Re: "Negotiating" a curve. EUREKA?

[John Denker <jsd@MONMOUTH.COM>]

At 10:02 AM 11/6/99 -0500, Ludwik Kowalski wrote:
> I think I found another misconception. An attempt to explain
> the centripetal force (on the center of mass of a turning car) in
> terms of an FBD (a free body diagram) is impossible, unless
> the wheels of the car are excluded.

I 99% disagree, for reasons given below.

> The implied assumption of any FBD is that we are dealing
> with a particle, or with a rigid body. Each turning wheel is
> a rigid object but it is not rigid with respect to the car.

I agree that the kinematics of nonrigid bodies have additional
complexities, but in this case
  a) the additional complexities are small in magnitude, and
  b) the additional complexities are easily handled exactly as a second
"refinement" step.

Specifically, the only complexity that the kinematics requires us consider
is the precession that results as we carry the angular momentum of the
wheels around the corner.
  a) This effect becomes quantitatively negligible if we consider the limit
of a heavy wagon with light wheels.
  b) This effect can be taken into account with *no* approximations with
little effort.

It may help to replace the ideal wheels with "ideal skates".  I define
ideal skates to have the property that you can choose the orientation of
the skate, whereupon the locus of the motion is constrained to be
tangential to the orientation of the skate.  There is no friction for
motion along the orientation of the skate, and "large enough" forces of
constraint in the two perpendicular directions.

If you wish to model the precession, you can always carry a gyroscope as
"cargo" in the wagon.  This combination of skates + cargo should model all
the essential kinematics of the wagon and wheels.


> Some may defned textbook writers by saying that every
> thing we teach is only an approcimation. True.

Agred.

> But not every elimination of a feature is a simplification.

Agreed.  Mustn't throw the baby out with the bathwater.

> Hiding an essential element is an error not a simplification. It
> makes the problem more difficult, not mor simple.

Agreed in principle.  But in this case there is no reason to believe that
disregarding the rotational properties of the wheels is anything but a
helpful simplification.

> To understand is to find a satisfactory causal relation.

Strongly disagree in principle.  To understand is to find the correct
relation.  Sometimes the correct relation is "causality".  Sometimes it
isn't.  Please read what Feynman says about this in _The Character of
Physical Law_.

______________________________________________________________
copyright (C) 1999 John S. Denker             jsd@monmouth.com