Can a String's Tension Exert a Torque on a Pulley?

摘要

Atypical textbook problem in rotational dynamics involves calculating the angular acceleration of a massive pulley due to a string, such as in the example shown in Fig. 1. The string is assumed to be massless and to move without slipping over the pulley, which is mounted on a frictionless axle. If T_L and T_R are the tensions pulling at the left and right edges of the pulley (see Fig. 1), respectively, the net torque on the pulley is then τnet = (T_L- T_R)R, where R is the radius of the pulley. (It is assumed that positive torque corresponds to the counterclockwise direction.) While this analysis, which is typical of what is found in many introductory physics texts,~1 is correct, it should raise several questions in the mind of a student. First, since most texts argue that the tension everywhere in a massless string is constant,~2why is T_L≠T_R? Second, since tension is an internal force (except at the ends of the string, which are obviously not tied to the pulley),~3 how can tension exert a force and torque on a pulley? In this paper, we will address these questions, which are overlooked in most textbook treatments of this problem whose approach appears inconsistent with the concepts pre-sented elsewhere in the text.