Limitations of Dimensional Analysis

Powerful as it is, dimensional analysis has real limitations, and it is important to know when it cannot be trusted to give a complete answer:

1. It cannot determine dimensionless constants. As seen in section 1.6.1(ii), dimensional analysis correctly gives the form $T \propto \sqrt{l/g}$ for a pendulum's period, but the actual numerical constant of proportionality ($2\pi$ in that case) can only be found by experiment (or by a full dynamical derivation) — it never falls out of a purely dimensional argument.

2. It cannot handle trigonometric, exponential or logarithmic functions. Quantities like $\sin\theta$, $e^{x}$ or $\ln x$ are themselves dimensionless (their arguments must be dimensionless too), so dimensional analysis carries no information about them and cannot be used to derive relations that genuinely involve such functions.

3. It fails whenever the constant of proportionality is itself dimensional (not a pure number). For example, Newton's law of gravitation states that the gravitational force between two point masses is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them, $F \propto \dfrac{m_1 m_2}{r^2}$, i.e. $F = G\dfrac{m_1 m_2}{r^2}$. Here the constant of proportionality $G$ (the universal gravitational constant) is not dimensionless — it carries dimensions of its own — so the ordinary dimensional-analysis method used in section 1.6.1(ii) cannot be applied to derive this relationship; $G$'s value and dimensions have to be found by other means (in this case, by experiment). …