Q: Imagine 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 the star is proportional to R^{-5/2}, then

(a)T^{2} is proportional to R^{2}

(b)T^{2} is proportional to R^{7/2}

(c)T^{2} is proportional to R^{3/2}

(d)T^{2} is proportional to R^{3.75}

Ans: (b)

Sol: $\large \frac{mv^2}{R} \propto R^{-5/2}$

$\large v \propto R^{-3/4}$

$\large T = \frac{2\pi R}{v}$

$\large T^2 \propto (\frac{R}{v})^2$

$\large T^2 \propto (\frac{R}{R^{-3/4}})^2$

$\large T^2 \propto R^{7/2}$