Question

If speed of light ( $c$ ), acceleration due to gravity $(g)$ and pressure $(P)$ are taken as fundamental units, the dimensions of gravitational constant $(G)$ are

• A. $\left[c^{0}g{P}^{-3}\right]$
• B. $\left[c^2{g}^3{P}^{-2}\right]$
• C. $\left[c^0{g}^2{P}^{-1}\right]$
• D. $\left[c^2{g}^2{P}^{-2}\right]$

Answer 1

Answer: (C)

Let $G=k{c}^x{g}^y{P}^z$ where $k$ is a dimensionless constant
$\therefore \left[M^{-1}{L}^3{T}^{-2}\right]={\left[L{T}^{-1}\right]}^x{\left[L{T}^{-2}\right]}^{\gamma }{\left[M{L}^{-1}{T}^{-2}\right]}^z$
$=\left[M^2{L}^{x+y-z}{T}^{-x-2y-2z}\right]$
Applying principle of homogeneity of dimensions, we get
$z=-\mathrm{1...}(i)$
$x+y-z=\mathrm{3...}(ii)$
$-x-2y-2z=-\mathrm{2...}(iii)$
On solving (i), (ii) and (iii), we get
$x=0,y=2,z=-1\therefore [G]=\left[c^0{g}^2{P}^{-1}\right]$