Units and Measurement: Principle of homogeneity of dimensions

Principle of homogeneity of dimensions

"A given physical relation is dimensionally correct if the dimensions of the each term on either side of the relation are the same."

This principle is based on the fact that only quantities of the same kind can be added or subtracted.  For example, velocity cannot be added to the force.

This principle is very helpful in checking the correctness of an equation.  If the dimensional formula of each term of the equation is not the same, the equation is wrong.

Example:  Consider the physical relation
s = ut + ½ at²  In this relation, dimensional formula of each term is same i.e.

Dimensional formula for s = L, ut = LT^-1T, ½ at²= LT^-2T² = L

Thus, we find that the given relation is dimensionally correct as each term has the same dimensional formula.  But we must note that dimensional consistency does not guarantee correct equation.

Reason: In dimensional analysis, it does not consider dimensionaless quantities, dimensionaless function (e.g. trigonometric functions, logarithmic functions, exponential functions etc.) pure numer, ratio of two similar quantities etc.

Conclusion: If an equation fails the consistency test, it is wrong.  But if it passes, it is not proved right.  Thus, a dimensionally correct equation may or may not be actually correct, but a dimensionally wrong equation must be wrong.

Uses of dimensional equation 

1. To check the correctness of a physical relation (using principle of homogenity of dimension)
2. To convert one system of units to another
3. To find the dimensions of constants in a given relation
4. To derive relationships between different physical quantities
5. To write a forgotten formula