Game Development Reference
F IGURE 10.1
A force generating zero torque.
in the direction of their axis. To get a counterclockwise torque, we simply flip the sign
of the axis.
Equation 10.1 provides our torque in the correct format: the torque is the vector
product of the force (which includes its direction and magnitude) and the position of
T HE M OMENT OF I NERTIA
So we have torque—the rotational equivalent to force. Now we come to the moment
of inertia: roughly the rotational equivalent of mass.
The moment of inertia of an object is a measure of how difficult it is to change that
object's rotation speed. Unlike mass, however, it depends on how you spin the object.
Take a long stick like a broom handle and twirl it. You have to put a reasonable
amount of effort into getting it twirling. Once it is twirling, you likewise have to apply
a noticeable braking force to stop it again. Now stand it on end on the ground and
you can get it spinning lengthwise quite easily with two fingers. And you can very
easily stop its motion.
For any axis on which you spin an object, it may have a different moment of iner-
tia. The moment of inertia depends on the mass of the object and the distance of that
mass from the axis of rotation. Imagine the stick being made up of lots of particles;
twirling the stick in the manner of a majorette involves accelerating particles that lie a
long way from the axis of rotation. In comparison to twirling the stick lengthwise, the
particles of the stick are a long way from the axis. The inertia will therefore be greater,
and the stick will be more difficult to rotate.
We can calculate the moment of inertia about an axis in terms of a set of particles
in the object:
m i d p i → a
I a =
where n is the number of particles, d p i → a is the distance of particle i from the axis
of rotation a ,and I a is the moment of inertia about that axis. You may also see this