The dimension of a physical quantity is defined as the power to which the fundamental quantities are raised to express the physical quantity. The dimension of mass, length and time are represented as [M], [L] and [T] respectively. For example:
We say that dimension of velocity are, zero in mass, 1 in length and -1 in time.
Dimensional formula of a Physical Quantity
The dimensional formula is defined as the expression of the physical quantity in terms of its basic unit with proper dimensions. For example, dimensional force is
F = [M L T-2]
It’s because the unit of Force is Netwon or kg*m/s2
Dimensional Formula of some Physical Quantities
Dimension formula of a physical quantity can only be written when its relation with other physical quantities is known. Some of the physical quantities with the dimensional formula are given below:
An equation containing physical quantities with dimensional formula is known as dimensional equation. Dimensional equation is obtained by equating dimensional formula on right hand side and left hand side of an equation.
Principle of Homogeneity of Dimensional Equation
According to this principle, the dimensions of fundamental quantities on left hand side of an equation must be equal to the dimensions of the fundamental quantities on the right hand side of that equation. Let us consider three quantities A, B and C such that C = A + B. Therefore, according to this principle, the dimensions of C are equal to the dimensions of A and B. For example:
Dimensional equation of v = u + at is:
[M0 L T-1] = [M0 L T-1] + [M0 L T-1] X [M0 L0 T] = [M0 L T-1]
Uses of Dimensional Equations
The dimensional equations have got the following uses:
- To check the correctness of a physical relation.
- To derive the relation between various physical quantities.
- To convert value of physical quantity from one system of unit to another system.
- To find the dimension of constants in a given relation.
Limitation of Dimensional analysis
Following are the limitations of the dimensional analysis.
- It does not give information about the dimensional constant.
- if a quantity depends on more than three factors having dimension, the formula cannot be derived.
- We cannot derive the formulae containing trigonometric function, exponential functions, logarithmic function, etc.
- The exact form of relation cannot be developed when there are more than one part in any relation.
- It gives no information whether a physical quantity is scalar or vector.