UNITS AND MEASUREMENTS

1. Units and measurements

Without a widely accepted system of units of measurement our civilization could not exist. Imagine the chaos if there were no standards for the measurement of mass, length and time, for example.

Scientists identify quantities which describe the natural world. In order to decide how much of a particular quantity a given object has, for example, MASS, it is necessary to compare that object to the masses of other objects which have known masses, defined in terms of an agreed standard. The instrument used is called a BALANCE, iand is shown here on the right. A PHYSICAL QUANTITY can be defined in terms of the operations necessary to measure it. For example, the length of an object can be determined by comparing it to an object of known length, such as a ruler (see examples of physical quantities).


A unit is an established standard for a physical quantity against which particular examples of that physical quantity can be compared. The act of comparing a physical quantity to a unit is called MEASUREMENT and the MEASURE of a particular physical quantity is the ratio of that physical quantity to the unit. For example, let us take the physical quantity, distance, with a value of 25 metres. The units are metres, and if we divide 25 metres by metres, we get the measure which is 25. The measure is a numerical value. When we perform calculations, we manipulate the measure, not the physical quantity.

2. Base and derived quantities and units

Physical quantities are related to one another by mathematical equations, for example:

velocity = displacement / time.

Some physical quantities are chosen as base quantities. Other physical quantities are obtained from the base quantities using the appropriate algebraic relationships and these are called derived quantities .

Base quantities are said to have base units and derived quantities have derived units. For example, if displacement and time are chosen to be base quantities, then velocity is a derived quantity. Note that metres/second is pronounced "metres per second".

By international convention the following seven physical quantities are chosen for use as dimensionally independent base quantities.

Physical Quantity Symbol for the Quantity
length l
mass m
time t
electric current I
thermodynamic temperature T
luminous intensity Iv
amount of substance n

SI base units

In South Africa, the INTERNATIONAL SYSTEM OF UNITS (which is abbreviated "SI", from the French "le Système International d' Unités") as defined by the International Standards Organization is used. SI units are widely accepted and are established by law in most countries (the United States being a notable exception). The SI base units are shown in the table below. (Clicking on a name of a unit will take you to its definition).

BASE UNITS
Physical Quantity Name of Unit Symbol
length metre m
mass kilogram kg
time second s
electric current ampere A
temperature kelvin K
luminous intensity candela cd
amount of substance mole mol


SI derived units:

All SI units that are not base units are expressed as combinations of the base units.

SOME EXAMPLES OF DERIVED SI UNITS
Physical Quantity SI Unit Symbol
angle radian rad
solid angle steradian sr
area square metre m2
volume cubic metre m3
density kilogram per cubic metre kg·m-3
speed metre per second m·s-1
acceleration metre per second squared m·s-2
concentration mole per cubic metre mol·m-3

Note the equivalent forms of writing down derived units: for example, mol·m-3 = mol/m3. Some derived units (shown in the table below) are given special names: (Clicking on a name will take you to its definition).

You are encouraged to use the "middle dot" notation to indicate the multiplication of units, rather than the full stop. Thus, the preferred way of writing the units for acceleration are m·s-2 rather than m.s-2 or m/s2.

SOME DERIVED SI UNITS WITH SPECIAL NAMES
Physical Quantity Name of Unit Symbol
energy joule J
force newton N
pressure pascal Pa
power watt W
electric charge coulomb C
electric potential difference volt V
electric resistance ohm W
frequency hertz Hz
Further information: http://en.wikipedia.org/wiki/SI_derived_units

SI prefixes

A SI prefix is a name that is added to the name of a basic unit and which indicates whether that unit is a multiple (or a fraction) of that unit. For example, the prefix "kilo" added to "meter" gives "kilometer", which is a unit 1 000 times LARGER than the base unit "meter". Similarly, the prefix "milli" added to "gram" gives "milligram", which is a unit 1 000 times SMALLER than the base unit "gram". The table shown below lists the names of approved SI prefixes.

SI Prefixes Remarks
Multiple Prefix Symbol
10-24
10-21
10-18
10-15
10-12
10-9
10-6
10-3
10-2
10-1
10
102
103
106
109
1012
1015
1018
1021
1024
xyocto
xzepto
xatto
xfemto
xpico
xnano
xmicro
xmilli
xcenti
xdeci
xdeca
xhecto
xkilo
xmega
xgiga
xtera
xpeta
xexa
xzetta
xyotta
xy
xz
xa
xf
xp
xn
xµ
xm
xc
xd
xda
xh
xk
xM
xG
xT
xP
xE
xZ
xY
Decimal multiples are formed by adding prefixes to the name of the SI unit. This avoids having to use cumbersome numbers of digits.

It is considered good practice to use prefixes representing 10 raised to a power which is a multiple of 3. For example, 100 ms is preferable to 10 cs or 1 ds.
Further information: http://en.wikipedia.org/wiki/SI_prefix

For historical reasons, some multiples of SI units are given special names. While these units are not part of the SI, they are precisely defined in terms of SI units, as shown in the table below.

Physical Quantity Name of Unit Symbol Definition Remarks
length ångstrom Å 10-10 m Prefixes are added to the unit name, and written as one word, e.g. megawatt.

Compound prefixes such as millimicrometres are not used.

In the case of derived units, only one unit takes a prefix. We write km·s-1 and not mm·µs-1.
volume litre l 10-3 m3
mass tonne t 103 kg
pressure bar bar 105 Pa
time minute
hour
day
min
h
d
60 s
3600 s
86400 s

Additional questions


Examples of physical quantities and their dimensions

Quantity Dimensions Quantity Dimensions
length [L] force [MLT-2]
area [L2] pressure [ML-1T-2]
volume [L3] energy/work [ML2T-2]
density [ML-3] power [ML2T-3]
speed [LT-1] electric charge [IT]
acceleration [LT-2] electric potential difference [ML2T-3I-1]
momentum [MLT-1] electric resistance [ML2T-3I-2]


SI base units definitions

SI unit of length: metre - The metre is the the length of the path travelled by light in a vacuum during a time interval of 1/299 792 458 of a second.


SI units of mass: kilogram - The kilogram is the mass of the international prototype kilogram (a platinum-iridium cylinder) kept at the Bureau International des Poids et Mesures at Sèvres in France.


SI unit of time: second - The second is the duration of exactly 9 192 631 770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the caesium-133 atom at a temperature of 0 K.


SI unit of electric current: ampere - The ampere is the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible cross section, and placed 1 metre apart in a vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per metre of length.


SI unit of thermodynamic temperature (or absolute temperature): kelvin - The kelvin is the fraction 1/273.16 (exactly) of the thermodynamic temperature of the triple point of water.


SI unit of luminous intensity candela - The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. .


SI unit of amount of substance mole - The mole is the quantity of substance of a system that contains the same number of elementary entities (atoms, molecules, ions, electrons or particles, depending on the substance) as there are atoms in 0.012 kilograms of pure, unbound carbon-12; this number (NA) is approximately equal to 6.02214199×1023.


SI unit of plane angle: radian - The radian is the angle subtended at the centre of a circle by an arc of the circumference equal in length to the radius of the circle. There are 2π radians in a circle.


SI unit of solid angle angle: steradian - The steradian is the solid angle subtended at the centre of a sphere of radius r by a portion of the surface of the sphere having an area r2. There are 4π steradians on a sphere.


SI derived units definitions

SI derived unit of energy: joule - The joule is the work done when the point of application of a force of one newton (1N) is displaced through a distance of one metre (1m) in the direction of the force.

Symbol Definition of SI Unit Equivalent Form of SI Unit
J m2·kg·s-2


SI derived unit of force: newton- The newton is that force which, when applied to a body with a mass of one kilogram (1kg), gives it an acceleration of one metre per second squared (1 m·s-2).

Symbol Definition of SI Unit Equivalent Form of SI Unit
N m·kg·s-2 J·m-1


SI derived unit of pressure: pascal - The pascal is the pressure which results when a force of one newton (1N), is applied evenly and perpendicularly to an area of one square metre (1 m2).

Symbol Definition of SI Unit Equivalent Form of SI Unit
Pa m-1·kg·s-2 N·m-2, J·m-3


SI derived unit of power: watt - The watt is the power which results in the production of energy at the rate of one joule per second (1 J·s-1).

Symbol Definition of SI Unit Equivalent Form of SI Unit
W m2·kg·s-3 J·s-1


SI derived unit of electric charge: coulomb - The coulomb is the quantity of electric charge transported in one second (1s) by a current of one ampere (1A).

Symbol Definition of SI Unit Equivalent Form of SI Unit
C s·A A·s


SI derived unit of electric potential difference: volt - The volt is the potential difference between two points of a conduction wire carrying a constant current of one ampere (1A), when the power dissipated between these points is one watt (1W).

Symbol Definition of SI Unit Equivalent Form of SI Unit
V m2·kg·s-3·A-1 J·A-1·s-1, J·C-1, W·A-1


SI derived unit of electric resistance: ohm - The ohm is the electric resistance between two points of a conductor when a constant potential difference of one volt (1V), applied between these two points, produces a current of one ampere (1A), the conductor not being the source of any electromotive force.

Symbol Definition of SI Unit Equivalent Form of SI Unit
Ω m2·kg·s-3·A-2 V·A-1


SI derived unit of frequency: hertz- The hertz is the frequency of a periodic phenomenon of which the period is one second (1s).

Symbol Definition of SI Unit Equivalent Form of SI Unit
Hz s-1