1
By the end of
this chapter
you should be
able to:
Physical quantities and units
嘼
嘼
嘼
嘼
嘼
explain what is meant by a ‘quantity’ in physics;
state the five fundamental quantities recognised and used in physics;
explain the need for units when dealing with physical quantities;
state how the base units used in this course are defined;
explain what is meant by derived quantities and obtain their units in
terms of base units;
嘼 recall and use the symbols for base units and derived units;
嘼 use multiples and submultiples of units;
嘼 do calculations using these multiple and submultiple units.
Concept map
physical quantities
fundamental quantities
derived quantities
base S.I. units
derived S.I. units
multiple and submultiple
base units
multiple and submultiple
derived units
Introduction
unit of measurement Measurement is something we use every day to find the value or size of
things.
We describe the results using a wide variety of units, depending on what it is
we are measuring, but the results always begin with a number usually followed
by the unit. For example, a cricket score might be 85 runs; a cake recipe may
mention 6 cups of flour; a salary may be 2500 dollars; and the size of a hotel
could be 100 rooms. Here the units of measurement are runs, cups, dollars and
rooms.
The units we use in physics are internationally agreed, and generally used,
particularly in science, industry and technology. They are called S.I. units. S.I.
stands for the French Système International (‘International System’). This system
of units was agreed at a conference of prominent scientists in France in 1960.
This chapter will introduce you to quantities measured in physics as well as
the units in which they are measured.
Physical quantities
fundamental quantity In a school physics laboratory, there are a host of different quantities we may
measure, from the length of a bench to the voltage supplied by a battery. In
physics, seven quantities are seen as fundamental. You will come across five
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A – Measurement and practical work
of the fundamental quantities in your course: mass (figure 1.1), length, time,
temperature and electric current. (The other two fundamental quantities are
‘luminous intensity’ and ‘amount of substance’.)
Each of these fundamental quantities is represented by a symbol, as shown
in table 1.1.
Table 1.1
Figure 1.1 Mass is a fundamental
quantity in physics. The kilogram
standard mass, shown here, is kept at
Sèvres.
base unit Fundamental quantity
Symbol for the quantity
mass
m
length
l
time
t
temperature
T
electric current
I
Units for fundamental quantities
When we measure a quantity, we express the value as a number followed by a
unit such as ‘metre’ or ‘second’.
Each of the fundamental quantities in physics has an S.I. base unit. For
example, the base unit of length is the metre. The base units are defined using
internationally agreed standards.
The five base units most often used in physics are shown in table 1.2 with
their symbols.
Table 1.2
In print, the symbols for quantities in physics are
shown as here, in italic, for example T, not T.
Symbols for units are never written in the plural.
For example, we would write 10 kg, not 10 kgs.
Five fundamental quantities and their symbols.
Five fundamental quantities and their S.I. base units.
Fundamental
quantity
Symbol for the
quantity
Base unit
Symbol for the
unit
mass
m
kilogram
kg
length
l
metre
m
time
t
second
s
temperature
T
kelvin
K
electric current
I
ampere
A
The units kelvin and ampere are
named after famous scientists.
The kilogram
The standards kept at the International
Bureau of Weights and Measures are ‘primary’
standards. Other ‘standards’, made in properly
equipped laboratories and based on those at
the International Bureau, are called ‘secondary
standards’.
kilogram standard 2
The kilogram is the base unit of mass.
The kilogram is defined as the mass
of a particular platinum–iridium
cylinder kept at the International
Bureau of Weights and Measures at
Sèvres, near Paris, in France, stored
under specified conditions (figure
1.1). This cylinder is called the kilogram standard. All other masses are
ultimately measured against this
standard (figure 1.2).
Figure 1.2 A high-precision balance.
Values for mass are ultimately based
upon the primary standard kilogram at
Sèvres.
1 – Physical quantities and units
Thus if we say that a certain mass is 40 kilograms, what we mean is that the
mass is 40 times that of the kilogram ‘standard’. The mass of a standard must
not change with time or with environmental conditions. The kilogram standard
is made from an alloy chosen for its resistance to corrosion and is kept under
very closely controlled conditions (figure 1.1).
The metre and the second
Figure 1.3
A caesium clock.
Caesium clocks are so constant that two of them
will agree with each other to within 1 second
in 300 000 years! This means that if the two
clocks were switched on at the same time,
then after they had been working for a period
of 300 000 years, the times they showed would
differ by no more than 1 second!
Measurements using a caesium clock show that
the Earth’s daily rotation is not constant, but is
very gradually slowing down.
The values of base units must remain constant, irrespective of the environment.
Because of this, the older definition of the metre, based on the separation of
two fine scratches on a bar of a particular alloy, has had to be abandoned. In
1983, the metre was redefined as the distance travelled by light in a vacuum in
1/299 792 458 of a second. (You do not need to remember this number!)
The older definition of the second, based on the rotation of the Earth on its
axis, has also been abandoned. The second was redefined in 1967, as the time
for 9 192 631 770 vibrations of a particular electromagnetic wave given off by
the atoms of caesium-133 (figure 1.3). (You do not need to remember that
number either!)
When standardised in this way, the values of the metre and the second are
not affected by environmental conditions. Further, these definitions allow the
standards to be reproduced in any properly equipped laboratory anywhere in
the world with the same accuracy.
The kelvin and the ampere
The standard definitions of the kelvin and the ampere are outside the
scope of our course, but are given here for completeness. You do not have to
remember them.
Zero kelvin (0 K) is the absolute zero of temperature. The kelvin (K) is defined
as 1/273.16 of the temperature at which water can coexist as liquid, solid and
gas. The ice point 0 °C = 273.15 K on the Kelvin scale.
The ampere is defined as the current which, if flowing in two straight and
infinitely long parallel wires 1 metre apart in vacuum, would produce a force
between them of 2 × 10–7 newtons per metre.
Multiple and submultiple units
‘Sub’ means ‘lower than’, ‘less than’, ‘below’ or
‘under’.
submultiple unit multiple unit Imagine that two students are asked to measure the thickness of a leaf of their
exercise books. One student gives the thickness as 0.2 millimetre while the other
expresses the result as 0.0002 metre. Which of these two statements gives one a
better idea of the thickness?
You may have a perfectly good idea of the size of a millimetre and of a metre,
but it is more difficult to visualise two ten-thousandths of a metre than twotenths of a millimetre. So there is a need for units smaller than the base unit,
and these are called submultiple units.
We also need units of length that are greater than the metre. Imagine you
are to run a marathon race. Is it easier to visualise a distance of 26 kilometres
rather than 26 000 metres? Units such as the kilometre, which are larger than
the fundamental unit, are called multiple units.
There is a need for both multiple and submultiple units of most quantities.
These units are the base unit multiplied by a power of 10. The factor by which
the base unit is multiplied is given by a prefix, as shown in table 1.3.
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A – Measurement and practical work
Table 1.3
Prefixes for multiple and submultiple S.I. units.
Prefix
Submultiples
The abbreviation da is hardly ever used. Hectoand deca- are also seldom used nowadays in
physics.
Multiples
A hectare is a unit of area used for land
measurement. One hectare is 10 000 m2, roughly
2.5 acres.
micron Abbreviation
Power of 10
pico
p
10–12
nano
n
10–9
micro
µ
10–6
milli
m
10–3
centi
c
10–2
deci
d
10–1
deca (or deka)
da
101
hecto
h
102
kilo
k
103
mega
M
106
giga
G
109
tera
T
1012
The unit ‘micrometre’ is sometimes called the micron, written as ‘µ’ (the
Greek mu), without the m for ‘metre’.
Liquid volumes in chemistry are commonly
measured in dm3.
1 decimetre3 = 1 dm3
= (10–1 m)3
= 10–3 m3
This is 1 litre (l). The litre is used in chemistry
and in commerce (figure 1.4).
ITQ1
How many cm3 are there in 1 dm3?
ITQ2
Express the following numbers in standard
form:
(i)
2000
(ii) 0.002 34
(iii) 3833.33
(iv) 0.000 000 02
(v) 123 456.789
ITQ3
Express
(i)
10 000 milliseconds in seconds
(ii) 2000 km in metres
(iii) 2000 km in megametres
(iv) 0.002 g in micrograms
Take care to write the symbols correctly.
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Figure 1.4 A filling station in Trinidad. What is the unit used on the pump for
measuring the quantity of gasoline bought?
Standard form
This is a convenient way of writing very large or very small numbers, by expressing them as a number between 1 and 9.9999 that is multiplied by a power of 10.
Standard form is also referred to as scientific notation. Examples are
6.02 × 1023
2.000 × 103
2 × 10–3
1 – Physical quantities and units
Derived quantities
derived quantity density Fundamental quantities can be multiplied or divided. For example, length (as
in distance travelled) may be divided by time to find a speed. The resulting
quantity, speed in this example, is called a derived quantity. Another derived
quantity is density, which is mass per unit volume.
density =
mass
volume
Some other examples of derived S.I. quantities are shown in table 1.4.
Table 1.4
derived unit Full stops are not used within units: we would
write 5 m s–1, not 5 m. s.–1. We also leave a space
between the m and the s, so we write m s–1, not
ms–1, because ‘ms’ means ‘millisecond’.
Some derived S.I. quantities.
Derived
quantity
Unit
Symbol
Derivation
acceleration
metre per second
squared
m s–2
area
metre squared
m2
density
kilogram per metre
cubed
kg m–3
electric charge
coulomb
C
1C = 1As
energy
joule
J
1J = 1Nm
force
newton
N
1 N = 1 kg m s–2
momentum
kilogram metre per
second
kg m s–1
potential
difference
volt
V
1 V = 1 J C–1
power
watt
W
1 W = 1 J s–1
pressure
pascal
Pa
1 Pa = 1 N m–2 = 1 kg m–1 s–2
velocity
metre per second
m s–1
volume
metre cubed
m3
for unit
The units used to measure derived quantities are called derived units. For
example,
derived quantity, speed =
distance travelled (length)
time taken (time)
derived unit of speed
metre
m
=
second
s
=
So the unit of speed is the derived unit, m/s, or m s–1. We say that the unit of
speed has the dimensions ‘metre/second’, or m s–1.
Any unit obtained by multiplying or dividing base units is a derived unit.
Units named after famous scientists
newton Units are often named after scientists who have made a significant contribution
to a particular field of study. For example, Isaac Newton did a lot of work in the
area of mechanics, which is mostly about the effect of forces, and so the unit of
force, the newton, has been named after him. The symbol for this unit is N.
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