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(A) The necessary centripetal force required for a planet to move around the sun is provided by the gravitational force.Let the gravitational force be

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$R^{\left( \frac{n+1}{2} \right)}$

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(A) The necessary centripetal force required for a planet to move around the sun is provided by the gravitational force.Let the gravitational force be $F = \frac{k}{R^n}$,where $k$ is a constant.The centripetal force is given by $F_c = \frac{mv^2}{R}$.Equating the two,we have $\frac{mv^2}{R} = \frac{k}{R^n}$.This simplifies to $v^2 = \frac{k}{mR^{n-1}}$,so $v \propto R^{-\frac{n-1}{2}}$.The time period $T$ is given by $T = \frac{2\pi R}{v}$.Substituting the expression for $v$,we get $T \propto \frac{R}{R^{-\frac{n-1}{2}}}$.Therefore,$T \propto R^{1 + \frac{n-1}{2}} = R^{\frac{2+n-1}{2}} = R^{\frac{n+1}{2}}$.

Suppose the gravitational force varies inversely as the $n^{th}$ power of distance. Then the time period of a planet in a circular orbit of radius $R$ around the sun will be proportional to

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