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Expression for time in terms of GG (universal gravitational constant), hh (Planck constant) and cc (speed of light) is proportional to :

A)$$\sqrt{\dfrac{Gh}{c^3}}$$

B)$$\sqrt{\dfrac{ch}{G^3}}$$

C) $$\sqrt{\dfrac{Gh}{c^5}}$$

D)$$\sqrt{\dfrac{G^5h}{c}}$$

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

Explaination::

$$F = \dfrac{GM^2}{R^2} \Rightarrow G = [M^{-1} L^3 T^{-2}]$$

$$E = hv \Rightarrow h = [ML^2 T^{-1}]$$

$$C = [LT^{-1}]$$

$$t \propto G^x h^y C^z$$

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

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) $$x = y = \dfrac{1}{2}, z = -\dfrac{5}{2}$$

$$t \propto \sqrt{\dfrac{Gh}{C^5}}$$

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