Mathematics-Physics High School SMILE Meeting
21 November 2000
Notes Prepared by Porter Johnson

Don Kanner (Lane Tech HS)
He first announced the answer to a question previously posed: for n equal resistors combined in irreducible planar networks, the number of different total resistances is 2n-1.

Then he began a discussion of centripetal [inward toward center] versus centrifugal [inertial forces that appear to act away from a center of rotation] forces.  When a mass is attached to a string held in the hand and swung in a circle, the mass experiences a centripetal [not centrifugal] force, being continually accelerated inward toward the center of the circle.  Conversely, the hand cannot swing the mass while being held perfectly still; in fact the hand moves (roughly) in a circle, so that the net force acting on it is centripetal, as well.

Comment by Porter Johnson [IIT]:  the earth and moon rotate about their center of mass, which lies inside the earth.  An astronomical investigation of this "earth wobble" was made in the first determination of the mass of the moon.

Bill Blunk [Joliet Central HS]
He used the mini-camera first presented at the ISPP Meeting a year ago and shown at a SMILE class last year [ph102699.htm].  He cut a circular piece of plywood and attached it to a lazy susan, attached the camera to the plywood to the table, and ran the cable up to the ceiling so that the system could rotate freely for several turns.  Then, he set various objects on the rotating table, and we saw their motion [as seen from the table] on the big TV screen. Specifically, he used these objects:

The effects shown in this live demonstration were nicely presented in the PSSC Movie entitled Frames of Reference.

Fred Schaal [Lane Tech HS]
displayed the output of his TI 92 Calculator on the screen. He chose the "geometry" option and drew a triangle.  Then he added a line, and reflected the triangle about the line.  He noted that the second triangle could not be superposed on the first one, because it was "left-handed".  Next, he added another [non-parallel] line, and made reflection of the second triangle.  He showed that the third triangle could be superposed by rotation onto the first one.  Thus, two reflections correspond to a spatial rotation.

Eduardo De Santiago [IIT Civil and architectural Engineering]
presented the Bridge Design Lecture, in preparation of the Bridge Contest [] to be held at IIT on 13 February 2001.  He said that the simplest type of bridge is the "plank bridge" bridging a gap while supported on both ends.  When you stop in the middle of the plank, it sags under your weight.  The most noticeable effect is that of the bending moment, which causes the plank to "curl up".  These bending moments are the most evident in bridge design, although shear forces [transverse action-reaction pairs at opposite ends of the board] are also important. The bending moment causes the top of the plank to be compressed and the bottom to be extended. The bending moment produces the greatest stresses at the top and bottom of the plank and decrease to zero at center.  Therefore, the material in the center of the plank is being "wasted", since the greatest stress [force per unit area] is at the top [compression] and bottom [extension].

We may make a bridge more efficient by building a hollow beam with a few vertical supporting members [like a ladder turned on its side].  This construction reduces the effect of the bending moment, but increases that of shear stress.  One may reduce the effect of shear forces by putting diagonal brace members into the network.

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The object of design of wooden bridges is to convert bending moments and shear forces into longitudinal or axial forces [extension or compression], because wood is very strong under these forces. Also, we reduce the effect of shear by using triangles rather than rectangles, because rectangles collapse easily under shear, whereas triangles do not. Thus, the bridge geometry should consist entirely, or almost entirely, or triangles. In the above figure, you can remove the end members, which are subject to practically no stress, to simplify the construction to the following, known as a truss
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A bridge consists of trusses on the sides, as well as a deck on the bottom. To avoid a collapse at the top, you should include bracing at the top, again consisting of triangles like the first figure. You should line up vertical joints exactly with horizontal joints to avoid "punch through". There should also be lateral bracing at the top, to avoid shear in the transverse structure.

The bridge should be "left-right symmetric", since is one side is weaker than the other it will break first. Remember that the weakest part of the bridge always breaks first under loading. These bridges are operating is a "near failure zone", which is not the regime in which large bridges are designed to operate. Engineers are necessarily conservative in their designs, and one should become an "anti-engineer" to win the contest.

These bridges may undergo "buckling", since wood is more resistant to tension than to compression. Under compression, a "slender" piece may buckle. Therefore, one should keep the compressional members short and fat.

Real bridges are also subject to "impact loads", produced by fast moving trains, trucks, winds, and even earthquakes. They can be ignored in the contest, so long as you remember "not to drop the weights on the platform", etc.

In the San Francisco Earthquake more than a decade ago, the lower deck of the Bay Bridge collapsed under action of a wave set up by the earthquake. The structural members of the bridge remained sound, however.

Notes taken by  Porter Johnson