If the gravitational force is assumed to vary as the nth power of the distance, then the time period of a planet round the sun will be proportional to :

Options

A. `R^(n)`
B. `R^(-n)`
C. `R^((n+1)/(2))`
D. `(R )^((n-1)/(2))`

Answer

C

Explanation

Gravitational force provides the necessary centripetal force
`therefore (mv^(2))/(R )=(GMm)/(R^(n)) rArr v= sqrt((GM)/(R^(n-1)))`
Now `T=(2pi R)/(v)= (2pi R)/(sqrt((GM)/(R^(n-1))))= 2pi sqrt((R^(n+1))/(GM))`
`rArr T prop R^(((n+1)/(2)))`
Thus correct choice is (c ).