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Calculating the moment of inertia of a solid sphere

  1. Apr 24, 2017 #1
    1. The problem statement, all variables and given/known data
    To calculate moment of inertia of a solid sphere of uniform density



    2. Relevant equations
    $$ I = \int r^2 dm$$

    The attempt at a solution
    upload_2017-4-24_11-47-28.png
    I consider an elemental disk of small thickness ##d\theta##
    ##dm = \frac{M}{4/3 \pi R^3}*\pi R^2\cos^2\theta* Rd\theta##
    Therefore ##dI = r^2 dm = R^2\cos^2\theta dm## r is the distance from the axis to the disc which is same as ## R\cos\theta##
    Then I do the integration ##I = \int_\frac{-\pi}{2}^\frac{\pi}{2} dI##

    As I am getting a ##\cos^4\theta## I am not able to get the answer ##2/5 MR^2##
    Can someone tell me if I have gone wrong anywhere in what I have written above?
     
  2. jcsd
  3. Apr 24, 2017 #2

    andrewkirk

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    Your first formula implies that ##R\,d\theta## is the height of the incremental disc. It is not. It is the length of the slanted edge of the disc. What do you have to multiply that length by to give the height?

    Also, the moment of inertia of the incremental disc is not ##r^2\,dm##. Look at this list of moments of inertia to see what it should be.
     
  4. Apr 24, 2017 #3
     

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  5. Apr 24, 2017 #4

    andrewkirk

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    Yes. You can use multiple angle formulas to do this.
    Alternatively, you might prefer to integrate with respect to ##z## instead of ##\theta##. That gives an easier integration - but the same result of course.
     
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