Game Development Reference
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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
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
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
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).
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