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Homework Help: Moment of Inertia tensor SETUP, not to difficult but cant figure it out

  1. Feb 16, 2008 #1
    [SOLVED] Moment of Inertia tensor SETUP, not to difficult but cant figure it out

    1. The problem statement, all variables and given/known data
    a) For a cylinder of mass M, radius R and height h, calculate the inertia tensor about the center of mass. What are the principal axes?

    2. Relevant equations


    3. The attempt at a solution
    I need help with setting up the integral. I placed the origin at the center of the cylinder. So and i set my boundaries as = z going from -h/2 to h/2, y going from -R to R and x going from -sqrt(R^2-y^2) to sqrt(R^2-y^2).

    However when i integrate Ixx I get zero and I know I am not suppose to. Is there a better choice for my boundaries?? My x boundaries are the problem here because they give me a ugly answer.


    Attached Files:

  2. jcsd
  3. Feb 16, 2008 #2
    Try to use cylindrical coordinates, this will simplify the problem.
  4. Feb 16, 2008 #3
    so for Ixx will the integral be
    r^2sin^2(theta) + h^2 r^2sin(theta)dr d(theta) dz??

    thank you
    Last edited: Feb 16, 2008
  5. Feb 16, 2008 #4
    Not shperical coordinates, cylindrical ones! :smile:

    [tex]x=r\,\cos\phi,\,y=r\,\sin\phi \Rightarrow d\,x\,d\,y\,d\,z=r\,d\,r\,d\,\phi\,d\,z[/tex]

    thus [itex]I_{xx}[/itex] reads

    [tex]I_{xx}=\int_V \rho(r,\phi,z)\,\left(r^2\,\sin^2\phi+z^2\right)\,r\,d\,r\,d\,\phi\,d\,z[/tex]
    Last edited: Feb 16, 2008
  6. Feb 16, 2008 #5
    got it!! I feel so dumb spending so much time trying to figure it out in cartesian coord. Thank you so much!!
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