Dimensional Analysis

Introduction

• Dimensional analysis is a mathematical technique used in physics and engineering to check the correctness of equations and derive relationships between physical quantities.
• It utilises the fact that every physical quantity can be expressed in terms of fundamental dimensions (mass, length, time, etc.).

Fundamental Dimensions

• Mass (M), Length (L), and Time (T) are typically considered the base dimensions.
• Other dimensions such as Current (I) or Temperature (Θ) can be included as needed.

Dimensional Homogeneity

• An equation is said to be dimensionally homogeneous if each term on both sides of the equation has the same dimensions.
• This is based on the principle that quantities with different dimensions cannot be equated or added together.

Using Dimensions in Analysis

• Dimensional analysis can be used to check the consistence or correctness of derived formulae.
• It is also used to convert units from one system to another (like from SI to CGS or vice versa).

Limitations of Dimensional Analysis

• Dimensional analysis cannot test numerical constants or dimensionless quantities which make it impossible to fully validate a formula.
• It also cannot generate formulae where quantities add or subtract because quantities with different dimensions cannot be added or subtracted.

Key Applications in Mechanics

• Dimensional analysis is particularly useful in the study of fluid mechanics, including deriving the dimensionless Reynolds number and Froude number.
• It also plays a significant role in the understanding of mechanical vibrations, particularly in working out ratios of lengths, chemicals and even populations.

Always remember that dimensional analysis is a powerful tool but, like all tools, it is most effective when used appropriately and understood fully. It cannot replace rigorous derivation, but it can often provide useful checks or insights.