Q 11-01-149JEE MainJEE Main 2023 (1 Feb, Shift 2)Medium
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
Answer: (D) $\left[h^{1/2}c^{1/2}G^{-1/2}\right]$
$[h]=\text{ML}^2\text{T}^{-1}$, $[c]=\text{LT}^{-1}$, $[G]=\text{M}^{-1}\text{L}^3\text{T}^{-2}$. Then
$$\left[\frac{hc}{G}\right]=\frac{\text{ML}^2\text{T}^{-1}\cdot\text{LT}^{-1}}{\text{M}^{-1}\text{L}^3\text{T}^{-2}}=\text{M}^2$$
so $M=\left[h^{1/2}c^{1/2}G^{-1/2}\right]$ (the Planck mass).
Solution by Sreeraj P, M.Sc Physics