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Moment of inertia of spherical shell

  1. Sep 20, 2010 #1
    1. The problem statement, all variables and given/known data
    Moment of inertia of spherical shell of radius R, mass M along its rotation axis is given by [tex]\frac{2}{3}MR^{2}[/tex]
    I am trying to calculate this

    2. Relevant equations
    3. The attempt at a solution
    This is my attempt but is unsuccessful,
    since the spherical shell is an assembly of rings (of varying radius), and the MI of a ring is
    Hence [tex]dI=y^{2}dm[/tex]
    [tex]I=\int y^2(2\pi \sigma ydz[/tex]
    Using [tex]y=Rsin\theta[/tex] and [tex]z=Rcos\theta[/tex]
    I get:
    [tex]I=2 \pi \sigma R^{4} \int sin^{4}\theta d\theta
    =2 \pi \sigma R^{4} \frac{3\pi}{8}
    =\frac{3\pi MR^{2}}{16}[/tex]
    which is incorrect.

    Which step I have gone wrong? Thanks
    Last edited: Sep 20, 2010
  2. jcsd
  3. Sep 20, 2010 #2


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    Homework Helper

    The constant surface charge refers to the spherical surface element which is R^2 sinθ dφ dθ in the spherical polar coordinates. After integrating for φ for a ring, it is dA=2πR^2 dθ. You have to multiply this by σ to get dm, and by the square of the distance from the axis, (Rsinθ)^2. So you have only sin^3 in the integrand.

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