Physical Quantities

Reading Quantities from Measurements

When we write a quantity value such as $0.60 \text{ m}$ , $2$ parts are working together.

A quantity value is the numerical value multiplied by the unit.

In $0.60 \text{ m}$ , $0.60$ is the numerical value and $\text{m}$ is the reference unit. The quantity is the property of an object or event being measured, such as length, mass, time, temperature, or speed. A complete measurement result can also carry uncertainty and other relevant information.

Without a named quantity, a number has no physical meaning. Without a unit, its scale cannot be compared.

Base and Derived Quantities

SI defines $7$ base quantities. Other physical quantities are derived from combinations of those base quantities.

Base quantity	Typical symbol	SI unit	Dimension
Length	$\ell,\ x,\ r$	$\text{m}$	$\mathrm{L}$
Mass	$m$	$\text{kg}$	$\mathrm{M}$
Time	$t$	$\text{s}$	$\mathrm{T}$
Electric current	$I$	$\text{A}$	$\mathrm{I}$
Thermodynamic temperature	$T$	$\text{K}$	$\Theta$
Amount of substance	$n$	$\text{mol}$	$\mathrm{N}$
Luminous intensity	$I_v$	$\text{cd}$	$\mathrm{J}$

A derived quantity appears when base quantities are combined. For example, area comes from length multiplied by width.

\begin{aligned} A &= l \times w \\ \text{unit of } A &= \text{m} \times \text{m} = \text{m}^2 \\ \dim A &= \mathrm{L} \times \mathrm{L} = \mathrm{L}^2 \end{aligned}

Speed is also a derived quantity because it comes from distance divided by time.

\begin{aligned} v &= \frac{s}{t} \\ \text{unit of } v &= \frac{\text{m}}{\text{s}} = \text{m s}^{-1} \\ \dim v &= \frac{\mathrm{L}}{\mathrm{T}} = \mathrm{L}\mathrm{T}^{-1} \end{aligned}

Dimensions and Formula Checks

A dimension records which base quantities and powers form another quantity. Units state a measurement, while dimensions test whether the quantities on both sides of a formula are compatible.

How to check a quantity

For example, the area of a desk and the area of a sheet of paper can have different values, but they are the same kind of quantity. Both are area, both use the SI unit $\text{m}^2$ , and both have dimension $\mathrm{L}^2$ .

On the other hand, length, width, height, diameter, radius, distance, and displacement can all use length units. They all have dimension $\mathrm{L}$ .

Calculating the Area of a Sheet of Paper

A sheet of paper has length $0.30 \text{ m}$ and width $0.20 \text{ m}$ . Its area is:

\begin{aligned} A &= l \times w \\ &= 0.30 \text{ m} \times 0.20 \text{ m} \\ &= 0.060 \text{ m}^2 \end{aligned}

The value $0.060 \text{ m}^2$ is the quantity value for area. Since area is built from $2$ length quantities, area is a derived quantity.

A derived quantity is defined by combining base quantities, and its unit is built from base units. Compare the dimensions on both sides of a formula to test whether the quantities are compatible.

The term quantity value used here follows the International Vocabulary of Metrology.