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# A light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force of attraction between the planet and star is proportional to $R^{\large\frac{-5}{2}}$ then $T^2$ is proportional to

$(a)\;R^3 \quad (b)\;R^{\frac{7}{2}} \quad (c)\;R^{\frac{3}{2}} \quad (d)\;R^{\frac{9}{2}}$

Gravitational force $\alpha\;R ^{\large\frac{-5}{2}}$
Centripelal force $\alpha\; R^{\large\frac{-5}{2}}$
$m R w^2 \;\alpha\; R^{\large\frac{-5}{2}}$
$w^2 \;\alpha \; R^{\large\frac{-5}{2} -1}$
$w^2 \;\alpha \; R^{\large\frac{-7}{2}}$
$\large\bigg(\frac{2 \pi}{T}\bigg)^2 \alpha\;$$R^{\large\frac{-7}{2}}$
$T^2 \alpha\; R^{\large\frac{7}{2}}$
Hence b is the correct answer.

edited Feb 17, 2014 by meena.p