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//3)`
(d) `T^(2)` is proportional to `R^(3.75)`.

Options

A. `R^(3//2)`
B. `R^(3//5)`
C. `R^(7//2)`
D. `R^(7//4)`

Answer

D

Explanation

According to the question, the gravitational force between the planet and the star is `Fprop 1/(R^(5//2))`
`:. F=(GMm)/(R^(5//2))`
Where M and m be mass of star and planet respectively for motion of a planet in a circular orbit,
`mRomega^(2)=(GMm)/(R^(5//2))`
`mR((2pi)/T)^(2)=(GMm)/(R^(5//2)) ( :' omega=(2pi)/T)`
`(4pi^(2))/(T^(2))=(GM)/(R^(5//7)) implies T^(2)=(4pi^(2))/(GM)R^(7//2)`
`T^(2) prop R^(7//2) or T prop R^(7//4)`