$A$ 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?

Consider a spherical gaseous cloud of mass density $\rho(r)$ in free space,where $r$ is the radial distance from its center. The gaseous cloud is made of particles of equal mass $m$ moving in circular orbits about the common center with the same kinetic energy $K$. The force acting on the particles is their mutual gravitational force. If $\rho(r)$ is constant in time,the particle number density $n(r) = \rho(r) / m$ is:
[$G$ is the universal gravitational constant]

$A$ particle of mass $M$ is at a distance $a$ from the surface of a thin spherical shell of equal mass $M$ and radius $a$. Which of the following statements is correct regarding the gravitational field and potential inside the shell?

If the radius of the Earth becomes $n$ times its present value,without any change in its mass,then the duration of the day becomes:

Assume that a tunnel is dug along a chord of the earth,at a perpendicular distance $(R / 2)$ from the earth's centre,where '$R$' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel,it will execute a simple harmonic motion with a time period:

State whether the following statements are true or false:
$(a)$ The value of acceleration due to gravity increases as we go to a depth $d$ below the Earth's surface.
$(b)$ If the escape velocity for an object of mass $m$ on the Earth's surface is $11.2 \, km/s$,then the escape velocity for an object of mass $2m$ is $22.4 \, km/s$.
$(c)$ If the Earth's gravitational attraction suddenly disappears,the mass of an object becomes zero,but its weight remains the same.