Period of oscillation of magnet levitated over another magnet

Question

The situation is similar to what we used to do as kids, take a vertical wood dowel, with a ring magnet placed at the bottom, and another ring magnet opposing it, floating on top.

More precisely, it would be a round or square magnet, levitated over a similar magnet fixed in place on the bottom. The levitated magnet would have horizontal and rotational motion constrained, so that only vertical motion is allowed.

The Levitated magnet is then hit so that it oscillates vertically. The levitated magnet also can have a weight placed on it.

I'm 30 years past my last college physics class and lack the background to approach this problem analytically. I've tried several experiments to get a handle on it from a practical standpoint, but can't explain what I'm seeing.

I guessed that adding more weight on top of the levitating magnet would decrease the period. But it appears to have little or no effect. (I suspect what's happening here is that the magnets get pushed closer together, and that somehow offsets the increase in weight)

Answer 1

You are kind of on the right path here. The forces between magnetic poles are not linear but are increasing roughly with a Coulomb kind of law: https://en.wikipedia.org/wiki/Force_between_magnets.

If we linearize such a force of the form

$F(r) = {{F_0r_0^2}\over {r^2}}$

around an equilibrium position $r_0$, then we get

$F(r_0+dr) \approx F_0({{r_0^2} \over {r_0^2}} - 2{{r_0^2dr}\over {r_0^3}})=F_0(1 - 2{{dr}\over {r_0}}$),

i.e. the first order restoring force term $-2F_0{{dr}\over {r_0}}$ gets stronger as the equilibrium distance decreased due to adding the weight. The increase in "spring constant" compensates for the increase in mass and the frequency of small oscillations stays roughly constant.