A satellite is launched into a circular orbit of radius $R$ around earth, while a second satellite is launched into a circular orbit of radius $1.02\, {R}$. The percentage difference in the time periods of the two satellites is -

A rocket of mass $M$ is launched vertically from the surface of the earth with an initial speed $V.$ Assuming the radius of the earth to be $R$ and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is

The height at which the weight of a body becomes $1/16^{th}$, its weight on the surface of earth (radius $R$), is

Figure shows the variation of the gravitatioal acceleration $a_g$ of four planets with the radial distance $r$ from the centre of the planet for $r\geq $ radius of the planet. Plots $1$ and $2$ coincide for $r\geq R_2$ and plots $3$ and $4$ coincide for $r \geq  R_4$. The sequence of the planets in the descending order of their densities is 

A skylab of mass $m\,kg$ is first launched from the surface of the earth in a circular orbit of radius $2R$ (from the centre of the earth) and then it is shifted from this circular orbit to another circular orbit of radius $3R$ . The minimum energy required to shift the lab from first orbit to the second orbit are

Imagine a light planet revolving around a very massive star in a circular orbit of radius $R$ with a period of revolution $T$. If the gravitational force of attraction between the  planet and the star is proportional to $R^{-5/2}$, then,