Question

Why in dimensional analysis we can only check the correctness of a physical quantity dimensionally not mathematically



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Answer 1

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often easier within the metric or SI system than in others, due to the regular 10-base in all units. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.[1][2][3]

The concept of physical dimension was introduced by Joseph Fourier in 1822.[4] Physical quantities that are of the same kind (also called commensurable) (e.g., length or time or mass) have the same dimension and can be directly compared to other physical quantities of the same kind (i.e., length or time or mass, resp.), even if they are originally expressed in differing units of measure (such as yards and meters). If physical quantities have different dimensions (such as length vs. mass), they cannot be expressed in terms of similar units and cannot be compared in quantity (also called incommensurable). For example, asking whether a kilogram is larger than an hour is meaningless.

Any physically meaningful equation (and any inequality) will have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.

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