Relation between g and G

Using F = ma we know that the force on a mass m at a point where the gravitational intensity of a planet is g is mg. But this force is also given by F = GMm/r² where M is the mass of the planet and r is the distance from its centre (See Figure 1).

So F = mg = GMm/r² and therefore





If we think about the surface of the planet then
g = g_o and r = R and so:

Weight

What we call the weight of an object at the Earth's surface (mg_o) can also be expressed as GMm/R²



Density of a planet and its surface gravity
We know that the surface gravity (g_o) of a planet is given by the formula:

g_o= GM/R²

but the density of the planet (r) = M/[4/3pr³] and so