If the distance of Mercury from the Sun is $0.4$ times the distance of the Earth from the Sun,find the solar constant of Mercury in $cal/min \cdot cm^2$. The solar constant of the Earth is $2 \, cal/min \cdot cm^2$.

Which one of the following plots represents the variation of gravitational field on a particle with distance $r$ due to a thin spherical shell of radius $R$? ($r$ is measured from the centre of the spherical shell)

Two bodies of equal masses are at some distance apart. If $20 \%$ of the mass is transferred from the first body to the second body,then the gravitational force between them

Two particles of equal mass $m$ move in a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle with respect to their center of mass is

When a satellite in a circular orbit around the earth enters the atmospheric region,it encounters small air resistance to its motion. Then

$A$ small ball of mass $m$ is released at a height $R$ above the earth's surface,as shown in the figure. The ball enters a narrow groove and reaches a maximum depth of $R/2$ inside the earth before coming to rest momentarily. The groove contains an ideal spring of spring constant $K$ and natural length $R$. Find the value of $K$ if $R$ is the radius of the earth and $M$ is the mass of the earth.