The model presented by Cross uses a different definition of the coefficient of rolling friction than ours. There are multiple published definitions for the coefficient of rolling friction; ones similar to our definition appear in Refs. 1 and 2. We wrote in our paper, “we use a phenomenological approach that presumes the magnitude of the rolling friction force fr is proportional to the normal force N and has a direction opposite to the motion.” Our model uses a work–energy theorem approach that does not refer to any forces present that do zero work (such as static friction). The term “rolling friction” related to energy loss is not the static friction required for rolling without slipping. Our model does assume that the object must be rolling without sliding, which requires a static friction to exist, but the magnitude of the static friction is not considered in our model.
This phenomenological model allows a generalized approach to considering rolling resistive friction, whereas a detailed force vector analysis is limited to specific instances. For example, the model provided by Rod Cross assumes that the surface is rigid and the rolling object is deformable. The forces in that model differ from the forces when a rigid object rolls on a deformable surface, or, as is surely more common, a case where both the object and the surface are deformable.
Our model starts with the statement made in Ref. 3 that the accounting of friction is in fact like a book-keeper's accounting: If some part of energy disappears in the useless form, we can call it friction. We are confident that this approach should not cause confusion.
The purpose of our original article was to describe a lab experiment for students to calculate a parameter to describe energy loss for rolling objects rather than a rigorous mechanism analysis, and we believe that our lab experiment achieves that goal.