Re: rolling +- conservation of mechanical energy

["John S. Denker" <jsd@MONMOUTH.COM>]

At 03:55 PM 6/26/01 -0500, lorinda.stinnett wrote:

> ... in order to use cons. of ME, we need to be certain that there is no
> work done by nonconservative forces.

OK.  (It might be slightly safer to phrase it in terms of "negligible work
done" rather than "no work done".)

> Forces on ball:
> 1.  Gravity - conservative - OK
> 2.  Normal - does no work on object ... - OK
> 3.  Static friction at the point of contact between ball and surface. ??
>
> I am believe I understand that the static friction does no work, I am
> looking for an explanation my students would understand.

You cannot in general assume that an object rolls without
slipping.  Sometimes slippage is negligible, but sometimes it is
not.  Bowling, for instance, involves nontrivial slipping.

Even in the absence of slip, rolling friction is not exactly static.  It is
safer to think of it as "locally quasi-static".

We therefore have three categories:
 -- Truly static friction, such as a stationary block on an inclined plane;
 -- Rolling friction; and
 -- Full-blown slipping / sliding friction.

In introductory high-school classes it is traditional to assume that wheels
have negligible slip and negligible dissipation.  There are many real-world
situations where these assumptions are grossly wrong.  For instance,
consider the tires on an airliner during landing.  The tires are initially
nonrotating, and a short time later they are rotating.  There is quite a
lot of slip and quite a lot of dissipation during this process.  This
produces so much heat that some of the rubber is incinerated, producing
conspicuous puffs of smoke.

Even in the absence of slip, you should not assume rolling friction is
dissipationless.  You should not try to "prove" it is dissipationless.  It
is bad luck to prove things that aren't true.

However, it is common knowledge that in many cases, rolling produces very
little dissipation -- orders of magnitude less than sliding.  The amount of
dissipation depends on the details of the materials, et cetera.  For
instance, a high-pressure tire has less dissipation than a low-pressure
tire, other things being equal.

This requires a lot of explaining.
 a) The wheel in general is moving in the X direction.
 b) We need a force in the X direction, to enforce the no-slip condition
if/when the center-of-mass speed of the wheel changes, yet
 c) this force must be applied "just right" so that there is no dissipation.

For students who are having a hard time imagining how such a thing could be
possible, you can start with the following conceptual model:  Imagine a
cog-wheel rolling along its cog-track.  The no-slip condition is enforced
by forces on the face of the gear-teeth.  These are forces of constraint,
normal to the face of each tooth.  The teeth are lubricated so that any
non-normal forces (gnashing of the teeth) are negligible.

For objects that don't have cog-teeth on them, you need to make a slightly
more subtle argument.  Consider a tire rolling on a ramp.  You need to
argue that, to a good approximation, the force has the following important
property:  For each patch of surface, if there is a force in the X
direction, there is no relative motion in the X direction, and vice
versa.  Specifically:
 -- for patches that are not in contact, there can be relative motion in
the X direction, but there will be negligible force;  meanwhile
 -- for patches that are in contact, there can be a force in the X
direction, but there is negligible relative motion.  That is, there is
negligible motion of the patch of ramp relative to the patch of tire that
is in contact ... even though other parts of the tire have plenty of
relative motion.

To reiterate:  You need to calculate F dot dX for each patch
separately.  It doesn't make sense to calculate (force over there) dot (dX
somewhere else).

We can now understand the physics of why underinflated tires are
lossy:  Consider the point where the ramp is tangent to the tire: the
infinitesimal patch of tire at that point is comoving with the
ramp.  Nearby patches are _almost_ comoving.  If the tire is underinflated,
these not-quite-comoving patches will make contact, resulting in a
not-quite-zero F dot dX.