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Rotational Energy Deriving

  1. Dec 7, 2008 #1
    Hello, I think I've got the right idea on how to perform this question but I just need a little bit of help.

    1. The problem statement, all variables and given/known data

    Show that for a rigid body rotating with angular velocity [itex]\omega[/itex] the energy of rotation may be written as:
    [itex]E = \dfrac {1}{2}I\omega^{2}[/itex]

    where the moment of inertia of the body about the axis of rotation is given by:
    [itex]I = \int dV \rho r^{2}[/itex]

    where [itex]r[/itex] is the distance from the rotation axis to the volume element [itex]dV[/itex] and [itex]\rho[/itex] is the density of the object in that region



    2. Relevant equations
    [itex]E = \dfrac {1}{2} mv^{2}[/itex]
    [itex]v = \omega r[/itex]
    [itex] I = \int r^{2} dm[/itex]


    3. The attempt at a solution

    I can firstly identify that

    [itex]E = \dfrac {1}{2} mv^{2}[/itex] which looks similar to the rotation energy equation.

    I know that
    [itex]v = \omega r[/itex]

    But what's confusing me is that usually, the moment of Inertia is represented as
    [itex] I = \int r^{2} dm[/itex] and I don't really know how to link the two.

    Substiting angular velocity into the energy equation

    [itex] E = \dfrac {1}{2} m(r\omega)^{2}[/itex]

    but where do I go now?

    Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 7, 2008 #2

    CompuChip

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    Science Advisor
    Homework Helper

    So if you have a little volume dV at radius r, its mass is rho*dV, right? What is the kinetic energy of this little piece? For the total energy you should then sum (read: integrate) all these pieces.
     
  4. Dec 8, 2008 #3
    Thanks for the reply CompuChip, sorry for this late reply. I took onboard what you said, perhaps you could verify my answer.

    I have [itex] v = \omega r[/itex]

    [itex]I = \int dV \rho r^{2}[/itex]

    [itex] dm = \rho dV[/itex]

    [itex] I = \int \dfrac {dm}{\rho} \rho r^{2}[/itex]

    [itex] I = \int dm r^{2}[/itex]

    [itex] I = r^{2} \int dm[/itex]

    [itex] I = r^{2}m[/itex]

    [itex] m = \dfrac {I}{r^{2}}[/itex]

    [itex] E = \dfrac {1}{2} m v^{2}[/itex]

    [itex] E = \dfrac {1}{2} m \omega^{2} r^{2}[/itex]

    [itex] E = \dfrac {1}{2} \dfrac {I}{r^{2}} \omega^{2} r^{2}[/itex]

    [itex] E = \dfrac {1}{2} I \omega^{2}[/itex]

    Is that the best way?
     
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