Question

Consider a hypothetical planet which is very long and cylindrical. The density of the planet is $\rho$ and its radius is $R$. 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 the planet is $\sqrt{x\pi G\rho }R$. The value of $x$ is ________.

Answer 1

Consider a closed cylindrical Gaussian surface of radius $r$ and length $l$ around the planet, as shown in the figure.

From Gauss's Law for masses,

$2\pi rlE=\frac{\rho \times \pi R^{2}l}{1/4\pi G}=\left(4\pi G\right)\left(\rho \pi R^{2}l\right)$

$\Rightarrow E=\frac{2\pi G\rho R^2}{r}$

To rotate a satellite around the planet in the orbit $r$, centripetal force is provided by the gravitational force.

$i.e.mE=\frac{m{v}^2}{r}$

$\Rightarrow m\times \frac{2\pi G\rho R^2}{r}=\frac{m{v}^2}{r}$

$\Rightarrow v=\sqrt{2\pi G\rho }R$

$i.e.$ independent of radius $r$.

$\Rightarrow v=\sqrt{2\pi G\rho }R=\sqrt{x\pi G\rho }R$

$\therefore x=2$