Title: Force and torque of a string on a pulley,
Abstract: Every university introductory physics course considers the problem of
Atwood's machine taking into account the mass of the pulley. In the usual
treatment the tensions at the two ends of the string are offhandedly taken to
act on the pulley and be responsible for its rotation. However such a free-body
diagram of the forces on the pulley is not {\it a priori} justified, inducing
students to construct wrong hypotheses such as that the string transfers its
tension to the pulley or that some symmetry is in operation. We reexamine this
problem by integrating the contact forces between each element of the string
and the pulley and show that although the pulley does behave as if the tensions
were acting on it, this comes only as the end result of a detailed analysis. We
also address the question of how much friction is needed to prevent the string
from slipping over the pulley. Finally, we deal with the case in which the
string is on the verge of sliding and show that this will never happen unless
certain conditions are met by the coefficient of friction and the masses
involved.