High School SMILE Meeting
21 February 2006
Information:
Fred Schaal (Lane Tech HS)
Constructing
Points on an Ellipse
Fred
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.
P
.
/ \
x / \ y
/ \
* *
F F
Larry Alofs (Kenwood Academy,
retired)
Pan Pipes, Fresnel Lenses, and Hall Effect Sensors
Larry
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
Walter
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
Roy described a useful way to teach the right hand rule.
Tres simple, non! Thanks, Roy.
Bill Blunk (Joliet Central HS,
retired)
Ping-pong Electrostatics
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.
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.
Bill 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!