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EEJ MAIN Physics QUESTION #7037
Question 1
Express time as a proportional relationship using the universal gravitational constant ($G$), Planck's constant ($h$), and the speed of light ($c$).
  • $\sqrt{\frac{hc^5}{G}}$
  • $\sqrt{\frac{c^3}{Gh}}$
  • $\sqrt{\frac{Gh}{c^5}}$✔️
  • $\sqrt{\frac{Gh}{c^3}}$
Correct Answer Explanation

Let $T \propto G^x h^y c^z$. Using dimensional formulas:

  • $[T] = [T^1]$
  • $[G] = [M^{-1} L^3 T^{-2}]$
  • $[h] = [M L^2 T^{-1}]$
  • $[c] = [L T^{-1}]$

Comparing powers of $M, L, T$:

  1. $-x + y = 0 \implies x = y$
  2. $3x + 2y + z = 0 \implies 5x + z = 0 \implies z = -5x$
  3. $-2x - y - z = 1 \implies -3x - (-5x) = 1 \implies 2x = 1 \implies x = 1/2$

Thus, $x=1/2, y=1/2, z=-5/2$. The expression is $\sqrt{\frac{Gh}{c^5}}$.