# Homework Help: Modeling unfolding paper cylinder under given external moment

1. Sep 1, 2010

### saxin81

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

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

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