Dimensions of Physical Quantities

Base Dimensions and the Idea of a Dimensional Formula

Every physical quantity that is not itself one of the seven SI base quantities can be

expressed as a combination of powers of the base dimensions — conventionally written as

mass $[M]$, length $[L]$ and time $[T]$ for the purely mechanical quantities covered in this

unit (electric current $[A]$, temperature $[K]$, amount of substance $[\text{mol}]$ and

luminous intensity $[\text{cd}]$ complete the full set for other branches of physics).

The dimensional formula of a quantity $X$ is written $[M^{a}L^{b}T^{c}]$, where the

exponents $a, b, c$ show exactly how the quantity depends on mass, length and time — regardless

of which actual unit system (SI, CGS, FPS) is used to express it numerically. Two quantities

that look different (say, mechanical stress and pressure) but reduce to the same combination

of exponents are, dimensionally, the same kind of quantity.

Building Dimensional Formulae from Definitions

The dimensional formula of any mechanical quantity is obtained directly from the defining

relation that produces it, substituting the known dimensional formulae of length $[L]$,

mass $[M]$ and time $[T]$:

For instance, since $\text{force} = \text{mass}\times\text{acceleration}$, and

acceleration itself is $\text{velocity}/\text{time} = (\text{length}/\text{time})/\text{time}$:

Dimensional Constants vs. Dimensionless Constants

Not every quantity that appears in a physics formula has the same status:

• A dimensional constant has a fixed dimensional formula and a numerical value that …