Game Development Reference

In-Depth Information

the center of mass of the body, but you want to know the moment of inertia,
I
, about an

axis some distance from but parallel to this neutral axis. In this case, you can use the

transfer of axes, or
parallel axis theorem
, to determine the moment of inertia about this

new axis. The formula to use is:

I = I
o
+ md
2

where
m
is the mass of the body and
d
is the perpendicular distance between the parallel

axes.

There is an important practical observation to make here: the new moment of inertia

is a function of the distance separating the axes squared. This means that in cases where

I
o
is known to be relatively small and
d
relatively large, you can safely ignore
I
o
, since

the
md
2
term will dominate. You must use your best judgment here, of course. This

formula for transfer of axes also indicates that the moment of inertia of a body will be

at its minimum when calculated about an axis passing through the body's center of

gravity. The body's moment of inertia about any parallel axis will always increase by an

amount,
md
2
, when calculated about an axis not passing through the body's center of

mass.

In practice, calculating mass moment of inertia for all but the simplest shapes of uniform

density is a complicated endeavor, so we will often approximate the moment of inertia

of a body about axes passing through its center of mass by using simple formulas for

basic shapes that approximate the object. Further, we will break down complicated

bodies into smaller components and take advantage of the fact that
I
o
may be negligible

for certain components considering its
md
2
contribution to the total body's moment of

inertia.

Figure 1-3
through
Figure 1-7
show some simple solid geometries for which you can

easily calculate mass moments of inertia. The mass moment of inertia formulas for each

of these simple geometries of homogenous density about the three coordinate axes are

shown in the figure captions. You can readily find similar formulas for other basic ge‐

ometries in college-level dynamics texts (see the
Bibliography
at the end of this topic

for a few sources).