A numerical model of a wheel consists of a series of particles evenly spaced around the rim of the wheel, each connected with both neighboring particles by a damped spring. In the simulation, the wheel rolls along the ground, allowing determination of position, velocity, deformation, and angular velocity of the wheel as the system evolves. The results of the simulation are consistent with what is observed experimentally with rolling friction: the rolling resistance goes up as pressure goes down, and it goes up as speed goes up. The simulation indicates that the majority of the kinetic energy lost from the wheel is dissipated in deformations of the wheel, i.e., in the dampers between the particles. Kinetic friction is negligible. Both static friction and the contact normal force vary over the region of contact between wheel and ground, with the effective position of the normal force acting forward of the center of the wheel. Static friction is the force responsible for the change in linear momentum of the wheel.

1.
A.
Quale
, “
On the use of a standard spreadsheet to model physical systems in school teaching*
,”
Phys. Educ.
47
,
355
–
365
(
2012
).
https://doi.org/10.1088/0031-9120/47/3/355
Google Scholar
Crossref
Search ADS
 
2.
J. A.
Rizcallah
, “
Approximating the linearly impeded projectile by a tilted idealized one
,”
Phys. Educ.
53
(
6
),
065012
(
2018
).
https://doi.org/10.1088/1361-6552/aadc26
Google Scholar
Crossref
Search ADS
 
3.
R.
Chabay
 and
B.
Sherwood
,
Matter and Interactions
, 4th ed. (
John Wiley & Sons
,
Henderson, NJ
,
2015
).
Google Scholar
 
4.
J.
Hierrezuelo
 and
C.
Carnero
, “
Sliding and rolling: The physics of a rolling ball
,”
Phys. Educ.
30
,
177
–
182
(
1995
).
https://doi.org/10.1088/0031-9120/30/3/009
Google Scholar
Crossref
Search ADS
 
5.
L.
Vozdecký
,
J.
Bartoš
, and
J.
Musilová
, “
Rolling friction—Models and experiment. An undergraduate student project
,”
Eur. J. Phys.
35
(
5
),
055004
(
2014
).
https://doi.org/10.1088/0143-0807/35/5/055004
Google Scholar
Crossref
Search ADS
 
6.
R.
Cross
, “
Rolling to a stop down an inclined plane
,”
Eur. J. Phys.
36
(
6
),
065047
(
2015
).
https://doi.org/10.1088/0143-0807/36/6/065047
Google Scholar
Crossref
Search ADS
 
7.
C. E.
Mungan
, “
Rolling friction on a wheeled laboratory cart
,”
Phys. Educ.
47
,
288
–
292
(
2012
).
https://doi.org/10.1088/0031-9120/47/3/288
Google Scholar
Crossref
Search ADS
 
8.
R.
Cross
, “
Coulomb's law for rolling friction
,”
Am. J. Phys.
84
,
221
–
230
(
2016
).
https://doi.org/10.1119/1.4938149
Google Scholar
Crossref
Search ADS
 
9.
R.
Cross
, “
Origins of rolling friction
,”
Phys. Educ.
52
(
5
),
055001
(
2017
).
https://doi.org/10.1088/1361-6552/aa77b4
Google Scholar
Crossref
Search ADS
 
10.
J.
Renart
 and
P.
Roura-Grabulosa
, “
Deformation of an inflated bicycle tire when loaded
,”
Am. J. Phys.
87
(
2
),
102
–
109
(
2019
).
https://doi.org/10.1119/1.5086008
Google Scholar
Crossref
Search ADS
 
11.
See supplemental material at https://doi.org/10.1119/1.5111940 for the code.
12.
E.
Jones
,
T.
Oliphant
,
P.
Peterson
 et al, “
SciPy: Open source scientific tools for Python
,” <http://www.scipy.org/> (accessed 16 August 2018).
13.
K.
Atkinson
,
W.
Han
, and
D. E.
Stewart
,
Numerical Solution of Ordinary Differential Equations
(
John Wiley & Sons
,
Henderson, NJ
,
2011
), Vol.
108
.
Google Scholar
 
14.
P. N.
Brown
,
G. D.
Byrne
, and
A. C.
Hindmarsh
, “
Vode: A variable-coefficient ode solver
,”
SIAM J. Sci, Stat. Comput.
10
(
5
),
1038
–
1051
(
1989
).
https://doi.org/10.1137/0910062
Google Scholar
Crossref
Search ADS
 
15.
R.
Cross
, “
The bounce of a ball
,”
Am. J. Phys.
67
,
222
–
227
(
1999
).
https://doi.org/10.1119/1.19229
Google Scholar
Crossref
Search ADS
 
16.
R.
Cross
, “
Why low bounce balls exhibit high rolling resistance
,”
Phys. Educ.
50
(
6
),
717
–
721
(
2015
).
https://doi.org/10.1088/0031-9120/50/6/717
Google Scholar
Crossref
Search ADS
 
17.
T.
Li
,
R.
Burdisso
, and
C.
Sandu
, “
Literature review of models on tire-pavement interaction noise
,”
J. Sound Vib.
420
,
357
–
445
(
2018
).
https://doi.org/10.1016/j.jsv.2018.01.026
Google Scholar
Crossref
Search ADS
 
18.
S. K.
Clark
, “
Rolling resistance of pneumatic tires
,”
Tire Sci. Technol.
6
(
3
),
163
–
175
(
1978
).
https://doi.org/10.2346/1.2151003
Google Scholar
Crossref
Search ADS
 
19.
D. E.
Hall
 and
J. C.
Moreland
, “
Fundamentals of rolling resistance
,”
Rubber Chem. Technol.
74
(
3
),
525
–
539
(
2001
).
https://doi.org/10.5254/1.3547650
Google Scholar
Crossref
Search ADS
 
20.
Y.
Xu
,
K. L.
Yung
, and
S. M.
Ko
, “
A classroom experiment to measure the speed-dependent coefficient of rolling friction
,”
Am. J. Phys.
75
,
571
–
574
(
2007
).
https://doi.org/10.1119/1.2432130
Google Scholar
Crossref
Search ADS
 
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2019
American Association of Physics Teachers

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