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Torsional oscillator with angular displacement

  1. Apr 24, 2013 #1
    1. A torsional oscillator of rotational inertia 2.1 kg·m2 and torsional constant 3.4 N·m/rad has a total energy of 5.4 J.
    What is its maximum angular displacement?
    What is its maximum angular speed?




    2. Relevant equations
    θ(t)=Acosωt



    3. The attempt at a solution
    still trying to think on how to use the energy given, I can't relate kinetic energy...
     
  2. jcsd
  3. Apr 24, 2013 #2
    I'm not sure if i completely understand the question, so please feel free to correct me.

    Now, consider a torsional oscillator: f.ex. a mass at the end of a wire: The mass has the angular inertia, [itex]I[/itex], and the wire it's torsional constant, [itex]\tau_{c}[/itex], with values as specified. Assuming the total energy is conserved (no friction or outside forces), the total energy will be the sum of: (I) the potential energy, [itex]E_{pot}[/itex], corresponding to the work needed to rotate the mass to some angular displacement. (II) the kinetic energy, [itex]E_{kin}[/itex], due to the mass' velocity and angular inertia.

    If you study the units of the torsional constant, [itex]\tau_{c}[/itex], you see that [N*m / rad] = [J / rad], applying that work is given from force times distance (newton meters). Hence, if the energy you submitted is the total energy, [itex]E[/itex], then the maximum angular displacement, [itex]\theta_{max}[/itex], will be when,
    [itex]
    E_{kin}=0 \\
    E_{pot}= E = 5.4 J
    [/itex]

    corresponding to the potential energy stored in the wire by 'twinning it up',
    [itex]
    \tau_{c} \cdot \theta_{max} = E_{pot}
    [/itex]

    As you'll easily calculate yourself, inserting correct values and solving for [itex]\theta_{max}[/itex].


    The maximum angular velocity, [itex]\omega[/itex], will then be at the point of rotation where,
    [itex]
    E_{pot}=0 \\
    E_{kin}= E = 5.4 J \\

    E_{kin} = \frac{1}{2} \cdot I \cdot \omega ^2
    [/itex]

    Also there are all necessary values given: just solve for [itex]\omega[/itex]. That should be it. Good Luck!
     
    Last edited: Apr 24, 2013
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