What is Dimension?

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 $\beta = 0$, $\gamma = -\tfrac12$, $\alpha = \tfrac12$:

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.