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The kinematics of a rolling wheel can get complicated quickly, especially for the accelerations which you need to solve kinetics (also called dynamics) problems. One of the nice things about this topic is that there are only two variations. The first is a wheel that is being driven by an applied torque (T) at its center, and the second is a wheel that is either being pushed or pulled by a horizontal force (P) at its center.
As always, start with a free-body diagram. Both free-body diagrams will have a weight (W) and normal force (N). Both will have a friction force (Ff), however, for the driven wheel, the friction force will be in the direction of motion, while the friction force for the push or pull wheel will be opposite to the direction of motion. The friction force for the driven wheel will cause linear acceleration, while the friction force for the push or pull wheel will cause angular acceleration.
Once the free-body diagrams have been drawn and the accelerations established, the equations of motion can be applied. For both wheels, 3 equations will emerge in 4 unknowns. A 4th equation will be obtained from kinematics, equating the linear acceration (ac) to the angular acceleration times the radius of the wheel (R x alpha). From there, these 4 equations can be solved simultaneously for whatever quantities are required.
Once I understood this process, the solution of rolling wheel problems became very straightforward. – Dr. Tom
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