$A$ spherical planet has a mass $M$ and diameter $D$. $A$ particle of mass $m$ falling freely near the surface of this planet will experience an acceleration due to gravity equal to:

Derive the equation for the variation of $g$ due to height from the surface of the Earth.

The maximum vertical distance through which a fully dressed astronaut can jump on the earth is $0.5\, m$. If the mean density of the moon is two-thirds that of the earth and the radius is one-quarter that of the earth,what is the maximum vertical distance through which he can jump on the moon and the ratio of the duration of the jump on the moon to that on the earth?

If the density of a small planet is the same as that of earth, while the radius of the planet is $0.2$ times that of the earth, the gravitational acceleration on the surface of that planet is (in $\,g$)

Find the magnitude of acceleration due to gravity at a height of $10 \, km$ from the surface of the earth.

The linear speed of a particle at the equator of the earth due to its spin motion is $V$. The linear speed of the particle at latitude $30^{\circ}$ is: