If $r$ represents the radius of the orbit of a satellite of mass $m$ moving around a planet of mass $M$,the velocity of the satellite is given by

The mean radius of the earth is $R$,its angular speed on its own axis is $\omega$,and the acceleration due to gravity at the earth's surface is $g$. The cube of the radius of the orbit of a geostationary satellite will be

Two small,equal masses are attached by a light rod. This object orbits a planet; the length of the rod is smaller than the radius of the orbit,but not negligible. The rod rotates about its axis in such a way that it remains vertical with respect to the planet. Determine the state of the rod (tension or compression) and the stability of the equilibrium with respect to a small angular perturbation.

$A$ planet moving along an elliptical orbit is closest to the sun at a distance $r_1$ and farthest away at a distance of $r_2$. If $v_1$ and $v_2$ are the linear velocities at these points respectively,then the ratio $\frac{v_1}{v_2}$ is

The distance between two stars of masses $3 M_S$ and $6 M_S$ is $9 R$. Here $R$ is the mean distance between the centers of the Earth and the Sun,and $M_S$ is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period $n T$,where $T$ is the period of Earth's revolution around the Sun. The value of $n$ is. . . . .

Express the constant $k$ of $T^{2}=k(R_{E}+h)^{3}$ in days and kilometres. Given $k = 10^{-13} \; s^{2} \; m^{-3}$. The moon is at a distance of $3.84 \times 10^{5} \; km$ from the earth. Obtain its time period of revolution in days.