Physical Quantities, SI Units and Prefixes – Introduction to Physics and Measurement of Physical Quantities (Senior 1 Physics) | Ganzaa

Introduction to Physics and Measurement of Physical Quantities

Lessons

Senior 1 · Physics

Introduction to Physics and Measurement of Physical Quantities

5 lessons

Lesson 2 of 5

Physical Quantities, SI Units and Prefixes

25 min

How long is your desk, in hand-spans?

Three pupils in a classroom in Nyagatare measured the same desk with their hand-spans. Then they measured it again with a metre rule.

Pupil	Length in hand-spans	Length with a metre rule
Aline	6.5 spans	1.20 m
Eric	7 spans	1.20 m
Uwase	8 spans	1.20 m

The hand-spans disagree because every hand is a different size. The metre rule gives everyone the same answer because the metre is the same for everyone. A measurement only means something when the unit is agreed by all. That is why a kilogram of beans at Kimironko market is the same amount as a kilogram in Rubavu, or in Nairobi.

Definition

A unit is an agreed standard amount of a quantity, used to measure it. Example: the metre (m) is the unit of length.

The SI (from the French Système International d'Unités) is the international system of units used by scientists and by almost every country, including Rwanda.

A fundamental (base) quantity is one of seven quantities from which all others are built. A derived quantity is made by multiplying or dividing base quantities. Example: speed $= \frac{\text{distance}}{\text{time}}$ is derived from length and time.

The seven SI base quantities

Base quantity	Usual symbol	SI unit	Unit symbol
Length	$l$	metre	m
Mass	$m$	kilogram	kg
Time	$t$	second	s
Electric current	$I$	ampere	A
Temperature	$T$	kelvin	K
Amount of substance	$n$	mole	mol
Luminous intensity (brightness of a light source)	$I_{v}$	candela	cd

The units are defined very precisely, so that a laboratory anywhere can reproduce them:

The second is the time for 9 192 631 770 vibrations of the radiation given out by a caesium-133 atom. Atomic clocks count these vibrations.
The metre is the distance light travels in empty space in $\frac{1}{299\,792\,458}$ of a second.
The kilogram has been defined since 2019 from a fixed value of a constant of nature, the Planck constant. Before that it was the mass of one metal cylinder kept near Paris.

Derived quantities

Every other quantity in this course is built from the base quantities. Here are the ones you will use most in Senior 1:

Derived quantity	Defining equation	SI unit
Area	$A = l \times w$	$\mathrm{m^2}$
Volume	$V = l \times w \times h$	$\mathrm{m^3}$
Density	$\rho = \frac{m}{V}$	$\mathrm{kg\,m^{-3}}$
Speed	$v = \frac{d}{t}$	$\mathrm{m\,s^{-1}}$
Force (and weight)	$W = m g$	newton, $1\,\mathrm{N} = 1\,\mathrm{kg\,m\,s^{-2}}$

Notice how the unit of a derived quantity comes straight from its equation. Density is mass divided by volume, so its unit is $\frac{\mathrm{kg}}{\mathrm{m^3}}$ , written $\mathrm{kg\,m^{-3}}$ . The small $- 3$ means "divided by $\mathrm{m^3}$ ".

Figure 2.1: A brick is a cuboid. Its volume is its length times its width times its height.

Worked example 1: the volume of a brick

A brick measures 20 cm by 10 cm by 5 cm. Find its volume in $\mathrm{cm^3}$ and in $\mathrm{m^3}$ .

V = l \times w \times h = 20\,\mathrm{cm} \times 10\,\mathrm{cm} \times 5\,\mathrm{cm} = 1000\,\mathrm{cm^3}

In metres the sides are 0.20 m, 0.10 m and 0.05 m, so

V = 0.20\,\mathrm{m} \times 0.10\,\mathrm{m} \times 0.05\,\mathrm{m} = 0.001\,\mathrm{m^3} = 1.0 \times 10^{-3}\,\mathrm{m^3}

Definition

A prefix is a word placed in front of a unit to multiply it by a power of ten. Example: kilo means $10^{3}$ , so $1\,\mathrm{km} = 10^{3}\,\mathrm{m} = 1000\,\mathrm{m}$ .

Metric prefixes in everyday life

Prefix	Symbol	Factor	Everyday example
giga	G	$10^{9}$	a phone with 64 GB of storage
mega	M	$10^{6}$	a hydropower station giving tens of megawatts (MW)
kilo	k	$10^{3}$	a 5 km run; a 2 kg bag of sugar
deci	d	$10^{-1}$	$1\,\mathrm{dm^3}$ is one litre
centi	c	$10^{-2}$	a 30 cm ruler
milli	m	$10^{-3}$	5 mL of syrup; a 500 mg tablet
micro	μ	$10^{-6}$	a red blood cell, about 8 μm across
nano	n	$10^{-9}$	a virus, about 100 nm across

To convert: replace the prefix symbol by its power of ten. For example, $3.5\,\mathrm{km} = 3.5 \times 10^{3}\,\mathrm{m} = 3500\,\mathrm{m}$ .

Worked example 2: converting with prefixes

(a) Convert 2.5 km to metres. (b) Convert 450 mg to kilograms.

(a) $2.5\,\mathrm{km} = 2.5 \times 10^{3}\,\mathrm{m} = 2500\,\mathrm{m}$

(b) First to grams, then to kilograms. Since $1\,\mathrm{g} = 10^{-3}\,\mathrm{kg}$ :

450\,\mathrm{mg} = 450 \times 10^{-3}\,\mathrm{g} = 0.450\,\mathrm{g} = 0.450 \times 10^{-3}\,\mathrm{kg} = 4.5 \times 10^{-4}\,\mathrm{kg}

Worked example 3: area and volume conversions

(a) A classroom floor is 8.0 m by 6.5 m. Find its area in $\mathrm{m^2}$ and in $\mathrm{cm^2}$ . (b) A jerrycan holds 20 L. What is this in $\mathrm{m^3}$ ?

(a) $A = 8.0\,\mathrm{m} \times 6.5\,\mathrm{m} = 52\,\mathrm{m^2}$ . Since $1\,\mathrm{m^2} = 10^{4}\,\mathrm{cm^2}$ , $A = 52 \times 10^{4}\,\mathrm{cm^2} = 5.2 \times 10^{5}\,\mathrm{cm^2}$ .

(b) $1\,\mathrm{L} = 1\,\mathrm{dm^3} = 10^{-3}\,\mathrm{m^3}$ , so $20\,\mathrm{L} = 20 \times 10^{-3}\,\mathrm{m^3} = 0.020\,\mathrm{m^3}$ .

Warning

Common mistake: converting $\mathrm{m^2}$ to $\mathrm{cm^2}$ by multiplying by 100. A square metre is 100 cm long AND 100 cm wide, so

1\,\mathrm{m^2} = 100\,\mathrm{cm} \times 100\,\mathrm{cm} = 10^{4}\,\mathrm{cm^2}, \qquad 1\,\mathrm{m^3} = 100\,\mathrm{cm} \times 100\,\mathrm{cm} \times 100\,\mathrm{cm} = 10^{6}\,\mathrm{cm^3}

Figure 2.2: A square metre holds 100 squares of 10 cm by 10 cm, each of 100 square centimetres, so one square metre is 10 000 square centimetres.

Tip

For the steps of ten between kilo and milli, remember King Henry Died By Drinking Chocolate Milk: kilo, hecto, deca, Base unit, deci, centi, milli. Each step to the right multiplies the number by 10.

Units you can trust: the Rwanda Standards Board

Rwanda uses SI units everywhere: road signs in kilometres, fuel sold in litres, milk collected at dairy cooperatives by the litre, and sugar and cement sold by the kilogram. The Rwanda Standards Board (RSB) works to keep these measurements honest. Its metrology service checks, or verifies, weighing scales used for trade and the pumps at fuel stations against reference standards, so that a customer who pays for 20 L of fuel receives 20 L.

Prefixes used with the base units (Alom Shaha). Watch for how each prefix is a power of ten, and try the conversions before he gives the answers.

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Practice

Give the SI unit and its symbol for: (a) mass, (b) time, (c) temperature, (d) electric current.
Convert: (a) 2.4 km to m; (b) 750 g to kg; (c) 45 cm to m; (d) 3.2 ms to s.
Sort these into base quantities and derived quantities: length, speed, area, mass, density, time.
Write with a suitable prefix: (a) 0.000 004 m; (b) 5 000 000 W.
A plot of land in Bugesera is 30 m by 20 m. Find its area in $\mathrm{m^2}$ and in $\mathrm{cm^2}$ .
A water tank is a cuboid 2.0 m long, 1.5 m wide and 1.2 m high. Find its volume in $\mathrm{m^3}$ and in litres.
A bottle holds 120 mL of medicine. How many 5 mL doses does it hold? Write 120 mL in $\mathrm{m^3}$ .
A pupil writes $3\,\mathrm{m^2} = 300\,\mathrm{cm^2}$ . Explain the mistake and give the correct value.

Show answersHide answers

(a) kilogram, kg; (b) second, s; (c) kelvin, K; (d) ampere, A.
(a) 2400 m; (b) 0.75 kg; (c) 0.45 m; (d) $3.2 \times 10^{-3}\,\mathrm{s} = 0.0032\,\mathrm{s}$ .
Base: length, mass, time. Derived: speed, area, density.
(a) 4 μm; (b) 5 MW.
$A = 30\,\mathrm{m} \times 20\,\mathrm{m} = 600\,\mathrm{m^2} = 600 \times 10^{4}\,\mathrm{cm^2} = 6.0 \times 10^{6}\,\mathrm{cm^2}$ .
$V = 2.0 \times 1.5 \times 1.2 = 3.6\,\mathrm{m^3}$ . Since $1\,\mathrm{m^3} = 1000\,\mathrm{L}$ , $V = 3600\,\mathrm{L}$ .
$\frac{120}{5} = 24$ doses. $120\,\mathrm{mL} = 120\,\mathrm{cm^3} = 120 \times 10^{-6}\,\mathrm{m^3} = 1.2 \times 10^{-4}\,\mathrm{m^3}$ .
The pupil multiplied by 100, as for length. For area the factor is $100 \times 100 = 10^{4}$ , so $3\,\mathrm{m^2} = 3 \times 10^{4}\,\mathrm{cm^2} = 30\,000\,\mathrm{cm^2}$ .

Key points

A measurement is a number and a unit. SI units are the same everywhere.
The seven base units are the metre, kilogram, second, ampere, kelvin, mole and candela. All other units are derived from them.
A prefix multiplies a unit by a power of ten: $\mathrm{k} = 10^{3}$ , $\mathrm{c} = 10^{-2}$ , $\mathrm{m} = 10^{-3}$ , $\mu = 10^{-6}$ .
For areas and volumes, square or cube the conversion factor: $1\,\mathrm{m^2} = 10^{4}\,\mathrm{cm^2}$ , $1\,\mathrm{m^3} = 10^{6}\,\mathrm{cm^3}$ .

Sources and credits

1.2: Physical Quantities and Units, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Prefixes used with the base units, Alom Shaha, embedded from YouTube.

The worked examples, diagrams, Rwandan applications and practice exercises are written by Ganzaa.

1How long is your desk, in hand-spans?

2The seven SI base quantities

3Derived quantities

4Metric prefixes in everyday life

5Units you can trust: the Rwanda Standards Board

How long is your desk, in hand-spans?

The seven SI base quantities

Derived quantities

Metric prefixes in everyday life

Units you can trust: the Rwanda Standards Board