Question #43675

A rigid, heavy deformed nucleus of mass $M$ is initially at rest. Its moment of inertia about an axis passing through its center of mass and perpendicular to the scattering plane is $I_0$. The surface of the nucleus in the scattering plane is given by the polar equation $r(\theta) = R_0(1 + \beta \cos 2\theta)$, where $\beta \ll 1$ is a dimensionless deformation parameter, and $\theta$ is the polar angle. An alpha particle of mass $m$ is fired along the radial line $\theta = \pi/4$ towards the center of the nucleus with an initial speed $v_0$. It strikes the frictionless surface of the nucleus. Assuming the collision is perfectly elastic, what is the angular velocity $\omega$ acquired by the nucleus after the collision? (Assume $I_0 \gg m R_0^2$ such that the rotational recoil energy of the nucleus is negligible, and neglect translational recoil).