Dimensional Analysis

Introduction to Dimensional Analysis

Dimensional Analysis is a mathematical technique used primarily in physics and engineering to check the validity of equations.
It involves comparing the dimensions (not the units) of the physical quantities on both sides of an equation to ensure they are the same.
Dimensions refer to the fundamental physical quantities such as Length (L), Time (T), Mass (M), and so on.

Basic Principle of Dimensional Analysis

The principle of homogeneity is the bedrock of dimensional analysis. It postulates that equations can only equate quantities of the same dimension.
If an equation is dimensionally homogenous, this implies that all its terms have the same dimensions. This, however, does not affirm that the equation is correct. An incorrect equation might also be dimensionally homogenous.

Applications of Dimensional Analysis

Checking equations: Dimensional analysis verifies the possible correctness of the derived equations by verifying the dimensions on both sides of the equation.
Deriving equations: It allows derivation of relationships between physical quantities in a problem.
Converting units: Dimensional analysis can be employed to convert units from one system (e.g., SI) to another (e.g., CGS).

Steps in Dimensional Analysis

Identify every quantity in the formula or equation and replace it with its dimension. For instance, if v = u + at (where v is final velocity, u is initial velocity, a is acceleration and t is time), replace v and u with “L.T^-1”, a with “L.T^-2”, and t with “T”.
Check each term on both sides of the equation for homogeneity. The terms are homogenous if they have the same dimensions.
If all terms match dimensionally, the equation passes the dimensional analysis. If not, it indicates a possible error in equation formulation.

Tips for Dimensional Analysis

Remember the dimensions for Length (L), Time (T), and Mass (M). For any other physical quantity examine its definition or how it is derived from basic quantities.
Be cautious about dimensionless quantities such as sine, cosine, tangent, exponential or logarithmic functions, constants, etc. These do not have dimensions.
Maintain accurate track of the dimensions throughout your analysis.
In case of complex equations, break down the process - analyse each side of the equation independently before comparing them.

Strong knowledge of dimensional analysis can serve as a vital tool for solving several problems in mechanics. It develops your skills in problem-solving and verifying your work, enhancing your overall understanding of the subject.