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