The dimension of a physical quantity can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally rational) power. This may give insight into the fundamental properties of the system, as illustrated in the examples below.
Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.Ī dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. The Buckingham π theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the rank of the dimensional matrix. The concept of physical dimension, and of dimensional analysis, was introduced by Joseph Fourier in 1822. 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. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. For example, asking whether a gram is larger than an hour is meaningless.Īny physically meaningful equation, or inequality, must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. metres and grams, seconds and grams, metres and seconds. Incommensurable physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are expressed in, e.g. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.Ĭommensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement e.g., metres and feet, grams and pounds, seconds and years. 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 current) and units of measurement (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed. Analysis of the relationships between different physical quantities
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