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 12:46 PM, Philip Keller wrote:

> But if anyone argues that force is a "bound vector", I am not seeing
> it.

You're not likely to see such an argument in high school ... not 
explicitly, not by that name anyway, because the students don't
know the name.  Be that as it may:
 *) Even at the high-school level, the concept exists in some
  vague, implicit guise ... even if the students cannot give
  voice to it.

 *) At the college level, there are textbooks and professors
  who teach about "bound vectors".  Sometimes this happens in
  not-very-sophisticated engineering courses ... and sometimes
  in seemingly-sophisticated math courses.

I've struggled with this for years, and my understanding is
undoubtedly still imperfect, but here's my current best take:

There is a pedagogical progression:
  a) Motion of a pointlike particle in one dimension.
  b) Motion of a pointlike particle in two dimensions.
  c) Rigid motion in two dimensions, including rotation.
  d) Non-rigid motion in one dimension (e.g. plane waves).
  *) etc.
  *) etc.
  *) etc.
  x) fluid dynamics
  y) quantum field theory
  *) etc.

My point for today is that the transition from (b) to (c) is quite
a big leap for the students.  For the folks on this list, it is easy
to see (b) and (a) as special cases of (c) ... but my point is that
going the other way is highly nontrivial;  (c) is not an obvious 
generalization of (b).  Ideas that were perfectly fine for (b) must
be unlearned.  Unlearning is always hard.

In particular, the "F=ma" equation is practically the emblem of basic
physics.  
  ++ It works fine for rigid objects in 1D and for pointlike 
   particles in 1D, 2D or higher.  If you know the force you
   know the acceleration, and vice versa.
  -- For rotatable or nonrigid objects, F=ma is not good enough.
   You have to now the force /and/ the point of application.

This becomes a problem almost immediately, because AFAICT we
don't have a good name for the concept of "force and point of
application".  Whether this is a big problem or a small problem
depends on the student.

In more detail, consider the following contrast:
  Level (b) concepts:  F (force) and Δr·F (work)                             [1b]
  Level (c) concepts:  F (force), r_i ∧ F_i (torque), and Δr_i · F_i (work)  [1c]

At level (c), there is a new wrinkle in the work concept:  there
are multiple forces and multiple points of application, as indicated
by the subscript i in equation [1c] ... and you have to match them
up properly.  In 1D the focus was on "net" force, but in 2D and
higher that's not good enough, and will have to be unlearned.
However, this is not my main point for today.

Let's focus instead on the torque, which is an entirely new
feature at level (c).  It means that F=ma does not tell the
whole story.  If you know the force, that is not sufficient
to predict the motion (or even the second derivative thereof).

This is /at best/ a terminology problem.  It is /at best/ a
trap for the unwary.  For example, here is a trick question.
There are two versions of the question:

1) We start with a shopping cart, supported by castering wheels
  in the usual way, initially at rest.  I attach a string and 
  pull for 3 seconds, starting at time t=0.  We do the experiment
  in 9 different ways, as shown in the diagram:
    https://www.av8n.com/physics/img48/shopping-carts.png

  The string is slightly stretchy.  In all 9 scenarios the
  string is stretched so that it is 10% longer than it would
  be in its natural, zero-tension state.  We take scenario 5 
  as the reference.  In which (if any) of the other 8 scenarios
  is the force on the cart the same as in scenario 5?


2) We start with a shopping cart, supported by castering wheels
  in the usual way, initially at rest.  I attach a string and 
  pull for 3 seconds, starting at time t=0.  We do the experiment
  in 9 different ways, as shown in the diagram:
    https://www.av8n.com/physics/img48/shopping-carts.png

  The string is slightly stretchy.  In all 9 scenarios the
  string is stretched so that it is 10% longer than it would
  be in its natural, zero-tension state.  We take scenario 5 
  as the reference.  In which (if any) of the other 8 scenarios
  is the motion of the cart the same as in scenario 5?

The only difference here is in the last sentence:  "motion of"
instead of "force on".

Version 2 is a relatively straightforward test of conceptual
understanding.  Version 1 is nasty, because it is at least as
much a test of terminology as of understanding.  It hinges on
a strict, technical definition of "force".

Some people would defend version 1 by saying you can't really
understand physics if you don't adhere to the strict definitions
... but I disagree.

   BTW, it should go without saying:  You must not give both
   versions of the question to the same student!  Give one
   version to half the class, and the other version to the
   other half.  Students aren't stupid;  the first thing they
   will do is compare the two questions and discover that you
   are harping on the words "force" and "motion" for some
   reason.

Just to be clear, I claim that according to standard definitions,
the answer to version 1 is different from the answer to version
2.  However, I know that some folks on this list would argue
that the two answers are the same.  That's because they consider
"force" to be a bound vector.  I don't see how they can reconcile
that with F=ma, given that acceleration is not a bound vector, 
but that doesn't seem to bother them.

> I am not sure I would discuss this with students if they didn't ask.

They're not going to ask.  For starters, they don't know 
enough terminology to frame the question, even if they 
wanted to.

Consider the contrast:

 -- For a known rigid body in one dimension, if you know the
  _force_ you can predict the motion (or the second derivative
  thereof).

 -- Fill in the blank:
  For a known rigid body in two dimensions, if you know the
  ________ you can predict the motion (or the second derivative
  thereof).


Again, there are some people on this list (and lots of others)
who would argue that "force" fills in the blank just fine,
treating "force" as a bound vector.  I would argue that this
is non-standard.

A second problem is that I don't know of any particularly 
concise standard term that could be used to fill in the
blank.  If anybody has a suggestion, please speak up!

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

Tangential remark:  The distinction between the two versions
of the shopping-cart question highlights one of the big problems
with the FCI.  Sometimes I think it should be called FVI :
force vocabulary inventory.  Too many of the questions hinge
on a legalistic interpretation of the terminology, rather than
a conceptual understanding of the physics.