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

Options

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)`

Answer

B

Explanation

(d) According to the question, the gravitational force between the planet and the star is ` F prop (1)/(R^(5//2))`
` :. F=(GMm)/(R^(5//2))`
where M and m be masses 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))`
`(4 pi^(2))/(T^(2))=(GM)/(R^(7//2)) implies T^(2)=(4 pi^(2))/(GM)R^(7//2)`
`T^(2) prop R^(7//2) or T prop R^(7//4)`