Dimensions: Measurement in Scientific Work - Physics (Grade 10)

Dimension Is Not Just Object Size

In everyday language, dimension often means spatial size such as length, width, and height. In physics, dimension has a more specific meaning: it is a code that shows which base quantities build a physical quantity.

For example, bolt length, nut diameter, marble radius, and travel distance can use different units, but they are still the same kind of quantity.

\text{length, diameter, radius, distance} \Rightarrow [\mathrm{L}]

A dimension does not tell us the measured value. It tells us the physical kind behind the measured value.

Seeing Dimension Powers

The visual below focuses only on the length dimension. Switch from length to area and then volume. Notice how one length factor becomes $[\mathrm{L}]$ , $[\mathrm{L}]^2$ , and $[\mathrm{L}]^3$ .

Length Dimension Builder

Choose a shape built from length factors to see its formula, unit, and dimension.

Formula: $s$
SI unit: $\text{m}$
Dimension: $[\mathrm{L}]$

The Alphabet of Base Quantities

The International System of Units or SI uses $7$ base quantities. In dimensional analysis, these base quantities work like an alphabet for building other quantities.

Base quantity	Example symbol	Dimension
Length	$\ell,\ x,\ r$	$[\mathrm{L}]$
Mass	$m$	$[\mathrm{M}]$
Time	$t$	$[\mathrm{T}]$
Electric current	$I$	$[\mathrm{I}]$
Thermodynamic temperature	$T$	$[\Theta]$
Amount of substance	$n$	$[\mathrm{N}]$
Luminous intensity	$I_v$	$[\mathrm{J}]$

Dimensions are written in square brackets so they are not confused with units. For example, $[\mathrm{L}]$ is the dimension of length, while $\text{m}$ is the unit meter.

Deriving Dimensions from Formulas

Dimensional analysis works by replacing every quantity in a formula with its dimension, then simplifying the powers.

\begin{aligned} [A] &= [l][w] = [\mathrm{L}][\mathrm{L}] = [\mathrm{L}]^2 \\ [V] &= [l][w][h] = [\mathrm{L}]^3 \\ [v] &= \frac{[\Delta s]}{[\Delta t]} = \frac{[\mathrm{L}]}{[\mathrm{T}]} = [\mathrm{L}][\mathrm{T}]^{-1} \\ [a] &= \frac{[\Delta v]}{[\Delta t]} = \frac{[\mathrm{L}][\mathrm{T}]^{-1}}{[\mathrm{T}]} = [\mathrm{L}][\mathrm{T}]^{-2} \end{aligned}

Force, work, and power can also be read as dimension structures.

\begin{aligned} [F] &= [m][a] = [\mathrm{M}][\mathrm{L}][\mathrm{T}]^{-2} \\ [W] &= [F][\Delta s] = [\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-2} \\ [P] &= \frac{[W]}{[t]} = [\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-3} \end{aligned}

A Formula That Passes Inspection

Dimensions can help check whether a formula form could be correct. Every term being added must have the same dimension.

Take the displacement formula for uniformly accelerated motion:

s = v_0 t + \frac{1}{2} a t^2

Check the dimensions:

\begin{aligned} [v_0t] &= [\mathrm{L}][\mathrm{T}]^{-1}[\mathrm{T}] = [\mathrm{L}] \\ [at^2] &= [\mathrm{L}][\mathrm{T}]^{-2}[\mathrm{T}]^2 = [\mathrm{L}] \end{aligned}

Both terms on the right have dimension $[\mathrm{L}]$ , so the formula passes the dimensional check. This check does not prove the formula is definitely correct, but it can quickly catch impossible formulas.

s + t \quad \text{is not valid because} \quad [\mathrm{L}] \ne [\mathrm{T}]

One Object, Many Dimensions

A bolt and a nut look like small objects, but measuring them can involve several different quantities. That is why one object sometimes needs more than one measuring tool.

What is checked	Suitable tool	Dimension
Bolt length	ruler or vernier caliper	$[\mathrm{L}]$
Outer diameter	vernier caliper or micrometer screw gauge	$[\mathrm{L}]$
Cross-sectional area	calculated from diameter	$[\mathrm{L}]^2$
Material volume	calculated from spatial size	$[\mathrm{L}]^3$
Mass	balance	$[\mathrm{M}]$
Density	mass divided by volume	$[\mathrm{M}][\mathrm{L}]^{-3}$

So, two measuring tools can measure quantities with the same dimension, but they serve different contexts. A ruler is enough for a large length that does not need high precision. A vernier caliper or micrometer screw gauge is better when a small diameter must be read more carefully.

OpenStax's concept reference for dimensional analysis can be opened through openstax.org.