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
$\frac {mgR}{6}$
$\frac {mgR}{12}$
${mgR}$
${mgR}$
Suppose the gravitational force varies inversely as the $n^{th}$ power of distance. Then the time period of a planet in circular orbit of radius $R$ around the sun will be proportional to
The orbital velocity of an artificial satellite in a circular orbit very close to earth is $v$. The velocity of a geo-stationary satellite orbiting in circular orbit at an altitude of $3R$ from earth's surface will be
The dependence of acceleration due to gravity $g$ on the distance $r$ from the centre of the earth assumed to be a sphere of radius $R$ of uniform density is as shown figure below
The correct figure is
Figure shows the orbit of a planet $P$ round the sun $S.$ $AB$ and $CD$ are the minor and major axes of the ellipse.
If $t_1$ is the time taken by the planet to travel along $ACB$ and $t_2$ the time along $BDA,$ then
A small ball of mass $'m'$ is released at a height $'R'$ above the Earth surface, as shown in the figure. If the maximum depth of the ball to which it goes is $R/2$ inside the Earth through a narrow grove before coming to rest momentarily. The grove, contain an ideal spring of spring constant $K$ and natural length $R,$ the value of $K$ is ( $R$ is radius of Earth and $M$ mass of Earth)