Suppose the gravitational force varies invers | vedclass

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(A) The gravitational force is given by $F \propto \frac{1}{R^n} = R^{-n}$.For a planet in a circular orbit,the centripetal force is provided by the g

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

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(A) The gravitational force is given by $F \propto \frac{1}{R^n} = R^{-n}$.For a planet in a circular orbit,the centripetal force is provided by the gravitational force:$M R \omega^2 = F \propto R^{-n}$.Since $\omega = \frac{2\pi}{T}$,we have $\omega^2 \propto \frac{1}{T^2}$.Substituting this into the force equation:$R \cdot \frac{1}{T^2} \propto R^{-n}$.$\frac{1}{T^2} \propto R^{-n-1} = R^{-(n+1)}$.$T^2 \propto R^{n+1}$.Taking the square root of both sides:$T \propto R^{\frac{n+1}{2}}$.

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

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