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Inertia tensor of a crosshanger

  1. Sep 5, 2014 #1
    I've done many exercises about inertia tensors of 3D bodies and sticks but now I have this exercise and I got stuck without any idea of how to do the integration to compute the inertia tensor. The statement is this:

    "Compute the inertia tensor of a cross-hanger consisting of 3 thin and linear wires, with mass M and lenght L, glued perpendicularly by their central parts which is placed at (x,y,z)=(0,0,0)."​

    As you know, the inertia tensor (or matrix) for an homogeneous body is constructed from

    [itex]I_{ij}=\int\rho(\delta_{ij}\sum_{k}x_{k}^{2}-x_{i}x_{j})dV[/itex]

    So that, for example,

    [itex]I_{11}=\int\rho(x_{2}^{2}+x_{3}^{2})dV[/itex]

    If it was a cube for example we could integrate the expression effortlessly with the integral limits in the 3 variables going from [itex]-L/2[/itex] to [itex]L/2[/itex]. But with this shape I'm not sure how to do it. Can you help me with the integration? Should I fix one variable at 0 and integrate the others?
     
    Last edited: Sep 5, 2014
  2. jcsd
  3. Sep 6, 2014 #2
    The inertia tensor of a body that consists of multiple parts is the sum of the inertia tensors of the parts.
     
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