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Re: [Phys-l] bound vectors ... or not

I am not sure it is necessary to specify the "point of application" of a force vector. I think it may be sufficient to specify a point in space that lies on the "line of action" aka the "line of force" of the force. The point of application could be the point that fixes the force vector in space, but requiring that specific point is probably overly restrictive. Once any point along the line of action is specified, (along with the magnitude and direction of the force vector itself), the specific point of application becomes irrelevant.

This is not my idea. It is something I recently discovered from reading textbooks in "engineering physics."

Bob Sciamanda (treborsci@verizon.net) mentioned "vector field." It might be helpful to start with that idea and then move from field vectors to field lines just like we can visualize an electric field as electric field vectors or as electric field lines. If we have a uniform force field represented by parallel straight lines, and we know the direction of these lines and the magnitude of the force, then we might visualize this force field as occupying the same region of space as some object that the force is going to act on. In order to determine how the object is going to behave (translate or rotate or both) we don't need to pick a point of attachment, we only need to pick a specific field line. We pick a specific field line by picking any point in space that the line goes through. The point we pick need not be the point at which the force attaches to the object. Indeed, the point specifying the line need not be within the object. Any point in space will specify the "line of action" and that will suffice to determine the outcome.

Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu



-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Tuesday, September 07, 2010 12:01 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] bound vectors ... or not

On 09/07/2010 06:02 AM, Folkerts, Timothy J wrote:

I'm saying if I tie Rope B to a Object A, the force F(on A due to B)
*is* applied to a specific point (or can at least be integrated to one
effective point).

We agree that the physics of the situation requires us
to specify the point-of-application as well as the
force-direction-and-magnitude.

The question is:
a) does the concept of "force" include just the vector
describing the direction-and-magnitude of the push, or
b) does the concept of "force" include /two/ vectors,
i.e. both the point-of-application and the push?

This strikes me as interesting and important question.
If you had asked me a couple days ago whether I knew what
a "force" is, I would have said sure, I know what a force
is. But now I realize that my concept of "force" has been
somewhat ambiguous all these years.

The vector we use to describe the magnitude and direction of the force
is not "localized' or physically bound anywhere. That same vector could
be used to describe the magnitude and direction of any number of other
forces (perhaps the force of you pushing on the other side).

Agreed.

But the question remains: Is this vector the same as "the"
force, or it just one of the two vectors that make up "the"
force?

To do vector addition, I can move the vectors around wherever I
want (but that doesn't move where the rope is tied).

Agreed.

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

Faced with an ambiguity of this sort, it may be helpful, at
least for the sake of discussion, to accept *both* possibilities
and give them distinct names. We can always throw one of them
away later if we don't like it.

So let us at least temporarily define
-- an unbound force Fu, and
-- a creature X which is an ordered pair
X = (r, Fu)
where r is the position vector specifying the
point-of-application.

Some folks would call X a "bound force" but others (including
Philip K.) would argue that this is a misnomer, i.e. that Fu
is "the" force and X is not a force.

I remark that X really is an ordered pair (r, Fu) and not
merely the wedge product (bivector) r /\ Fu [aka the cross
product (pseudovector) r x Fu]. Given an ordered pair (r, Fu)
we can always pick out the second element Fu, whereas given
only the product r /\ Fu we cannot reconstruct r or Fu.

It appears that equations such as
F = ma
F = dp/dt
F = - dE/dx
involve only Fu not X. So as far as the mathematical formalism
goes, it appears that F (usually?) means Fu ... and that oddly
enough, the concept X has heretofore gone without a symbol.

Bob S. pointed out that X is in some ways a watered-down version
of the _vector field_ F() or a selected point F(r) in the field.
This raises pedagogical issues, because we want to explain force
and point-of-application to students who are not yet ready to
cope with vector fields. The vector field concept is tried-and-
true, but it is overkill for simple point-particle mechanics
problems, in the sense that the vector field is defined for an
infinite number of points, be we only care about the handful
of points where the particles happen to be.

The same questions apply to lots of things, not just forces.
For each particle in a system of point-particles, and for each
parcel of fluid in a continuous system, again we have a choice:
a) Should we talk about the position (r), the net force (F),
the momentum (p), the angular momentum (L), the entropy (S),
et cetera, or
b) Should we talk about the so-called "bound force" (r,F),
the "bound momentum" (r,p), the "bound angular momentum"
(r,L), the "bound entropy" (r, S), et cetera?

Note that (r, S) cannot qualify as a "bound vector" since
S is a scalar.

It seems to me that the conventional approach is to describe
the parcel using one big "dynamical state variable" i.e. the
ordered tuple (r, F, p, L, S, ...).

To this way of thinking, the aforementioned "X" is seen as a
type of dynamical state variable. If two states X1=(r1, F)
and X2=(r2, F) share the same F, we can say that the force
F is the same, but the dynamical situation is different.

I'm not saying that's the only way of thinking about it, but
at first glance it seems attractive. It allows my head to
stop spinning, and (perhaps) allows me to talk about these
things without contradicting myself every few minutes.

To this way of thinking, we can say
-- A so-called "bound vector" is not a vector but rather
an ordered pair of vectors.
-- A so-called "bound force" is not a vector and is not
a force, but rather a dynamical state variable consisting
of a pair of vectors (r, F).

This way of looking at things has the advantage that it
uses the word "vector" in a way that is consistent with
the axioms of linear algebra.

Comments? Suggestions?
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