An earth satellite of mass $m$ revolves in a circular orbit at a height $h$ from the surface of the earth. $R$ is the radius of the earth and $g$ is acceleration due to gravity at the surface of the earth. The velocity of the satellite in the orbit is given by

Consider a star of mass $m_2 \ kg$ revolving in a circular orbit around another star of mass $m_1 \ kg$ with $m_1 \gg m_2$. The heavier star slowly acquires mass from the lighter star at a constant rate of $\gamma \ kg/s$. In this transfer process,there is no other loss of mass. If the separation between the centers of the stars is $r$,then its relative rate of change $\frac{1}{r} \frac{dr}{dt} \ (\text{in } s^{-1})$ is given by:

Suppose the gravitational force varies inversely as the $n^{th}$ power of distance. Then the time period of a planet in a circular orbit of radius $R$ around the sun will be proportional to

An artificial satellite is moving around the Earth in a circular orbit with a speed equal to one-fourth the escape speed of a body from the surface of the Earth. The height of the satellite above the Earth's surface is ............ ($R$ is the radius of the Earth). (in $R$)

Two satellites of earth,$S_{1}$ and $S_{2}$,are moving in the same orbit. The mass of $S_{1}$ is four times the mass of $S_{2}$. Which one of the following statements is true?

$A$ planet of mass $M$ has a satellite of mass $m$,revolving around the planet in a circular orbit of radius $r$ and time period $T$. The mass $(M)$ of the planet is