Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Angular Momentum and Principal Axes of Inertia

  1. Mar 29, 2010 #1

    I´m self-studying Alonso and Finn´s Mechanics and I have a question about this subject.

    Let a body rotate about an arbitrary axis P having angular momentum [tex]\vec L [/tex].
    Consider a referential with three perpendicular axes, [tex] X_{0} , Y_{0} , Z_{0} [/tex] , which are also principal axes of inertia.
    The book says we can write [tex] \vec L [/tex] as

    [tex] \vec L = \vec u_{x} I_1 \omega_{x0} + \vec u_{y} I_2 \omega_{y0} + \vec u_{z} I_3 \omega_{z0} [/tex]

    Does anybody how to derive this formula? The book usually explains things, but perhaps this is supost to be obvious.

    By the way, I already know how to derive [tex] \vec L = I \vec \omega [/tex] for a body rotating about a principal axis of inertia but I don´t know how to derive this one.

    Thank you​
    Last edited: Mar 29, 2010
  2. jcsd
  3. Mar 29, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper

    Generally, if a rigid body is rotating about an arbitrary axis, the angular momentum need not point in the same direction as the rotation axis, as it does when [itex]\vec L = I \vec \omega [/itex] (for rotation about a principal axis).

    The relation between [itex]\vec L[/itex] and [itex]\omega[/itex] is still linear, and I is generally a tensor quantity (the inertia tensor).
    An object always has three principal axes and in that coordinate system the inertia tensor is diagonal. This leads directly to:
    \vec L = \vec u_{x} I_1 \omega_{x0} + \vec u_{y} I_2 \omega_{y0} + \vec u_{z} I_3 \omega_{z0}
    It's really the only thing it can be if you know [itex]\vec L = I \vec \omega [/itex] holds for principal axes, there are three principal axes and the correspondence between [itex]\vec w[/itex] and [itex]\L[/itex] is linear.
  4. Mar 29, 2010 #3
    Hello Galileo,

    Thanks for the answer. Unfortunately, I couldn´t follow it because I don´t know what a tensor is. I´m still a high school student. I guess I´ll just have to use it without knowing how to derive it. which is something I really hate.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook