From: terry@asl.dl.nec.com
Subject: Re: Passive Magnetic Levitation (was: Science Fair Project)
Originator: terry@aslws01
Organization: (Speaking only for myself)
Date: Fri, 11 Dec 1992 17:03:37 GMT

In article <BCR.92Dec11013519@hfl3sn02.cern.ch>
bcr@cernapo.cern.ch (Bill Riemers) writes:

> In article <1992Dec10.221950.13548@asl.dl.nec.com>
> terry@asl.dl.nec.com writes:
>
> | Wow, strong magnets!  But I cannot see how that could possibly be right --
> | the [Bollinger] 3-pronger will either slide off in some fashion, or it
> | won't.  There isn't enough stored energy in the system to make it flip,
> | nor do the intense regions of the repulsive fields extend far enough out
> | to perform such a feat.
>
> Not true.  If there is enough force to levitate it then there must also be
> enough force to flip it.

Sure -- in the absence of gravity, my 3-pronger will eventually flip over.

However, when I said that there "isn't enough stored energy in the system to
make it flip," I was referring to the energy needed to raise it out of the
metastable state, not the net energy gain:

            Metastable   Energy Needed to     Stable
            Levitation    Rotate 3-Prong       Final
           Energy State  Around horiz axis     State
                                                       
                                --                     
                              -    -                   
                             -      -                  
                            -        -                 
                           -          -                
                          -             -              
                        -                -            
               -------                     -           
                                             -         
                                               ------  
                                 
     Peak_energy(theta) = f(arm_length) = ~ constant x arm_length

Would you agree that this diagram and the ballpark equation under it describe
the energetics of flipping my three-pronged device in gravity?  Note that if
you assume the (levitated) magnets to have constant mass and that materials
exist whereby stretching the length of the arms within plausible limits (say
up to a meter) adds no significant mass to the starting mass of the system,
you can *always* increase the flip-over stability of the system until the
likelihood of a flip-over around a horizontal axis becomes vanishingly small.

However, rotating the 3-prong device around a vertical axis (and/or moving
moving it in the horizontal plane) also produces a curve like the one above,
except that the "final" state just becomes removing the 3-prong and setting
it down on the table.  This is the potential energy curve that I would argue
is the relevant one for a 3-prong, because unlike the first curve there is
no obvious, easy way to increase the height of the potential energy peak
when going from the levitation state to the "flat on the table" state.

Thus again:  If it fails to work, it will be due to some kind of instability
that ultimately results in vertical rotation and/or translation in the
horizontal plane -- but not, I think, flip-over around a horizontal axis.
(Unless you can buid a quantum tunneling version!)

I might note that the main reason that I have never been able to figure out
any such instabilities is that the same sort of "extend the arm" arguments
I just used for flip-over also apply to the destabilizing effects of the
levitated magnets on each other.  If magnet A is close to magnet B and B
rises slightly, it can destabilize the magnet A by twisting it too far away
from horizontal.  Magnet A then becomes subject to the "torque instability"
I mentioned before, and lifts B through the addition of its new-found torque.
The consequences are apparent -- you've got a positive feedback look that
will not be satisfied until *everything* flips over.

But what if the arm between A and B is very, very long?  If B rises by the
same amount, the resulting angular motion of A will be greatly reduced,
and thus the torque instability (which for small angles is proportional
to the angle to which the magnet's axis has been twisted away from vertical)
will be greatly reduced, too.  Feed that back through the long leverage arm
between A and B and the return lift to B will be also be rather drastically
reduced.  You can get a general feel for what's going considering case of
what would happen if infinitely rigid, infinitely light arms were extended
to infinity; *any* twisting of *any* of the magnets would require infinite
energy to accomplish, no matter how small the angle involved.

But in all real cases there is still *some* of this twisting-and-feedback
effect, right?  Yes, but what I've had no luck at all if figuring out how
such greatly attenuated responses could destabilize the system.  The
problem is that once their impact falls to a level below the height of the
levitation-to-table energy barrier (the one provided ring magnets), they
no longe appear to be capable of "joining together" to make the overall
setup unstable.  If that is true, a real, bonafide passive "pizza box"
levitation arrangement should be attainable with the 3-prong.

Anyone notice any subtle instabilities, or errors in the above analysis?

(Dr. Carr, any comments?)

Any experiments yet?  It might make a nice paper if it actually works...

				Cheers,
				Terry Bollinger