Kepler's Laws and Newton's Law of Gravitation | JEE Main previous year questions with Solutions

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$T \propto m^{x} G^{y} a^{z}$

$T \propto M^{x} {[M^{- 1} L^{3} T^{- 2}]}^{y} {[L]}^{z}$

$T \propto M^{x - y} L^{3 y + z} T^{- 2 y}$

$x - y = 0 \Rightarrow x = y$

$- 2 y = 1 \Rightarrow y = - \frac{1}{2}, x = - \frac{1}{2}$

$3 y + z = 0 \Rightarrow z = - 3 y = \frac{3}{2}$

Hence, $T \propto m^{- 1 / 2} G^{- 1 / 2} a^{3 / 2}$

$T \propto {(\frac{a^{3}}{m})}^{1 / 2} \Rightarrow T = 4 \times {(\frac{2^{3}}{2})}^{1 / 2} = 8 s$

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