Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius 'R' around the sun will be proportional to

Options

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

Answer

A

Explanation

The necessary centripetal force required for a planet to move round the sun = gravitational force exerted on it
`(mv^(2))/(R) = (GM_(e)m)/(R^(n))` or `v = ((GM)/(R^(n-1)))^(1//2)`
As, `T = (2piR)/(v) = 2pi R xx ((R^(n-1))/(GM))^(1//2)`
`T = 2pi [(R^(((n+1))/(2)))/((GM_(e))^(1//2))]`
`T prop R^((n-1)//2)`