Question #42853

A massless disk of radius $R = 0.5 \text{ m}$ is pivoted at its center and can rotate freely in a horizontal plane. Two identical long strings, each of linear mass density $\mu = 0.1 \text{ kg/m}$ and under tension $T = 40 \text{ N}$, are attached to a pin on the rim of the disk. String 1 extends along the negative x-axis ($x < 0$) and String 2 extends along the positive x-axis ($x > 0$). The transverse displacement of the strings at $x=0$ is constrained to $y(0,t) = R\theta(t)$, where $\theta(t)$ is the angular displacement of the disk. The disk is subjected to a viscous rotational damping torque $\tau_d = -b \omega$, where $b = 1.0 \text{ N m s}$ and $\omega = \dot{\theta}$. Two rectangular wave pulses are sent towards the disk from opposite sides. The incident pulses are characterized by their constant transverse velocities $v_{y1} = \frac{\partial y_{i1}}{\partial t} = 0.4 \text{ m/s}$ (for Pulse 1 on String 1) and $v_{y2} = \frac{\partial y_{i2}}{\partial t} = 0.2 \text{ m/s}$ (for Pulse 2 on String 2). Both pulses last for a duration $t_0 = 0.5 \text{ s}$ and are timed such that they arrive at the disk simultaneously, overlapping completely. Calculate the total energy dissipated by the damping torque during this interaction.