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

Solution:

$F = \frac{GM^{2}}{R^{2}} \Rightarrow G = \left[M^{-1} L^{3} T^{-2}\right] $
$ E = hv \Rightarrow h = \left[ML^{2} T^{-1}\right] $
$ C= \left[LT^{-1}\right] $
$ t \propto G^{x}h^{y}C^{z} $
$ \left[T\right] = \left[M^{-1} L^{3} T^{-2} \right]^{x} \left[ML^{2} T^{-1}\right]^{y} \left[LT^{-1}\right]^{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 ....(i)
-2x - y - z = 1 $\Rightarrow$ 3x + z = -1 ...(ii)
on solving (i) & (ii) x = y = $\frac{1}{2}$ , z = $ - \frac{ 5}{2}$
$ t \propto \sqrt{\frac{Gh}{C^5}}$

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