Question

If the 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. $c^{0}g{p}^{-3}$
• B. $c^2{g}^3{p}^{-2}$
• C. $c^0{g}^2{p}^{-1}$
• D. $c^2{g}^2{p}^{-2}$

Answer 1

Answer: (C)

$c^0{g}^2{p}^{-1}$

Dimension of $G=M^{-1}{L}^3{T}^{-2}$

$\left[c\right]=\left[L{T}^{-1}\right]$

$\left[g\right]=\left[L{T}^{-2}\right]$

$\left[p\right]=\left[M{L}^{-1}{T}^{-2}\right]$

Equate dimensions

$\left[M^{-1}{L}^3{T}^{-2}\right]=\left[L{T}^{-1}{]}^{\alpha }\right[L{T}^{-2}{]}^{\beta }[M{L}^{-1}{T}^{-2}{]}^{\gamma }$

$-1=\gamma$

$3=\alpha +\beta -\gamma$

$\alpha +\beta =2$ (1)

And, $-2=-\alpha -2\beta -2\gamma$

$\alpha +2\beta =4$ (2)

Solving (1) and (2) we get,

$\alpha =0,\beta =2$

$\therefore Dimension=c^0{g}^2{p}^{-1}$