Dimensional Analysis

Dimensional analysis is the tool, which find or check the relations among physical quantities by using their dimensions in physics and science. The dimensions of a physical quantity is the combination of basic physical dimensions i.e. mass, length, time, electric charge and temperature describes it. For an example, speed has the dimension length per unit time and may be measured in meters per second, miles per hour or other units. The dimensional analysis is depends on the fact, that physical law must be independent of the units used to measure the physical variables.

Dimensional analysis is used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorize types of physical quantities and units based on their relations to or dependence on other units. Isaac Newton (1686) knew about the dimensional analysis. Who referred to it as the “great principle of similitude”. In establishing the modern use of dimensional analysis, James clerk Maxwell played a major role by distinguishing mass, length, and times as fundamental units. Based on the idea that physical laws like F=ma, the 19^th century mathematician Joseph Fourier made important contribution. Finally this led to the conclusion that meaningful laws must be uniformity in their various units of measurement, this eventually formalized in the Buckingham π theorem. This theorem will describe that how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n-m dimensionless parameters, where m is the number of fundamental dimensions used. Through the non dimensionalization, a dimensional equation can have the dimensions reduced or eliminated, which begin with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature.

Definition

The physical quantity of the dimensions are associated with the mixture of mass, length, time, electric charge, and temperature, represented sans-serif symbols M,L, T,Q, and O, respectively, each raised to rational powers. The term dimension is more abstract than scale unit. Mass is a dimension, while Kilograms are a scale unit in the mass dimension.

Commensurability

The basic outcome of dimensional analysis is, the quantities with the same dimensions may be compared, equated, added, or subtracted. But by multiplying or dividing the incommensurable quantities (quantities with different dimensions) the ratio can be taken.

Other uses

To derive the relationship between the physical quantities that are involved in a particular phenomenon, dimensional analysis is used. To understand why the sky is blue, Lord Rayleigh was used first time (pesic, 2005) in this way in 1872.

Questions:

• What is dimensional analysis?
• Mention the other use of dimensional analysis?