Question

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

Answer 1

Correct Answer - B
Gravitational force `(=(GMm)/(R^(3//2)))` providies the necessary
centripetal force i.e `mRomega^(2)`
so `(GMm)/(R^(3//2)=mR omega^(2)=mR(2pi)/(T)^(2)=(4pi^(2)mR)/(T^(2))`
or `T^(2)=(4pi^(2)R^(5//2))/(GM), i.e T^(2) prop R^(5//2)`