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Energy of a solid rotating sphere

  1. Oct 15, 2009 #1
    1. The problem statement, all variables and given/known data

    A system of particles, forming a sphere of uniform mass density [tex]\rho[/tex] and radius R, rotates around the axis of the sphere with angular velocity [tex]omega[/tex](t) calculate the energy of the system.



    2. Relevant equations

    we were told to solve the problem with this integral:

    E = [tex]\int[/tex] [tex]\rho[/tex] v2 (r) d3r


    3. The attempt at a solution

    I took [tex]\rho[/tex] = M/( 4/3 [tex]\pi[/tex] R3)

    v3 = r2 [tex]\omega[/tex]2

    and d3 r = r^2 sin [tex]\theta[/tex] dr d[tex]\phi[/tex] d[tex]\theta[/tex]

    integrating from 0 to R for r, from 0 to pi for theta, and from 0 to 2 pi for phi, I got

    3/5 M R^2 [tex]\omega[/tex]^2

    I think the answer ought to be 1/5 MR^2 w^2, right? Thanks for the help.
     
  2. jcsd
  3. Oct 16, 2009 #2

    tiny-tim

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    Welcome to PF!

    Hi wduff! Welcome to PF! :smile:

    (have a pi: π and an omega: ω and a theta: θ and a phi: φ and a rho: ρ and try using the X2 tag just above the Reply box :wink:)

    You've used v = ωr, but it's not the same r, is it? :wink:

    (and it's 2/5 … see http://en.wikipedia.org/wiki/List_of_moments_of_inertia)
     
  4. Oct 16, 2009 #3
    Aha! Thanks a ton tiny tim! And yeah sorry about the terrible formatting I'm sure I'll get the hang of it.
     
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