Gravitational potential energy between two points masses is
`U = -(Km_(1)m_(2))/(r^(n))`
where, `K` is a positive constant. With what power of 'r' time period of a satellite of mas 'm' varies in circular orbit if mass of planet is `M` ?

Options

A. `r^(3/2)`
B. `r^(1/2)`
C. `r^(2/3)`
D. `r^(2)`

Answer

A

Explanation

`U = -(KMm)/(r^(n))`
`F = -(dU)/(dr) = (KMmn)/(r^(n+1))` or `F prop r^(-n+1)`
Now, this force provides the necessary centripetal force.
`:. (m upsilon^(2))/(r) =(KMmn)/(r^(n+1))` or `upsilon^(2)prop r^(-n)`
`:. upsilon prop r^((-n)/(2))` Time period is given by
`T = (2pi r)/(upsilon)` or`T prop (r)/(upsilon)`
or `T prop (r)/(r^((-n)/(2))) rArr :. Tprop r^(1+(n)/(2))`
`U = (-GMm)/(r)`
So if, we compare with the given equation then, ` n=1`.
Now, ` Tprop r^(1+(n)/(2))` or `Tprop r^((3)/(2))` (for `n = 1`).