Quantities, Dimensions & Units

What are dimensions and units?

All physical quantities have a dimension. This is simply a statement of the origins of the quantity in question. Every quantity can be derived from a combination of one or more of the seven base quantities: length, mass, time, electric current, temperature, amount of substance and luminous intensity.

Determining the combination of these basic quantities into the others takes a bit of thought, but mostly common sense.

eg. length / time = speed or speed x time = length

The rules of algebra apply to physical dimensions: any dimensions can be multiplied together, but only like dimensions can be added.

This sort of calculation is perhaps easier to comprehend when the units of each quantity are considered. Units are simply a measurement of the magnitude of each quantity. Different units can be used to describe the same quantity.

eg. speed = length / time

Any measurements of length or time could be used here: metres per year, yards per hour, miles per nanosecond etc.

In this example, the dimension of speed is always length per unit of time, but the units are convertible.

Standard (SI) units

In order to avoid confusion and simplify calculations, an international system of standard units has been devised. All measurements should be given in the SI (System Internationale) units or multiples of them. There are seven fundamental quantities.

Length (distance): metre (m).

Throughout history, different civilisations and cultures have used different measures of length. These have all been based on some kind of arbitrary reference. The metre was originally decided to be 1/40,000,000 the polar circumference of the Earth. It was then given as the length of a piece of metal in Sevres, France. As technology developed, it was measured to be the distance travelled by light in a vacuum in 1/299,792,458 of a second.

Mass: kilogram (kg).

The kilogram was originally defined as the mass of a particular cubic decimetre (litre) of water in France; however the measuring cylinder used for this was later found to be 28 parts per million too large. As this mass had been used as the basis for some time, it was kept and the volume known as the litre was later redefined as being 1000.028 cm3.

(Interestingly, the kilogram is the only SI unit that contains a prefix, kilo = 103.)

Time: second (s).

Seconds were originally determined from the period of the rotation of the Earth. 1 second is 1/86,400 of a day. A second is now measured as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels in the ground state of caesium 133.

Temperature: kelvin (K).

One kelvin is 1/273.16 of the temperature of the triple point of water. (The triple point of water is the temperature and pressure at which water molecules can exist in all three forms: solid, liquid and gas. This is very close to, but not exactly 0°, the freezing point of water.) The definition of temperature above may seem a confusing definition, but it comes from the fact that there is an absolute zero of temperature: -273.16°C = 0K. This temperature is the temperature at which all molecular vibration and movement ceases (temperature is a measurement of the average amount of kinetic energy of the atoms in a substance). As a gas cools, it contracts. If the volume of an ideal gas at lower temperatures was to be graphed against it's temperature, the graph would be linear, corresponding to a volume of zero at -273.16° = 0K (Zero temperature means no molecular movement or vibrations). Thus the coefficient describing the change in volume is in fact 1/273.16 per degree of temperature.

This is one of the reasons why absolute zero can never be exactly reached, as if a gas was contracted as far as possible, the individual protons, neutrons and electrons of the atoms would still have some volume. A more obvious reason is that no perfect insulators exist, so even small amounts of heat can still be conducted though to the cooled material. Scientists have managed to cool gases within a fraction of a degree of the absolute zero.

The kelvin and celsius temperature units are both of the same magnitude ie. it still takes an increase of one hundred units to boil water. The celsius scale would say that the temperature has been increased from 0° to 100°, while the kelvin scale would say that the temperature has been raised from 273K to 373K. Measurements are easily converted: add 273 to a celsius measurement to put in the absolute kelvin scale, subtract to 273 from a kelvin measurement to make it a celsius measurement. To confuse the issue even more, many of the older (especially American) texts will use the fahrenheit scale, where

Tf = 9/5Tc + 32° or Tc = 5/9(Tf - 32°)

Amount of substance: mole (mol).

The mole is unit describing the number of atoms (or molecules) in an amount of given substance. This eliminates any problems from density differences in materials (ie one kilogram of hydrogen contains more than 200 times as many atoms as a kilogram of uranium, but a mole of hydrogen contains just as many atoms as a mole of uranium or any other element). One mole contains the same number of individual entities as there are atoms in 12 g of carbon 12.

The number of atoms in a mole of a substance is given the symbol NA. This is known as Avogadro's number and is equal to 6.02 x 1023 mol-1.

Luminous intensity: candela (cd).

This is defined as the intensity in a perpendicular direction of a surface of a black body with an area of 1/600 000 m2 at the temperature of freezing platinum under a pressure of 101.325 kPa. ( In case you ever wanted to know!)

Current: ampere (A).

An ampere is defined as the electrical current that would provide .2N of force between two parallel conductors one metre apart. Current describes the amount of electrical charge passing in a time.

Magnitudes and scales

Scientific notation and prefixes are often used to indicate the magnitude of the given quantity:
 

109
giga
G
106
mega
M
103
kilo
k
102
hecto
h
101
deka
da
10-1
deci
d
10-2
centi
c
10-3
milli
m
10-6
micro
m
10-9
nano
n

eg. 1 km = 103 m. The unit of kilometres is not an SI unit in the strict sense, although it is a simple multiple of one and is acceptable for use. However if it were to be used in an equation using SI values, it may need to be written as 103 m.

Derived quantities

While there are only seven fundamental quantities, all other quantities are derived from these.

eg. speed = length per time = metres/second = m/s

The notation used in this case to eliminate the need for division signs everywhere is

m/s = m s-1 as (remember from maths) 1/x = x-1

eg. acceleration = the change in speed over time = ms-1 x s-1 = m s-2

This statement says that acceleration is "the change over time in the change of distance over time."

Conversion factors

Units that measure the same quantity (have the same dimension) can be interchanged if the need arises. This is easily done by simple multiplications.

eg. The conversion of metres per second into miles per hour

1 mile = 1609 metres

1 hour = 3600 seconds

1 m s-1 x 3600 s hr-1 x 1/1609 miles m-1 = 2.23 miles hr-1

Using the method of this example, each factor of multiplication is dimensionless, as the units and magnitudes cancel (they are equal) eg. 3600 seconds divided by one hour is the same amount of time divided by itself. If you look at the algebra, you can seen that on the left hand side of the equation, all of the units cancel with the exception of those that we set out to convert to.

ie. s x s-1 = s0 = 1. (Algebra again! Anything to the power of 0 equals 1.)

All conversions between units can be calculated in this manner.

When using an equation, always be consistent in the units that use (SI units are the easiest).