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

[John Denker <jsd@av8n.com>]

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)

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

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.