Thanks to John Denker's careful explanations, I *think* I now understand
something that I didn't before. I hope John and Leigh, in particular,
will take a look at the following and see if they think I got it right.
The breakthrough--if it was that--required understanding that there are
*three* proposed definitions of weight being tossed around here--something
I think John tried to get me to see earlier, but which went over my head
at the time.
Here is what I think the three "camps" are saying:
Camp 1: (As exemplified, I think, by Robert Carlson, Jerome Epstein,
Michael Edmiston, and "the textbooks.")
Weight is what you get when you integrate the law of universal
gravitation over all matter in the universe
(or, loosely, "weight is GM/R^2")
Camp 2: (As exemplified, I think, by John Denker)
Weight is what you get when you multiply the mass of an object
by the acceleration of a freely falling object at the same
location *as measured within some chosen reference frame*
(or, loosely, "weight depends on the reference frame.")
Camp 3: (As exemplified by me and, I think, Leigh, but maybe Leigh really
agrees with John Denker. I might even come around myself.)
Weight is what you get when you multiply the mass of an object
by the acceleration of a freely falling object at the same
location *relative to the object*
(or, loosely, "weight is what a scale reads.")
Then there is the question of whether or not weight is a vector, which I
think is not central and will simply leave alone.
Finally, there is the question of what we mean by "the gravitational
force." I'll give my *preference*, but I don't want it to become the
central focus of this message. (Please understand that I am *not*, in
this message, arguing for any particular position. I am simply trying to
make it clear what those positions *are*.) In the company of introductory
students, I would say that "the gravitational force" is what camp 1 calls
"weight." In the company of other physicists, I would say it is what camp
2 calls "weight."
Perhaps the best way to illustrate my understanding of the three different
definitions is to show how they work out in a specific case. I offer the
following:
Consider a Newtonian universe in which the only significant matter is an
earthlike planet of radius R and spherically symmetric mass distribution M
that results in GM/R^2 = 10 N/kg.
Let the planet rotate in such a way that the magnitude of the centripetal
acceleration of observer A located at a point on the equator is 5.0 m/s^2.
Let observer B be riding in an elevator located adjacent to observer A and
let the elevator accelerate upward at 10 m/s^2 relative to observer A.
Let observer B be weighing a book of mass m = 1.0 kg on a properly
functioning, force measuring scale that is placed on the floor of the
elevator.
Let observer C be nonrotating and nonaccelerating relative to the center
of the planet.
Let observer D be on the equator 1/4 of the way around the planet from the
location of A and B.
Here's what I think the various camps would say about the weight of the
book.
Observer Camp 1 Camp 2 Camp 3
A 10 N 5 N 15 N
B 10 N 15 N 15 N
C 10 N 10 N 15 N
D 10 N 125^(1/2) N 15 N