Limitations of Dimensional Analysis
Dimensional Analysis can’t derive relation or formula if a physical quantity depends upon more than three factors having dimensions. It can’t derive a formula containing trigonometric function, exponential function, and logarithmic function and it can’t derive a relation having more than one part in an equation.
Although dimensional analysis is very useful it can’t lead us too far as,
1) If dimensions are given, physical quantity may not be unique as many physical quantities have same dimensions. For example if the dimensional formula of a physical quantity is \(\left[ M{{L}^{1}}{{T}^{-2}} \right]\) it may be work or energy or torque.
2) Numerical constant having no dimension (K) such as (1/2), 1 or \(2\pi \) etc. can’t be deducted by the methods of dimensions.
3) The method of dimension can’t be used to derive relations other than product of power functions. For example, \(s=ut+\frac{1}{2}a{{t}^{2}}\) (or) \(y=a\,\sin \,\omega t\) can’t be derived by using this theory. However, the dimensional correctness of these can be checked.
4) The method of dimension can’t be applied to derive formula if in mechanics a physical quantity depends on more than 3 physical quantities as then there will be less number of equations than the unknowns. However still we can check correctness of the given equation dimensionally. For example: \(T=\frac{2\pi }{\sqrt{I/mgl}}\) can’t be derived by theory of dimensions but its dimensional correctness can be checked.
5) Even if a physical quantity depends on three physical quantities, out of which two have same dimensions, the formula can’t be derived by theory of dimensions, e.g., formula for the frequency of a tuning fork \(f=\left( d/{{L}^{2}} \right)v\) can’t be derived by theory of dimensions but can be checked.