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# Wheel Climbing A Step

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Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 55

Wheel Climbing A Step

Introduction

The problem that follows was brought to my attention by a student at a great European university. It was a part of a much bigger problem with which the student was engaged, but this is quite enough to hold our interest. The results are differential equations of motion which could (at least in principle) be solved for the motion. For the purposes of the student, all that was of interest was the initial value of the angular acceleration of the wheel about the contact point on the step. On the one hand, it is an elementary problem, but it has a twist that makes it surprisingly interesting.
For this problem, we consider a wheel that rolls slowly across a horizontal surface to a vertical step as shown in Figure 1. All the dimensions and physical properties (mass, moment of inertia, etc.) are known. It is shown there just before the clock starts for this problem, that is, at t=0 . The velocity is zero (the approach is very slow), but the wheel will climb the step because it is driven by known forces (not shown in Figure 1),
Fx, positive to the right at Co,
Fy, positive upwards at Co,
T. a torque acting about Co

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