These are the properties of solids, liquids and gases that can be measured/quantified. Examples include the volume of a gas, temperature of a cup of tea, height of a building, etc.
Fundamental quantities
These are the physical quantities from which all other physical quantities are derived. According to the International System of Units (SI), there are seven (7) fundamental quantities:
| Quantity | Symbol | Unit | Abbreviation for unit |
|---|---|---|---|
| Mass | $m$ | kilogram | $kg$ |
| Length | $l$ | metre | $m$ |
| Time | $t$ | second | $s$ |
| Temperature | $T$ | Kelvin | $K$ |
| Amount of substance | $n$ | mole | $mol$ |
| Current | $I$ | ampere | $A$ |
| Luminous intensity | $I_v$ | candela | $cd$ |
Derived quantities
These are combinations of the fundamental quantities. The following are examples of derived quantities. Note the fundamental quantities which are involved in each. $$ \begin{equation}\begin{aligned} area=length\times width\\ \end{aligned}\end{equation} $$ Width is a length measurement and is essentially Length disguised under another word. This would mean that height, breadth, distance, displacement, perimeter and circumference are also length measurements (they fall under the fundamental quantity Length). $$ \begin{equation}\begin{aligned} volume&=length\times width\times height\\ OR&\\ volume&=area\times height\\ \end{aligned}\end{equation} $$ Notice that area is a derived quantity. Thus volume can be seen as the product of three fundamental quantities (length, width and height) or as the product of a derived quantity (area) and a fundamental quantity (height).
Let’s look at some other examples. $$ \begin{equation}\begin{aligned} speed&=\frac{distance}{time}\\ velocity&=\frac{displacement}{time}\\ acceleration&=\frac{velocity}{time}\\ force&=mass\times acceleration\\ pressure&=\frac{force}{area}\\ \end{aligned}\end{equation} $$
Magnitude and unit
Every physical quantity can be expressed as the product of a magnitude (how many) and unit (of what): $$ \begin{equation}\begin{aligned} measurement&\rightarrow magnitude\times unit\\ e.g.\ 3\ kg&\rightarrow 3\times kg\\ \end{aligned}\end{equation} $$
The exception to this rule are dimensionless quantities. These are physical quantities which do not have any units. They are often the ratio of two physical quantities with the same units.
Some examples of dimensionless quantities are:
- Relative density
- Refractive index
- Mechanical advantage
- Velocity ratio
We can use their formulae to determine the units: $$ \begin{equation}\begin{aligned} &MA=\frac{load}{effort}\rightarrow \frac{N}{N}\rightarrow no\ units (dimensionless)\\ &relative\ density=\frac{density\ of\ substance}{density\ of\ water}\rightarrow \frac{kgm^{-3}}{kgm^{-3}}\rightarrow no\ units\\ &refractive\ index=\frac{sin(\hat i)}{sin(\hat r)}\rightarrow \frac{no\ units}{no\ units}\rightarrow no\ units\\ \end{aligned}\end{equation} $$
Finding the units of derived quantities
We can find the units of derived quantities if we substitute the units of the other quantities which make it up: $$ \begin{equation}\begin{aligned} speed&=\frac{distance}{time}\\ &\rightarrow \frac{m}{s}\ OR\ ms^{-1}\\ \end{aligned}\end{equation} $$
Mission details
Find the units of the derived quantities:
- velocity
- acceleration
- force
- pressure
Some physical quantities have units which are placeholders for the fundamental units e.g. in measuring force, we use the Newton ($N$) acknowledging that this unit is just a placeholder for $kgms^{-2}$.