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Question

What is dimensional analysis? give example also.


Solution

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed. 

These are the uses of dimensional analysis:
a) To check the dimensional consistency of a formula:-
It is used to check the formula whether it is dimensionally correct. Here we check the dimensions of each term of the formula. For example:-
x = ut +0.5at2
[x] = L
[ut] = velocity × time = (length/time) × time = L
[0.5at2] = [ at2] = acceleration × (time)2
= (velocity/time) × (time)2
= (length/(time)2) × (time)2
=length
=L
Hence above formula is dimensionally correct.
b) Conversion of Units:-
force of 1N = ......?........ Dyne
[F] = MLT-2
so 1 Newton = (1kg) (1m) (1s)-2
1 Dyne = (1g) (1cm) (1s)-2
Thus, 1 Newton/ (1 Dyne) = (1kg/1g) ×(1m/1cm) ×(1s/1s)-2
= 10^3 (10)^2
=10^5
So, 1 Newton = 10^5 Dyne

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