Physical quantities possess units. For example, a meter is a unit of length, a kilogram is a unit of mass and a second is a unit of time. (There are also other units for length, mass and time, which we will discuss later in this chapter.) This principle transcends the various realms of physics – a meter is a unit of length in classical physics, general relativistic physics, and in quantum physics, even though the notions of length itself is not the same in these realms. The origin of units for physical quantities is, therefore, somewhat mysterious. (When you have difficulty falling asleep, ponder about the origin of units for physical quantities.)
The function served by units for physical quantities, to an extent, is also served by units defined for certain mathematical (or even unphysical) quantities. For example, it is sometimes useful to describe the number of apples in a basket and/or the number of bananas with the units of apples and bananas. In economics and commerce, pounds (£) and dollars ($) are units of currency, which are not based in any physical laws, and these units also serve many of the purposes described in this modules. Once the underlying principles are understood, they can also be applied to abstract mathematical problems.
I think the reason quantities have units is that those quantities are not interchangeable or even comparable with others (without suitable conversion, which we will learn about in this chapter). A meter cannot be interchanged with a second or a kilogram. A dollar is not interchangeable with a pound or an apple without an appropriate conversion factor. This seems like a very fundamental property.
Some mathematicians have already attempted to quantify Romeo’s love for Juliet (e.g. see here, here and, if all else fails, here). Love for Juliet is not interchangeable with meters, kilograms, seconds, apples, or dollars. So rest assured that even love (when expressed mathematically) has units. (Disclaimer: I neither approve nor disapprove of mathematically quantifying love. Please exercise own judgment in this matter.)
The focus in this module is, of course, on physical quantities, and the main function of units is to standardize measurement of the physical quantity. We start with the definitions of the Le Système International d’Unités, which is French for the International System of Units, and is abbreviated as SI units.
The SI system of units is the standard that most of the world has adopted for representing physical quantities. The system consists of the “Base units” shown in Table 2.1.
| Quantity | Unit | Symbol |
| Mass | kilogram | kg |
| Length | metre | m |
| Time | second | s |
| Electric Current | Ampere | A |
| Absolute Temperature | Kelvin | K |
| Luminous Intensity | candela | cd |
| Amount of substance | kilomole | kmol |
From these base units, all other units are derived based on definition of the derived quantities or based on some physical law. Here are two examples.
Determine the SI units of velocity based on the definition as length per unit time and acceleration defined as rate of change of velocity.
The derived units are constructed as below.
| Quantity | Unit | Symbol |
|---|---|---|
| Velocity = Length/Time | meters/second | m/s |
| Acceleration = Velocity/Time | ||
| = meters/second2 | = m/s2 |
Determine the SI units of force, work and power. Here you may use Newton’s second law of motion (force = mass acceleration), and the definitions of work (force distance) and power (work per unit time).
The derived unit is constructed as below.
| Quantity | Unit | Symbol |
| Force | kilogram metres/second2 = Newton | kg m/s2 = N |
| Work | Newton metres = Joule | N m = J |
| Power | Joule/second = Watt | J/s = W |
The definition of units has evolved from being based on physical artefacts to being based on physical constants. For example, from approximately 1800 to 1927, the definition of a metre was based on two fine marks made on an platinum-iridium bar stored in Paris, France. This artefact-based definition has disadvantages. The artefacts are not easy to reproduce nor can they be easily adapted for every kind of measurement. These disadvantages caused an evolution of the artefacts themselves and ultimately, in 1927 for the metre, a shift to artefact-free definition. Today, the metre is defined in terms of the speed of light (which is a physical constant) and the definition of a second. Here is one resource that provides all the information on the SI units and their history.
As we are all familiar, the metre is not the only unit of length, so is a yard, a foot, a furlong and a light year. Each unit of length is related to the other units of length by a scale factor. This also holds for all quantities. For example,
| (2.1) | |||||
| (2.2) |
These conversion factors define an identity in the following way:
| (2.3) |
Identities such as (2.3) allow us to convert from one units to another. (Note that there are no conversions between a metre and a second – a length cannot be expressed in units of time.)
An exception to this rule of scale factor is temperature. The conversion of temperature from Celsius to Kelvin uses a shift
| (2.4) |
Temperature is a quantity for which, in some instances, it is beneficial to set the zero by shifting the scale. This is in addition to the scale factor discussed above. That is how the Fahrenheit scale is related to the Celsius
| (2.5) |
A shift is permissible for scales of temperature but not for other units because in many situations involving temperature, it is only temperature differences that matter (and it is only in those situations that a shifted scale can be used). Note that temperature differences convert from one unit to another purely using scale factors, e.g.
| (2.6) |
The scale of absolute temperature is necessary for situations such as the ideal gas law
| (2.7) |
where is the pressure of a gas, is the volume it occupies, is the amount of the gas (e.g. in moles), is the gas constant, and is the absolute temperature. Values of temperature expressed in a shifted scale such as Celsius or Fahrenheit cannot be used in the ideal gas law.
A few words now about a few other common systems of units apart from the SI system in historical and present prevalence.
The base units in this system are the centimeter (length), gram (mass) and second (time).
Also known simply as the MKS system, this is the system prevalent before the adoption of the SI system. The base units of length, mass and time are the metre, kilogram and the second.
This is the first example we see where the base quantities are force, length, and time. The base units of these quantities are kilogram force (kgf), metre and second, respectively. A kgf is defined as the weight of 1 kilogram mass in the MKS mass system.
The base units of mass, length and time in this system are the pound (mass), foot and second.
Analogous to the MKS force system, the base quantities in this system are force, length and time, with units pound (force), foot and second. A pound force (abbreviated as lbf for clarity) is defined as the weight of the pound mass (abbreviated as lbm for clarity).
The identity involving scale factors can be used to convert derived units from one system of units to another. The following examples illustrate how.
(from Dr John P Longley’s Lecture Notes)
A typical car is 14 ft long. How long is it in metres?
| (2.8) |
(from Dr John P Longley’s Lecture Notes)
A typical car speed is 35 mph. How fast is that in m/s?
| (2.9) |
Note the use of multiple conversion factors to arrive at the desired units.
In this chapter, we introduced units, scales of units and conversion between units.