Write and explain principle of homogeneity. Check dimensional consistency of given equation.

Magnitude of physical quantities having same dimension can be added together or subtracted from one another.

This principle is called principle of homogeneity of dimension.

This rule is very useful to check dimensional consistency of given equation.

To check dimensional consistency of given equation all terms on both side of the equation should have same dimension.

Dimensional consistency do not guarantee correctness of equation.

It is uncertain to the extent of dimensionless quantities or function.

For example,

$x=x_{0}+v_{0} t+\frac{1}{2} a t^{2}$

Here, $x$ is distance covered by object in time $t$.

$x_{0}=\text { initial position of object during motion }$

$x=\text { final position }$

$v_{0}=\text { initial velocity }$

$a=\text { acceleration }$

$\text { LHS }=x=\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{0}$

In $RHS$ there are three terms,

$x_{0}=\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{0}$

$v_{0} t=\left[\mathrm{L}^{1} \mathrm{~T}^{-1}\right]\left[\mathrm{T}^{1}\right]=\mathrm{M}$$^{0} \mathrm{~L}^{1} \mathrm{~T}^{0}$

In $\frac{1}{2} a t^{2}$ is constant term which is dimensionless.

$\therefore a t^{2} =\left[\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{-2}\right]\left[\mathrm{T}^{2}\right]$

$=\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{0}$

Here all terms in given equation have same dimension. Hence, given equation,