
407Motion of Rigid Bodies
© 2010 Taylor & Francis Group, LLC
I
11
+ I
12
tan θ = I, I
12
+ I
22
tan θ = I tan θ.
Eliminating I from these two equations, we find
(I
22
− I
11
)tan θ = I
12
(tan
2
θ − 1)
from which we obtain
tan2
2
12
11 22
θ=
−
I
where we have used the identity tan2 θ = 2 tan θ/(1 − tan
2
θ).
In the interval 0° to 180°, there are two values of θ, differing by 90°, that satisfy Equation 12.76.
These two values of θ give the directions of the two principal axes in the xy-plane.
A body whose three principal moments of inertia are all different is called an asymmetrical top.
If two are equal, it is termed a symmetrical top. In this case, the direc