Applications of Dimensional Analysis
Dimensional Analysis is also called Factor Label Method (or) Unit Factor Method as we use conversion factor in order to get same units. The basic concept of dimensions is that we can add or subtract only those quantities which have same dimensions. Also, two physical quantities are equal if they have same dimensions.
1) To find the unit of a physical quantity in a given system of units:
To write the definition or formula for the physical quantity we find its dimension. Now in the dimensional formula replacing M, L and T by the fundamental unit of the required system we get the unit of physical quantity. However, sometime to this unit we further assign a specific name.
e.g. \(Work\left( W \right)=Force\left( F \right)\times Displacement\left( d
\right)\)
So its unit in CGS system will be \(g.c{{m}^{2}}/{{\sec }^{2}}\) which is called erg while in MKS system will be \(kg.{{m}^{2}}/{{\sec }^{2}}\) which is called joule.
2) To find dimensions of physical constant or coefficients:
As dimensions of a physical quantity are unique, we write any formula or equation incorporating the given constant and then by substituting the dimensional formulae of all other quantities, we can find the dimensions of the required constant of coefficient. For example:
Gravitational Constant: According to Newton’s law of gravitation,
\(F=G\frac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\Rightarrow G=\frac{F{{r}^{2}}}{{{m}_{1}}{{m}_{2}}}\)Substituting the dimensions of all physical quantities:
\(\left[ G \right]=\frac{\left[ ML{{T}^{-2}} \right]\left[ {{L}^{2}} \right]}{\left[ M \right]\left[ M \right]}=\left[ {{M}^{-1}}{{L}^{3}}{{T}^{-2}} \right]\)