
402 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
Grouping coefcients of α
2
, β
2
, γ
2
, and so forth, and using the denitions of moments and prod-
ucts of inertia, we obtain
I
λ
= α
2
I
xx
+ β
2
I
yy
+ γ
2
I
zz
− 2αβI
xy
− 2αγI
xz
− 2βγI
yz
. (12.61)
From Equation 12.61, we can calculate the moment of inertia about any axis through O if the
moments and products of inertia with respect to the coordinate axes are known. This, in turn, can
be related, with the aid of the parallel-axis theorem, to the moment of inertia about a parallel axis
passing through the CM.
Equation 12.61 has a geometrical interpretation. As we allow α, β, and γ to vary, we nd moments ...