Dimensional Analysis and its Applications

The Principle of Homogeneity

The most fundamental idea in dimensional analysis is that you can only add or subtract physical quantities that have the same dimensions. This is not a convention — it is a logical necessity. If you try to add a velocity to a force, the result is meaningless because the two quantities describe fundamentally different aspects of nature. The same restriction applies to equations: every term on both sides of a valid physical equation must have identical dimensions.

This principle is called the principle of homogeneity of dimensions. It provides a powerful and quick way to test whether an equation could possibly be correct. If the dimensions do not match on both sides, the equation is definitely wrong. If they do match, the equation may be correct — but it is not guaranteed to be correct, because dimensional analysis cannot detect errors involving dimensionless constants or functions. …