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For me, the justification for dealing the 'apparent' weight is that it helps
us deal with a whole range of _perceived_ forces--internally experienced by
people. We tend to interpret many forces _backwards_ and if we keep weight
as the 'downward' pull of the earth (or other large body) then weight is
also one of these backwards forces.
The apparent weight is then the force
that we perceive (to be our weight). To be sure, in _most_ cases, this is
what the bathroom scale reads, but what the bathroom scale reads is an
UPWARDS force.
Don't tell me that isn't going to be confusing to students
for whom weight has been _defined_ as what the bathroom scale reads but for
whom the 'sensation' is downward. Apparent weight is an attempt to deal
with this discrepency--perhaps not successfully for some, but it works for
me. ;-)