A closed horizontal tube is completely filled with an incompressible liquid of density $\rho_\text{L}$. A small ball of volume $V$ and density $\rho_\text{B}$ (where $\rho_\text{B} < \rho_\text{L}$) is placed inside the tube. The ball is attached to the left end of the tube by a massless spring of spring constant $k$, relaxed length $\ell_0$, and negligible volume. The entire tube is then rotated in a horizontal plane with a constant angular velocity $\omega$ about a vertical axis passing through its left end. Assuming the ball can slide without friction along the inside of the tube, what is its equilibrium distance $r$ from the axis of rotation?