Question

If the speed of light $c$ , acceleration due to gravity $g$ and pressure $p$ are takes 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

Hint
We have to find dimensions of gravitational constant in terms of our-defined fundamental units (gravity, pressure and speed of light). So we will first find the dimensions of gravitational constant, our-defined fundamental units in terms of real fundamental units (mass, length, etc) then we will balance the powers of every term to get our answer.

Complete Step by Step Explanation
We know the dimensions of $G = [{M^{ - 1}}{L^3}{T^{ - 2}}]$
Where $M$ denotes mass, $L$ denotes length and $T$ denotes time
We know the dimensions of $c = [L{T^{ - 1}}]$ because $c$ is velocity
We know the dimensions of $p = [M{L^{ - 1}}{T^{ - 2}}]$
We know the dimensions of $g = [L{T^{ - 2}}]$
Now we want $G$ in terms of $c$ , $g$ and $p$
So let $G = [{c^x}{g^y}{p^z}]$ ---------(1)
Now putting the dimensions of $G$ , $c$ , $g$ and $p$ in above equations, we get,
\[[{M^{ - 1}}{L^3}{T^{ - 2}}] = [{[L{T^{ - 1}}]^x}{[L{T^{ - 2}}]^y}{[M{L^{ - 1}}{T^{ - 2}}]^z}]\]
Simplify above equations, we get,
\[[{M^{ - 1}}{L^3}{T^{ - 2}}] = [{M^z}{L^{x + y - z}}{T^{ - x - 2y - 2z}}]\]
Now, comparing powers of $M$ we get,
$z = - 1$
comparing powers of $L$ we get,
$x + y - z = 3$
We got $z = - 1$ earlier, so above equation simplifies to
$x + y = 2$ ------------(2)
Comparing powers of $T$ , we get,
$ - x - 2y - 2z = - 2$
Putting $z = - 1$ and doing some simplifications in above equation we get,
$x + 2y = 4$ ---------(3)
Solving equations $2\& 3$ we get,
$x = 0$ and,
$y = 2$
Putting values of $x,y,z$ in equation $1$ , we get dimension of $G$ as,
$G = {c^0}{g^2}{p^{ - 1}}$
So the correct answer is option (C).

Note
We do not have to find dimensions of $G$ in terms of mass, length, time, etc. and other real fundamental units. We have to find dimensions of $G$ only in terms of given quantities and also we can not include any other fundamental quantity in our answer. We just used fundamental quantities to get our answer in required units.