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Help me sort this out

  1. Apr 29, 2009 #1
    NOTE: THIS IS NOT A HW PROBLEM
    While calculating the moment of inertia of a cone, I made a mistake and got confused. Although I juggled the numbers and got the right answer, I have a doubt.

    Say I have to calculate the centroid of a cone z2 = r2.

    0 < z < h

    It is quite obvious that the x and y co-ordinates of the centroid are zero. As for the z co-ordinate it can be calculated as follows:

    [tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r2 dV =

    [tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r2 r dr d[tex]\Theta[/tex] dz =

    [tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r3 dr d[tex]\Theta[/tex] dz = [tex]\Pi[/tex]h5/10

    But instead, in step 2 if I substituted r = z (from the equation of the cone), I get a completely different answer.

    [tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z2 z dr d[tex]\Theta[/tex] dz =

    [tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z3 dr d[tex]\Theta[/tex] dz = 2[tex]\Pi[/tex]h5/5

    I got a similar doubt while calculating the center of mass, wherein I plugged in z = r in the equation and got a different (and wrong) answer.

    Shouldn't I get the same answer either way? Why don't I get it?

    Anirudh
     
  2. jcsd
  3. Apr 29, 2009 #2

    tiny-tim

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    Hi Anirudh! :smile:

    (have a theta: θ and a pi: π :wink:)

    You've probably got the limits wrong …

    what limits did you use?
     
  4. Apr 29, 2009 #3
    :rofl:

    For the first one I used:

    [tex]\int[/tex][tex]^{2\Pi}_{\Theta = 0}[/tex][tex]\int[/tex][tex]^{h}_{z = 0}[/tex][tex]\int[/tex][tex]^{z}_{r = 0}[/tex] r3 dr d[tex]\Theta[/tex] dz

    For the second one I used:


    [tex]\int[/tex][tex]^{2\Pi}_{\Theta = 0}[/tex][tex]\int[/tex][tex]^{h}_{z = 0}[/tex][tex]\int[/tex][tex]^{z}_{z = 0}[/tex] z3 dr d[tex]\Theta[/tex] dz

    I used the same limits because r = z. Is that wrong? (Obviously, it MUST be as the answer isn't right)
     
  5. Apr 29, 2009 #4

    tiny-tim

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    hmm … looking again, these are different integrals …

    ∫∫∫ r2 is not the same as ∫∫∫ z2

    z2 = r2 only on the boundary, not everywhere in the middle …

    (and you can't transform away r dr anyway)
     
  6. Apr 29, 2009 #5
    Okay I understood that r = z only on the boundary. What does the above statement mean though?
     
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