High School Math Physics SMILE Meeting
02 May 2000
Notes by Porter Johnson

ANNOUNCEMENT:

Henry Heald Lecture
Professor Leon Lederman
Nobel Laureate in Physics and
Chair, Teachers Academy for Mathematics and Science

Monday, 22 May 2000
Perlstein Auditorium [33rd & State, NW Corner]

for a discussion about

The Essential Transition from 19th to 21st Century
High School Science Education

buffet supper 6 pm
discussion 7 pm
RSVP to (312) 567-8820 or parson@iit.edu

`        OUR NEXT MEETING......will be the start of the Fall semester on September 5, Tuesday.You will receive a reminder in the mail. You may already haveregistered...`

Bill Colson (Morgan Park HS)
handed out an article from the Chicago Reader, from the pages of the Mad Scientist, Spring, 1999, Zine-O-File, and an excerpt from Science Runs Amok? Time Travel - A Medical Nightmare? by Robert Vanderwoude. It points out the biological consequences of time travel, among them being that the antibiotic resistant microbes in time travelers of today could be transferred to people of, say, the 1950s. Also, an article "How Big Are the Planets," Parade Magazine, April 30, 2000, Marilyn Vos Savant -- giving an idea of scale. Use and Keep these in your file!

Fred Schaal (Lane Tech HS)
did the Law of Sines with us as an exploration by hand-held calculator. He drew a triangle on the board and identified its three angles as A,B,C, and the lengths of the sides opposite as a,b,c, as shown:

` `
Then

a/sinA = b/sinB = c/sinC.

Choosing values for A, and keeping c and B fixed but letting the side a increase with A, he obtained the following table of values, which he wrote on the board:

A a / sin A
6o 16.3
19o 14.4
46o 17.8
83o 15.4
99o 20.2
153o 16.3

Triangles were actually drawn for the various values of A, and a was then measured, and the above ratios were calculated based on these experimental measurements. Fred noted that the ratio does not change much, and wondered at this.

Fred set up the calculation of the curve for x(T), y(T), where

x = T[1+1/(2p) ] cos T
y = T[1+1/(2p) ] sin T
for 0 < T < 20p.

He displayed the result as a set of points with the aid of a TI graphing calculator, a transparency projector and LCD display. The graph looked like a roll of carpeting viewed end-on, a kind of spiral around the origin. Most interesting! A good example of how to connect the abstract to the concrete so as to make math "real" for students!

Bill Lilly (Kenwood HS)
gave us a handout titled "Pendulum," which was a lab exercise. Using a TI CBR Sonic Ranger, high frequency sound from the Ranger was reflected from a swinging balloon [held on the end of string by Pearline Scott (Franklin School)]. The output from Ranger (for 10 s) was stored in the calculator and shown on the projector screen as a graph with the aid of Fred Schaal's LCD display. It looked like a sine wave, and Bill described how students could analyze the data for frequency, effect of shortening string, lengthening string, amplitude effects, velocity minima and maxima with position, etc. Another beautiful example of concrete/abstract connections!

Roy Coleman (Morgan Park HS)
held up the power line cable that Earl Zwicker had discussed with us last meeting, and then he described a house wiring problem that he was involved with that had occurred many years ago. Strange things happened: The dog would do a flying leap over the threshold when coming back into the house, which necessitated his continuing on down the basement stairs, then back upstairs to get in. Why? When some lights were turned on, other lights would become dim. Why? A water pipe in the basement corroded and had to be replaced, and the problems became worsened. An electrician called in to find the problem gave up after 2-3 hours, shaking his head, and never sent a bill. Roy drew a circuit diagram on the board showing a stepdown power transformer and the low-voltage secondary with center-tap to ground and its + and - 120 VAC ends going to serve the house power. What they finally discovered could be seen only from a second-story bedroom window. The ground wire from the secondary of the stepdown transformer passed through a tree next to the house, and it had broken. Thus, the entire house wiring was "floating" with respect to ground, and any connections to the grounded power transformer primary were being made through grounded water pipes, etc. Com Ed was called, and they got out there and fixed it in record time once the problem and its consequences were made clear. Most interesting!

Bill Blunk (Joliet Central HS)
showed us a physics "toy" called a ROMP [Randomly Oscillating Magnetic Pendulum], which came from Edmund Scientific. It is item CR82-172, available for \$13.95. See the website http://www.edsci.com/. (It is also available at American Science Center. See the website http://sciplus.com/.) It has a flat metal base about 8 in square. A pendulum with a length of about 12 inches is suspended above the base so that, when at rest, its end is about 1 inch above the center of the base. The pendulum is made from a rigid, slender plastic rod, and it has a disk-shaped magnet (about 1 cm diam) at its end. There are 9 identical disk magnets stuck to the base by magnetic attraction; they can be arranged on the base in any pattern one desires. Bill placed them in two concentric circles with one at the center. When he held the pendulum out at an angle and released it, it swung in an erratic pattern over the magnets on the base. He used it as a model for nuclear forces. The magnet on the end of the pendulum he likened to a neutron, and the other magnets on the base represented particles in the nucleus. When he held the pendulum at a high angle to represent a neutron with high energy, the pendulum swung in its erratic pattern. But when he lowered the angle to represent a neutron with smaller energy, the "neutron" magnet became "captured", and centered itself over the magnet at the center of the base. Neat!

Fred Farnell (Lane Tech HS)
asks his students to estimate how fast an egg can be thrown into a vertically-held blanket without breaking the egg. He also asks from what height h could an egg be dropped onto a hard surface (about 5 cm) or a cushion (about 2 m) without breaking. What is the speed of the egg when it strikes the surface?

v = Ö(2gh)
gives an answer. What force is required to break an egg? To answer this last question, Fred showed us his egg tester. A flat piece of wood (about 4x6 sq in) with bolts sticking vertically up from its corners was placed on the table. An identical piece of wood had holes in its corners so it could move up-and-down freely, guided by the bolts. Fred placed an egg into a plastic bag and sealed it, then placed it between the wood pieces. With a set of weights handy, he asked us,"How much force will it take to break this egg?" Lee Slick thought about 10 Newtons, but others guessed lower. So - Fred stacked weights onto the top board; we were entranced as he added each 0.50 kg mass. The egg finally broke at 2.5 kg, or about 25 Newtons. Using Newton's second law, F = ma = m (dv/dt), Fred re-wrote it in the form Fdt = mdv, which led to the idea of "impulse," (Fdt), and showed that when the momentum of the egg is changed by mdv when being brought to rest, whether or not it would break would depend on how long (dt) it takes to bring it to rest. The longer dt, the smaller F, for a given change in momentum (mdv) (ie, dropping the egg from a given height h). By bringing it to rest relatively slowly, F would not exceed the F needed to break the egg! A great way to involve students with the concept of impulse. Nice, Fred!

Larry Alofs (Kenwood HS)
gave us a handout from Science News, April 1, 2000, describing the new golden dollar bearing the likeness of Sacagawea, "...the young Shoshone woman who guided Lewis and Clark..." The metallurgy involved was tricky; the new coin will not require the retooling of vending machines. The make-up of other US coins was listed out on the same page. A most useful reference for physics teachers. Thanks, Larry!

Arlyn van Ek (Illiana Christian HS)
showed us an interesting 50 minute video available from NOVA. See the website http://www.pbs.org/whatson/press/winspring/secrets_medsiege.html. Titled Medieval Siege, it sells for about \$26 (1-800-949-8670). He uses it when he must be out of the classroom. He showed us some excerpts about the WARWOLF [the atom bomb of the 14th century] and the Trebucket (like a catapult with a sling on its end), and how a group of people experimented building several models, many to full-scale, to test whether they had the range and power to knock down the thick defensive walls of a medieval castle. They had some problems until they put wheels on it, like old engravings showed them to be built. The wheels, counter-intuitively, increased the range of the projectiles and stability of the trebucket! Most interesting physics involved. Thanks, Arlyn!

Janet Sheard (West School, Glencoe)
showed us Build It, Move It, a kit on Force and Motion (ETA catalogs)/ 4th grade. She said the materials could be adopted to a wide range of levels. The inexpensive but effective apparatus came in a plastic storage box. Many investigations into forces: What are they? Sources of? Motion, Friction, Inertia, Machines. Janet had not yet used the kit, and we ran out of time, but maybe Janet will show us how some of it works next Fall. Essentially "Physics for Every Kid." Inspiring!

ANOTHER GREAT MEETING! SEE YOU NEXT FALL!

Porter Johnson

## Using EXCEL to Study Volume of Cut-out Cone See Notes for 28 March 2000 HS Math-Phys Class

If you take a sheet of paper with a circle of radius R, and cut out a sector of opening angle ko, as described in the 28 March 2000 SMILE HS Math Physics Notes, the volume of the cone formed by that sheet is

V = [p R3/3] * [ko/360o]2 * [ 1 - (ko/360o)2 ]1/2.

The formula may be simplifying by defining the ratio x = ko/360o, so that for radius R = 1

V(x) = [p x2 /3] Ö (1 - x2)

One may show by elementary calculus that the volume is maximized by the choice

x = Ö (2/3) = .861497,

corresponding to angle k = 293.9388o. The maximum volume is

Vmax = 0.403067.
We may use the graphics program EXCEL [a standard component of the Microsoft Office program package] to create a a data table, as well as a graph. An EXCEL data table is arranged in cells, which look somewhat like the following:
 *** A B C D 1 _____ _____ _____ _____ 2 _____ _____ _____ _____ 3 _____ _____ _____ _____ 4 _____ _____ _____ _____ 5 _____ _____ _____ _____
Let us identify these cells by giving the Column and Row; eg, C3 or A5. You may enter data into a given cell by moving the mouse to the cell in question, and clicking.

Here is a set of steps that will result in the data table and the graph: First, type these descriptive words in lines #2 and #3

• In Cell A2 type cone vol
• In Cell A3 type angle
• In Cell B3 type x
• In Cell C3 type angle
• In Cell D4 type volume
Now, type in these numbers in line #4
• In Cell A4 type 290
• In Cell B4 type = A4/360
• In Cell C4 type = A4
• In Cell D4 type = PI() * B4^2 / 3 * SQRT(1 - B4^2)
Now, type this number into line #5:
• In Cell A5 type 291
Your EXCEL table should look like this:
 *** A B C D 1 _____ _____ _____ _____ 2 cone vol _____ _____ _____ 3 angle x angle volume 4 290 .805556 290 .402645 5 291 _____ _____ _____
Note that the formulas typed on line #4 do not appear, but only numerical answers.

At this point you should save the EXCEL table [after all, you've worked hard to create it!].

• Take the mouse and "enclose" [i.e. "highlight"] both cells A4 and A5, which contain the numbers 290 and 291, respectively. If you pull downward on the lower right corner of the rectangle with a + sign there, you will get evenly spaced points as far down as you pull; say, 290 to 310.
• Now, enclose the cell B4, and pull down on the + sign at the lower right corner, so as to fill cells in column B as far down as column A is filled.
• Similarly, enclose cell C4, and pull it down.
• Finally, enclose cell D4 and pull it down.
Your table should look like this:
 *** A B C D 1 _____ _____ _____ _____ 2 cone vol _____ _____ _____ 3 angle x angle volume 4 290 .805556 290 .402645 5 291 .808333 291 .402830 6 292 .811111 292 .402963 7 293 .813889 293 .403042 8 294 .816667 294 .403066 9 295 .819444 295 .403035 10 296 .822222 296 .402946 11 297 .825000 297 .402798 12 298 .827778 298 .402589 13 299 .830556 299 .402320 14 300 .833333 300 .401986 15 301 .836111 301 .401588 16 302 .838889 302 .401123 17 303 .841667 303 .400590 18 304 .844444 304 .399987 19 305 .847222 305 .399313 20 306 .850000 306 .398564 21 307 .852778 307 .397739 22 308 .855556 308 .396837 23 309 .858333 309 .395855 24 310 .861111 310 .394791
The next objective is to make a graph of the volume versus angle using EXCEL. To do so, complete the following steps:
• Using the mouse, capture the data in column A, and then in column D.
• From the toolbar menu, click on the "chart wizard" ikon.
• Pick the "xy scatter" option
• Then pick the "smooth curve" option
• Include the "major grid lines" and "minor grid line" options for x and y
• Finish drawing the graph.
The result should look like the image shown here:
Here is the graph of cone volume over the full angular range 0o to 360o: