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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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1 Answer

 
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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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