A planet is revolving around a star in a circular orbit of radius R with a period T. If the gravitational force between the planet and the star is proportional to `R^(-3//2)`, then

Options

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

Answer

A

Explanation

The C.P. force is provided by the gravitational force.
`therefore (mv^(2))/(R )=(GMm)/(R^(3//2))`
`therefore mR omega^(2)=(GMm)/(R^(3//2))`
`therefore R omega^(2)prop R^(-3//2)` or `omega^(2)=R^(-5//2)`
`therefore ((2pi)/(T))^(2)prop R^(-5//2)`
`therefore (4pi^(2))/(T^(2))prop (1)/(R^(5//2))` or `T^(2)prop R^(5//2)" "` [Option (a)]