Some pipe sections are gently nudged from rest down an incline and roll without slipping a...

Some pipe sections are gently nudged from rest down an incline and roll without slipping all the way to a step of height b. Assume that each pipe section does not slide or rebound against the step, so that the pipes move as if hinged at the corner of the step. Modeling a pipe as a uniform thin ring of mass m and radius r, and letting d be the height from which the pipes are released, determine the minimum value of d so that the pipes can roll over the step. Hint: When a pipe hits the corner of the step, its motion changes almost instantaneously from rolling without slip on the ground to a fixed axis rotation about the corner of the step. Model this transition, using the ideas presented in Section 5.2 on p. 356. That is, assume that there is an infinitesimal time interval right after the impact between a pipe and the corner of the step, in which the pipe does not change its position significantly, the pipe loses contact with the ground, and its weight is negligible relative to the contact forces between the pipe and the step.

Figure P8.76