High School Mathematics Physics SMILE Meeting
19 November 2002
Notes Prepared by Porter Johnson
-
A meeting announcement was passed around about the Michiana Physics
Teachers Group, operated by Mike McFarland of Notre Dame University.
For more details about this organization, you may contact him at (574) 631-7057,
or by email at mmcfarla@nd.edu.
-
A new exhibit on Albert Einstein is opening at the American Museum of
Natural History in New York. To obtain more information, see their
website, http://www.amnh.org.
- Information on the website It's About Time
http://www.its-about-time.com
was distributed. This group develops curricula in Earth Science, Physics and Math curricula for middle
school through high school students
- The website
http://www.humorsphere.com/fun/color.php
contains an interactive test, in which you are to identify the colors
themselves, rather than names of colors. For example, if you see the word
"BLUE", written with red letters, you should answer
"red", rather than "blue". Check it
out, to see how long it takes you to master the puzzle! Thanks to Rudy
Keil for this!
Betty Roombos [Gordon Tech HS, Mathematics]
The Ballistic Cart Put on Television
Betty presented a 21st century adaptation of the standard ballistics car
demonstration, for which apparatus is available from, say, Sargent-Welch:
http://www.sargentwelch.com/ [for a
description of their ballistics car, see
http://sargentwelch.com/product.asp_Q_pn_E_WL0742%5FEA].
The idea is that, when a ball is shot straight up from the cart
while the cart is in
uniform motion, it lands back in the cart, just where it came from.As a modern variant of this standard demonstration, Betty
used her digital
camera, a Sony Mavica Model FD-88 with an 8X lens. It records 5,
10, or 15 second video images directly onto the 1.44 MB diskette that serves as the
"film". She recorded the video sequence before class,
and then played it back on her MacIntosh computer, using the "freeze
frame" option to show that the ball left the cart, went up, "stopped" in
mid-air, and returned to the cart. She used a grid on a transparency sheet
to obtain quantitative information on the position of the cart at various time
intervals. To show us how simple this is, she played her recorded
image back to us on our large TV Monitor, with impressive results. It was
suggested that she could relate the "frame rep rate" to real time by
recording the image of the second hand of a clock with her camera, either
separately or as part of the apparatus. You showed us how it really
should be done, Betty!
Fred Schaal [Lane Tech High School,
Mathematics] Coding / Decoding and Matrices
Fred showed us how to code and decode a message, based upon matrix
multiplication. He illustrated the procedure for encoding and decoding,
using the following highly significant message:
GO LANE HOBBLE THE MUSTANGSThis simple message consists entirely of letters
(to form words) and spaces (between words), without punctuation, capital
letters, numbers, or other symbols. Fred first encoded the message by
identifying the space [_] and the 26 letters of the alphabet with the numbers
0
through 26, as follows:
| _ |
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |
L |
M |
N |
O |
P |
Q |
R |
S |
T |
U |
V |
W |
X |
Y |
Z |
| 0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
The coded message can thus be converted into a list of numbers:
| G |
O |
_ |
L |
A |
N |
E |
_ |
H |
O |
B |
B |
L |
E |
_ |
T |
H |
E |
_ |
M |
U |
S |
T |
A |
N |
G |
S |
_ |
| 7 |
15 |
0 |
12 |
1 |
14 |
5 |
0 |
8 |
15 |
2 |
2 |
12 |
5 |
0 |
20 |
8 |
5 |
O |
13 |
22 |
19 |
20 |
1 |
14 |
7 |
19 |
0 |
The next step is to parse the numbers in pairs, to obtain the following sequence
of pairs:
| 7 15 | 0 12 | 1 14 | 5 0 | 8 15
| 2 2 | 12 5 | 0 20 | 8 5 |
0 13 | 22 19 | 20 1 | 14 7 | 19 0 |Fred then
asked us to invent a 2
dimensional matrix B of positive integers, with non-zero determinant. We
came up with the following modest example:
B = | 6 3 |
| 5 2 |
Then we multiplied each of the parsed pairs in the message [treated as a rows] by this matrix,
to obtain
new pairs. For example:
[ 7 15 ] | 6 3 | = [ 117 51] AND [ 0 12 ] | 6 3 | = [ 60 24]
| 5 2 | | 5 2 |
We continue with to multiply to obtain the coded message [first two words shown]:
| Original Alphabet Message |
G |
O |
| |
_ |
L |
| |
A |
N |
| |
E |
_ |
First Row |
| Original Message as Numbers |
7 |
15 |
| |
0 |
12 |
| |
1 |
14 |
| |
5 |
0 |
Second Row |
| Coded Message |
117 |
51 |
| |
60 |
24 |
| |
76 |
31 |
| |
30 |
15 |
Third Row |
How do we decipher the coded message contained in the third row? To do so, we
first construct the inverse of the matrix B, which is
A = | -2/3 1 | = B-1
| 5/3 -2 |
Note that A B-1 = I, the 2-dimensional identity or unit matrix. To decipher the message, multiply the matrix
A by the each of coded column pairs. For example
[ 117 51 ] | -2/3 1 | = [7 15] AND [ 60 24 ] |-2/3 1 | = [ 0 12]
| 5/3 -2 | | 5/3 -2 |
In summary, we recover the decoded numbers on the second row, and then may convert it back
to the letters of the first row. Fred said that it would be unwise for
potential spies to use
anything below a 5-dimensional matrix for transmitting coded messages. Here, as
in many contexts, bigger is better. This is a standard "matrix
code" which is simple to decode if you know the matrix, but takes some time to
crack for a large matrix. The matrix must frequently be changed, of course, to
guard against cracking.
Relatively primitive codes of this type were used for
communications in World War II. Porter Johnson mentioned the book
Between Silk and Cyanide: A Codemaker's War 1941-1945
by Leo Marks [Free Press 1999] ISBN 0-6848-64223, which describes the
experiences of a British cryptographer. His codes for communicating with
operatives behind Axis lines, based upon limericks and poems, were printed on parachute
silk (You can guess what the cyanide pills were for!). A
great way to motivate students to learn matrix operations,
Fred!
Professor Eduardo De Santiago [Civil and
Architectural Engineering, IIT] Bridge Design
Eduardo De Santiago [http://www.iit.edu/~santiago/]
made his fourth annual presentation before SMILE and guest students and
teachers on "How to be a structural engineer in
one lesson"! He began by posing the following difficult question:
When and where will a given contest bridge fail?
He remarked that the answer to this question depends upon the details of the contest
rules, craftsmanship in constructing the bridge, and other factors, although it seems that all good
bridges up to now have been truss bridges. We will not repeat the discussion of why truss bridges
are good, but refer to the relevant SMILE write-ups of 1999 [ph120799.htm],
2000 [mp112100.htm]
and 2001 [mp112001.htm]. [See
also the Bridge Building Contest Home Page: http://www.iit.edu/~hsbridge/database/search.cgi/:/public/index]
Instead,
we will simply list the relevant points that he made, in bullet form.
- To design a bridge for center loading, an optimal bridge will be symmetric about the center;
that is, if your bridge is not symmetric, you are wasting material. In general, you should have
a good understanding of the points at which the bridge may be loaded.
- When a bridge is supporting an external load, internal forces are developed
in various parts
of the bridge. Civil engineers analyze these forces in terms of bending moments and internal
shear forces.
A shear force tends to sever a beam, whereas a bending moment induces a deformation of
the beam into a "smile" or a "frown". In general, bending moments are more significant than
shears for bridge design.
- If you place a load on a horizontal plank placed between two abutments, the plank bows downward.
In this situation the top part of the plank is under compression, and the bottom part of the plank
is under tension. The "neutral axis" running horizontally along the center of the plank is under
relatively weak internal forces. This idea is the basis for the "I beam" [transverse cross-section
shaped like an I], in which the material is located primarily at the top and bottom of the beam,
where the greatest internal stresses are found.
- Suppose we make a bridge that looks like a ladder turned on its side:
This bridge will be resist bending, but will be very vulnerable to shear. We use
trusses [diagonals] to handle shear forces efficiently.
- Here is a typical truss bridge panel
The two good things about trusses are (1) that they can handle shear forces efficiently, and
(2) that truss bridges --- assuming ideal "pin connections" --- are completely solvable, as well as generally strong
structures. Note that the "triangulation" provides strength by preventing "buckling" of the bridge.
[Since the ends of the bridge are supported by the abutment and do not experience a bending moment,
a triangular portion at each end, being unnecessary, is removed.]
- A truss bridge is made by connecting two side panels, with cross-bracing and connections to provide
triangulation at the top. In addition, portal bracing is required at the top to eliminate
side-sway.
-
As a material, wood is strong under tension, but has a strong tendency toward
buckling under compression. Long, thin pieces of wood may be laminated by
gluing them together along their entire length. This is especially
important for the bottom members of the bridge.
-
You should minimize the total number of joints, and be sure that the joints fit
together snugly without gaps, before gluing them. Remember that you are
trying to obtain strength through triangulation. Alignment of joints is critical for building strong contest bridges.
-
In practice, gusset plates may be used for strengthening joints in steel
truss bridges, but these are probably not practical for contest bridges.
-
It is better to have "butt joints" with the full member resting on top
of the piece below, so that the wood, rather than merely the glue, is helping to
support the weight of the bridge.
-
You have to be sure that your weight platforms will support the weight by
themselves, since there is the possibility of a "punch out", in which
the bridge remains largely intact, while the platform punches through to release
the weights.
- Some experimentation in the Seattle area has suggested that the best glue
for contest bridges is ordinary wood glue [Elmer's Glue™?] , rather than the
more expensive varieties.
- If weight must be supported below the roadbed, you can build an inverted truss.
It is important to have bracing below ground level at the abutments. Remember that
the Romans understood the arching effect, and also learned [sometimes the hard way]
that you must have the arch well attached to prevent buckling. You can build an
inverted arch, as well as the usual kind.
- Structural engineers are required to over-design bridges by safety factors, so that
a 1000 kg load bridge will actually support 1500 kg, etc. Such caution is, or course,
a sure way to lose bridge contests. The perfect contest bridge would resemble the legendary
One Horse Shay [For details see THE DEACON'S MASTERPIECE OR, THE WONDERFUL "ONE-HOSS SHAY": A LOGICAL STORY
by Oliver Wendell Holmes
http://www.bartleby.com/102/103.html]
That is, it would shatter to smithereens when it failed, since all its members
would be equally pushed to the limit.
- Truss bridges are limited as to the distances they can span. For longer spans, either cable
stay bridges or suspension bridges are required.
- Good luck to one and all on your bridge building!
A great deal of information is provided at the West Point Bridge Design
Contest website, http://bridgecontest.usma.edu/index.htm.
In particular, you can design your bridge, and test it to find how and when it
will fail. Also, you can download the following packet from that website:
Designing and Building File-Folder Bridges: A Problem-Based
Introduction to Engineering by Stephen J Rossler
This book provides students with an opportunity to learn how engineers use
math, science, and technology to design real structures. It is intended
primarily for high school students, but those in lower grades should be able
to complete all but Learning Activity #3, which requires the application of
geometry, algebra, and some basic trigonometry. A windows-based software
package is also available at that website; see
http://bridgecontest.usma.edu/download.htm.
Ann Brandon, Larry Alofs and Bill Blunk were unable to do their presentations
due to lack of time, but will be scheduled for next time. See you then!
Notes taken by Porter Johnson