$A$ body situated on the surface of the earth becomes weightless at the equator when the rotational kinetic energy of the earth reaches a critical value '$K$'. The value of '$K$' is given by [$g$ = gravitational acceleration on earth's surface,$M$ = mass of the earth,and $R$ = radius of the earth].

Two spherical bodies of mass $M$ and $5M$ and radii $R$ and $2R$ are released in free space with initial separation between their centres equal to $12R.$ If they attract each other due to gravitational force only,then the distance covered by the smaller body before collision is (in $R$)

$(a)$ 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 $g = 9.8 \, m/s^2$?

Four particles,each of mass $M$ and equidistant from each other,move along a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle is

$A$ binary star system consists of two stars $A$ (mass $M_A = 2.2 M_S$) and $B$ (mass $M_B = 11 M_S$),where $M_S$ is the mass of the Sun. They are separated by a distance $d$ and rotate about their common centre of mass,which is stationary. What is the ratio of the angular momentum of star $A$ to the angular momentum of star $B$ about the centre of mass?

Particles of mass $100 \, g$ each are placed at the vertices of an equilateral triangle of side $20 \, cm$. How much work must be done to increase the separation between them to infinity?