Hole drilled through center of the earth

Suppose that a small hole is drilled straight through the center of the earth, thus connecting two antipodal points on its surface. Let a a particle of mass m be dropped at time t = 0 into this hole with initial speed zero. Find the period of the simple harmonic motion exhibited by the particle.

Look up (or derive) the period of a satellite that just skims the surface of the earth; compare with the previous result. How do you explain the coincidence. Or is it a coincidence?

My attempt:

I derived the formula for the orbital period:

$T=2\pi\sqrt{\frac{a^3}{GM}}$

where $a$ is the the semi-major axis, which is $R$ in this case.

But a path that goes through the center of the earth isn't an orbit, in the usual sense of the word. However, I was able to derive independently that the period in this case has the same formula. So, is that a coincidence?

Think about what happens to (say) the x co-ordinate of a body undergoing uniform circular motion over 1 complete circle.

But isn't it a coincidence that the motion of the particle through the hole mirrors the projection of the motion of the satellite along that dimension?

..

Note that if the density of the earth isn't uniform (and in real life it is very definitely NOT uniform), or if gravity didn't follow an inverse square law, then the motion of the object falling through the hole would not be simple harmonic.

If you have uniform density, and gravity follows an inverse square law, *then* it follows that the two periods must be the same.

I think whether you consider that a 'coincidence' is somewhat of a philosophical question.

But isn't it a coincidence that the motion of the particle through the hole mirrors the projection of the motion of the satellite along that dimension?

You could argue that its motion is an orbit, just with no lateral component. You can get extremely elliptical orbits which do little more than oscillate back and forth, so why not take it to an extreme and flatten that ellipse into a straight line?