A) \[{{R}^{\left( \frac{n+1}{2} \right)}}\]
B) \[{{R}^{\left( \frac{n-1}{2} \right)}}\]
C) \[{{R}^{n}}\]
D) \[{{R}^{\left( \frac{n-2}{2} \right)}}\]
Correct Answer: A
Solution :
The necessary centripetal force required for a planet to move round the sun = gravitational force exerted on it i.e., \[\frac{m{{v}^{2}}}{R}=\frac{G{{M}_{e}}m}{{{R}^{n}}}\,\Rightarrow \,v={{\left( \frac{G{{M}_{e}}}{{{R}^{n-1}}} \right)}^{1/2}}\] now \[T=\frac{2\pi R}{v}=2\pi R\times {{\left( \frac{{{R}^{n-1}}}{G{{M}_{e}}} \right)}^{1/2}}\] \[=2\pi {{\left( \frac{{{R}^{2}}\times {{R}^{n-1}}}{G{{M}_{e}}} \right)}^{1/2}}\,=2\pi \left( \frac{{{R}^{(n+1)/2}}}{{{(G{{M}_{e}})}^{1/2}}} \right)\]\[\Rightarrow \,T\propto {{R}^{\left( \frac{n+1}{2} \right)}}\]
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