What is Dimension? — Physics

Definition 1.5 University Physics — Year 1 · Chapter 1 — Units, Dimensions, and Measurement

The dimension of a quantity $X$ , written $[X]$ , records how $X$ is built from the seven base quantities, independently of the units chosen: length $L$ , mass $M$ , time $T$ , electric current $I$ , temperature $\Theta$ , amount of substance $N$ , luminous intensity $J$ . A speed has dimension $[v] = L\,T^{-1}$ , a force $[F] = M\,L\,T^{-2}$ , an energy $[E] = M\,L^2\,T^{-2}$ . A quantity of dimension $1$ (an angle, a ratio, a refractive index) is dimensionless.

Examples

Example 1.8 (The period of a pendulum)

A pendulum of length $\ell$ and mass $m$ swings in gravity $g$ . Suppose its period is $\tau = k\,\ell^\alpha m^\beta g^\gamma$ with $k$ dimensionless. Then $T = L^\alpha M^\beta (L T^{-2})^\gamma = L^{\alpha+\gamma} M^\beta T^{-2\gamma}$ , so $\beta = 0$ , $\gamma = -\tfrac12$ , $\alpha = \tfrac12$ :

\tau = k\sqrt{\ell/g} .

Without solving a single equation of motion: the mass drops out, and doubling the length multiplies the period by $\sqrt2$ . Dynamics (Chapter 14) will supply $k = 2\pi$ for small swings.

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Examples

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