Ivars Peterson's MathTrek

November 10, 1997

# Springing a Physical Surprise

The philosopher and mathematician Bertrand Russell (1872-1970) once remarked that "a logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science."

Russell was keenly interested in how we come to know what we know and in how we draw conclusions from evidence. For him, such questions required c lose study of logic puzzles and paradoxes to help expose potential cracks in structures of belief.

Similarly, physical paradoxes and surprises call attention to flaws in our reasoning about the world around us. Indeed, intuition can sometimes le ad us astray, and we may then encounter a paradox that forces us to recalibrate our thinking.

Consider a weight hanging from a spring, which in turn is suspended by a piece of string from an identical spring attached to the ceiling. Cutting the connecting string would send the weight and the lower spring plummeting to the floor.

Now add two "safety" strings to the original arrangement. One string joins the upper end of the lower spring to the ceiling. An identical string joins the lower end of the upper spring to the weight. Both safety strings initially hang limply. When the taut string in the middle is cut, the safety strings prevent the weight from plunging all the way to the floor.

Intuition suggests that, given the safety strings' slack, the weight will end up hanging somewhat lower than before. The surprise is that for certain combinations of sprin gs, string lengths, and weights, the opposite is true. Cutting the middle string and letting the safety strings carry the load actually pulls the weight above its initial position and closer to the ceiling.

With string lengths (in meters), spring constants (k), and weight (in newtons) as shown, cutting the middle string will raise the weight.

That physical paradox was originally developed by Joel E. Cohen, an applied mathematician at Rockefeller University in New York City. The idea for this startling demonstration arose out of Cohen's long-standing interest in mathematical models of biological competition, especially models that produce counterintuitive outcomes. For example, one model involving traffic flow, discovered in 1968 by the German researcher Dietrich Braess in the context of communications networks and now known as Braess's paradox, demonstrates that adding extra roads to a congested transportation network may actually increase the amount of congestion rather than relieve it.

As a step toward learning whether the same kind of surprising result could occur in a biological system, Cohen started by looking for a mechanical analog of the traffic paradox, and he came up with the string-spring arrangement described above. In fact, it's fairly straightforward to construct a working model of such an apparatus by using strings, rubber bands, and a plastic jug partially filled with water as a weight.

Cohen then turned to physicist Paul Horowitz of Harvard University for an electrical version of the original mechanical setup. Horowitz designed an electric circuit in which appropriate resistors replaced the springs and devices known as Zener diodes replaced the strings. In this case, adding extra current-carrying paths had the effect of reducing the current flow through the circuit. The same sort of behavior occurs in hydraulic systems in which lengths of tubing and pressure-relief valves replace springs and strings.

The strings-and-springs conundrum actually presents no surprise to anyone familiar with elementary mechanics. Once the slack safety strings are erased from the diagram in which they are not involved, it is obvious that the springs in one diagram are arranged in parallel and that those in the other diagram are in series. The weight on each spring in series is twice that in parallel. The effect on weight height is then not surprising in the least.

Nonetheless, when confronted by the initial spring-and-string configuration, most people are inclined to guess that the weight will go down. Few automatically predict that the weight will go up unless they have already worked out the problem or have seen it before. Only when someone knows the answer does it become "obvious." You can't rely on your intuition unless it already incorporates an understanding of springs in series and parallel arrangements.

Moreover, most people fail to recognize the same sort of behavior in analogous, though more complicated situations, such as transportation networks, and some even deny that it can occur in those cases. In one tragic situation, a routine engineering change during construction altered the arrangement of nuts and bolts supporting a set of balconies in a Kansas City hotel from a series to a parallel configuration. When the nuts failed to support the added weight of people gathered on the balconies, the balconies collapsed, leading to considerable loss of life.

The strings-and-springs mechanical analogy provides a useful hint about the kind of surprising behavior that may lurk in a wide range of networks. The general lesson is that physical networks may not necessarily behave as expected when paths or components are added.

### References:

Bass, Thomas. 1992. Road to ruin. Discover (May):56-61.

Blum, Fredric M. 1991. Retrospective chuckle. Science News 140(Sept. 21):179.

Cohen, Joel E. 1988. The counterintuitive in conflict and cooperation. American Scientist 76(November-December):577-584.

Cohen, Joel E., and Paul Horowitz. 1991. Paradoxical behaviour of mechanical and electrical networks. Nature 352(Aug. 22):699-701.

McKelvey, Steve. 1992. Braess's paradox: A puzzler from applied network analysis. UMAP Journal 13(No. 4):303-312.

Peterson, Ivars. 1997. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.

______. 1991. Strings and springs net mechanical surprise. Science News 140(Aug. 24):118.

Podell, Howard I. 1992. Real-life failure. Nature 355(Feb. 20):6832E