$A$ simple pendulum performing small oscillations at a height $R$ above the Earth's surface has a time period of $T_1 = 4 \ s$. What would be its time period $T_2$ if it is brought to a point at a height $2R$ from the Earth's surface? Choose the correct relation ($R =$ radius of Earth).

The variation of acceleration due to gravity $g$ with distance $x$ from the centre of the Earth is best represented by ($R \rightarrow$ Radius of the Earth):

Assume that the Earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the Earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is $100 \, g$. The time period of the motion of the particle will be (approximately) (take $g = 10 \, m/s^2$, radius of Earth $R = 6400 \, km$):

The acceleration due to gravity on the planet $A$ is $9$ times the acceleration due to gravity on planet $B$. $A$ man jumps to a height of $2 \,m$ on the surface of $A$. What is the height of jump by the same person on the planet $B$?

If the Earth loses its gravity,then for a body:

What should be the angular speed of the Earth about its axis so that the acceleration due to gravity at the equator becomes zero?