How does the acceleration due to gravity change with altitude?

Earth is assumed to be a sphere of radius $R$. If $g_{\phi}$ is the value of effective acceleration due to gravity at latitude $30^{\circ}$ and $g$ is the value at the equator,then the value of $|g - g_{\phi}|$ is ($\omega$ is the angular velocity of rotation of the Earth,$\cos 30^{\circ} = \frac{\sqrt{3}}{2}$).

Starting from the centre of the earth having radius $R$,the variation of $g$ (acceleration due to gravity) is shown by:

If the change in the value of $g$ at a height $h$ above the surface of the earth is the same as at a depth $x$ below it,then (both $x$ and $h$ being much smaller than the radius of the earth)

Calculate the difference in the value of $g$ at the equator and at the poles due to the rotation of the Earth.

If the radius of the Earth were to shrink by half,its mass remaining the same,what would be the weight of an object on its surface?