Question #42259

A planet of radius $R$ has a spherically symmetric, non-uniform mass density profile given by $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$, where $r$ is the distance from the center. A straight, smooth tunnel is excavated through the planet, passing at a perpendicular distance $d = \frac{R}{2}$ from the center. A particle of mass $m$ is placed in the tunnel. It experiences a viscous drag force $F_{\text{drag}} = -b v$ (where $v$ is its velocity) and is driven by an external periodic force $F_{\text{ext}} = F_0 \cos(\omega t)$ directed along the tunnel. Assuming small oscillations about the midpoint of the tunnel ($x \ll R$), what is the resonant frequency $\omega_r$ at which the steady-state amplitude of the particle's oscillation is maximized? (Assume the system is underdamped).