$A$ planet has a core of density $3\rho$ and an outer crust of density $\rho$. There is a small tunnel through the core. $A$ small particle of mass $m$ is released from end $A$. What is the time required to reach end $B$?

$A$ body projected vertically from the earth reaches a height equal to earth's radius before returning to the earth. The power exerted by the gravitational force is greatest

$A$ spiral galaxy can be approximated as an infinitesimally thin disc of uniform surface mass density located at $z=0$. Two stars $A$ and $B$ start from rest from heights $2z_{0}$ and $z_{0}$ (where $z_{0} \ll$ radial extent of the disc),respectively,and fall towards the disc,cross over to the other side,and execute periodic oscillations. The ratio of the time periods of $A$ and $B$ is

$A$ planet is orbiting the sun in an elliptical orbit. Let $U$ denote the potential energy and $K$ denote the kinetic energy of the planet at an arbitrary point on the orbit. Choose the correct statement.

Two particles,each of mass $m$,are moving in a circle of radius $R$ under the influence of their mutual gravitational attraction. What is their speed?

$A$ small point mass $m$ is placed at a distance $2R$ from the centre $O$ of a big uniform solid sphere of mass $M$ and radius $R$. The gravitational force on $m$ due to $M$ is $F_1$. $A$ spherical part of radius $R/3$ is removed from the big sphere as shown in the figure and the gravitational force on $m$ due to the remaining part of $M$ is found to be $F_2$. The value of the ratio $F_1: F_2$ is