$A$ particle is moving in an elliptical orbit as shown in the figure. If $\vec{p}$, $\vec{L}$, and $\vec{r}$ denote the linear momentum, angular momentum, and position vector of the particle (from focus $O$) respectively at point $A$, then the direction of $\vec{\alpha} = \vec{p} \times \vec{L}$ is along

$A$ satellite is to be placed in an equatorial geostationary orbit around the Earth for communication.
$(a)$ Calculate the height of such a satellite.
$(b)$ Find out the minimum number of satellites that are needed to cover the entire Earth,so that at least one satellite is visible from any point on the equator.
Given: $M = 6 \times 10^{24} \ kg$,$R = 6400 \ km$,$T = 24 \ h$,$G = 6.67 \times 10^{-11} \ N \cdot m^2/kg^2$.

If the height of a satellite from the earth is negligible in comparison to the radius of the earth $R$,the orbital velocity of the satellite is

The time period of a $1500 \,kg$ satellite is equal to the time period of rotation of the earth. The altitude of the satellite is nearly

If $\rho$ is the density of the planet,the time period of a nearby satellite is given by

Define the periodic time (orbital period) of a satellite.