Re: [Phys-L] fundamental notion of force --> using an arrow to represent something more than a vector

[John Denker <jsd@av8n.com>]

On 10/24/2015 07:03 AM, Philip Keller wrote:

> I think I am missing the point.

A problem exists in the world, as discussed below.  If this
problem has never affected you, that's good.  It means you've 
been smart, careful, and somewhat lucky.  However I predict
that sooner or later you -- and your students -- will run 
into the problem.

> A force is a vector.  But a rope is not.  And a "lever-arm" is a
> vector. But a beam is not.

Yes.
 
> When a rope pulls on a beam, we define the force vector with
> magnitude and direction. But the force vector is not a physical
> entity. It is a mathematical abstraction. And it does not have
> location. When we are working with those abstractions, we find it
> convenient to move them around, such as when adding them tip-to-tail.
> We can move the force but we are not moving the rope. We are moving
> them in an abstract vector space, not the real world. [Just this
> week, I have been teaching first year students how to take the forces
> on a free-body diagram and and add them tip-to-tail.]

Yes.

> Similarly, lever-arm is a vector. We can define it as the
> displacement vector from the chosen pivot axis to the point of
> application of the _rope_ (not "the force", the rope). Moving the
> real-world rope will change the lever arm vector. Moving the abstract
> (vector-space) force will not.

Yes.
 
> Then, we can define torque = r x F = |r| times |F| times sine of
> angle between them. Go ahead and move F or r wherever you
> want...torque won't change.

Yes.  That's all fine.

-----

The problem is, not everybody plays by those rules ... for
semi-understandable reasons.

We agree that the force vector does not tell us everything
we want to know about the physical situation.  However, the 
situation still exists.  The concept of "force with lever
arm" exists, and it would be nice to have a concise description
for it.  Here's one possible name:
     forque = (force, torque)
     forque = (force, line of action)

If you know both the force and the torque, you have a good
description of what the rope is doing to the beam.  So far
so good.

Problem #1:
 There exists another not-so-good name for the forque concept,
 namely "bound vector".  The name suggests that it is some 
 special type of vector, but really it is not a vector at all,
 because it does not uphold the vector-space axioms.  

 As I like to say: /Ideas/ are primary and fundamental. Terminology 
 is important only insofar as it helps us formulate and communicate 
 the ideas.  Conversely, if the terminology is unhelpful, we should
 switch to better terminology.

Problem #2:
 Some people are lazy, and say "vector" when they mean "bound
 vector".  This leads to situations where students think
 they are being told that
      ☠     force = (force, line of action)    ☠

 When you write it that way, it's obviously ridiculous, but
 situations like this, where the same word is used with two
 different yet deceptively similar meanings, are exceedingly 
 common.  For example, a lap in the swimming pool is conceptually,
 qualitatively, and quantitatively different from a lap on the 
 race track.  Ambiguities are common even within physics:
 acceleration, heat, gravity, photon, charge, spin, et cetera.

 I reckon a big part of my job description is identifying
 deceptive situations and finding ways to disambiguate them.

Problem #3:
 A partially-related problem has to do with the meaning of
 the symbols we use on diagrams.

 Again: /Ideas/ are primary and fundamental.  Symbolism and
 notation are important only insofar as they help us formulate
 and communicate the ideas.  Conversely, if the symbolism is
 unhelpful, we should switch to better symbols.

 This is relevant because normally an arrow on a diagram
 represents a vector, with magnitude and direction ... but
 *not* location.  However, sometimes you see diagrams where
 the arrow is not a vector, because it has a partially-
 significant location.

 Maybe you assiduously avoid drawing such diagrams, but
 nevertheless they exist in the world.  There are textbooks
 that feature such things.  I kid thee not.
       https://www.google.com/search?tbm=isch&q=%22bound+vector%22+%22free+vector%22+force+point
       https://www.google.com/search?tbm=isch&q=%22extended+free-body+diagram%22

 The idea of an arrow with length and orientation *and* 
 partially-significant location is useful, and it is not 
 going away anytime soon.  It's not crazy; it's just 
 ambiguous.  In expert hands, such symbolism is helpful
 ... but in less-skilled hands, it is a recipe for disaster.  

 Even if you and your students never draw such things, at
 some point you will encounter therm.  At that point, you
 need to recognize such things for what they are.  You need 
 to disambiguate the "vector arrows" from the "non-vector 
 arrows".  Otherwise such diagrams lead to profound 
 misconceptions about the meaning of "vector" and "force".

 So as to not be guilty of making the problem worse, I've
 adopted the practice of putting squiggles on the arrows
 that are not vectors. 
    https://www.av8n.com/physics/force-intro.htm#fig-crane-anvil-forques
 However, there remain lots of diagrams in the world where
 you simply cannot tell whether the arrows represent vectors
 or something else.

If you do not suffer from misconceptions about what's a vector
and what's not, that's good.  However there *are* some folks
out there, including some professors at Big Name universities,
who are confused about this at the most fundamental level.
This is the problem I'm trying to alleviate.