FHSST Physics/Units/Systems

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The Free High School Science Texts: A Textbook for High School Students Studying Physics
Main Page - Next Chapter (Waves and Wavelike Motion) >>
PGCE Comments - TO DO LIST - Introduction - Unit Systems - The Importance of Units - Choice of Units - How to Change Units - How Units Can Help You - Temperature - Scientific Notation, Significant Figures, and Rounding - Conclusion

SI Units (Système International d'Unités)[edit]

These units are internationally agreed upon and form the system we will use. Historically these units are based on the metric system which was developed in France at the time of the French Revolution.

Table 1.1: SI Base Units
Base quantity Name 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

All physical quantities have units which can be built from the 7 base units listed in Table 1.1 (incidentally the choice of these seven was arbitrary). They are called base units because none of them can be expressed as combinations of the other six. This is similar to breaking a language down into a set of sounds from which all words are made. Another way of viewing the base units is like the three primary colours. All other colours can be made from the primary colours but no primary colour can be made by combining the other two primaries.

Unit names are always written with lowercase initials (e.g. the metre). The symbols (or abbreviations) of units are also written with lowercase initials except if they are named after scientists (e.g. the kelvin (K) and the ampere (A)). An exception to this rule is the litre, which is abbreviated as either L or l.

To make life convenient, particular combinations of the base units are given special names. This makes working with them easier, but it is always correct to reduce everything to the base units. Table 1.2 lists some examples of combinations of SI base units assigned special names. Do not be concerned if the formulae look unfamiliar at this stage—we will deal with each in detail in the chapters ahead (as well as many others)!

It is very important that you are able to say the units correctly. For instance, the newton is another name for the kilogram metre per second squared (kg·m·s−2), while the kilogram metre squared per second squared (kg·m2·s−2) is called the joule.

Table 1.2: Some Examples of Combinations of SI Base Units Assigned Special Names
Quantity Formula Unit Expressed


Name of
    Base Units Combination
Force m·a kg·m·s−2 N (newton)
Frequency {1\over T} s−1 Hz (hertz)
Work & Energy F·s kg·m2·s−2 J (joule)
Electrical Potential W/A kg·m2·s−3·A−1 V (volt)

Another important aspect of dealing with units is the prefixes that they sometimes have (prefixes are words or letters written in front that change the meaning). The kilogram (kg) is a simple example: 1 kg is 1000 g or 1\times { 10^3\mbox{ g}} . Grouping the 103 and the g together we can replace the 103 with the prefix k (kilo). Therefore the k takes the place of the 103. Incidentally the kilogram is unique in that it is the only SI base unit containing a prefix.

There are prefixes for many powers of 10 (Table 1.3 lists a large set of these prefixes). This is a larger set than you will need but it serves as a good reference. The case of the prefix symbol is very important. Where a letter features twice in the table, it is written in uppercase for exponents bigger than one and in lowercase for exponents less than one. Those prefixes listed in boldface should be learnt.

Table 1.3: Unit Prefixes
Prefix Symbol Exponent Prefix Symbol Exponent
yotta Y 1024 yocto y 10-24
zetta Z 1021 zepto z 10-21
exa E 1018 atto a 10-18
peta P 1015 femto f 10-15
tera T 1012 pico p 10-12
giga G 109 nano n 10-9
mega M 106 micro µ 10-6
kilo k 103 milli m 10-3
hecto h 102 centi c 10-2
deca da 101 deci d 10-1

As another example of the use of prefixes,

1\times10^{-3} \mbox{ g} can be written as 1 mg (1 milligram).

The Other Systems of Units[edit]

The remaining sets of units, although not used by us, are also internationally recognised and still in use by others. We will mention them briefly for interest only.

CGS and MKS Units[edit]

In this system the basic measure of length is the centimetre, weight is in grams and time is in seconds. Later the metre is replaced the centimetre and the kilogram replaced the gram. The Second has remained the basic unit of time throughout. This is a simple change but it means that all units derived from these two are changed. For example, the units of force and work are different. These units are used most often in astrophysics and atomic physics.

When electromagnetism comes into play, there are three CGS systems, adapted to the fundamental equations each theory views as basic: The electric CGS, the magnetic CGS, and the combined Gaussian. The latter has the advantage that corresponding electric and magnetic phenomena have the same units and related equations.

It has the additional advantage that there is only one natural constant in the equations, the speed of light, where the SI system has two. And experience, i.e. measurements, has shown that there is only one constant. So the Gaussion system is a bit more 'right'.

These unit systems also show that the choice of base units is arbitrary. In SI, there is a base unit for the current, the ampere [A], derived from it the unit of charge, coulomb [C]. The Gaussian system does without a dedicated unit for electricity. It simply defines the factor in the law of force between two charged particles as one - and lo, the unit C disappears; the esu (electrostatic unit) can be derived from g, cm, s - the C cannot, it is As, and A is basic.

[The same could be done with mass, leading to kg vanishing, just by setting the gravitational constant in Newton's law to one. kg would then be replaced by a combination of m and s.

Imperial Units[edit]

These units (as their name suggests) stem from the days when the Roman Empire decided measures. Some of these were later altered by local rulers. As a result, different countries used different base units for each quantity (except for time). The British abandoned the Roman measurement and money system in 1972. There were 12 pennies or denaries in a shilling or solidus, and 20 shillings in a pound or libra ergo there were 240 'old pennies' and are now 100 new pennies in the pound sterling or GBP - which large unit was unchanged. The British also used both avoirdupois and troy weight and other capricious local measures, but following its integration in the EU, Britain now officially use decimal SI units for all measurements.

Although the British once used an imperial metric system similar to that in use in the US, it is important to know that there are some differences, because the colonists made certain incorrect assumptions, such as that because there were 16 ounces in a pound weight, there were also 16 fluid ounces in a pint of liquid, when the Romans and British defined 20 fl oz. This matters, because during World War II, for example, great fraud was perpetrated by the British selling the smaller American gallons (8 pints) at the price for the larger British measure!

The decimal metric system was invented in France in 1791, following the French revolution. This later became the MKS (Meter/Kilogram/Second) system and is now the System International (SI) system, which is still close that early French system. Using different units in different places would make effective scientific communication very difficult. That is why the scientific community has adopted SI units as its internationally agreed upon standard. Therefore the SI is overwhelmingly predominant for nearly all international scientific and technical use.

Natural Units[edit]

This is the most sophisticated choice of units. Here the most fundamental discovered quantities (such as the speed of light) are set equal to 1. The argument for this choice is that all other quantities should be built from these fundamental units. This system of units is used in high energy physics and quantum mechanics.