Re: definition of weight

["John S. Denker" <jsd@MONMOUTH.COM>]

At 03:20 PM 8/24/01 -0400, Carl E. Mungan wrote (off-list):
> ... Eq. (2) ... intended to be for the ideal case, while Eq. (4) was
> supposed to be for the real case

Psychological / pedagogical remark:  That seems to be the place where a lot
of students (and a lot of textbook writers) get stuck on the horns of a
dilemma:
  -- For "simplicity" they want (g) to be whatever force a unit mass
produces, as measured on a laboratory spring-scale, and then
  -- for "simplicity" they want (g) to be given by the mass and radius of
the earth, according to the law of universal gravitation.

Alas you can't have it both ways.  Close but not quite.

In a lecture on universal gravitation, there are seemingly-excellent
reasons for doing it one way, while in a lecture on scales and pulleys
etc., there are seemingly-excellent reasons for doing it the other
way.  Sigh.  The concepts are clear enough, but there is a lot of
inconsistent terminology running around.

Possibly constructive suggestion:  subscripts.
 -- Let (g_eff) refer to the acceleration of the lab frame relative to a
nearby freely-falling frame, no matter what causes the acceleration.  Throw
in a minus sign to agree with the conventional (downward) orientation of
(g).  Note that the spelling of "g" does !not! imply that only
gravitational contributions are allowed.
 -- Let (g_e_ideal) be the gravitational contribution from the ideal
isolated airless nonrotating homogeneous spherical earth.

For an introductory class, it should suffice to say
        "This classroom has g_eff = 9.8 m/s/s"
and leave it at that.  Freely-falling objects will accelerate relative to
the lab frame at g_eff, and objects constrained to be comoving with the lab
frame will exhibit a weight proportional to mass.  Understanding in detail
the origin of g_eff is of only secondary importance.

=================

For those who insist on examining the origin of g_eff, here is a possible
way to organize the ideas: Pretend we are calculating _ab initio_ what
force our spring-scale will attribute to various objects.  Tabulate various
contributions:
 1) Start with the g_e_ideal, the contribution from the ideal isolated
airless nonrotating homogeneous spherical earth, with some nominal mass and
radius.  Calculate the gravitational acceleration to 3 or 4 sig digs.
 2) Add in a correction for the fact that the earth's polar radius is
0.33% less than its equatorial radius.  Estimate the effect on polar and
equatorial spring-scale observations to an appropriate level of accuracy.
 3) Add in the centrifugal acceleration at sea level at the equator.
 4) Explain the "coincidence" between item (2) and item (3).  Make the
obvious generalization.
 5) Estimate the effect of buoyancy in air, as applied to weighing (with
the aforementioned spring scale) a kilogram of feathers and a kilogram of lead.
 6) Estimate the gravitational effect of the atmosphere.  Estimate how
this varies depending on weather conditions.
 7) Estimate the effect of relocating your lab from the bottom to the top
of the World Trade Center (i.e. moving from 15 to 430 meters above mean sea
level).
 8) Estimate the direct gravitational effect of the sun and the moon,
assuming the earth is rigid (no tides).
 9) Estimate the effect of the tides.
10) Estimate the effect of non-homogeneous distribution of heavy minerals, etc.
 *) Et cetera.

Note that all these items EXCEPT item (5) exhibit a (gravitational) weight
proportional to (inertial) mass.

==================

Also:  To give meaning to the equivalence principle, describe an
operational definition of mass.  To keep the logic non-circular, this must
be an inertial, non-gravitational definition.  Suggestion:  resistance of
an object to _horizontal_ acceleration when subjected to a known force
(generated perhaps by the spring scale).