A problem on the Atwood's machine is often introduced early in the teaching of dynamics to demonstrate the application of Newton's Laws to the motion of a compound system. In a series of preliminary studies, student understanding of the Atwood's machine was examined after this topic had been covered in a typical calculus-based course. Analysis of the data revealed that many students had serious difficulties with the acceleration, the internal and external forces, and the role of the string. The present study was undertaken to obtain more detailed information about the nature and prevalence of these difficulties and thus provide a sound basis for the design of more effective instruction. The context for the investigation is a group of related problems involving less complicated compound systems. Specific examples illustrate how this research, which was conducted primarily in a classroom setting, has served as a guide in the development of tutorial materials to supplement the lectures and textbook in a standard introductory course.