Q 11-01-120JEE MainJEE Main 2024 (30 Jan, Shift 2)Medium
If mass is written as $m = k\,c^{P}G^{-1/2}h^{1/2}$, then the value of $P$ will be: (Constants have their usual meaning with $k$ a dimensionless constant)
Answer: (A) $\dfrac12$
$[c] = \mathrm{LT^{-1}}$, $[G] = \mathrm{M^{-1}L^3T^{-2}}$, $[h] = \mathrm{ML^2T^{-1}}$.
$$G^{-1/2}h^{1/2} = \mathrm{M^{1/2}L^{-3/2}T^{1}}\cdot\mathrm{M^{1/2}LT^{-1/2}} = \mathrm{M\,L^{-1/2}T^{1/2}}$$
To be left with $\mathrm M$ alone, $c^P$ must supply $\mathrm{L^{1/2}T^{-1/2}}$, so $P = \dfrac12$.
Solution by Sreeraj P, M.Sc Physics