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Inertial tensor calculation

  1. Apr 20, 2005 #1
    my homework problem deals w/ rotation in x-y plane. so the tensor is only 2d. inertial tensor still seems obscure to me... my question for now is purely mathematical. assuming the basis are (x,y). I calculated the components of I, is the following correct?
    [tex]I_{xx} = m_i y_{i}^{2}[/tex]
    [tex]I_{yy} = m_i x_{i}^{2} [/tex]
    [tex]I_{xy} = I_{yx} = - m_i x_i y_i [/tex]

    for some reason my professor wrote...
    [tex]I_{xx} = m_i x_{i}^{2} [/tex]
    and i'm pretty sure its not right
    Last edited: Apr 20, 2005
  2. jcsd
  3. Apr 20, 2005 #2


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    You're fine (besides the missing summation symbols). Think of it this way: if you are rotating about the x-axis, would your inertia depend on where you are along the x-axis or how far away you are from the x-axis.
  4. Apr 20, 2005 #3
    what about the off diagonal terms? are they moments about x=y line?... also do you need orthogonal basis for calculating inertial tensor? from the general definition it doesn't look like a requirement.
  5. Apr 20, 2005 #4


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    Sorry, I don't have a good analogy in mind for the products of inertia; maybe someone else reading the thread can help us out. I guess it comes into play when you have rotation that is not along one of the principal axes.
    I suppose you could calculate the inertial tensor using a different basis, but wouldn't that just complicate the algebra(and unnecessarily at that)? I'm sorry if I haven't been much help.
  6. Apr 21, 2005 #5


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    The full tensor (diagonal and off diagonal terms) is important for rotations about any axis through the origin in the x.y plane, not just the y = x line. The tensor product of I with the angular velocity vector gives the angular momentum vector. When the angular velocity vector is along the x or y axes, the off diagonal terms do not contribute.
  7. Apr 21, 2005 #6
    I can see why angular momentum is in same direction as angular velocity if object rotates about principal axes only. My textbook derives the inertia tensor through rotational kinetic energy calculation. I can see that it relates energy w/ velocity... but i don't understand in what way products of inertia describe the relationship... perhaps i just need to do some problems and see how it all works out.
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