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One thing I want to point
out about my definition that I didn't emphasize: It is simpler
than Hecht's. Since it is correct (in that no cases are left
untreated or incorrectly treated), why is it not better than
Hecht's? Is not simpler better?
With my definition of weight the water is of no significance.
It constitutes a source of force in contact with the mass.
An object floating freely in water (no elevator to complicate
matters) is not weightless*. Are you intending to ask me the
question "What is the reading on the bathroom scale?"?
With my definition the weight in the elevator not accelerating
(not "at rest"; there is an important difference, as you will
recognize if you think carefully about it) is 1000 N. The
weight in the elevator accelerating upward is 1500 N, and the
weight in the elevator accelerating downward is 500 N.
Now let me answer the questions you probably meant to ask. The
scale will read the difference between the mass's weight and
the bouyant force. In case 1 the bouyant force is the weight
of the displaced water, or 200 N. In case 2 it is 300 N, and
in case 3 it is 100 N. Thust the scale readings will be: