Game Engine Architecture

643

Angular:

α ω

ωθ

() ()

() ();

t dt

dt

t

=

== t^

Linear:

a

v

vr

()

()

() ().

t

dt

dt

t

=

== t

12.4.5.3. Moment of Inertia

The rotational equivalent of mass is a quantity known as the moment of inertia.
Just as mass describes how easy or diffi cult it is to change the linear velocity
of a point mass, the moment of inertia measures how easy or diffi cult it is to
change the angular speed of a rigid body about a particular axis. If a body’s
mass is concentrated near an axis of rotation, it will be relatively easier to ro-
tate about that axis, and it will hence have a smaller moment of inertia than a
body whose mass is spread out away from that axis.
Since we’re focusing on two-dimensional angular dynamics right now,
the axis of rotation is always z, and a body’s moment of inertia is a simple
scalar value. Moment of inertia is usually denoted by the symbol I. We won’t
get into the details of how to calculate the moment of inertia here. For a full
derivation, see [15].

12.4.5.4. Torque

Until now, we’ve assumed that all forces are applied to the center of mass of a
rigid body. However, in general, forces can be applied at arbitrary points on a
body. If the line of action of a force passes through the body’s center of mass,
then the force will produce linear motion only, as we’ve already seen. Other-
wise, the force will introduce a rotational force known as a torque in addition
to the linear motion it normally causes. This is illustrated in Figure 12.24.
We can calculate torque using a cross product. First, we express the loca-
tion at which the force is applied as a vector r extending from the body’s center
of mass to the point of application of the force. (In other words, the vector r
is in body space , where the origin of body space is defi ned to be the center of

F 1

F 2

Figure 12.24. On the left, a force applied to a body’s CM produces purely linear motion. On
the right, a force applied off-center will give rise to a torque, producing rotational motion as
well as linear motion.

12.4. Rigid Body Dynamics