Question

If the speed of light $(c)$ , acceleration due to gravity $(g)$ and pressure $(p)$ are taken as the fundamental quantities, then the dimension of gravitational constant is

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

Answer 1

Answer: (B)

Let us assume, $G=c^x{g}^y{p}^z$
The dimensional formula of gravitational constant, $G=\left[M^{-1}{L}^3{T}^{-2}\right]$ , speed of light, $c=\left[M^0{L}^1{T}^{-1}\right]$ , acceleration due to gravity, $g=\left[M^0{L}^1{T}^{-2}\right]$ and pressure, $p=\left[M^1{L}^{-1}{T}^{-2}\right]$ .
Substituting the values in the above equation, we get, $\left[M^{-1}{L}^3{T}^{-2}\right]={\left[M^0{L}^1{T}^{-1}\right]}^x{\left[M^0{L}^1{T}^{-2}\right]}^y{\left[M^1{L}^{-1}{T}^{-2}\right]}^z$
$\left[M^{-1}{L}^3{T}^{-2}\right]=\left[M^{0+0+z}{L}^{x+y-z}{T}^{-x-2y-2z}\right]$ .
Now, comparing the coefficients, we get,
$z=-\mathrm{1...}\left(i\right)$
$x+y-z=\mathrm{3...}\left(ii\right)$
$-x-2y-2z=-\mathrm{2...}\left(iii\right)$
Solving the above three equations, we get, $x=0,y=2,z=-1$ .
Thus, the dimension of gravitational constant is, $G=\left[c^0{g}^2{p}^{-1}\right]$ .