Question #42410

A thin uniform ring of mass $m$ and radius $R$ is released from rest at a vertical height $h = 30 \text{ m}$ on a rough downward incline. It rolls without slipping until it reaches the lowest point of the track (height $0$). At the lowest point, the track smoothly transitions via a short frictionless circular fillet to a frictionless upward incline of angle $\theta = 30^\circ$ with the horizontal. The upward incline ends abruptly at a vertical height $H = 10 \text{ m}$ above the lowest point, from where the ring becomes a projectile in the air. Neglecting air resistance, find the total time of flight of the ring from the moment it leaves the track until it hits the horizontal ground (which is at height $0$). Take $g = 10 \text{ m/s}^2$.