Re: "Negotiating" a curve. EUREKA?

[John Denker <jsd@MONMOUTH.COM>]

I wrote:
> >.... I don't see a problem at all.
> >............
> > If this analysis isn't good enough, please explain
> > why not.  Please be as specific as possible.

And then at 03:37 PM 11/7/99 -0500, Ludwik Kowalski wrote:

> The question was presented like this. The pedal-less tricycle,
> at time zero, has its back wheels on the x axis and its front
> wheel on the y axis. At that moment the planes of all three
> wheels were parallel to the y axis and the tricycle travels
> along the y axis (north) with some speed. The net force, is
> the vector sum of three equal "rolling friction forces"; it is
> directed toward the south (-y).

That's fine as far as it goes, but it considers only a subset of the
conditions that might interestingly apply to the straight-ahead tricycle.

On two previous occasions I have asked you to consider what happens to such
a tricycle when a force *not* in the north/south direction is applied to
the straight-ahead tricycle.  You have not provided any feedback on this,
so I have no idea where we stand.

Learning proceeds from the known to the unknown.  If you want me to help
you, you really need to tell me how you would analyze this situation.

> Suppose we have two such, nearly identical ,tricycles  under
> identical initial conditions. The only difference between them
> is that one has the front wheel pointing north while the other
> has the front wheel turned west, by some angle. The first
> tricycle will be rolling forward (after t=0) while the other
> will be turning left. Why?

If we model the system using ideal wheels, there is an exceedingly large
"spring constant" for motions against the constraint.  If you imagine that
the  non-straight tricycle tries to violate the constraint by an
infinitesimal amount by continuing straight, then this immediately gives
rise to a non-infinitesimal force in the direction perpendicular to the
constraint.  To first order in theta (the steering angle) this force has a
component in the X (east/west) direction.

To second order in theta, this force has a component in the -Y (south)
direction.

We have now found the first two terms in a power-series expansion for a
rotation matrix which describes the rotation (turning) of the momentum vector.

> We agree that with locked wheels both tricycles would be
> rolling north and stop after their kinetic energies are
> thermalized.

This would happen virtually immediately.  For ideal wheels, the nonrolling
friction is exceedingly large and the rolling friction is exceedingly small.

> The road is horizontal. The rolling motion of
> the wheels seems to be an essential factor responsible for
> turning. Why?

Because it allows the coefficient of friction to depend (radically!) on
direction.

> I know that the so-called "rolling frictional" force is
> much smaller that the "sliding frictional" force because the
> mechanisms of thermalization are different.

Indeed.  Waaaaay different.  In fact we imagine that the force of
constraint is proportional to *distance* of constraint-violation (so it can
produce a finite force with zero velocity in that direction) and the force
in the unconstrained direction is proportional to velocity (independent of
position).  Either way you look at it, you have practically an infinite
ratio between the constrained and unconstrained directions.

> 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.

> To understand is to find a satisfactory causal  relation which is
> objectively correct.

I still disagree with that.  Suppose the correct statement about A and B is
that they are equivalent.  Then there is no causation
relationship.  Causation is directional.  Equivalence is
nondirectional.  There are formal definitions of these things, you know.

Why not just say that the goal is to find the correct relationship between
A and B, and leave it at that?