Two satellites of equal mass are revolving around Earth in elliptical orbits of different semi-major axes. If their angular momenta about the Earth's center are in the ratio $3:4$,then the ratio of their areal velocities is ........

If an artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of the escape velocity from the earth,the height of the satellite above the surface of the earth is

$A$ geostationary satellite orbits around the earth in a circular orbit of radius $36000\, km$. Then,the time period of a satellite orbiting a few hundred kilometres above the earth's surface $(R_{\text{Earth}} = 6400\, km)$ will approximately be ....... $hours$.

$A$ test particle is moving in a circular orbit in the gravitational field produced by a mass density $\rho(r) = \frac{K}{r^2}$. Identify the correct relation between the radius $R$ of the particle's orbit and its period $T$.

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. . . . .

If a satellite has to orbit the earth in a circular path every $6 \text{ hrs}$, at what distance from the surface of the earth should the satellite be placed (in $\text{ km}$)? (Radius of earth $R_e = 6400 \text{ km}$)
(Assume $\frac{GM}{4\pi^2} = 8 \times 10^{12} \text{ m}^3\text{s}^{-2}$, where $G$ and $M$ are the gravitational constant and mass of earth, and $10^{1/3} = 2.1$)