$A$ double star is a system of two stars rotating about their centre of mass only under their mutual gravitational attraction. Let the stars have masses $m$ and $2m$ and their separation be $l$. Their time period of rotation about their centre of mass will be proportional to:

What is the orbital velocity of a satellite? Derive its equation.

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.

Two satellites $A$ and $B$ of masses $200 \, kg$ and $400 \, kg$ are revolving around the Earth at heights of $600 \, km$ and $1600 \, km$ respectively. If $T_{A}$ and $T_{B}$ are the time periods of $A$ and $B$ respectively,then find the value of $T_{B} - T_{A}$.
[Given: Radius of Earth $R = 6400 \, km$,Mass of Earth $M = 6 \times 10^{24} \, kg$,$G = 6.67 \times 10^{-11} \, Nm^{2}/kg^{2}$]

Three equal mass satellites $A, B,$ and $C$ are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are $L_A, L_B$,and $L_C$. Which of the following statements is correct?

$A$ planet is revolving around the sun in an elliptical orbit. Its closest distance from the sun is $r_{min}$,and the farthest distance from the sun is $r_{max}$. If the orbital angular velocity of the planet when it is nearest to the sun is $\omega$,then the orbital angular velocity at the point when it is at the farthest distance from the sun is: