$A$ spherical body of radius $R$ consists of a fluid of constant density $\rho$ and is in equilibrium under its own gravity. If $P(r)$ is the pressure at a distance $r$ from the center $(r < R)$,then the correct option$(s)$ is(are):
$(A) P(r=0) = P_c$ (maximum pressure at center)
$(B) \frac{P(r=3R/4)}{P(r=2R/3)} = \frac{63}{80}$
$(C) \frac{P(r=3R/5)}{P(r=2R/5)} = \frac{16}{21}$
$(D) \frac{P(r=R/2)}{P(r=R/3)} = \frac{20}{27}$

Three particles of equal mass $m$ are situated at the vertices of an equilateral triangle of side $l$. They are moving in a circle under the influence of their mutual gravitational interaction. Then their time period of revolution is directly proportional to

Fill in the blanks:
$(a)$ The orbital period of Mars around the Sun is $8$ times the orbital period of Mercury. If the distance of Mercury from the Sun is $5.79 \times 10^{10} \, m$,then the distance of Mars from the Sun is approximately .......
$(b)$ If the mass of an object on Earth is $m \, kg$,then the mass of the same object on the Moon is ........... .
$(c)$ The height of a geostationary satellite from the Earth's surface is approximately ........ .
$(d)$ If the distance between two objects of mass $m_1 = m_2 = 1 \, kg$ is $1 \, mm$,then the magnitude of the gravitational force between them is ........... . $[G = 6.67 \times 10^{-11} \, SI \text{ units}]$

Three particles,each of mass $M$,are situated at the vertices of an equilateral triangle of side length $L$. The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original separation $L$. The initial speed that should be given to each particle is

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.

As shown in the figure,two spherical cavities are made in a uniform solid sphere of radius $R$. The boundaries of the cavities touch at the centre of the sphere. The centers of the cavities and the sphere lie on the $X$-axis. The mass of the solid sphere before the cavities were created was $M$. The gravitational force on a point mass $m$ at a distance $d$ away from the centre of the solid sphere is