High School Mathematics-Physics SMILE Meeting
04 December 2001
Notes Prepared by Porter Johnson

Porter Johnson (IIT Physics) Generator or Alternator?
passed out an article by Bob Weber [sobriquet Motormouth] that appeared in the Chicago Tribune [http://www.chicagotribune.com/] on 16 July 2001, entitled By alternate name, device still generates electricity.  The article began with the following question::

Q:  Years ago they were called generators, and now they are called alternators.  What is the difference?

 In answering the question, Bob / Motormouth had made the following points:

Bill Blunk (Joliet Central HS, Physics) Fan Car, Revisited
showed us a refinement of his presentation of the Fan Car on 20 November 2001 [see the write-up for the November 20 High School Math-Physics SMILE Meeting], in which he asked us whether we knew where to get flexible, braided wire with lower electrical resistance than his earphone cord.  He said that the answer is often right around us --- and especially around Christmas time!  He found that wire from burned-out Christmas tree lights worked very well.  Bill placed the fan car on an incline, and we observed a "stall angle" of 8° to 10°, as compared with a maximum of around obtained with smaller wires last time.  Thanks, Bill!  Persistence pays big dividends.

Bill also showed off his set of Choositz Decision Balls [which are also called "Happy" and "Sad" Balls], ordered from the Fall 2001 Educational Innovations Catalog.  You may call EI directly at the toll-free number 1 - 888 - 912-7474, or else visit their website, http://www.teachersource.com/.  Here is the information on this item that appears on their website:

 SS-3 Choositz Decision Balls

These are the largest "Happy-Sad" balls we have seen! They are over 1-1/4" in diameter! These two black rubber balls appear identical, but have extremely different physical properties. Dropped onto a hard surface, one ball bounces high while the other hits the floor and stops immediately. Show students that some properties cannot be observed without experimentation. Set includes two balls (one 'yes' and one 'no').

Qty Cost
1-10 $6.95
11 p $6.25

Bill Shanks  (happily retired from Joliet Central HS, Physics) Mixing of Colors 
is taking a course using software packages such as Adobe Photoshop® or Photo Deluxe® to edit digital photographic images.  He showed us some very impressive images, including a shot of the Sawtooth Mountains in Idaho.  The slide had been damaged by creases or scratches, but by digitizing the slide image using a scanner and repairing the damage using software, he produced a beautiful 8 ´ 10 print

Bill remarked that the "layering" of images in these photo editors is truly amazing, and raised the question of whether image overlays are treated "additively" or "subtractively".  He tested the question by forming yellow, cyan, and magenta circles in the image editor with a (grey background), and overlaying each pair of circles:  He held up a page showing us these results in for overlapping regions:

Subtractive Combination

Overlap colors Color produced
yellow + cyan green
cyan + magenta blue
yellow + magenta red

Bill led us to the conclusion that the colors are treated subtractively in the imaging program, as in the image of a color printer.  The inks in a desk-jet printer are yellow (red and green transmitted), cyan (blue and green transmitted), magenta (red and blue), and black (nothing transmitted).  Thus, when white light (all colors) is incident on a printed page, we obtain the above reflected colors, the rest being absorbed by the ink.  

By contrast, the electron guns in a color television picture tube strike phosphors in the screen that emit light of specific colors:  red, green, and blue.  The colors are produced additively, with following results:

Additive  Combinations

Overlap colors Color produced
red + green yellow
red + blue magenta
green + blue cyan

Porter Johnson mentioned that the "red" phosphors in picture tubes involve rare earth salts, and add significantly to the cost of the tubes.  Very interesting, Bill!

Fred Farnell (Lane Tech HS) Why is the flag on the Moon Wrinkled in the standard "moon landing" photo?
The standard image of the astronauts putting a flag on the moon shows the flag wrinkled and waving.  Why?  The "moon landing hoax" people have used this point, among others, to forward their case.  For a discussion of the scientific basis for this and other moon hoax effects, see the website http://www.badastronomy.com/bad/tv/foxapollo.html.  The NASA website [http://science.nasa.gov/headlines/y2001/ast23Feb_2.htm] contains specific discussion of how the astronauts managed to put a flag on the moon. Good Question, Fred!

Don Kanner (Lane Tech HS, Physics) Jury Duty Physics
described a [let us say, purely hypothetical] situation in which two occupants of the front seat of a car made 3 consecutive left turns in the car, while traveling at  a constant speed of about 15 miles/hour [5 meters/sec].  The passenger described the following sequence of events

  1. He was forcefully thrown forward
  2. He then got his sleeve caught in the door latch
  3. He then was forcefully thrown backward, and the door opened
  4. He then was forcefully thrown out of the car during the third left turn.

There was some discussion as to whether this sequence of events could have occurred as described.  Was either the passenger or the driver at fault?  What do you think about this purely hypothetical situation?

Bill Colson (Morgan Park HS, Mathematics) New Toys
used his $100 equipment allotment from CPS to obtain blackboard drawing materials from the K-12 Mathematics and Science Catalog for Fall 2001 of the EAI http://www.eaieducation.com/.

Eric Armin Inc (EAI) Education
567 Commerce Street
PO Box 644
Franklin Lakes NJ 07417-0644
1 - 800 - 770-8010
In particular, he purchased these items:

He showed us how well the compass worked on the blackboard; then we tossed the ball around for a while and answered the questions. Useful stuff. Thanks, Bill!

Leticia Rodriguez (Peck School) Experiment with Magnets, Continued
We divided into teams of two and performed an experiment using disc-shaped, pressed ceramic magnets and paper clips.  On a flat horizontal table, we brought the magnet and the paper clip [which lay directly away from the magnet] closer together, until the paper clip jumped over to the magnet.  Then each group recorded the distance of separation (jump distance) between the paper clip and the magnet at which the jump occurred. The experiment was repeated with 2, 3, and 4 magnets stacked on top of one another and one paper clip. The measurements were distributed in the following way.

Magnets and Paper Clip
Number of Magnets Distance (mm)
1 3- 5
2 6 - 9
3 10 - 13
4 12 - 15

We drew a graph of the jump distance versus the number of paper clips, which showed that with more magnets, the separation distance increases. We also experimented with the magnets as we moved them closer to one another. Do they attract or repel? This is a simple yet very interesting exercise in magnetism. Good work, Leticia!

Ann Brandon (Joliet West HS, Physics)
passed out a newspaper article describing an internet-based,  virtual bridge building contest sponsored by the U S Military Academy at West Point, NY,  using West Point Bridge Designer computer software available without cost at the contest website, http://bridgecontest.usma.edu/. Students may compete either individually or in teams in this contest, which marks the bicentennial of the USMA.  The prizes to winners are rather generous:

$15K [First], $10K [Second], and $5K [Third].

You may also obtain information by email: ic7097@usma.edu or by telephone at 1 - 845 - 938-2548. Thanks, Ann!

Hoi Huynh (Clemente HS) Areas of Polygons and the Pythagorean Theorem
showed us a balanced, right procedure for calculating the area of any polygon, using the standard formula for the area of a trapezoid:


Area = [ (top length + bottom length) /2 ] ´ height = [ (a + b)/2 ] ´ h

The balance comes in because one must "balance" the length of the top and the bottom  (a+b)/2, and the right comes in because one must use the "right angle" or perpendicular distance between the sides; ie, the height h  We can use the same principle for other figures


For the parallelogram the top and bottom sides are equal, and we get Area = a ´ h.


For the triangle the top side has length zero, and the bottom side has length a, so that the average is a/2, Thus, Area = a ´ h / 2.

By taking any polygon and cutting it into trapezoidal pieces, she showed us how to calculate its area.

Next she pointed out that the area of a circle of radius r [such as a pie] is equal to pie ´ r2, or p r2. [PJ comment:  You could slice the pie into ever smaller equal-sized pieces, and turn the pieces alternately "up" and "down", to make a figure that becomes a simple trapezoid in the limiting case. Half of the pie edge is on top, and half is on the bottom, so that the "height" of the trapezoidal figure is r, the circle radius.  Thus, by the trapezoidal formula, in the limiting case the area is   [ (p r+ p r) /2] ´ r, or p r2].

Next Hoi showed us how to prove the Pythagorean Theorem using area formulas.  She took a square of edge length (a + b), and divided each of the edges into components of lengths a and b, as shown:

The total area of the square is (a + b) 2.  Inside that square, there is a tilted smaller square of side c, with area c2.  In addition, there are four congruent right  triangles, each with sides a, b, and hypotenuse c.  Thus, the area of the larger square may be written as the sum of the area of the inner square plus the areas of the four (identical) triangles. ...

(a + b) =  c2 + 4 [a  ´ b / 2]
Writing out both sides of this expression, we get
a2 + 2a ´ b +b2 =  c2 + 2a  ´ b

a2 + 2a ´ b +  b2 =  c2 + 2a  ´ b

with the result
a2 + b2  =  c 
which is the Pythagorean Theorem.

Hoi told us that Mrs Pythagoras really invented the theorem while she was supposed to be knitting her husband's socks, and she let him pretend to discover it on his own. Perhaps this interesting story may not be completely accurate:

Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east of the heal of southern Italy) that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were [2]:-
  1. that at its deepest level, reality is mathematical in nature,
  2. that philosophy can be used for spiritual purification,
  3. that the soul can rise to union with the divine,
  4. that certain symbols have a mystical significance, and
  5. that all brothers of the order should observe strict loyalty and secrecy.
Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers. The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day. They were allowed their own possessions and were not required to be vegetarians.
Source: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

Very nice, Hoi!

Notes taken by Porter Johnson