High School SMILE Meeting
21 February 2006

Fred Schaal (Lane Tech HS)       Constructing Points on an Ellipse
showed us how to make points on an ellipse using the blackboard and only a (chalk) compass and a straight edge. An ellipse is defined as a geometric shape with two focal points (foci), so that the sum of the distances from any point on the ellipse to each  is always the same. Fred drew the two foci (F) on the board and then a third point (P), as shown.

/ \
x / \ y
/ \
* *
Distance between foci (*) =  2 e a
x + y = 2a
Eccentricity = e
Semi-major axis = a
This third point along with the two foci will allow us to unambiguously draw the ellipse which goes through this third point.  Fred used the compass to measure and lay out -- on a straight line -- the total distance (x + y) from the two foci to the point P. Then he used the compass to measure a new-x and a new-y which summed to the same, original distance 2a.  With the compass set at the new-x, Fred drew an arc at distance new-x from F.  Then he set the the compass at the new-y and drew an arc at distance new-y  from the other F. The point at the intersection of the arcs was thus located at distance x from F and at the same time at distance y from the other F -- so it was another point P on the same ellipse!  Repeating this process for different pairs of x + y = 2a located more points, all on the ellipse. And so the entire ellipse could be defined, 
point-by-point!  For details see the website  http://www.du.edu/~jcalvert/math/ellipse.htm. Neat, Fred!

Larry Alofs (Kenwood Academy, retired)             Pan Pipes, Fresnel Lenses, and Hall Effect Sensors
had made a Pan pipe by taping together 8 PVC  pipes (about 1 cm in diameter) and varying, in length, the shortest being about 6.5 cm  = 0.65m. Larry noted that the length of the tube should be one quarter of the wavelength (l) of the fundamental tone. For the 0.065 m tube, the wavelength would be l = 4 * 0.065m = 0.26 m. Now the frequency f of the tone will be given by f = V/l, where V = 350 meters/second is the speed of sound. Thus the frequency should be f = 1350 Hz.

Next, Larry held up - for all to see - a transparent plastic sheet, about 35 cm square. He showed us that it magnifies like a convex lens, despite being flat. Called a Fresnel lens, it has the advantages of being
flat, lighter weight, and less expensive than an equivalent convex lens of glass. A Fresnel lens may be thought of as formed from a convex glass lens. Imagine removing from its surface a narrow and thin ring of glass, concentric with its optical axis. Place the ring flat on a flat, transparent surface. Then remove the next larger ring and place it to surround the first ring. Continue this process until the entire glass lens surface has been placed on the flat, transparent surface as a series of thin glass rings, each having the curvature of the original convex lens surface from which it was removed. This would then be a Fresnel lens, and would focus light like the original convex glass lens.  The Fresnel lens was common in old light houses to make a focused, intense beam. For details see the Michigan Lighthouse  Conservatory website:  http://www.michiganlights.com/fresnel.htm.

Porter noted that Augustin Fresnel  1788-1827 [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fresnel.html] developed the wave theory of diffraction that led to the following striking prediction:

"Let parallel light impinge on an opaque disk, the surrounding being perfectly transparent. The disk casts a shadow - of course - but the very centre of the shadow will be bright. Succinctly, there is no darkness anywhere along the central perpendicular behind an opaque disk (except immediately behind the disk)."

When the existence of the bright spot was experimentally confirmed, the wave theory of light became accepted by virtually everybody.

Finally, Larry held up what we saw as a small (about a cubic inch) black object with wires coming out. "This is a Hall Effect Sensor," Larry  told us. He explained that he had a problem with a car that was hard to start once it had been warmed up and turned off, and he had traced the problem to the Sensor. Larry made a sketch on the board and explained how  the Sensor works. Suppose a strip of semiconductor conducts a current along  its length. If a magnetic field is produced transversely to the current, electrons are diverted toward one side of the strip, producing an electric field across the strip. This is the Hall Effect. For details see http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/hall.html.  In the car, a shaft (synchronized with the engine's crankshaft) rotates and moves a magnetic disk into a slot of the semiconductor strip, producing a Hall Effect electric field. A transistor detects the field and triggers other circuitry to fire a spark plug and to activate a fuel injector.  This Hall Effect Sensor replaces the points in old fashioned engines in delivering high voltage sparks; in addition, it controls the fuel injectors. For details see this Wikipedia website:  http://en.wikipedia.org/wiki/Hall_effect_sensor. The faulty sensor in Larry’s Saab apparently was misbehaving only when it got too hot, so that the semiconductor behaved more like a conductor. Larry worked very hard to take out the sensor from the Saab and made a circuit to test it.

Porter pointed out that modern Hall Effect Sensors use semiconductors rather than conductors because the rate of flow of individual current carriers (electrons) is really slow in conductors, and it is much faster in the semiconductors  -- because the latter have so many fewer current carriers, which travel at much greater drift speeds. This produces a "Hall voltage" that is large enough to detect. 

Fascinating stuff!  Thanks, Larry.

Walter McDonald (CPS substitute teacher and radiation technologist at the VA)                    Sangaku
told us about “sangaku”, Japanese temple geometry, which was a very popular pastime in Japan in the Edo Period (when there were Samurai), 1603 - 1867. See http://www.wasan.jp/english/.  The participants figured out solutions to geometrical problems and puzzles and then recorded the solutions on beautiful wooden tablets. These tablets were sometimes hung under the roofs of shrines and temples. Walter got the example from the book Play Thinks by Ivan Moscovich  [http://books.google.com/books/about/1_000_Play_Thinks.html?id=fBzVCzuFLWoC].  Walter gave us one such geometrical problem as an example. It is a problem similar in spirit to those given in high school geometry. See the Mathworld web page http://mathworld.wolfram.com/CircleInscribing.html for a discussion of this problem.

Roy Coleman (Morgan Park HS, retired!)                Torques
described a useful way to teach the right hand rule.

t = R ´ F
That is, the torque t is equal to the cross product of radius R and the force F. Let the radius R represent  “your right aRm”, the force F  “your Fingers”, and the torque t "your thumb". Point your right aRm in the direction of the first vector R and its bent Fingers in the direction of the second vector F; then the thumb will point in the direction of the torque (cross product)

Tres simple, non! Thanks, Roy.

Bill Blunk (Joliet Central HS, retired)                       Ping-pong Electrostatics
  had bought a gross of ping pong balls and sprayed them with silver conductive paint. He made a pair for everyone! With a  monofilament string attached to connect the pair of balls, he hung the balls from the ceiling -- like a pair of pendulums -- (hanging by about 2 meters) so that they were next to each other in contact. With this setup there are a lot of fun things to do!

Before experimenting with this setup, Bill rubbed a plastic rod with a piece of fur, and then rubbed a loop (about 25 cm diameter) made from a strip of light, plastic, packing foam. The resulting charges on the two pieces of plastic permitted Bill to levitate the plastic ring above the rod and move it around the room! Then he rubbed the  rod again and touched both ping-pong balls to it, giving them like charges so that they repelled each other, and served as an electroscope -- unlike most other electroscopes, the charge on the ping pong balls could be determined! The separation of the balls in equilibrium was 14 cm.

We can calculate the charge on the ping pong balls.

F = k Q1 Q2 / r2,
where F is the force, Q1 and Q2 are the charges, and r is the distance between the centers of  the two balls (r =14 cm = 0.14 m).  The other forces on the balls are their weights and the tensions in the strings. The three forces on a given ball sum to zero, as he illustrated in a free body diagram. This includes the known distance from the ceiling to the balls (about  2.0 m) and the assumption that that Q1 = Q2, which is reasonable since they are identical and were in contact with the same charged rod.

Bill constructed a homemade balance from a meter stick for arms and a block of wood for a fulcrum. He taped the ping pong ball to one end of the meter stick, and moved a nickel (mass =  5 gm) along the other arm of the meter stick until the balance was achieved; the nickel was a distance X = 29 cm from the fulcrum.  Bill determined the mass of the ping pong ball as the mass of the nickel multiplied by the ratio of distances X / 50 cm, obtaining 2.9 grams.  Then he calculated the charge on each ball,  Q1 = Q2, obtaining 46 nanoCoulombs.

An amazing tour de force!  Thanks, Bill.

Our next SMILE meeting will be on Tuesday March 07, 2006. See you there!

Notes prepared by Ben Stark and Porter Johnson.