What is the mass of the planet that has a satellite whose time period is `T` and orbital radius is `r`?

Options

A. `(4pi^(3)r^(3))/(GT^(2))`
B. `(4pi^(2)r^(3))/(GT^(2))`
C. `(4pi^(2)r^(3))/(GT^(3))`
D. `(4pi^(2)T)/(GT^(2))`

Answer

A

Explanation

Suppose that a satellite of mass `m` describes a circular orbit around a planetof mass `M`.
`F=(GmM)/(r^(2))`
This force must be mass times the cenripetal acceleration.
`:. F=(mv^(2))/r=momega^(2)=(4pi^(2))/(T^(2))r`
`:. M=(4pi^(2)r^(3))/(GT^(2))`