Dimensional Analysis and Its Applications

Three Practical Uses of Dimensional Analysis

Dimensional analysis is the technique of checking, converting, or even deriving physical

relations purely by tracking the dimensions $[M],[L],[T]$ (etc.) on both sides of an equation.

It rests on the principle of homogeneity of dimensions: for a physical equation to be

correct, every term added or subtracted in it must have the same dimensional formula, and

the two sides of the equation as a whole must also match.

1. Checking whether an equation could be correct. If the dimensions on the left- and

right-hand sides of a proposed equation do not match, the equation is definitely wrong (though

matching dimensions alone does not guarantee correctness — a purely numerical/dimensionless

factor like $2\pi$ could still be missing or wrong). For example, the equation

$v = u + at^2$ can be immediately rejected because $[at^2] = [LT^{-2}][T^2] = [L]$, which does

not match the dimension of a velocity, $[LT^{-1}]$; the dimensionally consistent form is

$v = u + at$.

2. Converting a measurement between unit systems. Because a dimensional formula shows

exactly how a quantity scales with mass, length and time, it can be used to convert a

numerical value from one system of units (say SI) into another (say CGS) without needing to

re-measure anything — only the conversion factors for mass, length and time need to be

substituted.

3. Deriving the form of a relation. If a quantity is believed to depend on a small number

of other quantities as a product of powers, e.g. $Q = k\, A^{x}B^{y}C^{z}$ for some

dimensionless constant $k$, then equating the dimensions of $Q$ to the combined dimensions of

$A^{x}B^{y}C^{z}$ produces simultaneous equations in the unknown exponents $x, y, z$. Solving

these gives the form of the relation, though dimensional analysis alone can never determine

the dimensionless constant $k$ itself — that must come from experiment or a fuller theoretical

derivation.