Question #43637

A uniform thin spherical shell of mass $M$ and radius $R$ is free to rotate about a fixed vertical axis passing through its center $O$. A particle of mass $m = \frac{16}{9}M$ is projected from infinity in the horizontal equatorial plane of the shell, with an initial speed $v_0 = \sqrt{\frac{GM}{R}}$ and an impact parameter $b = \frac{3R}{4}$. The particle enters the shell through a small hole. Inside the shell, there is a rigid diametric spoke of negligible mass lying in the equatorial plane. The spoke is oriented such that the particle strikes it exactly at the particle's point of closest approach to the center $O$, and gets embedded in it. Assuming the shell and spoke are initially at rest, and considering only the gravitational interaction between the particle and the shell, find the final angular velocity of the system.