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

The minimum and maximum distances of a planet revolving around the Sun are $x_{1}$ and $x_{2}$. If the minimum speed of the planet on its trajectory is $v_{0}$,then its maximum speed will be:

If a satellite has to orbit the earth in a circular path every $6 \text{ hrs}$, at what distance from the surface of the earth should the satellite be placed (in $\text{ km}$)? (Radius of earth $R_e = 6400 \text{ km}$)
(Assume $\frac{GM}{4\pi^2} = 8 \times 10^{12} \text{ m}^3\text{s}^{-2}$, where $G$ and $M$ are the gravitational constant and mass of earth, and $10^{1/3} = 2.1$)

Two satellites $A$ and $B$ are revolving around the earth in orbits of heights $1.25 R_E$ and $19.25 R_E$ from the surface of the earth respectively,where $R_E$ is the radius of the earth. The ratio of the orbital speeds of the satellites $A$ and $B$ is (in $: 1$)

$A$ satellite is in a circular equatorial orbit of radius $r = 7000 \, km$ around the Earth. If it is transferred to a circular orbit of double the radius $(2r)$,then its angular momentum will:

If due to air drag,the orbital radius of a satellite decreases from $R$ to $R - \Delta R$,where $\Delta R << R$,the expression for the change in orbital velocity $\Delta v$ is (mass of Earth is $M$):