1.1 Dimensional analysis  (Page 5/7)

An elaborate form of homogeneity principle is known as Buckingham π theorem. This theorem states that an expression of physical quantity having n variables can be expressed as an expression of n-m dimensionless parameters, where m is the numbers of dimensions. This principle is widely used in fluid mechanics to establish equations of physical quantities. A full treatment of this priciple is beyond the context of physics being covered in this course. Therefore, we shall use the simplified concept of homogeinity that requires that only dimensionally similar quantities can be added, subtracted and equated.

One interesting aspect of Buckingham π theorem is formation of dimensionless groups of variables. It helps us to understand relation among variables (physical quantities) and nature of dependence. Consider a simple example of time period of simple pendulum (T). Other relevant parameters affecting the phenomenon of oscillation are mass of pendulum bob (m), length of pendulum (l), acceleration due to gravity (g). The respective dimensional formula for time period is [T], mass of bob is [M], length of pendulum is [L]and acceleartion due to gravity is [ $L{T}^{-2}$ ].

We can have only one unique dimension-less group of these variables (quantities). Let us consider few combinations that aim to obtain one dimensionless group.

$$\frac{T^{2}g}{ML}=\frac{\left[T^{2}L{T}^{-2}\right]}{ML}=\left[M^{-1}\right]$$

This grouping indicates that we can not form a dimensionless group incorporating “mass of the bob”. A dimensionless group here is :

$$\left[\frac{T^{2}g}{L}\right]=\left[M^0{L}^0{T}^0\right]=\text{a constant}$$

Rearranging,

$$T=k\sqrt{\frac{l}{g}}$$

Thus, grouping of dimensionless parameter helps us to (i) determine the form of relation and (ii) determine dependencies. In this case, time period is independent of mass of bob. Formation of dimensionless parameter is one of the important considerations involved in the dimensional analysis based on Buckingham π theorem.

Transcendental functions

Many physical quantities are expressed in terms of transcendental functions like trigonometric, exponential or logarithmic functions etc. An immediate fall out of homogeneity principle is that we can only have “dimensionless groups” as the argument of these transcendental functions. These functions are expansions involving power terms :

$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\dots \dots .$$

$$\sin x=x-\frac{x^3}{3}+\frac{x^5}{5}-\dots \dots .$$

$$\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}\dots \dots .$$

If x is dimensional quantity, then each term on right hand sides have different dimensions as x is raised to different powers . This violates principle of homogeneity. For example, we know that trigonometric function has angle as its argument. Angle being dimensionless is permissible here as no dimension is involved. But, we can not have a dimensional argument like time to trigonometric functions. However, we know of the expressions involving physical quantities have transcendental terms. In accordance with homogeneity principle, argument to the functions needs to be dimensionless. Consider for example the wave equation given by :

$$y=A\sin (\mathrm{kx}-\omega t)$$

Here, both terms “kx” and “ωt” needs to be dimensionless.