Question #43521

A solid planet of radius $R \text{ (in m)}$ and total mass $M \text{ (in kg)}$ has a non-uniform, spherically symmetric density distribution $\rho(r) = \rho_0 \left(1 - \frac{r^2}{R^2}\right) \text{ (in kg m}^{-3}\text{)}$, where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. A narrow, frictionless tunnel is drilled along a diameter of the planet. A particle of mass $m \text{ (in kg)}$ is placed inside the tunnel at a distance $x_0 \text{ (in m)}$ from the center, such that $x_0 \ll R$, and released from rest. If $g \text{ (in m s}^{-2}\text{)}$ is the acceleration due to gravity at the surface of the planet, what is the time period $T \text{ (in s)}$ of the resulting small oscillations of the particle?