Mathematics Physics SMART High School Meeting
Website: http://www.iit.edu/~smile/
24 October 2000
Notes prepared by Earl Zwicker

Fred Schaal (Lane Tech HS)
set up the projector to display the output of a TI 92 calculator so we all could see it, and gave  the TI 92 to Betty Roombos to manipulate. Fred asked Betty to draw a triangle and label its vertices with  A,   B,   C and we saw it as it happened. Then he advised Betty what to do to construct the bisectors of the angles at the vertices A and B. The angle bisectors intersected at a point which was labeled F. Next, he had Betty construct a ray connecting vertex C with the point of intersection, F. The angles ACF and BCF were "measured" by manipulation of the TI 92 and labeled with their values, which both happened to be 37.05 o! This showed that the constructed ray was a bisector of the angle at vertex C, and was convincing evidence that the bisectors of all three angles intersect in a common point! Wow!

Then Fred asked Betty to "grab" the vertex point C and move it around. As this was done, we could see the shape of the triangle ABC change, and the values of the two angles also changed, but remained equal. Very nice! Similar exercises may be done to show that medians meet at a point; perpendicular bisectors of sides meet at a point. Porter Johnson asked: What about the "Nine Point Circle Theorem"? Anyone know? Check the website http://mathworld.wolfram.com/Nine-PointCircle.html.

A perfect complement to what you did last time, Fred! (...the actual construction on the white board, which did not display it well.)

John Bozovsky (Bowen HS)
asked us if anyone knows where to get replacement parts for a Pasco air track setup. Where can they be found. No one seemed to know, although there were suggestions. Porter Johnson suggested using Pasco's websites http://www.pasco.com and http://www.pasco.com/products/John responded that he had actually talked to them on the phone, without success. Maybe we'll find out next time!

Betty Roombos (Gordon Tech HS)
held up some Mr Coffee™ paper coffee filters. She picked one, held it out, then released it. It fell to the floor at what appeared to be a constant speed. (It turns out that terminal velocity is reached in a negligibly short time, so that it is indeed a good assumption that coffee filters fall with constant velocity.) So - Betty did a Drop Contest.  As viewed by us, she held up two coffee filters - nested together - on the left, and a single coffee filter on the right, both at the same height - one meter - above the floor. Which will reach the floor first? was her question to us. After we made our guesses, Betty released them simultaneously; the two nested filters reached the floor first. So that we all would have no doubts, Betty repeated the experiment twice more; same result.

Next, Betty asked, "How high above the floor must I hold the nested two, so that when I simultaneously release them and the single filter - still held one meter above the floor - they will reach the floor at the same time?"

(In what follows, we changed Betty's direct proportion notation into equalities following after Porter Johnson's notation, since it is easier to write up that way.) Betty used the relationship F = kv2, for an object acted upon by a constant force F, and falling through air with a constant speed v. For the coffee filter,

F   =   filter weight   =  mg,
so,
mg   =   kv2.
From this one may solve for v and construct the ratio,
v2/v1   =   (m2/m1)0.5.
Then the ratio of the mass of the nested 2 filters (m2) to the mass of the single filter (m1) is 2, Thus
v2/v1  =   20.5   =   1.41.
Since v  =  d / t, it follows that
d2/d1 =   v2/v1  =   1.41.
With d1 = 1 m, the distance d2 must be 1.41 m. With assistance from some of us to hold meter sticks at those heights, Betty held the single filter at 1 m above the floor, and the nested two at 1.41 m. Counting down - 3, 2, 1, blast off! --- Betty released them simultaneously, and they fell, reaching the floor exactly at the same time! Beautiful! She repeated this twice more, and we were convinced that it worked - and very well!

Someone pointed out that if a nested 3 filters were used, they would need to be released from a height above the floor of

30.5 m = 1.73 m.
So - she did this - and again, it worked perfectly. [Porter Johnson pointed out that these filters are very useful for dropping experiments, since they are flat on the bottom, but that cone-shaped filters are better for the purpose of filtering coffee!]

Great stuff, Betty

Larry Alofs (Kenwood HS)
asks his students, How high does a ping pong ball bounce? Students usually respond with, That depends on how high up you drop it. Larry's students then would find that beyond a certain height, no matter how much higher the point of drop was, a ping pong ball would not bounce any higher than about 3 meters! Similarly to a coffee filter, a ping pong ball reaches its terminal velocity fairly quickly (compared, say, to a baseball). Thus, no matter how much greater the drop height (beyond a certain height), the velocity at impact with the floor is the same terminal velocity, so the bounce height is the same.

Larry then went on to his presentation titled: Repulsive Fruit. He had fashioned out of copper wire (no.12?) and an inverted beaker a kind of balance. The wire behaved like a teeter totter, about 16 inches long, and supported at its center by the sharp point of one of its ends resting on the glass of the inverted beaker; this formed a low friction bearing. (The wire drooped down from the center toward both ends, giving it stability, with its center of gravity below the point of support.) While the wire was free to move up and down vertically, it was also free to rotate in a horizontal plane about the vertical axis through its point of support on the beaker.

Next, Larry took a grape from a cluster of grapes he had bought at a store. He stuck the grape onto one end of the wire teeter totter. The entire affair balanced about that central support point. Larry then took a small but very powerful, rare-earth magnet, and moved it carefully to within a millimeter or two of the grape. Any force acting between the magnet and the grape would result in rotation of the wire. The grape (and the wire) very slowly began to move away from the magnet, rotating in a circular path. Larry moved the magnet so that it continued to stay within a millimeter or so of the grape. Soon the wire teeter-totter-with-grape was obviously rotating in a circle, and it was clear that the magnet must have exerted a small repulsive force on the grape! To help convince us, Larry repeated the experiment, but this time to produce rotation in the opposite direction. Larry told us that a piece of apple worked also, and a rubber stopper worked, but with a noticeably weaker repulsive force. A glass marble, taped to the end of the wire, worked better than the rubber stopper, but not as good as the grape. The repulsive force is believed due to diamagnetism associated with the water molecules in fruit. See snacks on the Exploratorium Museum [San Francisco] website, and particularly the webpage http://www.exploratorium.edu/snacks/diamagnetism_www/index.htmlLarry gave us copies of that page.

He also drew a diagram on the board showing two horizontal slabs of bismuth (a diamagnetic material) separated by a small distance. An electromagnet placed above the upper slab produces a magnetic field in the space between the slabs, and a small but strong, permanent magnet may then be made to "float" when placed in the space between the slabs.

What a beautiful way to show diamagnetism in action! Thanks, Larry!

Don Kanner (Lane Tech HS)
told us about a puzzle he had thought of which seemed to tie physics and math together. (Handout). Given n identical resistors, if they are connected in various series/parallel combinations, [all planar, non-intersecting, irreducible] is there some sort of regular sequence of total resistance values that might be generated? For example, if each resistor has a value of 10 Ohms

:
 number series parallel other combinations 1 10 Ohms - - 2 20   " 5 - 3 30   " 3.3 15 6.7 4 40   " 2.5 16.7 13.3 ... 5 50   " 2.0 ...

Don said the real difficulties begin with n = 5. Equations used by electrical engineers called the Y-delta and delta-Y Transformation Equations would be needed to eliminate circuits with the same resistance. The number of possible combinations increases rapidly with n, making it difficult to investigate.

• Is there some sort of rigorous approach that might be used to go beyond the tedium of working out each possible combination?
• Could there be a Pascal's Triangle of values for such combinations?

Don next asked us the following trivia question:

Given the following sequence of numbers

777       _ _ _ _ _       999999,

what numbers belong in the blanks?

Interesting questions, Don. Answers, anyone?

Porter Johnson (IIT Physics)
gave us some insight into paramagnetism (materials with unpaired electrons). Such materials are weakly attracted to both poles of a magnet. Diamagnetic materials are weakly repelled by both poles of a magnet. Bringing a magnetic pole near creates electron orbital currents in such a direction as to set up an opposing magnetic field (Lenz's Law), producing repulsion. In ferromagnetic materials such as iron, cobalt, nickel, the magnetic fields of spinning electrons in the atoms become completely lined up within small regions called domains. Each domain is completely magnetized. But the domain directions of magnetization tend to be random, and their magnetic fields average out to zero in the material. However, if the material is placed in an external magnetic field, the domains re-orient so their directions of magnetization line up, resulting in very strong magnetism of the material. They then hold themselves in alignment after the external field is withdrawn, because of the strong internal and external field they produce. It is not until the temperature is raised to a certain point (the Curie Temperature) that the thermal motion of the atoms destroys the magnetic alignment within domains, and the strong magnetism disappears.

What a fascinating meeting!
Don't miss the next one!
SEE YOU THERE!!
Notes by Porter Johnson