|
| 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]
Not quite so fast:
Let's say I agree to this, (I'm inclined to do so)
What then should I call GMm/r^2, the force of . . .?
I think I either have to distinguish two gravity forces; the E type and
the N type much as you distinguish two g's
Or go to an awkard elocution similar to GMm/r^2 is that part of gravity
not due to W x W x v and W^2 r and W_dot and a_dot effects
Or ?? (suggestions)
| 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.
|
Important to distinguish whether by g you mean g_vec or |g|, but
regardless your not worrying about that in this post. (see John M's
definition)
| 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)
|
Agreed!
| 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.
Another very much life or death application are possible coriolis
effects any time you wish to lob an artillery shell a distance of many
miles and hit your target.
|
| 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.
|