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rotational plane of the tire. The tire as a whole is driven by the wheel, the axle, and
ultimately the drive train and engine to rotate. Because the point of contact resists
motion in the plane of the road, that frictional force propagates backward through
the system and produces a net linear acceleration of the car along the road. On ice
or in mud there is little friction between a tire and the road. In this case, the points
on the tire can move freely relative to the road, so the car does not move.
A similar situation applies to turning a car. At the point of contact, high fric-
tion along the axis of the axle resists motion along that axis, while lower friction
perpendicular to the axis of the car's wheel allows motion. This creates a net rota-
tion of the car. Thus, in the broader sense, friction can be essential for enabling
motion.
Frictional forces arise from electrostatic repulsion and attraction. In the repul-
sion case, molecules of adjacent surfaces collide. As we saw when studying
BSDFs, surfaces that are macroscopically planar may be microscopically rough.
Thus, what appears to be two smooth surfaces sliding parallel to each other may
actually more closely resemble the meshed teeth of two long linear gears. This is
why rougher surfaces increase friction. It is also why the sole of a new dress shoe
is slippery. That sole is a relatively smooth piece of leather. After the shoe has
been worn, say, by walking on concrete sidewalks for a few days, the leather sole
exhibits small bumps and cracks from uneven wear. These mesh with bumps on
the ground and create greater friction.
In the attraction case, materials of different surfaces chemically bond with
each other when in contact, and resist being pulled apart. The bonding is often
strongest when the two materials are the same. This is why smooth glass slides
easily on smooth wood but is hard to drag across another piece of smooth glass
(try this with two juice glasses at breakfast!).
There are case-specific models of friction for commonly arising scenarios. We
present two here.
Dry friction
occurs between solids moving parallel to the plane
of contact, and
drag
occurs between a solid and a fluid.
In the dry friction model, the force is factored into
static
and
kinetic
(a.k.a.
dynamic) terms. Let the two objects be numbered 1 and 2. We consider the force
on object 1 in the reference frame the combined system. That is, the center of mass
of both objects combined will always be at rest.
Static friction is zero when object 1 is moving, that is, when
x
(
t
)
=
0
in the
system's frame. When object 1 is still relative to the reference frame of the system
and is in contact with object 2 only along a surface, then we model static friction
as able to resist acceleration due to net forces up to
k
=
μ
s
||
F
n
||
(35.43)
parallel to the surface. That is, if the net force is
f
, then the object will experience
no acceleration in the plane of the surface if
k
.Otherwise,itwill
experience some acceleration and kinetic friction as described shortly.
The coefficient of static friction
||
f
||−|
f
·
n
| <
μ
s
is determined by the chemical composition
of the two surfaces, the amount of surface area in contact, and the microgeometry
of the surfaces. In other words, there are a lot of hidden parameters in the static
friction equation. However, the coefficient is often approximated as a constant
material property in simple simulations.
When two objects already have relative velocity with respect to each other in
the plane of their contact, we model kinetic friction. Kinetic friction has lower
magnitude than the static friction threshold. This is because objects that are
