Porter Johnson touted the book The Golden Ratio and Fibonacci Numbers {Springer Verlag 1997: ISBN 981-02-3264-D). In this invaluable book, the basic mathematical properties of the golden ratio and its occurrence in the dimensions of two- and three-dimensional figures with fivefold symmetry are discussed. In addition, the generation of the Fibonacci series and generalized Fibonacci series and their relation to the golden ratio are presented.Useful data for developing physics problems! Thanks, Roy!
Roy Coleman passed out information on weather and rain, that came from articles by Tom Skilling, which appeared in recent issues of the Chicago Tribune. They contained the following information:
- The earth's atmosphere weighs about 6 ´1012 tons, corresponding to 7 ´1015 kg.
- Within the United States, there are about 105 thunderstorms per year, generating about 9 ´ 107 lightning discharges.
- One inch of rain falling across metropolitan Chicago represents a weight of about 1012 pounds, corresponding to 5´1011 kg.
- Drizzle is defined as water droplets of diameter less than 0.02 inches (0.5 mm); they fall at speeds from 1-4 mph.
- Water droplets are found to be large in diameter as 0.2 inches (5 mm); above that size they become unstable and shatter into smaller droplets. They fall at 5-20 mph, with an average speed of 14 mph. The bigger they are, the faster they fall!
Properties of Typical Clouds:
Cloud Type Water vapor density Vertical height Horizontal spread Mass Cirrus 0.1 g/m3 1 km 25 km ´ 25 km 8 ´108 kg Cumulus 0.2 g/m3 1 km 1 km ´ 1 km 2 ´105 kg Cumulus Congestus 0.8 g/m3 5 km 2 km ´ 2 km 2 ´10 8 kg Cumulonimbus 1.0 g/m3 10 km 6 km ´ 6 km 4 ´109 kg
Roy Coleman [Morgan Park HS, Physics]
Estimating the Speed of Falling Rain
Drops
First, go out and buy a car in which the rear window slopes at a modest
angle
(say, q » 30°) to the
horizontal. Then, drive your car during a rainstorm, and
find the minimum speed v0 at which falling raindrops
do not
strike that rear window. If there is no wind, the speed of the
falling
raindrops vf should satisfy the relation vf
/
v0 = tan q. Simple,
non?
If the wind is blowing, repeat the observation while going in the
opposite
direction as well, and take the average. Be sure not to crash
into
anything while doing this experiment!
Now, there's a good reason to get rid of the old clunker and get a new
car. Nice work, Roy!
Bill Blunk [Joliet Central HS,
physics] Paper Match Rocket and More
Bill constructed a launching pad using a piece of cardboard paper,
and a paper clip bent to support a light
object leaning against it [a paper match stick -- wooden matches are
too heavy
to work].
First he tried to launch the paper match just by putting it on the
launching pad and
lighting it. The launch fizzled, because the match just sat still
and
burned. Why didn't it go flying away? After
some
discussion it was decided that there was no net impulse given to the
paper match in
this process, since the exhaust gases from the match were sent in all
directions. One must find a way to direct the flow of exhaust
gases to
provide a net impulse. So, Bill wrapped the head of the
paper match several times with a small piece of aluminum foil, pressed
it
tightly at the top so that exhaust gases would come only out the
bottom, and
placed the paper match head-up onto the launch pad. Bill
then lit another
match and held it under the wrapped head of the match on the launch
pad.
There were audible hissing sounds from inside the foil, as the match
head
ignited and (anaerobic) combustion began. A split second later,
the paper
match jetted off the launch pad, and bounced off the ceiling.
Details on construction and operation were given in last week's
summary: mp092303.html.
Bill, you really set things on fire, intellectually speaking!
Bill also presented an extension of last week's lesson on balancing an egg [mp092303.html] at the Autumnal equinox. In particular, he pretended to "balance" a golf ball on a horizontal board. Of course, one would not expect the ball to move, because it is round -- unlike an egg placed on end. and it did not move. Then, Bill slowly tilted the board up on one end, making an angle of about 30° to the horizontal -- and the golf ball still did not move! Amazing! After extensive cross-examination by the group, Bill finally admitted that the experiment was a hoax. Namely, the golf ball was spherical in shape, but its center of mass lay significantly below the geometrical center. Bill had made his annual pilgrimage to Amazing Toys in Great Falls MT. This item can also be ordered through their website http://www.amazingtoys.net. Very slick, Bill!
Siegerschnecke
-- which means Snail
Race auf Deutsch. Bill called attention to a very
important
race between trained snails which was held in Cremonia (Alpine
Italy)
last Summer. By holding a piece of lettuce and crawling in front
of the
snail, the winning snail trainer (female, age 11) had coaxed the snail
to travel
1 meter in 450 seconds, corresponding to an average speed of
about 2
mm/sec. As prize for this victory, she and her pet snail
received a
lettuce bowl. This speed is significant, in that it is
greater than a
typical drift velocity of electrons in a conducting wire, even
at relatively
high currents. And, think of how proud the winning team must be
in this annual
event, described in the (Deutsch) website (with pictures) given
here: http://www.toponline.ch/area-1.rub-39.art-39031.tce,
Fascinating
topics and spectacular stuff, Bill!
Imara Abdullah [Douglas Academy,
science]
Posters
Imara provided us with poster paper, colored markers, and tape,
and asked each of
us to prepare a poster to illustrate some concept or process in
mathematics or
science. We came up with the following displays:
Name | Display description | Concept or process illustrated |
Porter Johnson | blank sheet of paper | Vacuum, empty space, cosmic void |
Bill Colson | Flow chart 3 ® 1 | 3 conditions for triangle congruence |
Roy Coleman | I'm a p r2 (big wheel) | Area and circumference of circle |
Elizabeth Roombos | Rock hurled off cliff | Projectile motion |
Marilynn Stone | Click-clack apparatus | Momentum conservation |
Monica Seelman | 45°-45°-90° triangle | Pythagorean Theorem |
Earl Zwicker | Sequential images of ball on inclined plane |
Galileo experiments in mechanics |
Imara Abdullah | Walking dog around block | Perimeter |
John Bozovsky | Kneeling carpenter drilling into wall | Niels Bohr (kneeling and boring) |
Larry Alofs | Rectangle at new IIT student center | Golden rectangle -- or not? |
Jane Shields | Colored strips on paper | Northern lights |
Babatunde Taiwo |
Rocks thrown simultaneously up and down |
Do they hit the ground at the same time? |
Walter McDonald | Headlight beam image | Illumination: Inverse square law |
Rich Goberville | Projectile shot from cannon | Action-reaction Forces |
Bill Shanks | Plumb bob demons | Universal gravitation |
Fred Farnell | Light charged balls on strings | Coulomb's Law: Electrostatics |
Leticia Rodriguez | See-through skeleton | Systems in human body |
John Bozovsky | Truck accident | How the Mercedes bends |
Fred Schaal [Lane Tech HS,
mathematics]
Lubbock*** [TI-83] Helper
Fred used his calculator and plotter to extend his lesson of
29 February
2000 [mp022900.htm]
concerning tangent lines and asymptotic curves. This lesson had,
in turn,
been an extension of a lesson by Bill Colson on 11 November 1999
[ph110999.htm]
on producing intricate geometrical patterns. These comments by
Porter Johnson
are excerpted from the first lesson:
The formula for line segment lying in first quadrant and intercepting the x-axis at x = a and intercepting the y-axis at y = b - a, where 0 < a < b, is
Note: some browsers will incorrectly render the square root sign (Ö).
The curve is a parabola with the line of symmetry [axis] lying along the line x = y. Bill used the graphing calculator to extend the problem to an extended region: -a < x < a and -a < y < a. What did it look like? Amazingly, the original asymptotic curve (caustic) was symmetrically extended to all four quadrants:
How come?
Fred's trusty programmer, Bill Colson, developed and displayed the graphs for xp, where the real, non-integer variable p increases from 1 to 2. The curves all passed through x = 1, but became steeper below that point for x < 1 as p was increased. For x > 1, it was the other way around. Fred showed the graphs for negative x as well. Surprisingly, for p = 1.1, 1.3, 1.5, 1.7, and 1.9 the curves were negative, whereas for p = 1.2, 1.4, 1.6, and 1.8 they were positive. How come?
Comments by Porter Johnson: For negative x, there is a real, positive value of p whenever p is a rational fraction of the form a / b, where the integer b is even. Furthermore, there is a real, negative value of xp when p=a / b is rational, when b is odd. The gremlins and lubbockians*** of calculator programming were definitely on your side for this one, Fred.
Q: How many Texas Tech football supporters does it take to change a light bulb?Bigger plots are definitely better! Good, Fred!
A: Don't be silly; there's no electricity in Lubbock*** [home of Texas Instruments™]!
Ann Brandon [Joliet West HS,
physics] Inertia and Seat
Belts
Ann put a "bear crash doll" onto a physics collision
cart, and released
the cart at the top of a plank inclined at a moderate angle (about
30°) to the
horizontal table. The cart rolled down the plank, off its edge,
and onto
the table. At that point the bear-doll flipped out of the cart, and
landed on its head. This was a perfect illustration of Newton's
Second Law:
A force is needed to change the velocity of the bear. A force --
supplied
when the bear-doll struck the table -- is required
to change the direction of motion of the bear-doll; otherwise it would
just keep
going and going and going. It also illustrates the need to
wear seat
belts in an automobile. Next she strapped the bear into the
cart with a
rudimentary (rubber band) seat belt, and again released the cart from
the top of
the plank. This time, the bear whacked its head against the "dashboard",
and experienced severe whiplash. Again, Newton's Laws are at
work.
Also, the need to wear shoulder straps, as well as seat belts,
was shown
by Ann, using an additional rubber band.
The collision of the two Boston Red Sox outfielders [Johnny
Damon and Damian
Jackson] http://sports.espn.go.com/mlb/playoffs2003/news/story?id=1632209
chasing after a pop fly during the baseball playoff games was
discussed. One of
the players suffered a concussion, and was unconscious for a few
minutes,
although no brain damage apparently occurred as a result of the
collision.
A great danger in a concussion is brain swelling, which must be treated
quickly
to prevent permanent brain damage.
It was also mentioned in discussion that
Princess Diana and all other occupants of the car died as a result
of the
automobile crash in Paris -- except for the front "shotgun"
passenger, who was wearing a seat belt. For details see the
website Animation
of Princess Diana Collision: http://www.knottlab.com/aboutus.aspx?page=hi-profile%20cases&id=1
and Princess Diana: Cause of Death: http://www.dianaspeaks.info/AutopsySummary.html
Ann, you showed us the physics and importance of buckling up with these elegantly simple experiments!
Porter Johnson [IIT,
physics]
Physics of Baseball
Porter asked the following questions concerning baseball:
Go Cubbies!
Notes taken by Porter Johnson