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 ..........

$A$ rocket is fired vertically from the surface of Mars with a speed of $2 \; km/s$. If $20 \%$ of its initial kinetic energy is lost due to Martian atmospheric resistance,how far will the rocket go from the surface of Mars before returning to it? (Mass of Mars $= 6.4 \times 10^{23} \; kg$,radius of Mars $= 3395 \; km$,$G = 6.67 \times 10^{-11} \; N m^2 kg^{-2}$)

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

Match List-$I$ with List-$II$:

List-$I$	List-$II$
$(A)$ Kinetic energy of planet	$(1)$ $-\frac{GMm}{a}$
$(B)$ Gravitational potential energy of Sun-planet system	$(2)$ $\frac{GMm}{2a}$
$(C)$ Total mechanical energy of planet	$(3)$ $\frac{GM}{r}$
$(D)$ Escape energy at the surface of planet for unit mass object	$(4)$ $-\frac{GMm}{2a}$

(Where $a=$ radius of planet orbit,$r=$ radius of planet,$M=$ mass of Sun,$m=$ mass of planet)
Choose the correct answer from the options given below:

Two identical particles of mass $1 \, kg$ each go round a circle of radius $R$,under the action of their mutual gravitational attraction. The angular speed of each particle is:

The radius and mass of the Earth are both increased by $0.5\%$. Which of the following statements is true at the surface of the Earth?