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1. Why do you have one spring attached to a stage, the other two
attached to posts attached to the table?
That appears to be an
essential asymmetry in your setup.
I had understood that all forces
were measured with respect to the lab frame, therefore we want all
springs attached to fixed posts in the lab frame.
2. .... The particle is subjected to _multiple_ forces. We can
measure them. The particle experiences _one_ acceleration, we can
measure it.
......... But notice
here neither F nor m need be considered causative, we simply have a
correlation between them.
We can still add the forces as vectors in
special relativity, but we can't add relative velocities with our
non-relativistic formulas and although I haven't done it I would think
that relative accelerations would be even worse. Does this mean force
is more "basic" in some way than acceleration?