Question

In the expression $P=E{l}^2{m}^{-5}{G}^{-2}$ where $E,m,l$ and $G$ denote energy, mass, angular momentum and gravitational constant, respectively. Show that $P$ is a dimensionless quantity.

Answer 1

According to the problem,expression is $P=E{l}^2{m}^{-5}{G}^{-2}$
where E is energy $\left[E\right]=\left[M{L}^2{T}^{-2}\right]$, m is mass $\left[m\right]=\left[M\right]$
L is angular momentum $\left[L\right]=\left[M{L}^2{T}^{-1}\right]$, G is gravitational constant $\left[G\right]=\left[M^{-1}{L}^2{T}^{-2}\right]$
Substituting dimensions of each physical quantity in the given expression,
$\left[P\right]=\left[M{L}^2{T}^{-2}\right]\times \left[M{L}^2{T}^{-1}\right]\times [M{]}^{-5}\times [M^{-1}{L}^3{T}^{-2}{]}^{-2}$
$=\left[M^{1+2-5+2}{L}^{2+4-6}{T}^{-2-2+4}\right]$
$\left[M^0{L}^0{T}^0\right]$
This shows that P is dimensionless quantity.