1) The hard part of the recent discussion revolves around the
definition of g. There are disagreements about that.
By the way, there seems to be agreement that
force of gravity = m g [1a]
and some mild difference of opinion as to whether
weight = m g [1b]
In any case, I'm not gonna worry about [1b] right now.
As for g, here are two or three candidates on the table: the
Newtonian g
- G M
g_N = -------- r_vector [2]
r^3
and the Einsteinian g,
g_E = g_N + all other contributions to the [3]
acceleration of the reference frame
(relative to a freely-falling test particle)
Also: A third notion of g, namely the "standard" acceleration of
gravity, g_n (not to be confused with g_N) is not directly relevant
here. It can be considered a canonical value that approximates
the average value of |g_E|. http://physics.nist.gov/cgi-bin/cuu/Value?gn
Note: We need not assume that the equivalence principle holds
axiomatically, but we must agree that all evidence indicates that
it holds to a high degree of approximation: to one part in 10^12
or better. http://www.mazepath.com/uncleal/eotvos.htm (table V)
2) Recognize that in practical applications, when people talk about
gravity, they almost always mean Einsteinian gravity, not Newtonian
gravity. Swimming pool design is just one down-to-earth application
among many. The water doesn't care about your opinion (or your
textbook's opinion) about how to "define" g; the water is going to
distribute itself according to g_E, not g_N. The water distribution
is observable physics, not a matter of opinion.
You can "define" g to be g_N if you dare, at the risk of being
misleading, or at best irrelevant, because g_N is not useful in
ordinary practical applications.
If you design a pool using g = g_N, one side of the pool will be too
high by a couple of inches, for a typical 35-foot pool in temperate
latitudes. This would look really terrible.
Another relevant application involves calculating the force of gravity
that acts on an object (and which a stationary object, in turn, impresses
on its support). In the terrestrial lab frame, this force will be m g_E
... not m g_N.
3) Having learned the lesson in item (2), you can decide that g
is merely /approximated/ by g_N. After all, you're teaching an
introductory class, and you should have poetic license to approximate
things, especially when the approximation is good to a fraction of
a percent and/or a fraction of a degree in ordinary terrestrial
applications.
That's fine ... /so long as you don't abuse the license!/
(Attempting to treat the approximation as a definition would be the
epitome of abuse.)
4) Recognize that for the same reason that m g_N misstates the weight
by a fraction of a percent in the terrestrial lab frame, it misstates
the weight by 100% in the frame of a freely orbiting spacecraft.
So we see that understanding the difference between an approximation
and a definition is crucial to understanding the weightlessness of
astronauts. You can /sometimes/ get away with approxmating g = g_N
in the terrestrial lab frame, but you can't get away with it in the
spacecraft frame.
You can make all these problems go away by defining weight to be
m g_E. This works correctly for the swimming pool, correctly for
the spring scale, and correctly for the weightless astronauts.
No muss, no fuss.
If you want to approximate (m g_E) by (m g_N), subject to appropriate
conditions, that's fine ... but please remember that the latter is an
approximation, not a definition.
Bottom line:
-- You are free to define "weight" and "gravity" however you like ...
but if you choose wacky definitions, it's a disservice to your students.
-- You are free to define "weight" and "gravity" however you like ...
but if you want your definition to be *consistent* with practical
applications, including universally-accepted weighing procedures
and universally-accepted notions of horizontal and vertical, then
your choices are more limited.