1.1 Dimensional analysis  (Page 25/7)

We are familiar with dimensions of motion and motion related quantities. Often, we specify the description of physical process by numbers of coordinates involved – one, two or three. It indicates the context of motion in space. The dimension of physical quantities follows the same philosophy and indicates the nature of the constitution of quantities. In other words, dimension of a physical quantity indicates how it relates to one of the seven basic/ fundamental quantities. Basic quantities are the seven dimensions of the physical quantities.

Dimensions of a physical quantity are the powers with which basic quantities are raised to represent it. The dimension of a physical quantity in an individual basic quantity is the power with which that basic quantity is raised in the dimensional representation of physical quantity. We should be clear here that the dimension is not merely a power, but a combination of basic quantity and its power. Both are taken together and hence represented together. We may keep in mind that units follow dimensional constitution. Speed, for example, has dimension of 1 in length and dimension of -1 in time and hence its unit is m/s.

A pair of square bracket is used to represent the dimension of individual basic quantity with its symbol enclosed within the bracket. There is a convention in using symbol of basic quantities. The dimensions of seven basic quantities are represented as :

1. Mass : [M]
2. Length : [L]
3. Time : [T]
4. Current : [A]
5. Temperature : [K]
6. Amount of substance : [mol]
7. Luminous Intensity : [cd]

We can see here that there is no pattern. Sometimes we use initial letter of the basic physical quantity like "M", sometimes we use initial letter of basic unit like "A" and we even use abbreviated name of the basic unit like "mol".

Dimensions of derived quantities

The dimensions of derived quantities may include few or all dimensions in individual basic quantities. In order to understand the technique to write dimensions of a derived quantity, we consider the case of force. The force is defined as :

$$F=ma$$

Thus, dimensions of force is :

$$\Rightarrow \left[F\right]=\left[M\right]\left[a\right]$$

The dimension of acceleration, represented as [a], is itself a derived quantity being the ratio of velocity and time. In turn, velocity is also a derived quantity, being ratio of length and time.

$$\Rightarrow \left[F\right]=\left[M\right]\left[a\right]=\left[M\right]\left[v{T}^{-1}\right]=\left[M\right]\left[L{T}^{-1}{T}^{-1}\right]=\left[ML{T}^{-2}\right]$$

We read the dimension of force as : it has “1” dimension in mass, “1” dimension in length and “-2” dimension in time. This reading emphasizes the fact that the dimension is not merely a power, but a combination of basic quantity and its power.

Dimensional formula

The expression of dimensional representation is also called “dimensional formula” of the given physical quantity. For brevity, we do not include basic dimensions, which are not part of derived quantity, in the dimensional formula.

$$\text{Force},\left[F\right]=\left[ML{T}^{-2}\right]$$

$$\text{Velocity},\left[v\right]=\left[L{T}^{-1}\right]$$

$$\text{Charge},\left[q\right]=\left[AT\right]$$

$$\text{Specific heat},\left[s\right]=\left[L^2{T}^2{K}^{-1}\right]$$

$$\text{Gas constant},\left[R\right]=\left[M{L}^2{T}^{-2}{K}^{-1}{mol}^{-1}\right]$$

It is conventional to omit power of “1”. Further, in mechanics, we can optionally use all three symbols corresponding to three basic quantities, which may or may not be involved. Even if one or two of them are not present, it is considered conventional to report absence of the dimension in a particular basic quantity. It is done by raising the symbol to zero as :