Re: BEFORE "Negotiating" a curve.

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Tue, 9 Nov 1999, Ludwik Kowalski wrote:

> Suppose a cylindrical wheel is set to roll on a flat horizontal surface
> in a good vacuum. Its mass and radius are known and the kinetic energy,
> is initially 10 J. There is no sliding, only rolling. In the ideal world
> the CM would travel with a constant speed along the y axis. And the
> wheel would be turning at a constant angular velocity. The radius is
> such that 8 J is translational and 2 J is rotational.

Inconsequential nitpick: It is not the radius that determines the
division of KE between translational and rotational modes, but rather the
geometrical form of the mass distribution.

> But in the real world we have the process of thermalization and the
> kinetic energy, K, is found to be 6 J, after the distance d=5 m was
> covered (4.8 J translational + 1.2 rotational) . The kinetic energy
> of the CM was reduced by dK=(8-4.8)=3.2 J. The motion picture
> frames show that there were a constant acceleration (directed
> toward -y). Nothing unusual so far.
>
> According to the work-kinetic energy theorem the dK must be
> associated with (or caused by , as I would prefer) the "net work
> done on the center of mass". In other words, F*d must be 3.2 J
> and F=(3.2)/5=0.64 N. Right or wrong?

Fair enough.  Of course I would say "According to the pseudowork-kinetic
energy theorem ..." and I would want to be clear that the "F" in the above
is the magnitude of the net external force acting on the cylinder.
Finally, I would be less confident that the net force in a case of rolling
friction like this would be constant, but you claim that there is
experimental evidence to support that conclusion so I accept the verdict
of nature.

> My questions are:
> 1) Is there really a force F acting on the CM?

No.  The force F acts at the interface between the wheel and the ground.

> 2) What is its nature? Why is it directed along -y?
> 3) Where is it generated? How is it generated?

Rolling friction is complicated and involves deformations of the wheel
and/or ground at the interface.  In some cases, especially where the
ground deforms noticeably, you can think of it as the result of an
effective hill that the wheel has constantly to climb.  In any event, the
result is a component of the force from the ground that is opposite the
direction of motion of the wheel.

> 4) How is it transferred to the center of mass?

I don't think this is really a meaningful question.  The CM is not a
"thing" that a force can act on.  Even if one were talking about that part
of the system that is closest to the CM, an external force doesn't need to
be "transferred" to that part to produce the effect that it does on the CM
KE of the system.  For instance, imagine a system that consists of a large
number of noninteracting particles with one of them near the CM.  An
external force acting on some far flung particle will in no way be
"communicated to" the particle near the CM, but it will still produce the
same change in the system CM KE.  In the case of a rigid body, like the
cylinder, the external forces *do* cause internal forces to be set up that
insure that each part of the system "gets the message" and continues to
move as a rigid body, but this internal process is not at all essential to
the functioning of the pseudowork-kinetic energy theorem.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm