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Rotation of rigid body - wobbling

  1. May 7, 2008 #1
    When a rigid body experiences a rotation about an axis other than one of its "principal axis of rotation", it experiences a "wobble"

    I have been trying to understand why this is so (intuitively).

    Here is what I have come up with - please tell me if I have misconceptions or misunderstandings, or if I'm just full of crap :(

    I have come up with this qualitative explanation - the particles that comprise the rigid body are subject to two constraints - they must rotate about [tex]\omega[/tex], but at the same time they must stay fixed relative to each other. The latter constraint introduces the wobble when the rotation is not about a "principal axis of rotation".

    Is that a correct way of looking at it?

    A more mathematical explanation I have come up with is (tell me if any of these concepts are flawed... hopefully this makes some sense without a drawing) -

    Rigid body experiences a rotation about [tex]\omega[/tex], where [tex]\omega[/tex] is not parallel to a principal axis of rotation.

    Viewing the system from a coordinate system that is fixed with respect to the "inertial tensor" I of the rigid body,
    [tex]\omega[/tex] actually appears to rotate about a principle axis

    (imagine instead rotating the coordinate system about [tex]\omega[/tex] - [tex]\omega[/tex] effectivly rotates about a principal axis(s)).

    Therefore, because by definition

    L = I [tex]\bullet[/tex] [tex]\omega[/tex]

    coordinate system is fixed with respect to inertial mass tensor I
    so dI/dt = 0

    but [tex]\omega[/tex] is not fixed in the coordinate system so
    d[tex]\omega[/tex]/dt [tex]\neq[/tex] 0

    therefore, dL/dt = torque [tex]\neq[/tex] 0

    And this torque is responsible for the wobble.
  2. jcsd
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