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Moment of inertia tensor of three spheres

  1. Dec 7, 2011 #1
    Hi everyone

    1. The problem statement, all variables and given/known data

    I want to find out the moment of intertia tensor of the graphic below.



    2. Relevant equations

    parallel axis theorem



    3. The attempt at a solution

    We know the moment of inertia for one sphere, that's given, so I don't have to calculate it explicit.Now I have trouble understanding the moment of inertia tensor. It looks like the following:

    [tex]I=\begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{pmatrix}[/tex]

    I understand that Ixx, Iyy and Izz are the moments of inertia regarding to x y and z but I don't understand what Ixy Ixz Iyz etc. 'means'. I just can't picture it. Furthermore I tried to find the xx yy and zz, I found out :

    [tex] I_{xx}=I_{yy}=\frac {54} {5} MR^{2}+9r^{2}m[/tex]

    and

    [tex]I_{zz}=\frac 4 5 MR^{2}+(R+r)^{2}M[/tex]

    But how do I find the xy xz etc. I know I just have to calculate thre more, because the tensor is symmetrical. But I don't know how do to it. Is my solution right thus far?

    Thanks for your help
     

    Attached Files:

  2. jcsd
  3. Dec 7, 2011 #2

    vela

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    I take it M is the mass of the large spheres and m is the mass of the small sphere. How did you get 9mr2? Surely, the contribution of the small sphere must also depend on R. Also, you shouldn't have Ixx=Iyy as far as I can see.
    What happened to the small sphere? There's no m in your result.
    Crank out the integrals like
    [tex]I_{xy} = \int xy\,dm[/tex]
     
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