Question #43266

A spherical ferromagnetic rotor of volume $V = 1.0 \times 10^{-3} \text{ m}^3$ and moment of inertia $I = 5.0 \times 10^{-3} \text{ kg m}^2$ is free to rotate about a fixed vertical axis. A uniform horizontal magnetic field of magnitude $B = 2.0 \text{ T}$ is applied and forced to rotate about the same vertical axis at a constant angular velocity $\omega_B = 150 \text{ rad/s}$. Due to magnetic hysteresis, the rotor's magnetization vector $\vec{M}$ maintains a constant magnitude $M = 1.0 \times 10^4 \text{ A/m}$ but lags behind the rotating magnetic field $\vec{B}$ by a constant angle $\delta = 30^\circ$. The rotor experiences a viscous drag torque $\tau_d = -b \omega$, where $b = 0.2 \text{ N m s}$ and $\omega$ is the rotor's angular velocity. Once the rotor reaches its steady-state terminal angular velocity $\omega_0$, what is $\omega_0$ and the rate of internal heat generation $P_h$ within the rotor strictly due to magnetic hysteresis?