1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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?

    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


    User Avatar
    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook