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# Suppose the gravitational force varies inversely as the $n^{th}$ power of distance, then the time period of planet in circular orbit of Radius 'R' around the sun will be proportional to

$(a)\;R^n \quad (b)\;R^{\bigg(\Large\frac{n-1}{2}\bigg)} \quad (c)\;R^{\bigg(\Large \frac{n+1}{2}\bigg)} \quad (d)\;R^{\bigg(\Large \frac{n-2}{2}\bigg)}$

$F=\large\frac{K}{R^n}$
Centripetal force $=MRw^2$
$MRw^2=\large\frac{K}{R^n}$
$w^2=\large\frac{K}{M}$$R^{-(n+1)} w\; \alpha\; R^{-\bigg(\Large\frac{n+1}{2}\bigg)} w=\large\frac{2 \pi}{T} \therefore \large\frac{2 \pi}{T} \;$$\alpha\; R^{-\bigg(\Large\frac{n+1}{2}\bigg)}$
$T \;\alpha\; R^{\bigg(\Large\frac{n+1}{2}\bigg)}$
Hence c is the correct answer.

edited Feb 17, 2014 by meena.p