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Principle of Homogeneity

Expression fo...

Question

Expression for time in terms of $G$ (universal gravitational constant ), $h$ (Planck's constant) and $c$ (speed of light ) is proportional to

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Solution

Let $t \propto G^{x} h^{y} c^{z}$
Dimensions of $G = [M^{- 1} L^{3} T^{- 2}]$
$h = [M L^{2} T^{- 1}]$ and
$c = [L T^{- 1}]$
$[T] = [M^{- 1} L^{3} T^{- 2}]^{x} [M L^{2} T^{- 1}]^{y} [L T^{- 1}]^{z}$
$[M^{0} L^{0} T^{1}] = [M^{- x + y} L^{3 x + 2 y + z} T^{- 2 x - y - z}]$
By comparing the powers of $M, L, T$ both the sides
$- x + y = 0 \Rightarrow x = y$
$3 x + 2 y = z = 0 \Rightarrow 5 x + z = 0 . . . (i)$
$- 2 x - y - z = 1 \Rightarrow 3 x + z = - 1 . . . . (i i)$
Solving eqns. (i) and (ii),

$x = y = \frac{1}{2}, z = - \frac{5}{2} ∴ t \propto \sqrt{\frac{G h}{c^{5}}}$

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