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Modeling unfolding paper cylinder under given external moment

  1. Sep 1, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider a cylinder (roll) of paper rotating about it's axis. Paper density - [tex]\rho[/tex], cylinder length - [tex]L[/tex] and cylinder initial radius - [tex]R_0[/tex]. External torque - [tex]\tau(t)[/tex] is applied in the direction of cylinder axis. Given initial angular velocity - [tex]\omega_0[/tex] find the evolution of [tex]R(t)[/tex] and [tex]\omega(t)[/tex]

    2. Relevant equations
    Conservation of angular momentum, [tex]\frac{d}{dt}\left(Iw\right)=\tau(t)[/tex].
    Moment of inertia, [tex]I(t)=\frac{\pi L\rho R^4}{2}[/tex].
    Kinematic Relation, [tex]R(t)\omega(t)=V(t)[/tex]. (here [tex]V(t)[/tex] is the unwinding velocity of the paper sheet)

    3. The attempt at a solution
    I reach the following DAE, [tex]2\frac{\dot{R}}{R}+\frac{\dot{\omega}} {2 \omega} = \frac{\tau}{\omega R^4 \pi L \rho}[/tex]
    It seems I'm missing some relation between [tex]\omega(t)[/tex] and [tex]I(t)[/tex].
    Help is appreciated !
  2. jcsd
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