Question

In a new system of units energy,density,and power are taken as fundamental units, then the dimensional formula of universally gravitionalconstant "G" will be?

Answer 1

$$\begin{gathered} energyE\hspace{0.17em}=\left[M{L}^2{T}^{-2}\right] \\ density=d=\left[M{L}^{-3}\right] \\ power=P=\left[M{L}^2{T}^{-3}\right] \\ G\hspace{0.17em}=N{m}^2/k{g}^2=\left[M^{-1}{L}^3{T}^{-2}\right] \\ G=E^a{d}^b{P}^c \\ \left[M^{-1}{L}^3{T}^{-2}\right]={\left[M{L}^2{T}^{-2}\right]}^a{\left[M{L}^{-3}\right]}^b{\left[M{L}^2{T}^{-3}\right]}^c \\ \left[M^{-1}{L}^3{T}^{-2}\right]=\left[M^{a+b+c}{L}^{2a-3b+2c}{T}^{-2a-3c}\right] \\ a+b+c=-1 \\ 2a-3b+2c=3 \\ -2a-3c=-2 \\ a=-2 \\ b=-1 \\ c=2 \\ G=E^{-2}{d}^{-1}{P}^2 \end{gathered}$$