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

Answer 1

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

According to question

$t\propto G^x{h}^y{c}^z\dots \left(i\right)$

The dimensions of the given quantities are,

$G=\left[M^{-1}{L}^3{T}^{-2}\right]$

$h=\left[M{L}^2{T}^{-1}\right]$

$c=\left[L{T}^{-1}\right]$

$t=\left[T^1\right]$

Now from equation $\left(i\right)$,

$\left[M^0{L}^0{T}^1\right]$

$=\left[M^{-1}{L}^3{T}^{-2}{]}^x\right[M{L}^2{T}^{-1}{]}^y[L{T}^{-1}{]}^z$

$\left[M^0{L}^0{T}^1\right]=\left[M^{-x+y}{L}^{3x+2y+z}{T}^{-2x-y-z}\right]$

On comparing the powers of $M,L,T$

$-x+y=0$

$\Rightarrow x=y$

$3x+2y+z=0$

$\Rightarrow 5x+z=0\dots \left(i\right)$

$-2x-y-z=1$

$\Rightarrow 3x+z=-1\dots \left(ii\right)$

On solving $\left(i\right)$ and $\left(ii\right)$

$x=y=\frac{1}{2},Z=-\frac{5}{2}$

Hence , $t\propto \sqrt{\frac{Gh}{c^5}}$

Final Answer:(d)