1.1 Dimensional analysis  (Page 2/7)

$$\left[v\right]=\left[M^{0}L{T}^{-1}\right]$$

It should be noted here that dimensional representation of a physical quantity does not include “magnitude (n)” of the physical quantity. Further, dimensional formula does not distinguish nature of quantity. For example, the nature of force (or types of force) has no bearing on its dimensional representation. It, then, also follows that variants of a physical quantity bears the same dimensional representation. For example, velocity – whether instantaneous, average, relative – has the same dimensional representation $\left[L{T}^{-1}\right]$ . Moreover, dimensional representation of a vector and its scalar counterpart is same. For example, the dimensions of velocity and speed are same. Further, dimensions of the difference of a physical quantity are same as that of physical quantity itself. For this reason, dimensions of velocity and difference of two velocities are same.

A physical quantity need not have dimensions in any of the basic quantities. Such is the case, where physical quantities are equal to the ratio of quantities having same dimensions. Take the case of an angle, which is a ratio of arc(length) and radius(length). The dimensions of such physical quantity are zero in each of the basic quantities.

$$\text{Angles},\left[\theta \right]=\left[M^0{L}^0{T}^0\right]$$

Similar is the case with Reynold’s number, used in fluid mechanics. It also does not have dimensions in basic quantities. Thus, a physical quantity can be either “dimensional” or “dimensionless”.

On the other hand, there are numerical constants like trigonometric function, pi etc, which are not dependent on the basic physical quantities. The dimensions of such a constant are zero in each of the basic quantities. They are, therefore, called “dimensionless” constants.

$$\mathrm{pi},\left[\pi \right]=\left[M^0{L}^0{T}^0\right]$$

However, there are physical constants, which appear as the constant of proportionality in physical formula. These constants have dimensions in basic quantities. Such constant like Gravitational constant, Boltzmann constant, Planck's constant etc. are, therefore, “dimensional” constant.

In the nutshell, we categorize variables and constants in following categories :

• Dimensional variable : speed, force, current etc.
• Dimensionless variable : angle, reynold’s number, trigonometric, logarithmic functions etc.
• Dimensional constant : universal gas constant, permittivity, permeability
• Dimensionless constant : numerical constants, mathematical constants, trigonometric, logarithmic functions etc.

Dimensional equation

Dimensional equation is obtained by equating dimension of a given physical quantity with its dimensional formula. Hence, followings constitute a dimensional equation :

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

$$\left[v\right]=\left[M^{0}L{T}^{-1}\right]$$

Dimensions and unit

Dimensions of a derived quantities can always be expressed in accordance with their dimensional constitutions. Consider the example of force. Its dimensional formula is $ML{T}^{-2}$ . Its SI unit, then, can be “kg-m/s2” $\mathrm{kg-m}/s^2$ . Instead of using dimensional unit, we give a single name to it like Newton to honour his contribution in understanding force. Such examples are prevalent in physics.