Halliday, Resnick & Walker — Rolling, Torque, and Angular Momentum: Problem 13

13 Nonuniform ball. In Fig. 11-29, a ball of mass \(M\) and radius \(R\) rolls smoothly from rest down a ramp and onto a circular loop of radius \(0.48 \text{ m}\). The initial height of the ball is \(h = 0.36 \text{ m}\). At the loop bottom, the magnitude of the normal force on the ball is \(2.00Mg\). The ball consists of an outer spherical shell (of a certain uniform density) that is glued to a central sphere (of a different uniform density). The rotational inertia of the ball can be expressed in the general form \(I = \beta MR^2\), but \(\beta\) is not \(0.4\) as it is for a ball of uniform density. Determine \(\beta\).