I'm working on a problem on relating gravitation and simple harmonic motion. The idea is that a mass dropped in a hole drilled through the earth will oscillate (no friction, etc).(adsbygoogle = window.adsbygoogle || []).push({});

The question asks this:

"Show that the motion of the mass is simple harmonic motion and find a formula for r(t)"

So, I'm starting with the basic r(t) equation for simple harmonic motion, r(t) = Acos(wt) where A is the total radius of the earth (the amplitude), t is the time and w is the angular frequency. My problem is that any attempt to find an angular frequency seems to involve dictating the distance from the center, r, which is what I'm trying to find as a function of time.

The best I've been able to come up with is w = sqrt(k/m) where k is the "gravitational spring constant" involving (4/3)pi * p * R * G, R being the radius that I'm trying to find.

How should I find w in this case?

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# SHM and Gravitation

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