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Derivation of moment of inertia of a solid sphere

  1. Dec 18, 2015 #1
    1. The problem statement, all variables and given/known data
    I need to know why my derivation does not work. I am attempting to derive I=2/5 mR^2

    2. Relevant equations
    I have seen people derive it using disks but my question is why do the shells not work? Where in my set up did I go wrong??? Thanks

    3. The attempt at a solution
    I am attempting to use shells to do this.
    Ro=p=m/v so pdv=dm
    moment of inertia (I) =definite integral from r=0 to r=R (4p(pi)r^4dr)

    When I solve this integral it becomes (3/5)mR^2 and not (2/5) mR^2
  2. jcsd
  3. Dec 18, 2015 #2


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    Hello, and welcome to PF!

    What is the moment of inertia of a single thin shell of mass dm and radius r? (Hint: It is not r2dm)
  4. Dec 18, 2015 #3
    Well in general moment of inertia R^2dm and I am using the thin shells to eventually make up the entire volume so I can sub out for dm in terms of r
  5. Dec 18, 2015 #4


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    If all the mass in the element dm is at the same distance r from the axis of rotation, then the moment of inertia about that axis will be r2 dm. For a spherical shell, is all of the mass in the shell at the same distance from the axis?
  6. Dec 19, 2015 #5
    oh I see. thank you
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