Q 11-01-213JEE MainJEE Main 2019 (9 Jan, Shift 2)Easy
Expression for time in terms of $G$ (universal gravitational constant), $h$ (Planck constant) and $c$ (speed of light) is proportional to:
Answer: (C) $\sqrt{\dfrac{Gh}{c^5}}$
Use $[G] = M^{-1}L^3T^{-2}$, $[h] = ML^2T^{-1}$, $[c] = LT^{-1}$ and let $t \propto G^x h^y c^z$.
$M$: $-x + y = 0$
$L$: $3x + 2y + z = 0$
$T$: $-2x - y - z = 1$
With $y = x$: $z = -5x$ from the $L$ equation, and then $-3x + 5x = 1$ gives $x = \tfrac12$, so $y = \tfrac12$, $z = -\tfrac52$.
$$t \propto \sqrt{\frac{Gh}{c^5}}$$
Solution by Sreeraj P, M.Sc Physics