system
Given the following post, respond as if you're commenting on Reddit:
human
If it were possible to put a pipe straight through the earth, from north to south pole and you dropped a ball down the pipe what would happen?
gpt
Ignoring air resistance and assuming the earth is a homogenous sphere,

The force F acting on the ball while within the earth is described by:

**F** = -GMm/R^(3)**r** where r is the distance from the centre of the earth to the ball. 

For those that have studied springs, this equation is of the form:

**F** = -k**r** where k = GMm/R^(3)

Or equivalently:

**a** = -ω^(2)**r** where ω^2 = k/m = GM/R^3

ω is the angular frequency of such a system, and is also given by ω = 2π/T where T is the period of oscillation. 

Given those equations, we can calculate T to be:

T = 2π(R^(3)/GM)^(1/2) ~= 84 minutes. 

That's the time it takes to fall right through the earth and come back to its starting position. Coincidentally, that's also the time it takes to make a full orbit at the surface of the earth (it's not exactly a coincidence, since simple harmonic motion along one axis is simply circular motion projected onto that axis).

Since there is no air resistance, the ball will fall right through to the other side, reverse direction, then fall back to its starting point indefinitely in simple harmonic motion. 

If there is air resistance, the amplitude of this oscillation will decay exponentially and eventually the ball will come to a rest at the centre of the earth.