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Torsional pendulum period

  1. Oct 18, 2009 #1
    1. The problem statement, all variables and given/known data
    I have to show the formula derivation of this:
    [tex]T=2*pi*\sqrt{\frac{2IL}{pi*r^{4}*G}}[/tex]

    based on the fact that I know this:
    [tex]\tau=I\alpha=\frac{pi*G*r^{4}}{2L}\theta[/tex]


    2. Relevant equations
    See above


    3. The attempt at a solution

    Well, I know T=2pi/[tex]\omega[/tex] and [tex]\alpha=\Delta\omega/\Delta(t)[/tex]

    So I decided to just get an equation for omega from the expression for tau.
    So I had:
    d[tex]\omega[/tex]/dt=[tex]\frac{pi*G*r^{4}}{2IL}\theta[/tex]
    Which looked promising until I integrated both sides wrt 't' and got:
    [tex]\omega=\frac{pi*G*r^{4}}{2IL}\theta*t[/tex]

    And this really gets me nowhere and I don't know what else to do. Thanks in advance for the help!
     
  2. jcsd
  3. Oct 18, 2009 #2
    You mixed up two VERY different [tex]\omega[/tex]'s

    One is angular frequency and the other is angular velocity. They are completely unrelated.

    Look at your net torque equation, it is a differential equation of the form [tex]\ddot x=-kx[/tex] (Remember that it is a restoring torque, so you missed a negative sign)

    You should be very familiar with the general solution to that equation.

    I suggest that you use [tex]\Omega[/tex] for angular velocity instead, to prevent further mixups.
     
  4. Oct 18, 2009 #3
    Well, it's familiar and it's from SHM.

    There was the d''f(t)/dt''=-omega^2 * f(t) when f(t)=Asin(omega*t+phi)

    If I make theta(t)=f(t), then that k would equal omega^2 (the angular frequency omega)

    Is that correct though?
     
  5. Oct 19, 2009 #4
    That is 100% correct. :)

    Once you have the differential equation:

    [tex]\ddot \theta=-\Omega^2 \theta[/tex]

    The solution should be something that immediately pops into your head:

    [tex]\theta (t) = A\cos{(\Omega t +\phi)}[/tex]

    And the period for a harmonic function is something you can easily find,

    [tex]T=\frac{2\pi}{\Omega}[/tex]

    On a side note, when you tried to integrate:

    [tex]d\omega = -k\theta \cdot dt[/tex]

    You overlooked the fact that [tex]\theta[/tex] is a function of time. That was the source of your error. I was mistaken in thinking you got angular velocity and frequency mixed up.
     
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