A) \[{{R}^{(n-1)/2}}\]
B) \[{{R}^{(n+1)/2}}\]
C) \[{{R}^{n-1}}\]
D) \[{{R}^{n+1}}\]
Correct Answer: B
Solution :
In gravitational field of sun, gravitational force will provide the necessary centripetal force to the planet. That is, centripetal force = gravitational force i.e., \[\frac{m{{v}^{2}}}{R}=\frac{GMm}{{{R}^{n}}}\] \[\Rightarrow \] \[v=\sqrt{\frac{GM}{{{R}^{n-1}}}}\] Time period of planet in circular orbit of radius R around sun will be \[T=\frac{2\pi R}{v}\] or \[T=\frac{2\pi R}{\sqrt{\frac{GM}{{{R}^{n-1}}}}}=\frac{2\pi }{\sqrt{GM}}{{R}^{1+\left( \frac{n-1}{2} \right)}}\] or \[T=\frac{2\pi }{\sqrt{GM}}{{R}^{\frac{n+1}{2}}}\] or \[T\propto {{R}^{\frac{(n+1)}{2}}}\]
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