A simple pendulum has a periodic time ' $\mathrm{T}_1$ ' when it is on the surface of earth of radius ' $R$ '. Its periodic time is ' $\mathrm{T}_2$ ' when it is taken to a height ' $R$ ' above the earth's surface. The value of $\frac{T_2}{T_1}$ is

The minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ in a circular orbit at an altitude of $2 R$ is

The density of a new planet is twice that of earth. The acceleration due to gravity at the surface of the planet is equal to that at the surface of earth. If $R$ is the radius of earth, then radius of the planet would be

The weights of an object are measured in a coal mine of depth ' $h_1$ ', then at sea level of height ' $h_2$ ' and lastly at the top of a mountain of height ' $h_3$ ' as $W_1, W_2$ and $W_3$ respectively. Which one of the following relation is correct? [h $h_1 \ll R, h_3 \gg h_2=R, R=$ radius of the earth ]