High School Mathematics-Physics SMILE Meeting
25 March 2003
Notes Prepared by Porter Johnson
Fred J Schaal [Lane Tech HS, Mathematics]     Laptop Capers:  TI Interactive
showed the TI Interactive Software, which emulates a TI-83 Calculator on a Windows-based PC.  With the assistance of Monica Seelman, Fred went into the graphics mode, and first displayed a graph of the function y = x2 - 8, using the ranges (-10 £ x , y £ +10). A very nice upward parabolic curve appeared, which left the region of graphing at x » ± 4.2. Next Fred plotted the graph of y = 8 - x2, which corresponded to the same parabolic curve, but inverted. The two curves intersected at the line y = 0, for x = ± Ö» ± 2.8. Finally, Fred asked for the area between the two curves, which can be formally written as the integral
4 ò0Ö 8 dx [ 8 - x2] = 4 ´ [8 x - x3 / 3] (x = Ö 8) = 4 ´ 8 ´ 2/3 ´ Ö 8 » 60.33

Very visual and user-friendly, Fred!

Larry Alofs [Kenwood HS, Physics]        Totally Tubular!
placed 8 pieces of 1.25" [3 cm] diameter PVC Tubular Pipe, so they stood vertically on their ends on the table.  He arranged the pipes in a row in order of decreasing height.  He had cut them to length to produce an octave of major notes. He modeled the design of the apparatus on that shown in the Educational Innovations Catalog [http://www.teachersource.com/].

Specifically, the [BOM-100 Basic Boomwhacker Set - $31.95]:  These eight labeled tubes produce the C-Major Diatonic Scale. The end-caps lower the tones by one octave. Included in package: eight tubes, long with removable end caps.

"These brightly colored, tuned percussion tubes are great for teaching students, of any age, about sound. When whacked against your knee or the floor each produces a particular note. The longer the tube, the lower the note. Each tube is color-coded and labeled with its precise note. When the tube is closed at one end with a cap (available with tubes and separately, see below), the note shifts an octave lower. Boomwhackers were invented by Craig Ramsell when he noticed that cardboard tubes from wrapping paper could be used to produce music. These tubes are amazing, loads of fun and very educational. Put a class set together and compose your next science sound lesson!"

Larry first calculated the length of pipe necessary to make Middle C [f = 261 Hz, corresponding to a wavelength l = v / f =( 345 meters/sec) / 261 Hz = 1.32 meters].  Larry anticipated that the open-ended pipe should be of length L = l /2 = 0.66 meters. However, because of end effects, he found through acoustic tuning that a length of 0.63 meters was needed.  Larry then presented the following table of lengths:
Note Length Calculated Length after Tuning
C (63.0 cm) 63.0 cm
C# - Db - -
D 56.1 cm 56.1 cm
D#- Eb - -
E 50.0 cm 49.8 cm
F 47.2 cm 46.9 cm
F#- Gb - -
G 42.0 cm 41.5 cm
G#- Ab - -
A 37.5 cm 36.9 cm
A#- Bb - -
B 33.4 cm 32.2 cm
C 31.5 cm 30.3 cm

 The calculated lengths were obtained from the first number (63.0 cm) by dividing (once or twice, as appropriate) by the factor 2 1/12 = 1.05946 ... , as required for the Chromatic Scale.  There was a lot of discussion as to how to include end effects.  For a pipe with one end closed, the traditional expression for the effective length of a pipe, Leff,  is given in terms of the pipe length L and pipe diameter D as Leff = L + 0.4 D.  

Larry next pointed out that, for pipes of resonating air with one end closed, the pipe length is given in terms of the wavelength  l  by L = l /4.  In other words, for a given pipe, the wavelength would be reduced by a factor of 2, and the frequency f would double, in going from two open  ends to one open end.  By striking the end of the pipe against his hand, Larry demonstrated this octave shift.  While several of us held the pipes, Larry played the tune Mr Frog, which was the first piece he learned to play on a piano.  Not to be left behind in this musical extravaganza, Don Kanner illustrated the West African Shantu [http://www.billabbie.com/nigeria/music.htm] instrument, hitting the pipe on his thigh.  It produces an interesting sound, but it seems likely to leave bruises. For details see Exploring Music: The Science and Technology of Tones and Tunes by Charles Taylor [Institute of Physics, 1992, ISBN: 0-7503-02135]

Larry, you make "fairly" beautiful music while showing very beautiful physics!

Katherine Hocker [Bloom Trail HS, Physics]     Home-made Spectroscope
expressed frustration with traditional spectroscopes, in that students took almost a full lab period to be able to see simple diffraction images, etc.  She passed out several of her home-made spectroscopes, with a CD serving both as a diffraction grating, and as the base of a closed cylinder constructed from card stock.  A 3 cm x 0.5 cm slit is cut at the edge of the horizontal, circular top end of the cylinder, the short dimension being tangential and the long dimension radial.  A similar slit is cut at the bottom of the lateral cylindrical surface, just under the first slit and oriented with the long side up.  When we looked through the bottom slit while just under a bright ceiling light, we could see the  diffracted image, which contained the full visible spectrum For more details see the website The Compact Disc as Diffraction Gratinghttp://www.scitoys.com/scitoys/scitoys/light/cd_spectroscope/spectroscope.html.   Wow!

Pretty stuff, Katherine!

Ben Butler [Laura Ward Elementary School, Science Teacher]        What's a Million?
showed several exercises that he has presented to his students.

  1. First he showed us two capped containers [about 2 gallons or 10 liters] that contained colored, tiny plastic beads.  He remarked that each container contained 1 million individual pieces. The container with yellow beads contained one black bead.  Surprisingly, it was fairly easy to find that bead, since it migrated to the top as we shook the container.  Ben shook it to the tune of the chorus [Bounce-Bounce-Bounce- ... ] of the R Kelly rap song, Ignition. without the lyrics. [Ben occasionally does this chant in class, to let the students know that he is not totally ignorant of their world.]  Ben passed around another container with a million blue plastic pieces, and one black one, which is much harder to find.
  2. Ben next showed us the mechanism for a bar stool turntable.  First he used it  to demonstrate the relation between the radius R and circumference c of a circle: c = 2 p R.  He measured the radius (6" or 15 cm) with a ruler, and then calculated the circumference.  He demonstrated the expression by putting 3 sheets of notebook paper [11" or 33 cm each] around the edge, and then showing that he needs just a little more to make the circumference [37.7" or 96 cm]
  3. Ben next had a volunteer to stand on the mechanism, and Ben rotated him around several times. He asked us how far the edge of the mechanism had moved in, say, 5 complete revolutions --- more than 15 feet or nearly 5 meters.  The participant got very dizzy while being spun around, for some strange reason!
  4. The volume of a cylinder of radius R and height H is V = p R2 H, and the area of its lateral surface is A = 2 p R H. Starting with two  8.5" ´ 11" transparency sheets, Ben folded one into a long, 11" tall cylinder, and the other into an 8.5" short cylinder. With their bottom ends blocked off, which way  cylinder would hold the greater volume?  Most students expect that the taller cylinder will have a greater volume than the shorter one.  Ben stood both cylinders inside a large transparent container, with the shorter one encircling the taller one.  Then Ben showed us the answer by using Uncle Ben's Rice™ to fill the long cylinder completely. He then lifted the long cylinder, so that the rice inside it spilled into the shorter cylinder  --- which was then only partially filed with rice. Ben was able to add quite a bit more rice in filling the shorter cylinder! In the interest of full disclosure, Ben pointed out that he has no relation to either Uncle Ben™ or his rice!

A good set of ideas, Ben!

Arlyn van Ek [Illiana Christian HS, Physics]      Magnetic Fluids
  show us a Magnetic Fluid [Chemical Demonstration Kit, Catalog Number AP4681 about $20], which he had recently obtain from Flinn Scientific Inchttp://flinnsci.com/.  The material is a liquid that is pushed to move easily between two transparent cylinders by plungers, until a small magnet is brought close to the narrow tube joining the cylinders.  The fluid quickly "freezes", and cannot be pushed. Here is some information provided with the apparatus:

"The material in this fluid device is a Magneto-Rheological Fluid, or MR Fluid.  MR fluid is a suspension of micronized, magnetically susceptible (iron/steel) particles in water with suspension additives.  Under normal conditions, MR fluid is a free-flowing liquid with a consistency similar to that of very thick motor oil.  Thus, the fluid can flow freely between the syringes under pressure from your hand on the piston.  Exposure to a magnetic field, however, can transform the magnetically susceptible particles into a near-solid mass in just milliseconds.  The solid forms when the particles in the fluid align with the magnetic field lines of the magnet.  The space between the particles is diminished and the fluid cannot flow, taking on the properties of a solid mass.  The fluid can be returned to its liquid state with the removal of the magnetic field."

These magnets don't really "go with the flow", Arlyn!  Fascinating!

Hoi Huynh [Mathematics Teacher]      Maintaining Balance in Mathematics
demonstrated that, when a standing cylinder  has better balance, it will have greater volume.  In the previous example in making the lateral surface of a cylinder with an 8.5" ´ 11" sheet of transparency paper, Hoi pointed out that the shorter cylinder has better "balance" than the longer one.  She said that this was connected to the fact that the top (circular) edge was closer to its geometrical center for the shorter cylinder than for the longer cylinder.

Hoi pointed out that, of all quadrilaterals that have a given perimeter p , the square has the greatest area, A = p2/16.  For all shapes of a given perimeter p, the circle has the greatest area, A = p2/(4p)For three dimensional bodies of surface area A, the sphere has the greatest volume, V = [ A /(4p)]1.5 /3.

Hoi also mentioned that the formula for the Area of a Trapezoid of bases b1 and b2 and height h,

Area = (b1 + b2) /2 ´ h
yields the areas of a rectangle (b1 = b2) and a triangle {b2 =0) as special cases.

Thanks for the insights, Hoi!

Don Kanner [Lane Tech HS, Physics]      Proclamation Concerning Areas and Volumes
Don remarked that, because the lateral surface area of a cylinder of radius R and height H is A = 2 p R H, whereas its volume is V = p R2 H, it should follow that the cylinder of greatest volume for a given lateral area should be one of large radius R and very small height H. Do you believe this?

Don promised to prove it next time! We await edification, Don!
See you at our next meeting, 08 April 2003!
Notes prepared by Porter Johnson