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When they rotate one mass moves down the other moves up so they balance and the energy of the system should remain constant.
In the vertical alignment the system weighs more which represents an increase in energy of the system.
I believe that they should weigh the same
You assume I have an error, interesting.
When you say center of gravity what do you mean? The center of the gravitational potential or the center of gravitational potential energy.
My question is, if you weighed the two systems would they weigh the same or not?
If not, would taking a relativistic approach predict that they should weigh the same?
With the weights vertical about the same fixed fulcrum point. The higher weight now weighs less but has more GPE. The lower weight now weighs more but has less GPE.