Question

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