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





-----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 2:31 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] bound vectors ... or not

On 09/07/2010 10:46 AM, Folkerts, Timothy J wrote:
I am also seeing a pedagogical point here too.

If I have a car,
Alan can apply a (horizontal) force on the back bumper on the right
side.
Bob can push on the trunk in the middle.
Carl could open the driver's door and push on the frame.
Dan could grab the passenger door handle.

I suspect that students will intuitively think of these as
"different
forces" even if the people are pushing "as hard" as the others.
F(A),
F(B), F(C) or F(D) could each be applied independently of the other.
Each has the same effect, but it clearly matters to Alan if he is
actually pushing or not. Heck, we even gave them different names --
"clearly" they are different forces.

But the effect of all of these "different forces" is the same. The
car
will accelerate the same in each case (ignoring minor details of the
wheels, suspension, etc). The power of vector notation is that we
have
a formalism for saying all these forces are identical. We can draw
the
vector anywhere we want and it produces the same effect. Thus all
of
these "different vectors" are indeed the same at some level because
they
produce the same result. Once we establish that all of these are
"the
same" we can draw the vectors to suit our convenience -- for
example,
tail to tail to emphasize what object they are acting on, or tip to
tail
to emphasize the net force.

I agree with about 98% of that ... but that's not the whole
story.

In particular, what's true for a car with the steering wheel
locked or for a railroad car would not be true for a cart with
castering wheels, or barge, or hovercraft, or aircraft.

We agree that all of the _forces_ mentioned are the same when
considered as _forces_. But knowing the force isn't the whole
story.

Until somebody comes up with a better word for it, I'm going
to talk about "dynamical state". The scenario painted above
mentioned three different dynamical states. The force is the
same in each case, but the dynamical state is different because
the line of action (and indeed the point of application) is
different.

dynamical state = ordered pair
= (point of application, force)

I think that although I haven't seen it defined as an ordered pair, the
concept already has a few names, two of which would be load and loading,
as in the loading on the object due to the rope can be approximated as a
concentrated force exerted at point A in the direction in which the rope
extends away from point A.

concentrated load due to ____ = (point of application, force vector)

In Engineering Mechanics Statics, 6th Edition, by RC Hibbeler, the
author uses the word force for what you are trying to define here in
which case your word force (the second element in your ordered pair)
would have to be replaced with something like the direction and
magnitude or the force vector. In section 1.2 of that textbook Hibbeler
writes "In any case, a force is completely characterized by its
magnitude, direction, and point of application." For instance, your
equation could read:

force = (point of application, direction and magnitude)

Clearly, several authors have decided to name the bundle consisting of
the force vector and the point of application of the force as a "bound
vector". That seems to me to be an unfortunate name in that I don't
think that the entity in question is a vector.

I tend to go with such and such a force (vector) applied at such and
such a point. In other words, you have a force vector and you have a
point of application. They are two different things and I don't think
it is important to bundle them together. If I had to go with one of the
above, I think I would go with "concentrated load due to ______" rather
than "dynamical state" in that I think dynamical state is already used
for something else and why invent a new name for a concentrated load
when an acceptable one already exists.

I think we are in general pretty loose with the word force. Sometimes
it is just the vector and sometimes it is the vector and its point of
application. I asked before what the problem is with being imprecise
here. What space telescope has been ruined because of a lack of clarity
in our speech here? What predictions on the results of physical
processes do students/engineers/physicsists get wrong because of a lack
of clarity in our speech here? Are students unable to add force vectors
because they think you can't "slide them around" because of the language
we use in discussing them? This thread alone is going to make me more
self conscious of it (thanks for starting the thread John D.) but if
someone could point out the damage caused by our lack of clarity of
speech in this area I think I would make it a higher priority to work on
improvements in this area.



***********************************

As a related pedagogical issue: We need to talk about
_Free Body Diagrams_.

FWIW when I was in school, I never saw or even heard of
a Free Body Diagram ... even though I got a very good
education, beyond what most people can even imagine.

One salient thing about Free Body Diagrams is that you are
supposed to draw the "force vector" in the "right place"
i.e. at a place that corresponds (to the extent possible)
to its point of application (or, failing that, at least
somewhere along the correct line of action, assuming a
rigid body).

So, it would seem, students who are taught to draw Free
Body Diagrams are at risk of developing a notion of "force"
(and of "vector" in general) that does /not/ uphold the
principle that such things have a direction and magnitude
but not a location.

I'm not sure how to proceed on this. I reckon Free Body
Diagrams are still a good thing, but we need some way
to deal with the idea that the arrows on such a diagram
are not, strictly speaking, force vectors. They're
something else. This seems like it "should" be an
easy-to-solve problem, but I need to think about it
some more.
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