The height at which the weight of a body becomes $\frac{1}{16}^{th}$ of its weight on the surface of the Earth (radius $R$) is: (in $R$)

The angular velocity of the Earth at present is $\omega$. With what angular velocity should it rotate so that the weight of a body at the equator appears to be zero (in $\omega$)? (Express the answer as a multiple of $\omega$)

The weights of an object are measured in a coal mine of depth $h_1$,then at sea level of height $h_2=0$,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 relations is correct? [$h_1 \ll R, h_3 \ll R, R=$ radius of the earth]

Give the value of acceleration due to gravity at height $12\, km$ from the surface of earth. (in $, m/s^2$)

Assertion $(A)$: $A$ particle of mass $m$ dropped into a hole made along the diameter of the Earth from one end to the other possesses simple harmonic motion.
Reason $(R)$: Gravitational force between any two particles is inversely proportional to the square of the distance between them.

Which of the following statements are true about the acceleration due to gravity of the Earth (where $r$ is the distance from the center of the Earth and $R$ is the radius of the Earth)?
$(a)$ $g$ decreases when moving away from the center, if $r > R$.
$(b)$ $g$ decreases when moving away from the center, if $r < R$.
$(c)$ $g$ is zero at the center of the Earth.
$(d)$ $g$ at the equator decreases, if the Earth stops rotating about its axis.