Consider a hypothetical planet which is very long and cylinderical. The density of the planet is `rho`, its radius is `R`.

What is the possible orbital speed of the satellite in moving around the planet in circular orbit in a plane which is perpendicular to the axis of planet?

Options

A. `Rsqrt(piGrho)`
B. `2Rsqrt(piGrho)`
C. `Rsqrt(2piGrho)`
D. `Rsqrt((Grho)/(2pi))`

Answer

C

Explanation

consider a closed cylinderical Gaussian surface of radius `r` and length `l` around the planet. Gauss's law gives
`2pirlE=(4piG)rholpiR^(2)E=(2piGrhor^(2))/r`
to rotate a satellite around the planet in the orbit radius, `r` centripetal force is provided by the Gravitational force. i.e. `(2piGrhoR^(2))/rm=(mv^(2))/r,v=sqrt(2piGrho)R`
i.e., independent of radius `r`.