The density of a planet is $3$ times that of Earth and its radius is $2.5$ times that of Earth. If $g_p$ and $g_e$ represent the acceleration due to gravity on the planet and Earth respectively, then the ratio of $g_p$ to $g_e$ is:

At what height from the ground will the value of $g$ be the same as that in a $10 \, km$ deep mine below the surface of the Earth?

The weight of a body of mass $m$ decreases by $1\%$ when it is taken to a height $h$ above the surface of the Earth. What is the percentage decrease in its weight when it is taken to a depth $h$ below the surface of the Earth?

Obtain an expression for the variation in effective gravitational acceleration $g'$ with latitude due to the Earth's rotation.

The height $h$ from the surface of the earth at which the value of $g$ will be reduced by $64 \%$ from the value at the surface of the earth is ($R=$ radius of the earth).

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.