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Inertia matrix of a homogeneous cylinder

  1. Nov 7, 2016 #1
    1. The problem statement, all variables and given/known data

    3d45da.png

    2. Relevant equations
    N/A

    3. The attempt at a solution
    What I am confused about is where they got the (1/4)mR^2 + (1/12)ml^2 and (1/2)mR^2 from? I am guessing that these came from the integral of y'^2 + z'^2 and x'^2 +y'^2 but I don't understand how this happened exactly? Could someone point me in the right direction?

    Thanks
     
  2. jcsd
  3. Nov 7, 2016 #2

    Orodruin

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    Did you try actually computing the integrals? What did youget? Please show your work.
     
  4. Nov 8, 2016 #3
    What are y'^2 and z'^2? I don't know what to sub in for these (not sure what they represent?) so can't do the integral until I know them.
     
  5. Nov 8, 2016 #4

    Orodruin

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    They are coordinates as shown in the figure.
     
  6. Nov 8, 2016 #5
    I understand that but I mean how would one integrate those terms with respect to m? They aren't constants?
     
  7. Nov 8, 2016 #6

    Orodruin

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    They depend on where in the body you are.
     
  8. Nov 23, 2016 #7
    Sorry for the late reply. I am still confused. I just want to know how they (mathematically) got from the left hand side to the right. I don't understand how integrating the left hand side yields the right hand side.
     
  9. Nov 23, 2016 #8

    Orodruin

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    Use ##dm = \rho \, dx'dy'## and integrate over ##x'## and ##y'## with the appropriate boundaries. The value of ##\rho## in terms of the total mass ##m## can be inferred by computing the volume ##V## of the body and using ##\rho V = m##.
     
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