Dimensional equation

An equation is dimensionally correct if the dimensions of the various terms on either side of the equation are the same. This is called the principle of homogeneity of dimensions. This principle is based on the fact that two quantities of the same dimension only can be added up, the resulting quantity also possessing the same dimension. The equation A + B = C is valid only if the dimensions of A, B and C are the same.

Uses of dimensional analysis

The method of dimensional analysis is used to

• convert a physical quantity from one system of units to another.
• check the dimensional correctness of a given equation.
• establish a relationship between different physical quantities in an equation.

To check the dimensional correctness of a given equation

Let us take the equation of motion s = ut + (1/2)at² Applying dimensions on both sides, [L] = [LT^-1] [T] + [LT^-2] [T²] (1/2 is a constant having no dimension) [L] = [L] + [L] As the dimensions on both sides are the same, the equation is dimensionally correct.

Limitations of Dimensional Analysis

• The value of dimensionless constants cannot be determined by this method.
• This method cannot be applied to equations involving exponential and trigonometric functions.
• It cannot be applied to an equation involving more than three physical quantities.
• It can check only whether a physical relation is dimensionally correct or not. It cannot tell whether the relation is absolutely correct or not.