The density of a planet is `rho` The orbital period of a satellite revolving around it with remaining close to its surface is ..........

Options

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

Answer

B

Explanation

`implies` For planetary motion
centripetal force = Gravitational force between planet and satellite
`(mv^2)/R = (GMm)/(R^2)`
`:.v = ((GM)/R)^(1/2)`
`"Rw" = ((GM)/R)^(1/2) " " [ :. v = "rw"]`
but , M = volume `v xx` density `rho`
`=4/3 piR^3 rho`
`:. Rw=((G xx4/3piR^3rho)/R)1/2`
`:. w = (4/3 pi Grho)^(1/2)`
`:. T = (2pi)/w=(2pi)/((4/3piGrho)^(1/2))`
`:. T = ((3pi)/(Grho))^(1/2)`