Question

If the velocity of light $c$, universal gravitational constant $G$ and Planck's constant $h$ are chosen as fundamental quantities. The dimensions of mass in the new system is :

• A. $\left[h^1 c ^1 G ^{-1}\right]$
• B. $\left[h^{-1 / 2} c^{1 / 2} G^{1 / 2}\right]$
• C. $\left[h^{1 / 2} c^{1 / 2} G^{-1 / 2}\right]$
• D. $\left[h^{1 / 2} c^{-1 / 2} G^1\right]$

Answer 1

Answer: (C)

Say dimensional formale of mass is $H ^{ x } C ^{ y } G ^z$
$ M^1=\left(M L^2 T^{-1}\right)^x\left(L^{-1}\right)\left(M^{-1} L^3 T^{-2}\right)^Z$
$M^1 L^0 T^0=M^{x-z} L^{2 x+y+3 z} T^{-x-y z}$
on comparing both side
$ x-z=1 $
$ 2 x+y+3 z=0 $
$ -x-y-2 z=0$
On solving above equations we get
$x =\frac{1}{2} y =\frac{1}{2} z =\frac{-1}{2}$