The stability of the static equilibrium state of a mechanical system is not always obvious. Such a case can present a problem that is not only interesting but can be readily demonstrated in the classroom as well. For example, Figs. 1 and 2(a) show a rectangular picture frame that is suspended by a cord that passes over a fixed pulley. The equilibrium is stable when the longer string is used and unstable with the shorter string.1

Topics
Equilibrium thermodynamics, Education
1.
The problem of the stability of a hanging picture, with the answer that the system is unstable for α > β, is presented in: David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of Physics, 4th ed. (Wiley, New York, 1993), p. 371. The problem does not appear in the 5th and 6th editions.
2.
There are two analytical methods for investigating the stability of a system with more than one degree of freedom. One method is to determine each small-amplitude normal mode where the system responds with a single frequency. The system is stable if the values of the square normal mode frequencies are all non-negative. The other method is to determine whether the potential energy matrix yields positive values for all small-amplitude displacements. Refer to: Grant R. Fowles and George L. Cassiday, Analytical Mechanics, 6th ed. (Saunders, Fort Worth, 1999), ch. 11. We have performed the potential energy matrix method for the rectangular frame problem and have confirmed the results of this paper.
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© 2005 American Association of Physics Teachers.
2005
American Association of Physics Teachers
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