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

Nonuniform cylindrical object. In Fig. 11.17, 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 \(\beta\).