Re: Bar magnets

["John S. Denker" <jsd@MONMOUTH.COM>]

At 11:54 PM 2/16/01 -0500, I wrote a bunch of nonsense.

What I SHOULD have said is something like this:

1) First, consider the energy point of view.

1a) The energy of the spring is proportional to the square of the
extension.  Therefore this system is isomorphic to the potential energy of
a marble rolling in a parabolic potential.  The parabola is concave upwards.

1b) The energy of the magnets goes roughly like 1/x, where x is the
separation.  If the divergence bothers you, write it as 1/(x + epsilon) but
the epsilon is so small it doesn't affect any of the results.  This energy
profile is concave downwards.

2c) Now, considering the combined system:

For strong springs, large separations, and/or weak magnets, the energy
profile of the combined system looks roughly like this:

.   \
.    \          _
.     \       /   \                     (case 1)
.       \ _ /      \
.                   \

and marble can sit in the bottom of the well, in stable equilibrium.  There
is also an unstable equilibrium on top of the hump.

For floppier springs, smaller separations, and/or stronger magnets, the
profile becomes:

.   \
.    \
.     \       _                 (case 2)
.       \ _ /   \
.                \


And if we continue to yet-floppier springs, smaller separations, and/or
stronger magnets, the profile becomes:

.   \
.    \
.     \                                 (case 3)
.       \ _
.           \

... which describes a set of conditions (a set of measure zero) where there
is an equilibrium with zero stability.

And if we continue to even-yet-floppier springs, smaller separations,
and/or stronger magnets,

.   \
.    \
.     \                                 (case 4)
.        \
.           \
.            \


in which case there is no equilibrium whatsoever.

================

2) Now let's consider the force point of view.

2a) The force profile of the spring is a straight line.  For
positive  extensions the force tends to reduce the extension, so the line
has negative slope.

2b) The force profile of the magnets is essentially 1/r^2.  In the geometry
in question, it tends to increase the extension of the spring, so the curve
lies in positive territory and has positive slope.

2c) For the combined system, the equilibrium condition requires that the
two force contributions balance, i.e. they add to zero.  That means that
one must equal the negative of the other.  So let's negate one of the
curves, and see where the curves cross.

We choose to negate the spring curve.

In the following figure, the straight line represents the spring, and the
curvy line represents the magnets.  The places where they cross are
equilibrium points.

.                  |/
.                 /|
.               /  |
.             /   /
.           /    /
.    _ _ _/ _  /
.       /                increasing extension of spring ------->
.     /                  == decreasing separation of magnets

This corresponds to energy profile case 1 above.

The leftmost (large-separation) crossing is a _stable_ equilibrium.  That's
because the curve that we inverted has the larger slope;  if you consider
the relationship of the force curves to the energy curves you find that
this corresponds to upward curvature of the energy profile;  the curvature
of the stable (upward-facing) parabola dominates the curvature of the other
curve.

The other crossing corresponds to the unstable equilibrium.

Now, suppose we move the magnets closer, holding constant the other
parameters (magnet strength, spring constant, et cetera).

.                  |         /
.                  |       /
.                  |     /
.                 /    /
.                /   /
.    _ _ _  _  /   /
.                /       increasing extension of spring ------->
.              /         == decreasing separation of magnets


In this case, there is no equilibrium.  The curves don't cross.

This corresponds to energy profile case 4 above.

There is an intermediate case (not shown) where the force curves are just
tangent, which corresponds to energy profile case 3 above.


======================================================

At 11:54 PM 2/16/01 -0500, I foolishly wrote:

> There undoubtedly was equilibrium.

That's false.  In the troublesome case, there was no equilibrium.

> it was an unstable equilibrium.

Also false.  If an equilibrium exists, there exist two equilibria;  one
stable, and one unstable (except on a set of measure zero in parameter
space, where these two equilibria become degenerate, as shown in case 3 above).


> Constructive suggestions:
>   1) Try using a spring-scale with a shorter natural length... a length
> comparable to the gap between the magnets.

That's not quite right either.  You can use a shorter spring, and/or a
stiffer spring (perhaps made of multiple springs in parallel).  The thing
that matters is not the natural length, but the extension of the spring at
the operating point.

> See
>   http://www.monmouth.com/~jsd/how/htm/equilib.html
> for a definition of terms, and general discussion, with illustrations.

That's correct at least.