Suppose the gravitational force varies inversely  as the nth power of distance. Then the time period of 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

`implies` For circular motion centripetal force = Gravitational force
`mRomega^2 = (GMm)/(R^n)`
`:. omega = sqrt((GM)/(R^(n+1)))`
`:. (2pi)/T = sqrt((GM)/(R^(n+1)))`
`: T = 2pisqrt((R^(n+1))/(Gm))`
`= (2pi)/(sqrt(GM))xx(R^(n+1))^(1/2)`
`= (2pi) / sqrt(GM)xx R ^((n+1)/2) " in " (2pi)/(sqrt(GM))` constant
`:. T prop R^((n+1)/2)`