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Feb19-11, 06:36 PM
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The attractive force towards the centre is proportional to the distance from the centre.
Easy to prove.
The there is no net force due to any of the Earth at a bigger radius than where you are so it's only the mass in the sphere beneath your feet that counts - that is proportional to the cube of the distance from the centre. The gravitational effect of that mass is proportional to one over the radius squared. Taken together, that gives the attractive force as proportional to the distance from the centre (cube/square).
You might notice that this force variation produces simple harmonic (oscillatory) motion and the period is the same for all amplitudes of oscillation. It's also the same period as for a Low Earth orbit.
It seems, btw, that the microgravity when you're falling through the Earth tends to compress objects whereas the microgravity when you're in orbit is tending to stretch things. Strange.