Question

Expression for time in terms of $G$ (universal gravitational constant), $h$ (Planck constant) and $c$ (speed of light) is proportional to :

• A. $\sqrt{\frac{Gh}{c^3}}$
• B. $\sqrt{\frac{h{c}^5}{G}}$
• C. $\sqrt{\frac{c^3}{Gh}}$
• D. $\sqrt{\frac{Gh}{c^5}}$

Answer 1

The correct option is D $\sqrt{\frac{Gh}{c^5}}$

$F=\frac{G{M}^2}{R^2}\Rightarrow G=\left[M^{-1}{L}^3{T}^{-2}\right]$
$E=hv\Rightarrow h=\left[M{L}^2{T}^{-1}\right]$
$C=\left[L{T}^{-1}\right]$
$t\propto G^x{h}^y{C}^z$
$\left[T\right]=\left[M^{-1}{L}^3{T}^{-2}{]}^x\right[M{L}^2{T}^{-1}{]}^y[L{T}^{-1}{]}^z$
$\left[M^0{L}^0{T}^1\right]=\left[M^{-x+y}{L}^{3x+2y+z}{T}^{-2x-y-z}\right]$
on comparing the powers of M, L, T
$-x+y=0\Rightarrow x=y$
$3x+2y+z=0\Rightarrow 5x+z=0$
$-2x-y-z=1\Rightarrow 3x+z=-1$
on solving (i) and (ii) $x=y=\frac{1}{2},z=-\frac{5}{2}$
$t\propto \sqrt{\frac{Gh}{C^5}}$