Re: "Negotiating" a curve. NOT SIMPLE

[Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>]

I need time to think about what John_D wrote. But I can make
two immediate comments.

1) There were no external northwest push on the tricycle, the
     only external forces are coming from the road. That is why
     I did not try to follow the problem you wanted me to analyze
     John. It was a good problem, but more complicated. Let us
     deal with three forces first, then add another force. OK?

2) What I really want is an explanation which is acceptable
    at the level of an introductory physics course. The problem
    is taken from a textbook for such course and I feel that the
    authors, like many before them, are presenting it in a way
    which  is difficult to explain. Are there any lurking authors
    there to comment on this observation? Or from somebody
    else who knows how to teach "the flat road negotiation"?

The words such as tensor, eigenvalue, etc., are not acceptable
in the introductory physics course. (Even a teacher, like me,
only vaguely remembers what stands behind such words.
This is likely to be typical). The only thing the kids know
about frictional forces is that their directions are opposite to
velocities. The concepts of energy, angular velocity, angular
momentum and torque can not be used, they will be introduced
in later chapters. What kind of courses do you teach, John?

By the way, I will return to a generic (and informal) definition
of understanding in my signature file. Your definition of
"scientific understanding" is certainly acceptable.
Ludwik Kowalski
*********************************************
   To understand is to find a satisfactory causal relation.
   To explain is to express that understanding.
   To teach is to promote understanding.
*********************************************

John Denker wrote (in part):

> > But the directions of both forces are always opposite
> > to the direction of v. Right or wrong?
>
> Wrong.  That may be the key misconception.  So let's back up.  Learning
> proceeds from the known to the unknown.  I'm hoping you know about the
> tensor of inertia and its role in angular momentum.  Suppose I have an
> oblong object such as a disk.  Its tensor of inertial looks something like
>
>      1 0 0
>      0 1 0
>      0 0 2
>
> If I spin it around the X axis, I get a nice steady rotation around the X
> axis, with a certain angular momentum.  If I spin it around the Z axis, I
> get a nice steady rotation with a certain angular momentum.  But if I
> skewer the disk (through the center) with an axle in some cockeyed
> direction in the XZ plane and rotate it around that axis, it will wobble
> like crazy.  The angular momentum vector will *not* be aligned with the
> angular velocity vector.  If you doubt me, do the experiment.  Get a disk,
> drill a cockeyed hole, glue in an axle, and try to turn it.  See what happens.
>
> Now popping back from analogy-land to tricycle-land:  We have effectively a
> tensor of friction.  It has a really small eigenvalue in the direction of
> rotation, and a really big eigenvalue in the crosswise direction.  When the
> wheel is turned, the initial (northward) velocity is *not* an eigenvector
> of this matrix.  Consequently the frictional force is *not* opposite to v
> -- indeed not even close.