Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 606

mass of the rigid object, if the center of mass is moving.) There is only one
direction that is simultaneously perpendicular to all the velocities of all the
particles, and this direction is the axis of rotation.
We started with “a simple example” of an object rotating about an
axis that passes through its center of mass, but, as it turns out, this is
the general case, at least if we consider instantaneous velocity. The only
simplification we made is to fix the center of mass, but, in general, an object
can translate as well as rotate. It is somewhat surprising to realize, when
you imagine an object tumbling through space, that it will always rotate
about an axis passing through the center of mass (though the axis can be
arbitrarily oriented). When an object receives a force that induces rotation
(known as a torque, to be discussed shortly), the induced rotation will
always occur about the center of mass. In fact, to rotate an object about
an axis that does not pass through the center of mass requires continual
application of some sort of constraint force. In the absence of any external
torque (say, the constraint force is removed), the object will rotate about
an axis passing through its center of mass, and the angular velocity will
be constant—the axis of rotation will not change direction, and the rate of
rotation will not change. We are getting a bit ahead of ourselves talking
about torques, but we wanted to make it clear that this situation of angular
velocity is, in fact, the only situation we need to understand.
Of course, if torques are acting on the object, then the axis and rate of
rotation will change over time. This leads us to consider angular accelera-
tion, which is a vector quantity that we denote α. Angular velocity was not
simply the derivative of orientation, as one might naıvely expect by analogy
with the linear counterparts. However, the analogy does work for angular
acceleration, which is the vector time derivative of angular velocity:

α(t) = ω(t).

The analogy to the linear equation a (t) = v (t) is clear.

12.5.2 2D Rotational Dynamics

Now that we've defined the simple kinematics quantities involved—which
was primarily an exercise in notation and reusing the ideas developed
elsewhere—let's consider the dynamics of rotation. We first simplify the
situation to the case of rotation in the plane (or alternatively, we can think
about this as fixing the axis of rotation). In this situation, the angular ve-
locity and acceleration are scalar rather than vector quantities, since there
is only one degree of freedom. After we develop some basic ideas in two
dimensions in this section, we extend these ideas into three dimensions in
Section 12.5.3.