Imagine a narrow tunnel between two diametrically opposite points of the Earth. $A$ particle of mass $m$ is released in this tunnel. The time period of oscillation is ..........

Suppose the acceleration due to gravity at the earth’s surface is $g \text{ m/s}^2$ and at the surface of the moon it is $g' \text{ m/s}^2$. An $M \text{ kg}$ passenger travels from the earth to the moon in a spaceship moving with a constant velocity. Neglecting all other celestial objects, which curve best represents the net gravitational force on the passenger as a function of time?

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 ........... . $[G = 6.67 \times 10^{-11} \, SI \text{ units}]$

$A$ spherical hole of radius $R/2$ is excavated from an asteroid of mass $M$ and radius $R$,as shown in the figure. The gravitational acceleration at a point on the surface of the asteroid just above the excavation is:

$A$ 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)
$(B) \frac{P(r=3R/4)}{P(r=2R/3)} = \frac{63}{80}$
$(C) \frac{P(r=3R/5)}{P(r=2R/5)} = \frac{16}{21}$
$(D) \frac{P(r=R/2)}{P(r=R/3)} = \frac{20}{27}$

If two bodies $A$ and $B$ of equal masses $M$ are situated in air at a distance $d$ and the gravitational force between them is $F$. Now,$50 \%$ of the mass is transferred from body $A$ to $B$ and the distance between them is reduced to $\frac{d}{2}$. If the space around them is now filled with a liquid of specific density $3$,what will be the new gravitational force?