# Apparatus Competition

2007 AAPT Summer Meeting

Greensburo, NC

Coupled Oscillations with a Suspension Bridge Model

Department of Physics

Kenyon College

Gambier, Ohio 43022

740 437-2989

Abstract

A flexible metal sash chain is suspended between supports, and hanging weights are suspended uniformly at horizontally-spaced locations along the chain. The original intent was to show that the catenary formed by the freely-hanging cable is transformed into the parabola of the uniformly-loaded bridge. Serendipitously, I discovered that pairs of equal length weight suspensions symmetrically located on either side of the center acted as coupled pendulums.

Construction of Apparatus:

The model bridge was first constructed for an enrichment lecture during the 1972-3 academic year at the University of the West Indies, Kingston, Jamaica campus. The lecture on suspension bridges was then taken on tour to high schools in Jamaica and the Cayman Islands. The picture below shows it with some of my Jamaican students, and was published in an article on suspension bridges in the January 1984 issue of The Physics Teacher.

A length of about 4 m of metal sash chain is suspended freely from laboratory stands on either end. At equal distances, as measured along the horizontal, eleven identical weights are suspended from strings attached to opened-up paper clips passing through the links of the chain. The lengths of the strings are adjusted to make all of the masses the same height above the table, thus simulating the deck of the bridge.

Today I would estimate that the cost of the sash chain would be no more than \$5. Instead of the collection of identical hooked weights that I had available, I would use large steel nuts, which might cost \$0.50 apiece.

Use of Apparatus:

In my original lecture demonstration I used the light from a slide projector to cast a shadow of the freely-hanging chain onto the blackboard, and traced it with chalk. The students could then see that the parabola formed by the loaded bridge was slightly deeper in the center than the catenary of the same length.

When I set this demonstration up in early May 2007 for my annual end-of-the-year demonstration lecture at, I noted that setting the extreme left-hand suspension string swinging as a pendulum in a direction perpendicular to the length of the string caused only the corresponding pendulum on the right-hand side to start swinging. Clearly this was an example of resonant coupled oscillations, with only the pendulum of the proper resonant length responding. As expected, the two pendula came alternately to rest and then started swinging once more. This behavior was noted with all of the longer pendula.

So, here we have examples of geometric form (catenaries and parabolas), coupled oscillations and resonance – three demonstrations for the price of one!