Consider a light planet revolving around a massive star in a circular orbit of radius $r$ with time period $T$. If the gravitational force of attraction between the planet and the star is proportional to $r^{-7/2}$,then $T^2$ is proportional to:

Obtain an equation for the orbital time period of a satellite revolving around the Earth.

Using the data given below,find the height at which a communication satellite can reside. $(G=6.67 \times 10^{-11} \text{ N-m}^2 \text{ kg}^{-2}, M=5.98 \times 10^{24} \text{ kg}, R=6.4 \times 10^6 \text{ m})$ (in $\text{ km}$)

The period of moon's rotation around the earth is nearly $29$ days. If moon's mass were $2$ fold, its present value and all other things remained unchanged, the period of moon's rotation would be nearly ......... $days$.

Two satellites are launched simultaneously into orbits of radius $R$ and $4R$. At an instant,the two satellites are on the same radial line. The time after which they have maximum separation is (mass of Earth = $M_e$). Assume the same sense of rotation.

$A$ geostationary satellite is orbiting around an arbitrary planet $P$ at a height of $11R$ above the surface of $P$,where $R$ is the radius of $P$. The time period of another satellite in hours at a height of $2R$ from the surface of $P$ is $........$. The planet $P$ has a rotation period of $24\, \text{hours}$.