$A$ tunnel is dug along the diameter of the Earth of mass $M$. $A$ mass $m$ is released at a distance $X$ from the center,and the time period of oscillation is $T$. If a mass $4m$ is released from the same point,what will be the time period of oscillation?

$A$ small point mass $m$ is placed at a distance $2R$ from the centre $O$ of a big uniform solid sphere of mass $M$ and radius $R$. The gravitational force on $m$ due to $M$ is $F_1$. $A$ spherical part of radius $R/3$ is removed from the big sphere as shown in the figure and the gravitational force on $m$ due to the remaining part of $M$ is found to be $F_2$. The value of the ratio $F_1: F_2$ is

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 ..........

$A$ person whose mass is $100\, {kg}$ travels from Earth to Mars in a spaceship. Neglect all other objects in the sky and take acceleration due to gravity on the surface of the Earth and Mars as $10\, {m/s^2}$ and $4\, {m/s^2}$ respectively. Identify from the below figures,the curve that fits best for the weight of the passenger as a function of time.

$A$ satellite of mass $m$ is at a distance $a$ from a star of mass $M$. The speed of the satellite is $u$. Suppose the law of universal gravity is $F = -G \frac{Mm}{r^{2.1}}$ instead of $F = -G \frac{Mm}{r^2}$. Find the speed of the satellite when it is at a distance $b$ from the star.

The orbital velocity of a body near the surface of a planet $A$ is equal to the escape velocity of a body from the planet $B$. If the masses of planets $A$ and $B$ are the same,the ratio of their radii is