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)

$\left[h^{1 / 2} c^{1 / 2} G^{-1 / 2}\right]$

Answer 2

Say dimensional formale of mass is HxCyGz
M1=(ML2T−1)x(L−1)(M−1L3T−2)Z
M1L0T0=Mx−zL2x+y+3zT−x−yz
on comparing both side
x−z=1
2x+y+3z=0
−x−y−2z=0
On solving above equations we get
x=21​y=21​z=2−1​