Two point masses of mass $4m$ and $m$ respectively,separated by a distance $d$,are revolving under their mutual force of attraction. The ratio of their kinetic energies is:

$A$ 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.

$A$ body of mass $m$ is moving in a circular orbit of radius $R$ about a planet of mass $M$. At some instant,it splits into two equal masses. The first mass moves in a circular orbit of radius $\frac{R}{2}$,and the other mass moves in a circular orbit of radius $\frac{3R}{2}$. The difference between the final and initial total energies is

$A$ particle of mass $m$ is dropped from a height $R$ equal to the radius of the earth above the tunnel dug through the earth as shown in the figure. Hence the correct statement is

The solar constant on the surface of the earth is $S$. What will be its value on the surface of another planet which is about $5.3 \, AU$ away from the sun?

Four identical particles of mass $M$ are located at the corners of a square of side $a$. What should be their speed if each of them revolves under the influence of the others' gravitational field in a circular orbit circumscribing the square?