Torsional oscillator with angular displacement

[Robertoalva]

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?

Homework Equations

θ(t)=Acosωt

The Attempt at a Solution

still trying to think on how to use the energy given, I can't relate kinetic energy...

 

[mhsd91]

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, I, and the wire it's torsional constant, \tau_{c}, 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, E_{pot}, corresponding to the work needed to rotate the mass to some angular displacement. (II) the kinetic energy, E_{kin}, due to the mass' velocity and angular inertia.

If you study the units of the torsional constant, \tau_{c}, 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, E, then the maximum angular displacement, \theta_{max}, will be when,
<br /> E_{kin}=0 \\<br /> E_{pot}= E = 5.4 J<br />

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

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


The maximum angular velocity, \omega, will then be at the point of rotation where,
<br /> E_{pot}=0 \\<br /> E_{kin}= E = 5.4 J \\<br /> <br /> E_{kin} = \frac{1}{2} \cdot I \cdot \omega ^2<br />

Also there are all necessary values given: just solve for \omega. That should be it. Good Luck!