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beam is not a single solid piece, but instead it is really two beams that
have been bolted together? Then really what is happening is that forces
push down on the top part of the beam, which pushes down on the bottom
part of the beam, which pushes down on the Earth. Likewise, the Earth is
pushing back up on the bottom part of the beam, which pushes up on the
top part of the beam.
But why stop there? Isn't any object actually composed of not just two
or three pieces, but trillions of molecules? How can we possibly calculate
all these complicated quantum-electrical forces? This is where Newton's
third law comes in. We are justified in the treatment of this spliced beam
as a single rigid body, and we can ignore all the internal forces, provided
that the body stays rigid, which means that all pairs of points within the
object maintain a fixed distance from each other. In this situation, the
parts are not accelerating relative to each other, and this means that the
internal forces must be exactly balanced. In other words, all the internal
forces cancel each other out and thus make no contribution to the net force,
which is why we can ignore them. Of course, to the extent that the pieces
do accelerate relative to each other, any calculations we make ignoring the
internal forces will be inaccurate. If the bending or compression of the
object is very slight, then our calculations will not be perfect, but they
will be very close; if the object breaks apart, then our calculations will be
meaningless.
We can generalize arguments such as this even further to the case where
the parts are moving relative to each other. Of course, an object with
moving internal parts is the opposite of a rigid body; however, we'll see
that in many respects we are still able treat these complicated systems as
“particles.”
Section 12.3
discusses this idea and how it allows us to resolve
the conundrum of Moe and his box.
12.2
Some Simple Force Laws
Many different types of forces are at work in our universe.
6
In a real-time
simulation, we often ignore certain forces, make approximations to them,
and even invent fictional
7
forces to achieve a desired effect (such as forcing
a trajectory to obey an animator's constraints, or helping the AI or the
6
Actually, this is not true. At the time of this writing, physicists believe that there are
four fundamental forces. Almost all forces caused by matter bumping up against other
matter are essentially electrical because it is electrical forces that keep atoms separated
from one another. However, the situations in which matter pushes away from other
matter are so diverse that at the macroscopic level it is useful to have many different
force laws to describe the behavior.
7
As opposed to frictional.
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