Re: rolling

["Carl E. Mungan" <mungan@USNA.EDU>]

Gary wrote:

> Carl,
> I'm a little confused by your statement about friction and normal forces.
> For no slipping, we have static friction, which is not proportional to
> normal force anyway.  Did you intend that the maximum static friction is
> not proportional to the normal force?
>
> > Nothing wrong. See:
> > http://usna.edu/Users/physics/mungan/Publications/TPT.pdf
> > and references therein. It comes about because you simultaneously
> > have to consider forces and torques, because linear acceleration =
> > radius * angular acceleration for no slipping. Thus we do *not* in
>  >general have the condition that the friction is directly proportional
> > to normal force and other "common" rules about friction.
> >
> > This particular example is a favorite counter-example of mine to the
> > question: Is friction always opposite to the direction of an applied
> > force (in the absence of horizontal forces other than friction and
> > the applied force)?
> >
> > This problem is also a wonderful case study in the careful
> > distinction between pseudowork and real work. See Chabay & Sherwood's
> > text. Carl

Sorry for being cryptic. If you look in standard textbooks, you will
find that any quantitative problem involving static friction almost
invariably ends up requiring a calculation of the maximum static
friction, presumably to give one a way of calculating its value
without needing to know the magnitudes of the other horizontal
forces. This is why I put "common" in quote marks. Students hear us
say that only *maximum* static friction is mu*N. But since that's
what static friction always seems to equal in actual problems, they
quickly forget this qualifier. Of course it's easier to memorize a
formula f = mu*N, especially since that's the formula for kinetic
friction. This issue has been discussed previously on the list.
Someone even suggested a problem where if you used mu*N you end up
with an erroneous negative acceleration of a box to which a small
horizontal force is applied. But the feeling of the list was that
this is a sort of sneaky "trick" question to spring on students in an
exam.

BUT the one exception to this "rule" of static friction equaling mu*N
is the chapter on rolling. Suddenly the static frictional force of a
cylinder rolling down an incline is no longer mu*N. Even worse, one
encounters the "rolling paradox" of zero static friction on an ideal
cylinder freely rolling on an ideal flat floor.

Brian queried:

> If a constant force is applied tangentially to a rolling cylinder in the
> direction in which the cylinder is rolling by means of a horizontally
> unrolling line wrapped round the cylinder,
> won't that line move twice as fast as a line which applies a force of
> of the same magnitude in the same direction (say by a wire halter and axle)
> at the height of the center of the cylinder?
> If the distance of application is doubled in unit time for unit force
> that indicates more power?....

If someone beat me to the punch (I'm on the digest), my apologies.
As I said, it's a nice case study contrasting pseudowork and real work.
The *real* work you have to do *is* double because the hollow
cylinder ends up with equal amounts of translational and rotational
kinetic energy.
But the *pseudowork* you have to do is only half the real work
because it only considers the translation of the center of mass.
Friction does the other half of the pseudowork of getting the
cylinder moving.

That is, if you want to know how tired you're going to get when you
pull on the string, ask about the real work.
If you want to know how fast the cylinder is going to end up moving,
ask about the pseudowork. Carl
--
Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
      U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040
mailto:mungan@usna.edu       http://usna.edu/Users/physics/mungan/