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But your _sensation_ of your weight IS the force UP.
At least you can maintain a Newtonian view if you always analyze
rotating systems from the stationary observer and then try to explain
the sensations of someone in that rotating system.
Otherwise, as far as I can tell, you lose the the third law since
there is no object acting on you outwards to be the agent of the
I think this may be why some people (I did when teaching) go for the
'apparent weight' approach.