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I want that anyone do a very simple experiment:
Put the base of the levitron in a scales. Then put the levitron in the base and then the levitron floats.
I want to know if the weight that you can read in the scales is higher when you push down the levitron.
Don't have one to test with, but I would say that the scale will measure more weight if you push down on the top because you will be increasing the pressure on the system.
Sure the weight will increase! You don't need a levitron for that. Just take two opposing magnets...