Question #43305

A ferromagnetic solid sphere of moment of inertia $I = 0.1 \text{ kg m}^2$ is driven by a motor providing a constant external torque $\tau_0 = 0.53 \text{ N m}$ in a uniform magnetic field. The rotation induces eddy currents, dissipating power $P_e = k \omega^2$, where $k = 0.02 \text{ N m s}$ and $\omega$ is the angular velocity. Additionally, magnetic hysteresis dissipates an energy $E_h = 0.06\pi \text{ J}$ per complete revolution. If the sphere starts from rest, what is its steady-state angular velocity $\omega_0$ and the time $t$ taken to reach $\omega = \frac{\omega_0}{2}$?