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?

In the following four periods:
$(i)$ Time of revolution of a satellite just above the earth's surface $({T_{st}})$
$(ii)$ Period of oscillation of mass inside the tunnel bored along the diameter of the earth $({T_{ma}})$
$(iii)$ Period of simple pendulum having a length equal to the earth's radius in a uniform field of $9.8 \; N/kg \; ({T_{sp}})$
$(iv)$ Period of an infinite length simple pendulum in the earth's real gravitational field $({T_{is}})$

$A$ uniform sphere $A$ with radius $R$ exerts a force $F$ on a small particle $B$ situated at a distance $2R$ from the centre of the sphere. $A$ spherical portion of diameter $R$ is cut from the sphere $A$ as shown in the figure. If $F^{\prime}$ is the new gravitational force between the remaining part of the sphere $A$ and the particle $B$,then find the correct relation between $F$ and $F^{\prime}$.

$A$ 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:

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 $2 \, cal/min \cdot cm^2$.

$(a)$ Earth can be thought of as a sphere of radius $6400 \, km$. Any object (or a person) is performing circular motion around the axis of the Earth due to the Earth's rotation (period $1 \, \text{day}$). What is the acceleration of an object on the surface of the Earth (at the equator) towards its centre? What is it at latitude $\theta$? How do these accelerations compare with $g = 9.8 \, m/s^2$?
$(b)$ The Earth also moves in a circular orbit around the Sun once every year with an orbital radius of $1.5 \times 10^{11} \, m$. What is the acceleration of the Earth (or any object on the surface of the Earth) towards the centre of the Sun? How does this acceleration compare with $g = 9.8 \, m/s^2$?