While using addition and subtraction for two or more physical quantities, we notice that only those quantities can be added or subtracted that have similar dimension. On the other hand, in the case of multiplication and division, when two quantities are multiplied, along with their magnitudes, the dimensions are also treated accordingly, in order to find the dimension of the product.
In mathematical equations, the physical quantities on either side of the equation are required to be of the same dimension. The dimensional analysis is very important when dealing with these physical quantities.
Checking the dimensional consistency
Only similar physical quantities can be added or subtracted, thus two quantities having different dimensions cannot be added together. For example, we cannot add mass and force or electric potential and resistance. For any given equation, the principle of homogeneity of dimensions is used to check the correctness and consistency of the equation. The dimensions of each component on either side of the sign of equality are checked, and if they are not the same, the equation is considered wrong.
Let us consider the equation given below,
The dimensions of the LHS and the RHS are calculated
The dimensions of the LHS and the RHS are the same, hence, the equation is consistent.
Deducing the relation among physical quantities
Dimensional analysis is also used to deduce the relation between two or more physical quantities. If we know the degree of dependence of a physical quantity on another, that is the degree to which one quantity changes with the change in another, we can use the principle of consistency of two expressions to find the equation relating these two quantities.
Example: Derive the formula for centripetal force F acting on a particle moving in a uniform circle.
The centripetal force acting on a particle moving in a uniform circle depends on its mass m, velocity v and the radius r of the circle. Hence, we can write
Writing the dimensions of these quantities,
As per the principle of homogeneity, we can write,
a = 1, b+c = 1 and b = 2
Solving the above three equations we get, a = 1, b = 2 and c = -1.
Hence, the centripetal force F can be represented as,