# Physics of donut magnets levitating vertically on a pencil?

When you put identical donut magnets on a pencil horizontally, the magnets all repel from each other in a way that creates equal spacing between each magnet. However, spacing is not equal when the pencil is held vertically and spacing increases from the bottom up so it looks like this:

I think that the weight of each magnet has something to do with this, but I cannot figure out exactly why. Help?

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Here's a related question: consider a chain (or any other rope whose linear mass density is non-negligible) suspended from the ceiling with lower end free. What is the tension on this chain as a function of the distance from the ceiling? –  josh Jun 7 at 3:50
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## 1 Answer

This is simply a matter of force balancing. The only forces on each magnet are gravity and the magnetic repulsion from the neighbors.

The top magnet must be repelled from below with a force equal in magnitude to the force of gravity on it.

The next magnet down has not only its own weight pushing down, but the weight of the one above it as well. This is an example of Newton's Third Law: If the second-from-the-top magnet is pushing on the top one, then the reverse must be true with an equal force. Thus this second magnet must be pushed up with twice the force required to levitate the top magnet. It reaches equilibrium closer to the repelling magnet below.

This trend continues, with the $n$-th magnet bearing about $n$ times the weight of one of them. The spacings decrease as a result. Note that the force of repulsion depends strongly on distance, so the space below the $n$-th magnet does not have to be $1/n$ the spacing below the first magnet for equilibrium to be achieved.

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Got it, thanks! –  anon-eye Jun 7 at 3:54
@anon-eye Good to know. BTW, a good practice on this site is to wait a day or two before accepting answers. There's always the chance someone will come along with an even better answer, or will point out a flaw in what I wrote. Having an accepted answer tends to indicate the thread is no longer active and so discourages further contributions. (Not that I don't appreciate the vote of confidence :) –  Chris White Jun 7 at 4:00
an interesting demonstration of what happens microscopically ( pressure decrease with height in water for example) –  anna v Jun 7 at 4:29
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