We report on an investigation of student ability to account for static equilibrium in the simple familiar case in which an object is balanced on a frictionless pivot or fulcrum. Written questions were administered to more than 1000 university students who had completed the relevant instruction in introductory calculus-based physics. Almost all the students were able to answer questions about simple systems composed of point-like objects. However, when the mass distribution was continuous, most students attributed equilibrium to forces of equal magnitude applied on both sides of the fulcrum. Moreover, many students treated horizontal and tilted bodies, even if they were at rest, as distinct cases. The difficulties we identified were very persistent. Hands-on activities that were not influenced by research results had no discernible effect on student performance. Direct attempts to address specific difficulties using lecture demonstrations based on the research tasks described in this article led to some improvement. Greater success has been achieved by using a tutorial in which students work in small groups on experiments and exercises suggested by research findings.