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[Phys-l] Weightlessness in 4 easy steps




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.