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Re: little gee and its sign



At 05:33 PM 9/10/01 -0500, Herb Schulz wrote:

the calculated x-position of an object at a certain
moment in time might be +23.5m in one coordinate system while it is
-4.7m in another coordinate system;

OK. But neither of those can be the final answer in any
physically-relevant calculation.

the Physical Position is the same
in both cases (as you implied) but the Numbers used to Describe that
x-position depend upon the coordinate system chosen and you must know
that coordinate system for the numbers to have Physical Meaning.

IMHO those numbers do not, _per se_, have any Physical Meaning. They may
acquire physical meaning if there are some other numbers in the same
coordinate system to which they can be compared -- but then it is the
comparison, not the coordinate-dependent numbers, that has the Physical
Meaning.

In my world, any question that asks "what is the X-position" is not well
posed. The answer is manifestly non-invariant with respect to a change of
coordinates, and no physically-significant quantity can behave this way.

If somebody asks me
What is the Z-component of the base of the flagpole?
I will respond by saying something like
That can't possibly be what you really want to know.
If you tell me what the real question is, I'll try to answer it.
In contrast, a question like
What is the angle subtended by the flagpole, as seen
from the front door?
does not suffer from this problem; the answer is independent of the choice
of coordinate system.

This is not a trivial style point; this is an important physics principle,
on a par with dimensional analysis. You should train yourself, and your
students:

If you ever see a physics question where the answer would depend on the
choice of coordinate system, your hair should stand on end. Whoa!!! This
is a sure sign that the question is unphysical.

(Intermediate results in a calculation might be coordinate-dependent, but
later in the calculation the coordinate-dependence must drop out.)