Game Development Reference

In-Depth Information

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

its application.

10.2.2

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:

n

m
i
d
p
i
→
a

I
a
=

i

=

1

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