Explain the principle of homogeneity of dimensions. What are its uses? Given example.

Explanation

Principal of homogeneity of dimensions:
It states that the dimensions of all the terms in a physical expression should be the same.
For example
`v^(2)=u^(2)+2as`
`[LT^(-1)]=[LT^(-2)]^(2)+2[LT^(-2)][L]`
`[L^(2)T^(-2)]=[L^(2)T^(-2)]+2[L^(2)T^(-2)]`
This method is used to
(i) Convert a physical quantity from one system of units to another.
(ii) Check the dimensional correctness of a given physical equation.
(iii) Establish relations among various physical quantities.
To convert a physical quantitiy from one system of units of another
(i) This is based on the fact that the product of the numerical values (n) and its corresponding unit (u) is a constant i.e. n `[u]=` constnat (or ) `n_(1)[u_(1)]=n_(2)[u_(2)]`.
(ii) Consider a physical quantity which has dimension a in mass, b in length and c in time. If the fundamental units in one system are `M_(1),L_(1)` and `T_(1)` and the other system are `M_(2),L_(2)` and `T_(2)` respectively then we can write `n_(1)[M_(1)^(a)L_(1)^(b)T_(1)^(c)]=n_(2)[M_(2)^(a)L_(2)^(b)T_(2)^(c)]`
(iii)We have thus converted the numerical value of physicalk quantity from one system of units into the other system.