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Moment of Inertia of a solid sphere

  1. Nov 3, 2014 #1
    1. The problem statement, all variables and given/known data

    Taylor, Classical Mechanics Problem 10.11 **
    a) Use the result of problem 10.4 (derivation of the general integral for a moment of inertia of a continuous mass distribution in spherical coordinates, using point particles) to find the moment of inertia of a uniform solid sphere for rotation about a diameter.
    b) Do likewise for a uniform hollow sphere whose inner and outer radii are a and b. [One slick way to do this is to think of the hollow sphere as a solid sphere of radius b from which you have removed a sphere of the same density but radius a.]

    2. Relevant equations

    ## I = \int r^2 dm ## - 1

    ## dm = \rho dV ## - 2

    ## \rho = \frac{M}{\frac{4\pi R^3}{3}} = \frac{3M}{4\pi R^3} ## - 3

    ## dV = r^2 sin \theta dr d\theta d\phi ## - 4

    3. The attempt at a solution

    I'm limiting it to part a for now, since that's where I got stuck. The problem statement is "any diameter", and I'm going to center the sphere on the origin and have it rotate about the z-axis, for convenience. The previous problem it references is a derivation of the general integral in spherical coordinates, so that's the system I'll be using here.

    First, combining the equations to make a volume integral:

    ## I = \frac{3M}{4\pi R^3} \iiint r^4 sin \theta dr d\theta d\phi ##

    Where the integration bounds are:

    ## r: 0 \rightarrow R ##
    ## \theta: 0 \rightarrow \pi ##
    ## \phi: 0 \rightarrow 2 \pi ##

    First integrating phi:

    ## I = \frac{3M}{2 R^3} \iint r^4 sin \theta dr d\theta ##

    And then theta:

    ## I = \frac{3M}{R^3} \int r^4 dr ##

    And finally r:

    ## I = \frac{3M}{R^3} \frac{R^5}{5} = \frac{3}{5}MR^2 ##

    But I know this is wrong, as the answer I learned in freshman mechanics was

    ## I = \frac{2}{5} MR^2 ##

    And I can't seem to figure out why the answer is incorrect.
     
  2. jcsd
  3. Nov 3, 2014 #2

    Orodruin

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    ##r## in the definition of the moment of inertia is the distance from the axis of rotation, not the spherical coordinate ##r##.
     
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