Two stars of masses $m$ and $2m$ at a distance $d$ rotate about their common centre of mass in free space. The period of revolution is

$A$ satellite of mass $m$ is revolving close to the surface of a planet of density $d$ with time period $T$. The value of the universal gravitational constant $G$ in terms of $d$ and $T$ is given by:

Which of the following quantities remains constant for a satellite in an orbit?

What is the ratio of the time periods of a satellite $A$ of mass $m$ at a distance $r$ from the center of the Earth and a satellite $B$ of mass $2m$ at a distance $2r$ from the center of the Earth?

$A$ particle of mass $m$ is under the influence of the gravitational field of a body of mass $M (\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then,the particle is subjected to an additional central force,corresponding to the potential energy $V_c(r) = m \alpha / r^3$,where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_c(r)$,but with a new time period $T_1$,then $(T_1^2 - T_0^2) / T_1^2$ is given by [ $G$ is the gravitational constant.]

The motion of planets in the solar system is an example of the conservation of