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. Orisit a coincidence?

My attempt:

I derived the formula for the orbital period:

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

where $\displaystyle a$ is the the semi-major axis, which is $\displaystyle R$ 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?