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Unformatted text preview: following results can easily be verified:
(502)
(503) It follows from Equation (502) that
 and . In other words, is the angle of inclination between the axes. Finally, because the total angular velocity can be written (504) Equations (497), (499), and (501)(503) yield (505)
(506)
and The angles (507) , , and precession about the
body frame, and are termed Euler angles. Each has a clear physical interpretation:
axis in the fixed frame, is minus the angle of precession about the is the angle of inclination between the components of the angular velocity vector  and is the angle of
axis in the  axes. Moreover, we can express the in the body frame entirely in terms of the Eulerian angles, and their time derivatives [see Equations (505)(507)].
Consider a freely rotating body that is rotationally symmetric about one axis (the axis). In the absence of an external torque, the angular momentum vector is a constant of the motion [see Equation (455)]. Let
point along the axis. In the previous section, we saw that the angular momentum vector subtends a
constant angle with the axis of symmetry; that is, with the http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node71.html#e9.89 axis. Hence, the time derivative of the Page 3 of 5 Euler angles 11/6/13 1:39 AM Eulerian angle is zero. We also saw t...
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This document was uploaded on 01/22/2014.
 Winter '14
 mechanics, Inertia

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