$A$ 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 $B$?

Let $\omega$ be the angular velocity of the earth's rotation about its axis. Assume that the acceleration due to gravity on the earth's surface has the same value at the equator and the poles. An object weighed at the equator gives the same reading as a reading taken at a depth $d$ below the earth's surface at a pole $(d << R)$. The value of $d$ is

$A$ spacecraft travels in a straight line from the Earth to the Moon. How will its weight change during this journey?

$A$ spacecraft which is moving with a speed $u$ relative to the earth in the $x$-direction,enters the gravitational field of a much more massive planet which is moving with a speed $3u$ in the negative $x$-direction. The spacecraft exits following the trajectory as shown below. The speed of the spacecraft with respect to the earth a long time after it has escaped the planet's gravity is given by

Two particles of equal mass $m$ each move in a circle of radius $r$ under the action of their mutual gravitational attraction. The time period of each particle is

$A$ mass $m$,travelling at speed $V_0$ in a straight line from far away,is deflected when it passes near a black hole of mass $M$ which is at a perpendicular distance $R$ from the original line of flight. $a$,the distance of closest approach between the mass and the black hole,is given by the relation: