SI Units
Learning Outcome
When you complete this module you will be able to:
Perform simple calculations involving SI units.
Learning Objectives
Here is what you will be able to do when you complete each objective:
1. List basic SI units and their symbols.
2. Identify and list symbols for unit prefixes.
3. Perform unit analysis in simple problems.
4. List derived SI units and their symbols.
5. Perform conversions between SI and Imperial units.
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PHYS 6001
INTRODUCTION
Various systems of measurement have been used in different parts of the world
for centuries. However, because each system was designed with its own base
units, conversions from one system to another presented many problems. It
became obvious that a standard system would have to be developed. This system
would have to be precise, to allow for accurate measurement, yet simple enough
to allow for conversions from one unit to another within the system.
Such a system was developed in France in the seventeenth century, and was
gradually adopted by other countries. This system was called the Metric System.
In 1960, the latest version of the Metric System was developed, and named Le
Systeme International d’unites (the International System of Units), more
commonly called SI.
BASIC UNITS
There are seven base units in SI, as shown in Table 1:
Quantity
Name of Base Unit
Symbol
metre
m
Mass
kilogram
kg
Time
second
s
Electric current
ampere
A
Thermodynamic temperature
kelvin
K
Amount of substance
mole
mol
candela
cd
Length
Luminous intensity
Table 1
SI Base Units
Note: The base unit kilogram is the only base unit with a prefix. The kilogram
was selected as a base unit since the gram was considered to be too small
to be functional.
Several SI base units may be familiar to you, and others may not. In any case,
you must know this chart thoroughly before proceeding with this module.
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PHYS 6001
Multiples and Submultiples of Base Units
One area in which SI is simple to work with, compared to other systems, is that of
converting small units to large, and vice versa.
For example, suppose we wish to convert a distance, given in Imperial units, to a
value with smaller units. Let’s do a conversion from miles to inches. First, the
distance in miles would be converted to yards, or feet, using 1 mile = 1760 yards,
or 1 mile = 5280 feet. Then, using 1 foot = 12 inches, or possibly 1 yard = 3 feet,
we could arrive at the correct value and units. However, the conversion factors
have to be exact, and an error could occur simply because the conversion may
require several steps to complete.
In SI, conversion factors are not required. Changing from a larger unit to a
smaller one, or vice versa, requires only that you multiply or divide the unit by 10
or a multiple of 10 (i.e., 100, 1000, etc.) This can be done simply by moving a
decimal point, in most cases.
A specific set of prefixes is used to denote the resulting units after a decimal point
has been moved. Table 2 indicates the various prefixes, and the asterisks indicate
those most commonly used.
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PHYS 6001
Prefix
Symbol
Factor by Which Unit is Multiplied
18
exa
E
1 000 000 000 000 000 000
10
peta
P
1 000 000 000 000 000
10
tera
T
1 000 000 000 000
10
giga
G
1 000 000 000
10
*
mega
M
1 000 000
10
*
kilo
k
1 000
10
hecto
h
100
10
deca
da
10
10
1
10
15
12
9
6
3
2
1
0
-1
deci
d
0.1
10
*
centi
c
0.01
10
*
milli
m
0.001
10
*
micro
µ
0.000 001
10
nano
n
0.000 000 001
10
pico
p
0.000 000 000 001
10
femto
f
0.000 000 000 000 001
10
atto
a
0.000 000 000 000 000 001
10
-2
-3
-6
-9
-12
-15
-18
Table 2
Metric Prefixes
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To illustrate the use of prefixes, let’s consider the unit of length in SI, the metre,
with some prefixes.
Name
Symbol
Meaning
Multiply Metres By
megametre
Mm
one million metres
1 000 000
kilometre
km
one thousand metres
1 000
hectometre
hm
one hundred metres
100
decametre
dam
ten metres
10
decimetre
dm
one tenth of a metre
0.1
centimetre
cm
one hundredth of a metre
0.01
millimetre
mm
one thousandths of a metre
0.001
micrometre
µm
one millionth of a metre
0.000 001
Table 3
Metric Prefixes
Although any multiple or submultiple of a unit may be used, it is recommended
that prefixes representing 10 raised to the power of a multiple of 3 (i.e. 10-3, 103,
10-6, 106, 109, etc.) are selected. For example, kilometre (km = 103 x m) is
preferred to hectometre (102 x m).
The following are examples of conversions from one prefix to another:
Example 1:
Convert 0.723 km to metres.
Solution:
1 km = 1000 m, or there are one thousand metres per kilometre, written as 1000
m/km. For calculation purposes, we could write 1000 m/km as 1000 m
km
(The reason for this is explained shortly.)
Then:
0.723 km = 0.723 km × 1000 m
km
= 723 m (Ans.)
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PHYS 6001
Example 2:
Convert 0.045 m to millimetres.
Solution:
Since 1 m = 1000 mm
0.045 m = 0.045 m x 1000 mm
m
= 45 mm (Ans.)
Example 3:
Convert 109 mm to centimetres.
Solution:
Since 1 cm = 10 mm
109 mm = 109 mm x 1 cm
10 mm
= 10.9 cm (Ans.)
Unit Analysis
Notice that in the above examples, the units are included in the solutions. Units
should be included in all calculations if possible. If you conduct a "unit analysis"
for a solution, you will find that when the units for the final solution are proven to
be correct, then the numeric result is usually correct. This is assuming, of course,
that you have not made any math errors.
For example, a unit analysis for Example 3 would be:
cm = cm
mm × mm
The millimetres cancel out, leaving only cm in the final answer.
cm
mm
– – × mm
– – = cm
If you had solved the problem this way:
109 mm × 10 mm = 1090 cm
1 cm
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PHYS 6001
A unit analysis would reveal your error:
mm 2
mm × mm
cm = cm , which is not a unit of length
For simple problems, such as Example 3, do a unit analysis at the same time as
you solve the problem:
109 mm × 1 cm = 10.9 cm
10 mm
For more advanced calculations, you may find it easier to perform a separate unit
analysis. The important thing to remember is: Always do a unit analysis.
Example 4:
Find the area of a rectangle that is 1.5 m long, and 93 cm wide. Use the formula,
A = L x W.
In cases like this, where we are going to multiply units together, we will have to
change one, so that both units are the same. In this example, the final unit was not
specified, so there are various ways of solving the problem.
Solution 1:
Change both dimensions to centimetres.
A = LxW
= (1.5 m x 100 cm/1 m) x 93 cm
= 150 cm x 93 cm
= 13 950 cm2 (Ans.)
Solution 2:
Change both dimensions to metres.
A = LxW
= 1.5 m x (93 cm x 1 m/100 cm)
= 1.5 m x 0.93 m
= 1.395 m2 (Ans.)
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Alternate Solution 2:
From Solution 1:
A = 13 950 cm2
Since: 1 m2 = 1 m x 1 m
= 100 cm x 100 cm
= 10 000 cm2
Then:
1.395 m2 = 1.395 m2 x (10 000 cm2/1 m2)
= 13 950 cm2 (Ans.)
Example 5:
Convert 278 827 cg to kg.
Solution:
278 827 cg x (1 g/100 cg) x (1 kg/1000 g)
= 2.788 27 kg (Ans.)
Example 6:
Convert 1 000 000 mm to km.
Solution:
1 000 000 mm x (1 m/1000 mm) x (1 km/1000 m)
= 1 km (Ans.)
Notice in these examples that the conversion was done in two steps. Sometimes it
is easier to convert a value to unity (1 kg, 1 m, etc.) and then to convert to the
final unit.
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PHYS 6001
Writing SI Symbols
You should now be ready to attempt some problems using SI units and symbols.
However, before you do, learn the following basic rules for writing SI symbols
and for writing numbers with SI units. These rules must always be followed:
1. Always use the correct symbol. For example, use kg for kilogram, rather than
something like klg, which does not exist.
2. Symbols must always be written vertically. For example, use kg, not kg.
3. Symbols are lower case letters, unless they are derived from a proper name.
For example, use s (second), not S. (One exception to this rule will be
mentioned later.)
4. Symbols are always singular. For example, 75 kg, not 75 kgs.
5. Do not use a period after a symbol, unless the symbol is at the end of a
sentence. For example, use "...27 m long.", not "...27 m. long."
6. Do not leave a space between a prefix and a unit symbol. For example, use
kg, not k g.
7. Use only the symbol with numerals. Use the full name of the unit when the
number is written out. For example: Use 5 m, not 5 metres. Use five metres,
not five m.
8. Leave a space between a number and a symbol. For example, use 18 kg, not
18kg. An exception to this would be when a letter does not follow a number,
such as in 18°C.
9. When writing numbers, use decimals rather than fractions. For example, use
1.5 kg, not 1 1/2 kg.
10. If a number is less than one, use a zero before the decimal point. For
example, use 0.8 m, not .8 m.
11. Numbers must be separated into blocks of three digits each, instead of using
commas. For example, use 78 232 456.738 92 mm, not 78,232,456.73892
mm. However, if a number has only four digits, a space is optional. For
example, use 3.1416 cm or 3.141 6 cm.
12. When multiplying, use the multiplication sign (x) instead of a dot. For
example, use 178 x 97.3 m, not 178 • 97.3 m.
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PHYS 6001
MEASURING PHYSICAL QUANTITIES
The following is an introduction to several other units that are used in the SI
system. The topics are discussed in greater detail in other modules.
Length
Although the SI unit of length is the metre (m), multiples or submultiples are
often used in everyday situations. Engineering drawings frequently use
millimetres (mm), the textile industry often uses centimetres (cm), and highway
distances may be given in kilometres (km). For navigational purposes, the
nautical mile has been in use for a long time, and is presently included in SI. A
nautical mile is a distance of 1852 m.
Area
In SI, the product of any two quantities produces the unit of the resultant quantity.
This simply means, for example, that unit length (1 m) multiplied by unit width (1
m) equals unit area (1 m2). Although the square metre (m2) is the unit of area, the
square centimetre (cm2) and the square millimetre (mm2) are often used.
When measuring land area, such as a farm, the hectare (ha) is often used.
1 ha = 1 hm2
= 1 hm x 1 hm
= 100 m x 100 m
= 10 000 m2
For measuring extremely large geographical areas, the square kilometre (km2) is
used.
Volume
The cubic metre (m3) is the unit of volume in SI.
1 m3 = 1 m x 1 m x 1 m
The cubic centimetre (cm3) is often used for laboratory work.
1 cm3 = 1 cm x 1 cm x 1 cm
The cubic decimetre (dm3) is also used for measuring solids, liquids, or gases.
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PHYS 6001
1 dm3 = 1 dm x 1 dm x 1 dm
= 0.1 m x 0.1 m x 0.1 m
= 0.001 m3
The cubic decimetre is given the name "litre". The symbol for the litre is L. (If
you refer back to the rules on writing of SI symbols, the litre (L) is the one
exception referred to in Rule 3).
Since,
1 dm3 = 1 L
and,
1 L = 0.001 m3
then,
1000 L = 1 m3
This fact can be useful when dealing with problems, and you should become very
familiar with the litre (L).
The millilitre (mL) is often used as a unit of volume in medicine, cooking, and
laboratories.
Speed
Speed is defined as the distance a body travels in a unit of time. (Speed and
velocity are not indentical, although their units are.) The unit for distance is the
metre (m), and the unit for time is the second (s), as we already know.
In SI, the quotient of any two quantities produces the unit of the resultant
quantity. In the case of speed, a distance of one metre (1 m) divided by a time of
one second (1 s) results in a unit of speed of one metre per second (1 m/s).
The kilometre per hour (km/h) is often used to describe automobile and airplane
speeds.
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PHYS 6001
Ships and aircraft often use the knot (kn) as a measure of speed. One knot is a
speed of one nautical mile per hour.
1 kn = one nautical mile per hour
= one thousand eight hundred and fifty-two metres per hour
(1852 m/h)
Since,
1 hr = 3600 s (60 s/min x 60 min/h)
Then,
1 kn = 1852 m/h x (1 h/3600 s)
= 0.514 m/s (approximately)
Hopefully, you paid particular attention to the manner in which the unit for speed
(m/s) was written. When two symbols are combined to form a unit, the following
rules must always be applied:
1. Use a slash (oblique stroke) with symbols rather than the word "per". For
example, use km/h, not km per h.
2. Use the word "per" when writing full names of symbols rather than a slash.
For example, use kilometres per hour, not kilometres/hour.
Acceleration
Acceleration is defined as the rate of change of velocity. You have learned that
the unit of velocity is m/s. The word "rate" indicates a unit of time, the second
(s). So acceleration deals with change in velocity (m/s) per unit of time, the
second (s), or acceleration is metres per second, per second. The unit becomes
m/s/s. Then,
metres per second, per second = m/s
s
= (multiply top and bottom by 1/s, which does
not change the value).
= m/s × 1/s
s × 1/s
= m/s × 1/s
1
= m/s2
The unit for acceleration then is m/s2.
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PHYS 6001
Mass
The mass of a body refers to the quantity of matter that it contains. (The term
"weight" is not the same as mass, and the difference between the terms is
explained in another module.) In everyday use, the term "weight" is frequently
misused to mean "mass" when working in SI.
The unit of mass is the kilogram (kg). Large masses may be expressed in
megagrams (Mg), equal to one thousand kilograms. This is also called a metric
ton or tonne (t).
1000 g = 1 kg
1000 kg = 1 Mg
= 1t
Density
Density refers to the mass of a unit volume of a substance. Since mass is in
kilograms (kg), and volume is in cubic metres (m3), it is obvious that the unit of
density is kilograms per cubic metre, kg/m3.
Temperature
This topic is discussed in detail in another module. For now, let’s just state that
the degree Celsius (°C) is the unit encountered in everyday use. For scientific
work, the unit of temperature is the kelvin (K). The terms "Celsius degree",
"degree Kelvin", "°K", and "deg" are all incorrect, and must not be used when
expressing a temperature.
Force
Force is covered in detail in another module, so the definition will not be given
here. However, part of the definition of force involves the idea of imparting
acceleration (m/s2) to a mass (kg). The force required is calculated using the
following formula:
Force = mass x acceleration
= kg x m/s2
= kgm/s2
The unit kgm/s2 is called a newton (N). This unit is very important, because it is
used to form several other units as well.
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PHYS 6001
Work
Work is done when a force causes a body to move through a distance. The
following formula can be used:
Work = force x distance
= Nxm
= Nm (newton metre)
The newton metre (Nm) is called a Joule (J).
Energy
The term energy means the ability to do work. In other words, energy can be
converted to work. Since the unit of work is the joule (J), then energy must also
be expressed in joules, or a multiple of joules.
There are many forms of energy, and you will discover that various formulae are
used to calculate various forms of energy. As you encounter each form of energy,
do a unit analysis for each formula, to prove that the units are in joules or a
multiple of joules.
Power
Power is defined as the rate of doing work. As was mentioned earlier, the word
"rate" indicates a unit of time, the second (s). Power, then, is a measure of work,
in joules (J), over a given time period in seconds(s).
Power = work/time
= J/s
When one joule of work is done per second, we say the power developed is one
watt (W).
J/s = W
The kilowatt (kW) and megawatt (MW) are usually used to indicate larger values
of power.
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PHYS 6001
Pressure
Pressure is defined as the force exerted over a unit area of a surface.
Pressure = force/area
= N/m2
A force of 1 N acting on an area of 1 m2 is called a pascal (Pa)
Most equipment such as boilers, pumps, and compressors develop pressures
which are expressed in kilopascals (kPa) or megapascals (MPa), because the
pascal (Pa) is a relatively low pressure.
A unit of pressure sometimes associated with SI is the bar.
1 bar = 100 kPa
However, the bar is not a recognized unit in SI.
A millibar (mbar) is equal to a pressure of one hundred pascals (100 Pa). It may
be used, but only when performing international meteorological work.
Note: You probably have noticed that some of the units introduced in the last
few topics have been given a symbol beginning with a capital letter.
These symbols are derived from a proper name. Some examples are
newton (N), joule (J), watt (W), and pascal (Pa). Refer to the section on
"Writing SI Symbols", Rule 3.
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PHYS 6001
Table showing commonly used units.
Quantity
Unit
Symbol
Derivation
Area
square metre
m
2
mxm
Volume
cubic metre
m
3
mxmxm
Speed
metre per second
m/s
Acceleration
metre per second squared
m/s
Density
kilogram per cubic metre
kg/m
Specific volume
cubic metres per kilogram
m /kg
volume of a unit mass
Frequency
hertz
Hz
l/s; reciprocal time
Force
newton
N
kg•m/s ; mass times
acceleration
Pressure
pascal
Pa
N/m ; force per area
Energy (work)
joule
J
N•m; force times
displacement
Power
watt
W
J/s; energy per time
Concentration
mole per litre
mol/L
mole per unit volume
Molar mass
gram per mole
g/mol
(kg/kmol)
mass per unit mole
Molar volume
cubic metre per mole
m /mol
volume per unit mole
Mass flow rate
kilogram per second
kg/s
kilogram per unit time
distance per unit time
2
change in speed per
unit time
3
3
mass per unit volume
2
3
2
Table 4
Derived Units
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PHYS 6001
UNIT CONVERSIONS
There may be occasions when units from one system may have to be converted to
units of another system. The conversion chart shown below can be used to
convert from SI units to Imperial units, and vice versa.
Length
1 in
1 cm
1 ft
1m
1 mile
=
=
=
=
=
2.54 cm
0.3937 in
0.3048 m
3.28 ft
1609 m
Area
1 in2
1 cm2
1 ft2
1 m2
1 sq. mile
1 km2
=
=
=
=
=
=
6.45 cm2
0.155 in2
0.093 m2
10.75 ft2
2.59 km2
0.386 sq mile
Volume
1 in3
1 cm3
1 ft3
1 m3
=
=
=
=
16.39 cm3
0.061 in3
0.0283 m3
35.336 ft3
Capacity
1 qt
1L
1 gal
1L
=
=
=
=
1.136 L
0.88 qt
4.546 L
0.22 gal
Mass
1 lb
1 kg
=
=
0.454 kg
2.2 lb
Force
1 lb
1N
=
=
4.448 N
0.225 lb
Pressure
1 lb/in2(psi)
1 kPa
1 bar
1 bar
1 psi
=
=
=
=
=
6.895 kPa
0.145 lb/in2(psi)
100 kPa
14.51 psi
0.069 bar
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PHYS 6001
Energy
1 ft lb
1J
1 Btu
1 kJ
1 kcal
1 Btu
1 kcal
1 kJ
1 hp-hr
1 MJ
1 watt-hr
1 kJ
=
=
=
=
=
=
=
=
=
=
=
=
1.356 J
0.737 ft lb
1.055 kJ
0.948 Btu
3.968 Btu
0.252 kcal
4.186 kJ
0.239 kcal
2.685 MJ
0.372 hp-hr
3.6 kJ
0.278 watt-hr
Power
1 hp
1 kW
=
=
0.746 kW
1.34 hp
Example 7:
Convert 22 miles to kilometres.
Solution:
From the chart,
1 mile = 1609 m
= 1.609 km
22 miles = 22 miles x 1.609 km/1 mile
= 35.398 km (Ans.)
Example 8:
Convert 13 790 kPa to pounds per square inch.
Solution:
From the chart,
1 psi = 6.895 kPa
13 790 kPa = 13 790 kPa x 1 psi/6.895 kPa
= 2000 psi (Ans.)
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PHYS 6001
As you become familiar with SI, some of the commonly used conversions will
become familiar to you. For example, you will probably remember that 1 inch =
2.54 cm, or 25.4 mm. Another common conversion is 1 psi = 6.895 kPa.
Once you have memorized some of the common conversions, you should be able
to derive some of the others. For example, suppose a cube has sides of 3.75 feet,
and you wish to calculate the number of square metres in the cube. You could
solve the problem this way, without referring to a conversion chart, knowing from
memory only that 1 inch = 2.54 cm:
Each side length = 3.75 feet x 12 inches/1 foot
= 45 in
Side length in SI = 2.54 cm/inch x 45 inches
= 114.3 cm
114.3 cm x 1 m/100 cm = 1.143 m
Volume of cube = (length of side) 3
= (1.143 m) 3
(Notice that the unit, m, is also cubed.)
= 1.49 m3 (Ans.)
(Rounded off to 3 significant digits).
As you become more proficient with conversions, you will probably do the same
problem more quickly, such as this:
Side length = (3.75 ft x 12 in/ft) x 2.54 cm/in x 1 m/100 cm
= 1.143 m
Volume = (1.143 m) 3
= 1.49 m3 (Ans.)
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PHYS 6001
Self Test
After completion of the self test, check your answers with the answer guide that
follows.
1. Name the SI base units for the following, and in brackets behind each one,
write its symbol: (a) length (b) mass (c) time.
2. Write the symbol for each of the following: (a) millimetre
(b) megagram (c) decisecond (d) microgram (e) decametre
(f) centimetre (g) millisecond (h) kilometre.
3. Convert the following to metres: (a) 289 cm (b) 1828 mm (c) 1.45 km
(d) 17 dm (e) 17 dam.
4. Convert the following to centimetres: (a) 131 m (b) 2.7 km (c) 14 dm
(d) 25.4 mm (e) 118.3 dam.
5. Convert the following to kilograms: (a) 450 g (b) 7.3 Mg (c) 3921.2 mg (d)
145 hg (e) 10 000 dg.
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PHYS 6001
Self Test Answers
1. (a) metre (m)
(b) kilogram (kg)
(c) second (s)
2. (a) mm (b) Mg (c) ds (d) µ g (e) dam (f) cm (g) ms (h) km
3. (a) 289 cm x 1 m/100 cm = 2.89 m
(b) 1828 mm x 1 m/1000 mm = 1.828 m
(c) 1.45 km x 1000 m/1 km = 1450 m
(d) 17 dm x 1 m/10 dm = 1.7 m
(e) 17 dam x 10 m/1 dam = 170 m
4. (a) 131 m x 100 cm/1 m = 13 100 cm
(b) 2.7 km x 1000 m/km x 100 cm/m = 270 000 cm
(c) 14 dm x 1 m/10 dm x 100 cm/1 m = 140 cm
(d) 25.4 mm x 1 cm/10 mm = 2.54 cm
(e) 118.3 dam x 10 m/dam x 100 cm/1 m = 118 300 cm
5. (a) 450 g x 1 kg/1000 g = 0.45 kg
(b) 7.3 Mg x 1000 kg/1 Mg = 7300 kg
(c) 3921.2 mg x 1 g/1000 mg x 1 kg/1000 g = 0.003 921 2 kg
(d) 145 hg x 100 g/hg x 1 kg/1000 g = 14.5 kg
(e) 10 000 dg x 1 g/10 dg x 1 kg/1000 g = 1 kg
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Notes:
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