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Moment of inertia of a torus

  1. Apr 17, 2009 #1

    rock.freak667

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

    Find the moment of inertia of a torus if mass is m and density [itex]\rho[/itex] is constant.
    The cross-sectional radius is 'a' and the radius is R.

    2. Relevant equations

    [tex]I= \int r^2 dm[/tex]

    3. The attempt at a solution

    Well I looked up the answer to be

    [tex]I_z= m(R^2 + \frac{3}{4}a^2)[/tex]

    But I am not sure how to start. Can someone just point me in the right direction?
     
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  3. Apr 19, 2009 #2

    tiny-tim

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    Hi rock.freak667! :smile:

    (have a rho: ρ :wink:)

    Do you mean the moment of inertia about its axis of rotational symmetry? And is R the internal radius, or the average radius?

    Hint: divide the torus into cylindrical slices of thickness dr, and integrate between R ± a :wink:
     
  4. Apr 21, 2009 #3

    rock.freak667

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    Yes R is the internal radius.


    So if I am considering cylindrical shells of thickness dr.

    if I draw it in 2d, it makes a circle such that [itex]x^2+z^2=a^2[/itex]


    the volume of a cylindrical shell is

    [tex]dV= \pi z^2 dr[/tex]

    dV=pi z2 dr

    so the moment of inertia of the small cylindrical element is

    [tex]dI_c = \frac{1}{2} (\rho \pi z^2 dr) z^2[/tex]

    dIc= (1/2) (p*pi*z2 dr)z2

    But this would give me the inertia not about the z-axis right but the axis perpendicular to the cylindrical shell. Which is not about the z-axis.

    and also I would be integrating z w.r.t. r
     
    Last edited: Apr 21, 2009
  5. Apr 21, 2009 #4

    tiny-tim

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    Hi rock.freak667! :smile:
    I'm not sure what your slices are, but that looks like the volume of a cylinder.

    A cylindrical shell is the (thickened) surface of a cylinder. :wink:
     
  6. Apr 21, 2009 #5

    rock.freak667

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    so dV= (2pi*z)x dr ? Not sure on the surface area of cylindrical shell that I'm considering
     
  7. Apr 21, 2009 #6

    tiny-tim

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    (have a pi: π :wink:)
    (2π*z)x dr ? :confused:

    what are z and x ?

    slice it with a cookie cutter of radius r, then slice it again with a cookie cutter of radius r + dr :smile:
     
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