Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 571

all of the forces depicted in the free-body diagrams in Figure 12.4 remain
the same. Doing this would not change the angle at which the bowl would
begin to slide! Although it might seem like a larger surface area would give
the objects more to “grab” with, this is offset by the decrease in pressure,
since the same total normal force is now distributed over a smaller contact
area. Now, a very tall bowl may begin to tip over before it begins to slide.
But this is a matter of rotation, the increase in the tendency to rotate being
caused by an increase in the lever arm resulting in a greater torque. We
cover these issues in Section 12.5.
12.2.3 Spring Forces
There's one more class of force that is important enough to discuss in its
own section: the forces exerted by a spring disturbed from its equilibrium
position. Why do we discuss this admittedly peculiar class of force? Have
springs suddenly become prominent features in video games and their accu-
rate simulation an important gameplay feature? Actually, yes. Even if you
don't see very many literal springs in a video game, there are likely very
many “virtual springs” at work. Springs exhibit a general behavior that is
very useful for enforcing constraints, preventing objects from penetrating,
and the like.
This section presents the classic equations of motion for damped and un-
damped oscillation. It covers undamped oscillation first, and then damped.
It's often the case in a video game that programmers use a virtual spring
(often in the form of a spring-damper system) when really what they are
using is a control system. There are certain advantages to be had when
the physical nature of the problem is dropped and we think of it purely
in mathematical terms. (Indeed, many times the problem was never really
physical to begin with, and was only recast in physical terms so that the
spring-damper apparatus could be applied.)
Like the friction law, the force law for springs is a surprisingly accurate
approximation to the macroscopic behavior that is the result of complicated
microscopic interactions. Consider a spring with one end fixed and the other
end free to move in one dimension. When the spring is at equilibrium with
no external forces on it, it has a natural length, called the rest length. If
we stretch the spring, then it will pull back to try to regain its rest length.
Likewise, if we compress the spring, it will push back. But how do we know
the strength of the force in each case? That's what the force law tells us.
The force law for springs is known as Hooke's law, and it basically says
that the magnitude of the restorative force is proportional to the difference
from the current length and the rest length (provided the force does not
exceed a value called the elastic limit, which varies with the material used
to construct the spring). If we let l be the current length of the spring and