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In this paper we present two methods, the nonholonomic method and the vakonomic method, for deriving equations of motion for a mechanical system with constraints. The resulting equations are compared. Results are also presented from an experiment for a model system: a ball rolling without sliding on a rotating table. Both sets of equations of motion for the model system are compared with the experimental results. The effects of various forms of friction are considered in the nonholonomic equations. With appropriate friction terms, the nonholonomic equations of motion for the model system give reasonable agreement with the experimental observations.