Halliday, Resnick & Walker — Rolling, Torque, and Angular Momentum: Problem 16

16 Nonuniform cylindrical object. In Fig. 11-32, a cylindrical object of mass $M$ and radius $R$ rolls smoothly from rest down a ramp and onto a horizontal section. From there it rolls off the ramp and onto the floor, landing a horizontal distance $d = 0.506 \text{ m}$ from the end of the ramp. The initial height of the object is $H = 0.90 \text{ m}$; the end of the ramp is at height $h = 0.10 \text{ m}$. The object consists of an outer cylindrical shell (of a certain uniform density) that is glued to a central cylinder (of a different uniform density). The rotational inertia of the object can be expressed in the general form $I = \beta MR^2$, but $\beta$ is not 0.5 as it is for a cylinder of uniform density. Determine $\beta$.