Suppose the gravitational force varies inversely as the `n^(th) `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

Gravitational force , `F prop (1)/(r^(n))`
`F = (k)/(r^(n))` where k is a constant .
For a planet , moving in a circular orbit of radius R , F = `(k)/(R^(n))`
But , `F = m omega^(2) R `
`implies (k)/(R^(n)) = mR* ((2pi)/(T))^(2) `
`implies T^(2) prop R^(n + 1)`
`therefore T prop R^((n+1)/(2))`