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The normal force is very different from the other forces that we've consid-
ered. It depends not only on the state of the system, but also on
the other forces
.
For example, when considering spring forces, we can put two masses near each
other, attach some springs, and consider their change in velocity and position
based solely on the state of the objects. But if we stack ten topics on top of a table
and ask what the contact forces are between them, then the question doesn't even
make sense. The notion of “compute all of the other forces first” implies an acyclic
ordering, which doesn't exist in this case. Each book within the stack presses both
up and down. Therefore, the friction of the normal force breaks down and there
are cases where we simply can't compute reasonable normal forces. Even this
simple case of rigid bodies with stacking has proven remarkably challenging—
stable, efficient algorithms for solving it were introduced only in the past few
years [GBF03, WTF06]. When objects become heavily articulated or have com-
plex contact constraints like the pieces of a completed jigsaw puzzle, simple mod-
els like “normal force” break down.
In other words, we've changed modes. We have to model something that is out-
side of our Newtonian model proper, so we're extending it with a hack. That will
be a point where things go wrong. It is ironic that resting contact is surprisingly
hard (even before we consider stacking) to simulate compared to the ballistics of a
fast-moving bullet. Friction, which relies on normal force, has the same problems.
As we discussed in Chapter 1, science depends on the art of modeling what you
need for the level of accuracy desired. The simple, first-year Newtonian physics
model of normal forces is efficient and accessible, but it is a poor model for com-
plex interactions of many bodies. One can live within those limitations and avoid
complex interactions, attempt to patch over the model to hide its failures, or incor-
porate a more sophisticated model that is probably more expensive to compute
and to integrate.
As an example of developing a more sophisticated model, consider that the
circular relationships of contact forces are very similar to those studied in light
transport. What we need is a steady state solution to an integral equation, and as
with light transport there are many mathematical models for numerically approx-
imating that steady state. For this chapter, we will leave this particular problem at
that analogy in the interest of returning to efficient simulation of simple systems.
35.6.4.5 Friction and Drag
Friction is a description of a set of forces that lead to the common phenomenon of
deceleration. In general, we call any force frictional that is negatively proportional
to velocity. This is why friction is described as a force that “opposes motion.”
However, “motion” depends on your reference frame.
For example, a car could not move or turn without friction. When a car accel-
erates linearly, the drive shaft rotates the axles, which then rotate the wheels.
Friction between the tires covering the wheels and the road resists the rotation
of the wheels. This causes the car to move forward relative to the road, or the
road to move backward relative to the car, depending on the reference frame we
choose. For ideal tires, this completely eliminates motion in the axis along the
road between the point of contact of the tire and the road itself. Thus, friction is
indeed opposing some motion. However, there's little friction perpendicular to the
road, so the point of contact is still able to move outside the plane of the road. In
this case, it moves upward a moment after it breaks contact. Ignoring deformation,
every point on the tire has instantaneous velocity along a tangent to the tire, in the
