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I think there is no unique solution for this. The experiment you cite can
have many results depending on how the forces are applied. With the three
men, my personal experience tells me that as we carry such a load, the
forces vary as we move along the ground. We can all apply different forces
and still have the beam in equilibrium.
I am thinking of your experiment. A beam resting on three scales. If a slip
of paper is placed at one contact point, I suspect the three readings will
readjust. If the three men are all of different height......
I'll be interested in other's responses
Ken fox