The earth (mass $M = 6 \times 10^{24} \ kg$) revolves around the sun with an angular velocity $\omega = 2 \times 10^{-7} \ rad/s$ in a circular orbit of radius $R = 1.5 \times 10^8 \ km$. The force exerted by the sun on the earth in newtons is:

Earth revolves around the sun in a circular orbit of radius $R$. The angular momentum of the revolving Earth is directly proportional to:

$A$ 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: $M = 6 \times 10^{24} \ kg$,$R = 6400 \ km$,$T = 24 \ h$,$G = 6.67 \times 10^{-11} \ N \cdot m^2/kg^2$.

The planet Mars has two moons. If one of them has a period of $7\, \text{hours}, 30\, \text{minutes}$ and an orbital radius of $9.0 \times 10^{3}\, \text{km}$, find the mass of Mars. $\left\{\text{Given}: \frac{4 \pi^{2}}{G} = 6 \times 10^{11}\, \text{N}^{-1} \text{m}^{-2} \text{kg}^{2}\right\}$

If the gravitational constant $G$ is decreasing with time,what will remain unchanged for a satellite orbiting around the Earth?

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