Brian Whatcott says the infinite-mu limit might be more interesting than
the zero-mu limit because the infinite-mu limit is "not impossible to
realize."
What strikes me as more interesting is the sudden transition between two
values of v_f as we move from infinite-mu to any mu between zero and
infinity. I think this results from the mass of the earth being
essentially infinite compared to the tire so we don't have to bring
momentum into the picture. It also assumes that some slippage occurs up
to the point that v reaches v_f.
If the tire does not slip (infinite mu) then all initial rotational
energy stays with the tire, partly as rotational and partly as
translational. No energy shows up as thermal energy (assuming no
slippage and also no thermal energy from stretching). And no energy
gets imparted to the second body (the earth) because the the second body
has "infinite" mass.
On the other hand, if slippage occurs up to the point that the final
velocity is reached, then the part of the original energy that shows up
as thermal energy is 0.5m(v_f)^2 regardless of what mu is.
So the final energy in the rolling tire is either 0.5I(w_0)^2 (no
slippage; infinite mu) or it is [0.5I(w_0)^2 minus 0.5m(v_f)^2] for any
mu less than infinity as long as slippage occurs until v_f is reached.
Now, does it really slip until v_f is reached, or do we switch over to a
static-friction case at some point before v_f is reached?
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.