# Gravitational attraction in different points in a uniformly dense sphere

Suppose we lived on the surface of a spherical planet, with uniform density. How would the gravitational force we experience change if we drilled a hole and descended towards the centre of the planet?

The magnitude of gravitational attraction between two objects is G=m1*m2/d^2. When considering any two points, this force is 0 (as a point has no mass). For one point in relation to the infinitude of other points in the sphere, there would be a gravitational force. The question is this: for a point with a distance x from the centre of the sphere, what is the sum of the gravitational attraction to all other points in the sphere - that is, what is the net force experienced.

Clearly, at the centre of the sphere (x=0) there is no net force experienced; while at the surface there is a force experienced (towards the centre of the sphere)

• There is an analogous question on the electric field inside a uniformly-charged ball. The simplest approach in that case is to use Gauss's law for electrostatics. If you can deduce the gravitational analogue, that can also be used here. Jun 23, 2016 at 11:52

As noted in a comment, the most straightforward approach is through Gauss's law. The force on a body outside a spherical shell is the same as if the shell were concentrated at the centre, and the force on a body inside a spherical shell vanishes. Thus, at a distance $x$ from the centre of a sphere of radius $r\ge x$ and mass $m_1$, a body of mass $m_2$ experiences a force
$$G\frac{\left(\frac{x^3}{r^3}m_1\right)m_2}{x^2}=Gm_1m_2\frac x{r^3}\;,$$