Consider a long hypothetical cylindrical hollow planet of radius R and uniform mass per unit area $\sigma$. If a satellite revolves around the planet in circular orbit of radius r in a plane perpendicular to the axis of the planet, then :-

 

Intensity of gravitational field due to long hollow cylinder can be found using Gauss law, field and hence Force will be inversely proportional to radius of the orbit and directly proportional to mass per unit length of the planet

$$\begin{gathered} F=\frac{2GM\text{'}m}{r} M\text{'}=\sigma \pi R^2 \\ \frac{m{v}^2}{r}=\frac{2G\sigma \pi R^{2}m}{r}\Rightarrow V=\sqrt{2G\sigma \pi R^2} \\ T=\frac{2\pi r}{v} \end{gathered}$$