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 ..........

The gravitational force acting on a particle,due to a solid sphere of uniform density and radius $R$,at a distance of $3 R$ from the centre of the sphere is $F_1$. $A$ spherical hole of radius $(R / 2)$ is now made in the sphere as shown in the figure. The sphere with hole now exerts a force $F_2$ on the same particle. The ratio of $F_1$ and $F_2$ is

Every planet revolves around the sun in an elliptical orbit. Which of the following statements are correct?
$A.$ The force acting on a planet is inversely proportional to the square of the distance from the sun.
$B.$ The force acting on the planet is inversely proportional to the product of the masses of the planet and the sun.
$C.$ The centripetal force acting on the planet is directed away from the sun.
$D.$ The square of the time period of revolution of a planet around the sun is directly proportional to the cube of the semi-major axis of the elliptical orbit.
Choose the correct answer from the options given below:

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

Two particles of equal mass $(m)$ each move in a circle of radius $(r)$ under the action of their mutual gravitational attraction. Find the speed of each particle.

$A$ 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].