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 Feynman: http://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
Bill 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 image. Bill 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:
===================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!
right side eye left side
X E X
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\=========/ mirror
\ /
\ /
\ /
\ /
I
image of eye
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:
Beautiful phenomenological physics, Ann!
Betty Roombos [Gordon Tech HS,
Physics]
ESPN Sports Figures Videos: http://www.amazon.com/ESPN-Sports-Figures-Makes-Physics/dp/B000NPGLS2
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