Halliday, Resnick & Walker — Rolling, Torque, and Angular Momentum: Problem 80

Rolling into a loop. In Fig. 11.39, three objects will be released from rest at height \(h = 41.0 \text{ cm}\) to roll smoothly down a straight track and into a circular loop of radius \(R = 14.0 \text{ cm}\). The objects are (a) a thin hoop, (b) a solid, uniform disk, and (c) a solid, uniform sphere, each with radius \(r \ll R\). Determine if the object will reach the top of the loop (without falling off the track) and calculate its speed there.