Gravitational field strength inside the Earth

Question

please could someone give me a relatively straight forward answer why the gravitational field strength follows a straight line (linear) relationship with distance from the center of the Earth but when you get to the surface it follows a 1/r^2 relationship?

Answer 1

Could someone give me a relatively straight forward answer why the gravitational field strength follows a straight line (linear) relationship with distance from the center of the Earth but when you get to the surface it follows a 1/r^2 relationship?

Basically, it doesn't.

A better model is that gravitation is more or less constant at 10 m/s² to halfway down toward the center of the Earth, and then drops linearly toward zero at the from that point inward.

An even better model is the Preliminary reference Earth model, Dziewonski and Anderson, "Preliminary reference Earth model," Physics of the Earth and Planetary Interiors 25.4 (1981): 297-356. This 30 year old model of the Earth's interior gives a good view of the interior of the Earth. The tabular data that go into the model are freely available at http://geophysics.ou.edu/solid_earth/prem.html.

A planet with a uniform density would exhibit a gravitational force that falls linearly toward zero at the center. The problem is that the Earth doesn't have anything close to a uniform density. It has a very dense iron/nickel core surrounded by a rather dense rock (the mantle), which in turn supports the relatively light crust.

Answer 2

In spherical symmetry, the gravitational force only depends on the mass inside of a sphere of radius r, in an image where you pretend it was concentrated in the center.

The gravitational force falls off as $\frac{1}{r^{2}}$, always. When you are inside the sphere, though (if you assume constant density) then the mass included increases by a factor of $r^{3}$, so you pick up more mass a factor of $r$ more quickly than the force falls off.