⁻².

   Reason

   The prefix belongs to both lengths that make up the area.

   Working

   Answer

   3.6 cm² = 3.6 × (10⁻²)² m² = 3.6 × 10⁻⁴ m²

2. Build the spring constant's unit

   Working

   Answer

   k = F/x, so its unit is N m⁻¹ = (kg m s⁻²)/m = kg s⁻².

Check your understanding 1 Explain your reasoning

The spring above obeys F = kx. Use its defining equation to work out the SI base units of k.

The spring constant in base units, with the unit calculation

Show answer

k = F/x, so its unit is N m⁻¹ = kg m s⁻²/m = kg s⁻². Compare your defining equation, division by length and cancellation of the metre factors.

Check your understanding 2 The ohm

Use R = V/I and the table to express the ohm in SI base units.

Build the ohm from the units of voltage and current

Show answer

Ω = V A⁻¹ = (kg m² s⁻³ A⁻¹)/A = kg m² s⁻³ A⁻²

Check the division by current changes A⁻¹ to A⁻²; show the defining equation rather than just a memorised unit.

Checking that an equation is homogeneous

You can only add, subtract or equate quantities of the same kind: a length plus a time means nothing. So in a correct equation, every term that is added or subtracted, and the two sides of the equation, must have the same base units. Such an equation is homogeneous.

A homogeneity check can prove an equation wrong: if the base units do not match, it cannot be right. It cannot prove an equation right. Pure numbers such as 1/2 or 2π have no units, so a missing factor does not show up, and an equation can have the right units but describe the wrong physics.

Check your understanding 3 Can this equation be right?

A learner writes v = u + at², where v and u are velocities, a is an acceleration and t a time. Use base units to decide whether the equation can be right.

Compare the units of every term

Show answer

v and u have unit m s⁻¹, but at² has unit (m s⁻²)(s²) = m. A velocity cannot be added to a length, so the equation is wrong.

Compare the units of v, u and at², then state why unlike terms cannot be added.

A homogeneity check can also find the units of an unknown constant: whatever units are missing to make both sides match must belong to the constant.

Worked example 2

Units of a constant in a drag equation

Problem

The power needed to push a car through the air is modelled as P = kρ Av³, where ρ is the density of air, A the frontal area of the car and v its speed. Find the SI base units of k.

Show full solution

1. Write each quantity in base units

   Method

   Replace every known quantity by its base units.

   Reason

   The constant can only be isolated once both sides are in the same units.

   Working

   Answer

   P: kg m² s⁻³. ρ: kg m⁻³. A: m². v³: m³ s⁻³.

2. Combine the known factors

   Method

   Multiply the units of ρ, A and v³.

   Reason

   Multiplying quantities adds the powers of each base unit.

   Working

   Answer

   (kg m⁻³)(m²)(m³ s⁻³) = kg m² s⁻³

3. Compare with the left-hand side

   Working

   Answer

   ρ Av³ already has the units of power, so k has no units. The equation passes the check, but that does not show that the model, or any value of k, is correct.

Check your understanding 4 A different drag model

Another student proposes P = Cρ Ar²v³, where r is a length. Find the SI base units that C must have for the equation to be homogeneous.

Find the units of C and show the unit product

Show answer

ρ Ar²v³ has units (kg m⁻³)(m²)(m²)(m³ s⁻³) = kg m⁴ s⁻³. Power is kg m² s⁻³, so C must have unit m⁻².

Check that the added r² contributes m²; the constant must cancel that factor.

Exam-style question 1

The constant in a spring equation

4 marks

Examination question

The period T of a mass m oscillating on a spring of spring constant k is thought to be T = C square root of (m/k), where C is a constant. Use SI base units to show whether C has units, and state one thing that this check cannot tell you. [4 marks]

Find the units of k first

Write your four-mark answer

Show solution step by step

1. Units of k

   1 mark

   Method

   Use F = kx.

   Reason

   The spring constant is force per unit extension.

   Working

   Answer

   k = F/x: (kg m s⁻²)/m = kg s⁻²

2. Units of the square root

   1 mark

   Method

   Simplify m/k, then take the square root.

   Reason

   C must supply whatever is needed to turn square root of (m/k) into seconds.

   Working

   Answer

   m/k: kg/(kg s⁻²) = s², so square root of (m/k) has unit s.

3. Conclude

   2 marks

   Method

   Compare with the unit of T, and state the limit of the check.

   Reason

   Homogeneity is necessary for a correct equation, but not enough.

   Working

   Answer

   square root of (m/k) already has the unit of T, so C has no units. The check cannot give the value of C (it is 2π), and it cannot show that the equation describes a real spring.

Self-mark with the mark scheme

Tick each mark point that your answer includes.

Award one mark for each point.

Mark points

1. 1 mark: Units of k are kg s⁻².
2. 1 mark: m/k has unit s², so its square root has unit s.
3. 1 mark: C has no units.
4. 1 mark: The check cannot find the value of C or prove the equation correct.

Calculate your mark

Common mistakes

• Treating the gram as the base unit of mass. It is the kilogram.
• Converting cm² or cm³ with 10⁻² once, instead of squaring or cubing it.
• Mixing up m (milli) and M (mega).
• Saying an equation is correct because it is homogeneous. Homogeneity can only show that an equation is wrong.