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... We choose a reference frame and then work accordingly.
I don't recommend we confuse students by switching reference frames
without telling them.
Again, this is very much like the issue of centrifugal force. It
exists in one frame of reference but not in another.
For that matter, we could really confuse them by telling them that
when they move in a circle, they have a horizontal weight that
depends on how fast they're rotating. After all, we can put a scale
on the wall of a rotating cylinder.
you only make measurements in the frame you are in.
So let me add my voice to the hopeless cause of getting the ambiguity
out of our understanding of gravity and weight.