Broadly speaking, dimension is the nature of a Physical quantity. Understanding of this nature helps us in many ways.
Following are some of the applications of the theory of dimensional analysis in Physics:
(i) To find the unit of a given physical quantity in a given system of units:
By expressing a physical quantity in terms of basic quantity we find its dimensions. In the dimensional formula replacing M, L, T by the fundamental units of the required system, we get the unit of physical quantity. However, sometimes we assign a specific name to this unit.
Force is numerically equal to the product of mass and acceleration
i.e. Force = mass x acceleration
or [F] = mass x velocity/time= mass x displacement/(time)2 ) = mass x
= [M] x [LT-2] = [MLT-2]
Its unit in SI system will be Kgms-2 which is given a specific name “newton (N)”.
Similarly, its unit in CGS system will be gmcms-2 which is called “dyne”.
(ii) To find dimensions of physical constants or coefficients:
The dimension of a physical quantity is unique because it is the nature of the physical quantity and the nature does not change. If we write any formula or equation incorporating the given physical constant, we can find the dimensions of the required constant or co-efficient.
From Newton’s law of Gravitation, the exerted by one mass upon another is
F=G (m1 m2)/r2 or G=(Fr2)/(m1 m2 )
or [G] = ([MLT]2][L-2]) / ([M][M]) = [M-1 L3 T-2 ]
We can find its SI unit which is m3/Kgs2.
(iii) To convert a physical quantity from one system of units to another:
This is based on the fact that for a given physical quantity, magnitude x unit = constant
So, when unit changes, magnitude will also change.
Convert one Newton into dyne