Almost all introductory physics classes will, at some point, include springs and elastic forces. When studying such topics, it is interesting to consider the spring system shown in Fig. 1. In this system, two identical springs are arranged with the top of one spring anchored to the ceiling and the bottom of the second spring attached to a hanging weight. The two springs are then connected by a “middle” string, c. There are two other strings that are not tensioned (strings d and b), one connecting the ceiling and the bottom spring, and the other connecting to the hanging weight and the top spring. As indicated in Fig. 1, these two “support” strings have some slack in them.
REFERENCES
1.
J. E.
Cohen
and P.
Horowitz
, “Paradoxical behavior of mechanical and electrical networks
,” Nature
352
, 699
–701
(1991
).2.
C. M.
Penchina
and L. J.
Penchina
, “The Braess paradox in mechanical, traffic, and other networks
,” Am. J. Phys.
71
, 479
–482
(May 2003
).3.
L.
Baker
, “Removing Roads and Traffic Lights Speeds Up Urban Traffic
,” Scientific American website, www.scientificamerican.com/article.cfm?id=removing-roads-and-traffic-lights (Jan. 28, 2009
).4.
D.
Braess
, A.
Nagurney
, and T.
Wakolbinger
, “On a paradox of traffic planning,” Transportation Sei.
39
(4
), 446
–450
(2005
).5.
T.
Roughgarden
, Selfish Routing and the Price of Anarchy
(MIT Press
, Cambridge, MA
, 2005
).6.
B.
Calvert
and G.
Keady
, “Braess paradox and power-law nonlinearities in networks
,” ANZIAM J.
35
, 1
–22
(1993
).7.
Sarah's research paper with all data can be found at sp.eths. k12.il.us/vondracekm/Shared%20Documents/Student%20 Research%20Papers/Counterintuitive%20Behavior%20in%20 Mechanical%20Systems%20(networks)%20−%20Sarah%20Peters.pdf.
© 2012 American Association of Physics Teachers.
2012
American Association of Physics Teachers
AAPT members receive access to The Physics Teacher and the American Journal of Physics as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
