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Re: Apparent weight



A. R. continues:
Well then, according to your weaker criterion, forces become
accelerations; why use the two distinct terms? Why teach our students
two distinct terms if they really always mean the same thing? How
would you distinguish the two terms pedagogically? I am really asking
this to gain enlightenment on how to teach these matters in a practical
way.

These are good and fair questions. To answer the first part, I prefer
to use the concept of forces as the mathematically functional 'causers'
of accelerations in any given frame. Regarding the substantive last
question, the way I see the situation is that the answer is related to a
frequent/usual reason why the postulated existence of any forces are
invoked in the first place. Usually the way mechanics is presented to
students is in the context of the presentation of Newton's laws. First,
we tend to emphasize to students via N1 that the natural condition for
influence-isolated objects to be in is a state of coasting. That is,
things tend to keep doing what they have been doing unless compelled to
do otherwise by some external agency. We then call forces those external
influence agencies that act to change the state of motion of objects.
The details of how these forces act to change the state of motion of
objects are given by N2. Thus, forces are those (hypothetically
postulated causal) entities that act locally, immediately,
proportionately, and directionally parallel to the rate of change of the
state of motion of the object. Thus in this presentation N1 and N2
take precedence over N3 in the underlying reason for and definition of
forces.

Things may be different in a presentation of mechanics that started with
physical mutual interactions and N3 (rather than N1 and N2) which
considered forces as the ontologically real physical differential means
by which pairs of physical objects influence each other in such a way as
to transfer momentum between them in such a way that its total remains
manifestly conserved. In the N2-before-N3 presentation senario forces
are things the act on individual objects and describe perturbative
influences on objects from their natural N1 (coasting) state. In the
N3-before-N2 presentation senario forces are things that always act
pairwise between objects. If we get our idea of forces from the
N2-before-N3 senario we are naturally led to the idea of extra
(frame-generated) forces appearing in accelerated frames when the time
comes for such frames to be considered. If we get our idea of forces
from the N3-priority-over-N1-&-N2 senario then we are led to your idea
that N1 and N2 merely fail in accelerated frames while N3 remains valid
by our prior commitment to the very nature of what a real force has to be.
In such a description we must suffer conceptually bizarre prospects
such as objects remaining at rest when net unbalanced forces are acting
on them (e.g. the previous example of a mass pulled on by a taut string
with no other forces acting on it remaining at rest) and other objects
accelerating in the absence of motivating forces.

When relativity is later considered both N2 & N3 are violated by the
principles of relativity. It seems that the violation of N2 for the
N1 & N2 priority senario is less pedagogically problematic than the
violation of N3 is for the N3 priority senario. This is because for
(special) relativity a modified N2 law can still be proposed which
accounts for the relativistic effects by merely changing the definition
of the momentum while keeping the idea that the force on an object is what
tends to change the object's momentum where we keep the equation
dp/dt = F. The violation of N3 in a relativistic world where influences
between separated objects propagate with a nonzero time delay completely
destroys any N3 picture that includes only the objects themselves. The
way that physicists handle the problem is to require that the mechanism
of influence propagation be included in the system's description in terms
of dynamical force fields that both locally influence and are influenced by
their sources. Now instead of dealing with a system of a few particle
degrees of freedom we are suddenly confronted with a system which includes
a continuously infinite number of field degrees of freedom as well--all of
whose behavior needs to be solved for by solving a set of partial
differential equations coupled to the ordinary DEqs that describe the
original objects' degrees of freedom. If we, instead, use the idea of a
force as something which acts on an individual object (rather than always
between dynamical objects) then we can treat the motion of a relativistic
particle by solving the (modified N2) eq. of motion for that particle which
is locally forced by a prespecified background force field whose dynamics
is no longer part of the problem to be solved for, but is part of the
problem specification. This is allowable since we do not have to require
all momentum exchanges to dynamically satisfy N3 locally in terms of
action/reaction force pairs throughout all of space. This, it would seem,
makes understanding and dealing with the motion of relativistic objects
conceptually and practically easier for the students.

Since I don't take the ontology of forces that seriously in the first
place, I tend to gravitate toward the functionalism of the N1 & N2
priority over N3 senario, where forces are just the *mathematically
functional* motivators of accelerations rather than actual necessarily
ontologically existing influences between pairs of pieces of matter and
between different spatially adjacent regions of interaction fields. IOW,
I don't necessarily see forces as actual causes of anything, just as
mathematically useful constructs that mathematically function as such
causes on individual objects.

David Bowman
dbowman@gtc.georgetown.ky.us