Two satellites of masses $m_1$ and $m_2$ $(m_1 > m_2)$ are revolving around the Earth in circular orbits of radii $r_1$ and $r_2$ $(r_1 > r_2)$ respectively. Which of the following statements is true regarding their speeds $v_1$ and $v_2$?

$A$ satellite of mass $1000 \ kg$ is launched to revolve around the earth in an orbit at a height of $270 \ km$ from the earth's surface. Kinetic energy of the satellite in this orbit is . . . . . . $\times 10^{10} \ J$. (Mass of earth $= 6 \times 10^{24} \ kg$,Radius of earth $= 6.4 \times 10^6 \ m$,Gravitational constant $= 6.67 \times 10^{-11} \ Nm^2 \ kg^{-2}$)

Two point masses of mass $m_1 = fM$ and $m_2 = (1 - f)M$ $(f < 1)$ are in outer space (far from the gravitational influence of other objects) at a distance $R$ from each other. They move in circular orbits about their centre of mass with angular velocities $\omega_1$ for $m_1$ and $\omega_2$ for $m_2$. In that case:

The ratio of kinetic energy of a planet at perigee and apogee during its motion around the sun in an elliptical orbit of eccentricity $e$ is ..........

If the Earth is considered a point mass of $6 \times 10^{24} \ kg$ revolving around the Sun at a distance of $1.5 \times 10^8 \ km$ in a period of $T = 3.14 \times 10^7 \ s$,then the angular momentum of the Earth around the Sun will be:

$A$ double star is a system of two stars rotating about their centre of mass only under their mutual gravitational attraction. Let the stars have masses $m$ and $2m$ and their separation be $l$. Their time period of rotation about their centre of mass will be proportional to: