Physical quantities, SI units and prefixes: WASSCE Physics

Every answer you write in a WASSCE Physics paper is a physical quantity: a number and a unit. "The speed is 20" earns nothing; "the speed is 20 m s⁻¹" earns the mark. This lesson gives you the language the rest of Physics is written in.

Learning objectives

By the end of this lesson you should be able to:

define a physical quantity and distinguish fundamental from derived quantities
state the seven SI base quantities with their units and symbols
derive the SI units of common quantities from their defining equations
use prefixes and standard form, and convert between units correctly

What is a physical quantity?

Definition

A physical quantity is any property of a body or a phenomenon that can be measured and expressed as a number multiplied by a unit.

Length, mass, time, temperature, force and electric current are physical quantities. Beauty, kindness and taste are not: they cannot be measured on an agreed scale.

Every physical quantity has two parts:

\text{quantity} = \text{numerical value} \times \text{unit}

\begin{aligned}&\text{quantity} \\ &= \text{numerical value} \times \text{unit}\end{aligned}

For example, in $m = 65\ \text{kg}$ , the number is 65 and the unit is the kilogram. Change the unit and the number changes too: $65\ \text{kg} = 65\,000\ \text{g}$ . The quantity itself has not changed.

Fundamental and derived quantities

Definition

Fundamental (base) quantities are quantities that are not defined in terms of other quantities. Derived quantities are quantities obtained by combining fundamental quantities through multiplication or division.

The International System of Units (SI) uses seven fundamental quantities:

Fundamental quantity	SI unit	Unit symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Thermodynamic temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

The first three, length, mass and time, appear in almost every WASSCE question. Learn all seven: a common objective question simply asks which one of four quantities is fundamental.

Common mistake

Weight is not a fundamental quantity. Weight is a force ( $W = mg$ ), so it is derived. Mass is the fundamental one. Likewise, temperature is fundamental but heat (an energy) is derived.

Deriving SI units

The unit of a derived quantity comes straight from its defining equation. Replace each quantity by its SI unit and simplify.

Derived quantity	Defining equation	SI unit	Special name
Area	length × breadth	m²	—
Volume	length × breadth × height	m³	—
Density	mass ÷ volume	kg m⁻³	—
Speed, velocity	distance ÷ time	m s⁻¹	—
Acceleration	change in velocity ÷ time	m s⁻²	—
Force	mass × acceleration	kg m s⁻²	newton (N)
Momentum	mass × velocity	kg m s⁻¹	(= N s)
Pressure	force ÷ area	N m⁻² = kg m⁻¹ s⁻²	pascal (Pa)
Work, energy	force × distance	N m = kg m² s⁻²	joule (J)
Power	work ÷ time	J s⁻¹ = kg m² s⁻³	watt (W)
Frequency	1 ÷ period	s⁻¹	hertz (Hz)
Electric charge	current × time	A s	coulomb (C)

Worked example

Find the SI base units of pressure.

\text{pressure} = \dfrac{\text{force}}{\text{area}} = \dfrac{\text{mass} \times \text{acceleration}}{\text{area}}

\begin{aligned}&\text{pressure} \\ &= \dfrac{\text{force}}{\text{area}} \\ &= \dfrac{\text{mass} \times \text{acceleration}}{\text{area}}\end{aligned}

\text{unit} = \dfrac{\text{kg} \times \text{m s}^{-2}}{\text{m}^2} = \text{kg m}^{-1}\ \text{s}^{-2}

\begin{aligned}\text{unit} &= \dfrac{\text{kg} \times \text{m s}^{-2}}{\text{m}^2} \\ &= \text{kg m}^{-1}\ \text{s}^{-2}\end{aligned}

So $1\ \text{Pa} = 1\ \text{N m}^{-2} = 1\ \text{kg m}^{-1}\ \text{s}^{-2}$ .

Worked example

Express the joule in base units.

Work = force × distance, so $\text{J} = \text{N} \times \text{m} = (\text{kg m s}^{-2}) \times \text{m} = \text{kg m}^2\ \text{s}^{-2}$ .

Check with kinetic energy, $\tfrac{1}{2}mv^2$ : $\text{kg} \times (\text{m s}^{-1})^2 = \text{kg m}^2\ \text{s}^{-2}$ . The same unit, as it must be.

Prefixes: very large and very small numbers

Physics deals with the width of an atom and the distance to the Sun. Prefixes stop us writing long strings of zeros.

Prefix	Symbol	Multiplier	Example
giga	G	$10^{9}$	1 GHz = $10^{9}$ Hz
mega	M	$10^{6}$	1 MW = $10^{6}$ W
kilo	k	$10^{3}$	1 km = 1000 m
centi	c	$10^{-2}$	1 cm = 0.01 m
milli	m	$10^{-3}$	1 mA = 0.001 A
micro	μ	$10^{-6}$	1 μF = $10^{-6}$ F
nano	n	$10^{-9}$	1 nm = $10^{-9}$ m
pico	p	$10^{-12}$	1 pF = $10^{-12}$ F

In Sierra Leone

Bumbuna hydroelectric station was built to deliver about 50 MW: $50 \times 10^{6} = 5 \times 10^{7}$ W. A phone charger delivers about 10 W. The station's output is roughly five million phone chargers running at once.

Standard form (scientific notation)

Definition

A number is in standard form when it is written as $A \times 10^{n}$ , where $1 \le A < 10$ and $n$ is a whole number.

$384\,000\,000\ \text{m} = 3.84 \times 10^{8}\ \text{m}$ (distance to the Moon)
$0.000\,52\ \text{m} = 5.2 \times 10^{-4}\ \text{m}$

Converting units without mistakes

Write the conversion as a multiplication by a fraction equal to 1, so the unwanted unit cancels.

Worked example

Convert 72 km h⁻¹ to m s⁻¹.

72\ \dfrac{\text{km}}{\text{h}} \times \dfrac{1000\ \text{m}}{1\ \text{km}} \times \dfrac{1\ \text{h}}{3600\ \text{s}} = \dfrac{72\,000}{3600}\ \text{m s}^{-1} = 20\ \text{m s}^{-1}

\begin{aligned}&72\ \dfrac{\text{km}}{\text{h}} \times \dfrac{1000\ \text{m}}{1\ \text{km}} \times \dfrac{1\ \text{h}}{3600\ \text{s}} \\ &= \dfrac{72\,000}{3600}\ \text{m s}^{-1} \\ &= 20\ \text{m s}^{-1}\end{aligned}

Shortcut worth remembering: divide km h⁻¹ by 3.6 to get m s⁻¹.

Worked example

A cube of side 2 cm has a mass of 21.6 g. Find its density in kg m⁻³.

Volume $= (2\ \text{cm})^3 = 8\ \text{cm}^3 = 8 \times 10^{-6}\ \text{m}^3$ (because $1\ \text{cm}^3 = (10^{-2}\ \text{m})^3 = 10^{-6}\ \text{m}^3$ ).

Mass $= 21.6\ \text{g} = 0.0216\ \text{kg}$ .

\rho = \dfrac{m}{V} = \dfrac{0.0216}{8 \times 10^{-6}} = 2700\ \text{kg m}^{-3}

\begin{aligned}\rho &= \dfrac{m}{V} \\ &= \dfrac{0.0216}{8 \times 10^{-6}} \\ &= 2700\ \text{kg m}^{-3}\end{aligned}

That is the density of aluminium.

Common mistake

Converting areas and volumes: $1\ \text{m} = 100\ \text{cm}$ , but $1\ \text{m}^2 = 10\,000\ \text{cm}^2$ and $1\ \text{m}^3 = 1\,000\,000\ \text{cm}^3$ . Square or cube the conversion factor too.

Key points

A physical quantity = number × unit; always write the unit.
Seven SI base quantities: length (m), mass (kg), time (s), current (A), temperature (K), amount of substance (mol), luminous intensity (cd).
Derived units come from defining equations: N = kg m s⁻², J = kg m² s⁻², W = kg m² s⁻³, Pa = kg m⁻¹ s⁻².
Prefixes: G $10^9$ , M $10^6$ , k $10^3$ , c $10^{-2}$ , m $10^{-3}$ , μ $10^{-6}$ , n $10^{-9}$ , p $10^{-12}$ .
km h⁻¹ ÷ 3.6 = m s⁻¹.

WASSCE examination focus

WAEC tests this

Objective questions ask you to pick the fundamental quantity from a list, to identify the unit of a derived quantity (often in base units, e.g. "the unit of power is equivalent to kg m² s⁻³"), or to convert units. In theory papers, every final answer must carry the correct SI unit: a correct number with a wrong or missing unit usually loses the mark. Show the conversion line, especially for cm³ to m³ and km h⁻¹ to m s⁻¹.

Quick check

Which of these is a fundamental quantity: force, weight, mass, pressure?
Express the watt in SI base units.
Convert 54 km h⁻¹ to m s⁻¹.
Write 0.000 0067 m in standard form.

Answers: 1. Mass. 2. $\text{W} = \text{J s}^{-1} = \text{kg m}^2\ \text{s}^{-3}$ . 3. $54 \div 3.6 = 15\ \text{m s}^{-1}$ . 4. $6.7 \times 10^{-6}$ m.

Physical quantities, SI units and prefixes

Learning objectives

What is a physical quantity?

Fundamental and derived quantities

Deriving SI units

Prefixes: very large and very small numbers

Standard form (scientific notation)

Converting units without mistakes

Key points

WASSCE examination focus

Quick check

Keep going with physical quantities and units

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