Question #43509

A particle of mass $m$ is constrained to move inside a narrow, frictionless tube that extends radially outward from the center of a spherical planet of mass $M$. The tube rotates with a constant angular velocity $\omega$ in the equatorial plane of the planet. The particle is initially at rest relative to the tube at a radial distance $R_0$ from the center of the planet. It is given that the mass of the planet is $M = \frac{8\omega^2 R_0^3}{G}$, where $G$ is the universal gravitational constant. An instantaneous radial impulse $J$ is applied to the particle. What is the minimum magnitude of $J$ required for the particle to reach $r \to \infty$?