High School Mathematics-Physics SMILE Meeting
23 March 2004
Notes Prepared byPorter Johnson
Bill Colson [Morgan Park, Mathematics}         Euler's Disk
showed us a carefully designed coin-shaped disk.  Known as Euler's Disk, it is about 1 cm thick and 8 cm in diameter, with one side mirror-like and the other side with a bright and colorful iridescent pattern.  He set it spinning on a smooth glass surface in the shape of a round, slightly concave mirror about 25 cm in diameter.  It seems like it would never stop, only gradually losing energy because of friction and vibration.  As time goes by, it begins to "hum" with a progressively higher pitch and louder sound.  Lee Slick reflects a laser beam off the disk and onto the ceiling.  We can follow the motion of the disk more easily by watching that reflection, a circle that gradually decreases in diameter as the disk loses energy. Eventually, it simply stops, abruptly -- at which point the small circle on the ceiling condenses to a small dot.  Amazing, but why does it happen?

For additional details see The Official Euler's Disk website:  http://www.eulersdisk.com/.  In particular, for information on the physics of this object see the page http://www.eulersdisk.com/physics.html, from which the following is excerpted:

"When Euler's Disk is spun, the disk contains both potential and kinetic energy. The potential energy is given to the disk when it is placed upright on its side. The kinetic energy is given to the disk when it is spun on the mirrored base. Euler's Disk would spoll (i.e., spin and roll) forever if it were not for friction and vibration. ... "

For insights gained by Richard Feynman concerning a plate wobbling on a table, see the following excerpt from his book Surely You're Joking, Mr Feynmanhttp://www.amazon.com/Surely-Feynman-Adventures-Curious-Character/dp/0393316041.

Euler's Disk was named in honor of the famous mathematician Leonhard Euler (1707-1873) [ http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html], who maintained a life-long interest in puzzles.  His puzzle concerning the crossing of the 7 bridges over the river Pregel in the city of Konigsberg [ http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Konigsberg.html] represented the beginning of modern topology.  Euler's beautiful equation e ip + 1 = 0 connects the five most important numbers in mathematics --- two of which were invented by Euler himself (i and e).  For additional details see Tour of the Calculus by David Berlinski [ISBN 0-679-74788-5].  

An anonymous student once said "infinity is where things happen that don't".

Thanks for these delightful math and physics insights, Bill!

Bill Blunk [Joliet Central, Physics]           Plane Mirrors
attached a mirror to the blackboard with magnets, and stood in front of it.  He noted that, in the mirror, "left" and "right" appear to be interchanged.  In particular, when he held up his right hand, his image lifted its left hand, etc.  To clarify the situation, Bill recruited Ann Brandon to play the role of his mirror image.  Bill (facing North )and Ann (facing South) were looking directly at each other, while the imaginary mirror between them was vertical and in the East-West plane.  When Bill lifted his right hand, Ann lifted her left hand, as expected for a mirror imageBill then pointed out that when he moved his Eastern hand, Ann also moved her Eastern hand.  However, when Bill pointed his finger North (toward Ann), Ann pointed her finger South (toward Bill)Bill then said, "Isn't a mirror image just a front-to-back reversal?"  In other words, both the object and image agree on the directions parallel to the mirror (East or West, up or down), but disagree as to the direction perpendicular to the mirror (North - South).  

As a further test of these ideas, Bill showed us front-to-back reversal by mirror writing.  For more information see the website Mirror Writing: http://www.hawaii.rr.com/leisure/reviews/handwriting/2002-04_mirrorwriting.htm.  The following information on Leonardo Da Vinci is excerpted from that page:

" ...He is also the most celebrated mirror writer to date. Most of his manuscripts, letters and meticulously illustrated notebooks were written in mirror image. No one knows why he wrote this way but two theories suggest convenience and security. ..."
Next Bill asked the following question:  We often "back away" from a mirror to get a more complete view of ourselves.  Does this work?  To find the answer, Bill stood in front of the mirror on the blackboard about 1.5 meters away, a held a meter stick vertically alongside his head.  He noted the positions of the top and bottom edges of the mirror, as read on the reflected image of the meter stick. Then, he backed to a distance of about 3 meters from the mirror.  Guess what?  He could see the same amount of the meter stick in the mirror.  How come?

Bill then drew the following sketch of rays passing into the observer's eye after being reflected by the mirror:

right side eye left side
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\=========/ mirror
\ /
\ /
\ /
\ /
image of eye
Houston, we have a problem! It seems from the diagram that we can see a width (of ourselves) that twice the width of the mirror, and only that much --- no matter how far we are located from the mirror. Now, just why do we instinctively "back up" from a mirror in order to see more? Bill suggested that, in fact, our early experiences may have been with the mirror on mother's dresser.  We would get a better view of ourselves by backing up, to avoid blockage by the dresser itself. Very interesting, and perhaps even true!

You took us through the Looking Glass so that we could see our own world more completely! Great job, Bill.

Ann Brandon  [Joliet West, Physics]           Current without Batteries??
Ann began by outlining the approach she took in teaching magnetism in 6 class periods, according to the following plan:

  1. Day 1: She placed a permanent magnet under a sheet of paper, and placed small compasses on top of the paper.  The compasses pointed in the direction of the magnetic field at their locations, and thus she could indicate the pattern of magnetic fields surrounding the magnet.  Then, she sprinkled iron filings onto the paper.  The filings also indicated the direction of the magnetic field, by forming a pattern that revealed the lines of force.
  2. Day 2: She borrowed a plastic funnel with a long stem from the Chemistry laboratory, and turned it upside down on the table.  Then she put ring magnets onto the stem, with adjacent poles repelling one another.  She measured the distances between the magnets.  Then she determined the downward force on each magnet, since a given magnet would have to support the weight of all magnets above it. The students drew a graph of the force F versus the separation D between magnets.  They obtained an inverse square law F = K / D2.  But, why?? 
  3. Day 3: Build an electromagnet and hold up 10 paper clips with it, using wire, a nail, and a battery.  
  4. Day 4: Build a Rudy Keil electric motor using a battery and magnet
  5. Day 5: Summary; review; example problems. 
  6. Day 6: Producing Current in a Coil of Wire (Look Ma! No Batteries!)
    2 small coils of wire (Gilley Coils), galvanometer (micro-Ammeter), 2 different bar magnets, large coil of wire (air core solenoid).
    Label one end of each magnet as North. Connect the galvanometer to one of the small coils.
    Move the N pole of one of the magnets through the center of the coil. What happens to the galvanometer needle?
    Pull the magnet back out of the coil.  What happens to the needle now?
    Try the South Pole this time.  What happens to the needle this time?
    Instead of moving the magnet, move the coil.  What happens to the needle?
    What seems to be required to produce current?
    Try the other magnet.  What differences do you observe?
    Move the best magnet in and out faster than before.  What differences do you observe?
    Connect the large coil to the galvanometer.
    Try dropping the best magnet through the large coil.  CATCH THE MAGNET!  What difference did this make?
    If the bar magnets were replaced by an electromagnet, how could you increase the current through the galvanometer?
    Conclusion:  Which things affect the current in the galvanometer?
    If you have time, connect two small coils together in series with the galvanometer.  Place the two coils so that their center holes line up.
    Move your best magnet into the center of the two coils.  Pull it out fast.  What happens to the galvanometer?
    Turn one of the coils around (180°).  Place the best magnet into the center of the two coils.  Pull it out fast. What happens to the galvanometer this time?
    Why should you have different results?

Beautiful phenomenological physics, Ann!

Betty Roombos  [Gordon Tech HS, Physics]          
ESPN Sports Figures Videos: 
Betty showed us the ESPN video THE MU YOU DO, which concerned friction, especially in the context of NASCAR automobile racing.  The tires are typically filled with nitrogen. and it is frequently said that "tires win the race".  Specifically, the cars are run on a closed track with significant curves, so that  static friction between the tires and the track helps the automobile stay on the track, rather than sliding over to the outer wall.  The coefficient of static friction, mS = fs / N is about 1.5, according to measurements made in the video. Thus, the maximum speed at which the car could go through a curved, level track without slipping is vmax = Ö(mgR) , where g is the acceleration due to gravity, and R is the radius of the track. For a track or radius R = 200 meters, we get vmax = Ö (1.5 10 200) = Ö (3000) = 55 meters/second.

The set of 7 videos on other topics can be ordered online at http://www.amazon.com/ESPN-Sports-Figures-Makes-Physics/dp/B000NPGLS2 Neat stuff, Betty!

Fred Schaal [Lane Tech, HS Mathematics]           Of All the Crust!
Fred continued his discussion of slicing pie, which was begun at the last SMILE meeting  [mp030904.html]. He had shown that the ratio, R, of crust area to total area is given by R = 1 - (sin q) / q. (q in radians)

He programmed this formula on his TI-83 graphing calculator, and projected on the screen at the front of the room the graph of R versus q for various ranges.  In particular, he noted that for q greater than 180° or p radians R became greater than 1.  Here is a summary of results:

q: degrees   q: radians  R = 1 - (sin q) / q
60°   1.047 0.173
120°   2.094 0.586
180°   3.142 1.000
270°   4.712 1.212
360°   6.284 1.000
450°   7.854 0.873
630° 10.995 1.090
810° 14.137 0.927

It seems that the ratio R is approaching 1 at large slice angle q. Do we get more pie by circling many times?!

You can have your  p-pie and eat it, too!  Good, Fred!

We ran out of time before Don Kanner could make his presentation on the Vandegraaff Electrostatic Generator.  It will be scheduled at the beginning of our next class, Tuesday 06 April 2004 See you there!

Notes prepared by Porter Johnson