High School Mathematics-Physics SMILE Meeting
20 November 2001
Notes Prepared by Porter Johnson

Larry Alofs (Kenwood HS, Physics)  [Reveille]
Larry
passed out copies of  music for the standard bugle calls, as obtained from the website of the Boy Scouts of America, http://www.usscouts.org/mb/bugle_calls.html.. (Sound files of the various calls can be obtained at the website  http://www.kmialumni.org/bugle_calls.html.)

Larry asked why only certain notes appear in these calls --- only those notes that can be played on a bugle.  What is special about those notes?  To address the issue, he took an electric shaver, with a fairly loud characteristic 120 Hz hum, and placed it against his cheek with his mouth open.  By changing the position of his mouth, he was able to excite those same notes, and thus to play the familiar bugle calls.  How come?

As an additional exercise, he put a plastic tube inside a water-filled cylinder, and moved the tube up to produce a column of air about 50 cm long.  Then, he played the bugle calls using the hum from the electric shaver, varying the size of the tube opening with his hands.  Next, he took a tuning fork, and showed that the loudest resonance of the fork [440 Hz] corresponded to a column of air of about a quarter wavelength, or about 20 cm.  The other resonances of the column of air, at wavelengths l = 4 L/(2n+1), are those excited by the bugle. The mouth is more of a broad Helmholtz resonator [http://www.phys.unsw.edu.au/~jw/Helmholtz.html], which does have such a simple spectrum of resonances, but it is recognizably close to the linear resonator.

Brass Instruments:  "The pitch of a brass instrument depends on the volume of air that is vibrating, as well as the speed at which the player's lips vibrate. The volume of air depends on the length of the tube; a longer tube means a larger volume of air, hence lower pitch. By buzzing her lips faster or slower, the player can cause the air in the tube to resonate at different harmonics (see the discussion of harmonics and overtones in the physics section). With a single-length tube this yields only the notes found in bugle calls. To get all 12 notes of the chromatic scale, the player needs to change the length of the tube, as on the trombone, or play through different lengths of tubing, as on the brass instruments with valves."
Source:  http://exhibits.pacsci.org/music/Instruments.html.

Natural Trumpet: "The natural trumpet differs from the modern trumpet in two crucial ways: firstly, it is twice the length and secondly, it has no valves and is therefore unable to play more than a limited number of notes. These properties make the natural trumpet very distinctive. The military connotations of the trumpet made it one of the most prized instruments at court, often the trumpet corps outnumbered all the other courtly musicians put together; the Charamela Real corps in Lisbon called for 24 trumpets for one of its fanfares." Source:  http://www.iquint.co.uk/instruments.html.

It was pointed out that Rudy Keil had once used an electric shaver in a SMILE class to produce vibrations in a string, which were studied with a strobe light.

Bill Blunk (Joliet Central HS, Physics) Air-driven Cart
Bill
showed a fan cart [http://store.pasco.com/pascostore/showdetl.cfm?&DID=9&Product_ID=51492&Detail=1] that ran on a PASCO [http://www.pasco.com/] ramp [http://store.pasco.com/pascostore/showdetl.cfm?&DID=9&Product_ID=50994&Detail=1]. He turned the fan to the maximum setting, and tilted the ramp to determine the "stall angle". He used an electronic protractor to get an angle of 3.5°Lee Slick objected to the value, and determined the angle by measuring the "rise" and "run" of the ramp to be 7.5 cm and 120 cm, respectively. He then calculated the angle to be arcsin (7.5/120) = 3.58°, verifying the original result.  Bill then calculated the thrust produced by the air cart, which is equal to the component along the track of the weight of the cart, corresponding to F = 580 gram sin 3.5° = 35 "gram weights", or about 0.35 Newtons.  He then carefully leveled the track on the laboratory table, and set up a pulley system using a light string with a total of 3.5 grams suspended at the end of the track.  He released the air cart with the motor turned on, and showed that the forces were balanced. Great!

Bill next calculated that, if the cart of mass of 0.580 kg were released on the track with the fan running, it would accelerate at a » 0.35 nt / 0.58 kg = 0.6 m/s2.  In other words, it would travel 1 meter in a time of about Ö(2 d/a) = Ö(2.0/0.6) » 1.8 seconds.  As we watched, Bill set up the experiment, and we measured with a stopwatch to be 1.75 seconds.  Physics works!

Finally, Bill speculated that the air cart would go faster if he replaced the heavy batteries inside with a lead to an external battery pack.  However, he was only able to increase the stall angle to about .  How could he do it better, and where can he get better wires to conduct electricity into the fan?

Eduardo de Santiago (Civil and Architectural Engineering, IIT)
Bridge Design Lecture for 2002 IIT Bridge Contest
[http://bridgecontest.phys.iit.edu/]
Eduardo said that the goal of a Structural Engineer is to predict the forces acting on structures, and to determine whether those structures will collapse.  He limited the discussion to Truss Bridges, addressing these basic questions:

• Why do they look the way they do?
• How do we make them stronger?

An old-fashioned bridge design might amount to putting a plank [or a tree] across a gap between two supports, as shown here:

This is not a very good bridge design, as can be seen in the "worst case" scenario by putting a significant load at the middle of the bridge.  The bridge will bow in the middle if the load is substantial enough, because of the Shearing Force and the Bending Moment.

• The shearing force on a small segment acts "up" at one end and "down" at the other end, and tends to "slice through" the segment.
• The bending moment on a small segment acts clockwise on one end and counterclockwise on the other end, and tends to "bend" the segment.

The Bending Moment is most evident in practice; the plank bends as you walk across it. As viewed by a termite inside the middle of the plank the force changes gradually from compression to extension as you go through the plank from top to bottom, as shown:

From an engineering viewpoint, the material along the edges of the plank is under the greatest distress [stress], so that it would constitute an improvement to "hollow out the beam":

However, in such a case, the top part of the beam would carry all the load, and the bottom part would support nothing.  Therefore, we insert vertical supports to transfer the load from the top to the bottom:

The shear forces would then cause a problem, and we must add diagonal members to transfer both horizontal and vertical forces. It is the vertical components that serve to reduce shear forces:

Craftsmanship is important in preparing these joints, in that it is important that the pieces fit together tightly, and that the joint members line up so that their centers meet at a point.  The fundamental principle of Truss Design is to replace all shear and bending forces with compression and extension forces, and to reduce the structure to a series of triangles.  There are several different types of basic bridge designs, such as these:

Pratt Design
Source: http://www.geocities.com/Baja/8205/truss.htm

Warren Design
Source: http://www.geocities.com/Baja/8205/truss.htm

Sunshine Skyway Cable Stay Bridge
Source:  http://www.pbs.org/wgbh/nova/bridge/meetcable.html

A great deal of information is provided at the West Point Bicentennial Engineering 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.

Eduardo mentioned that cross-bracing between trusses is required at their tops and bottoms. Eduardo gave the following tips and pointers:
• Make as few joints as possible.
• Be sure that there is a good fit at all joints.
• For crossed pieces, it is better to notch them slightly and glue them for a better fit, but don't make another joint there.
• Be sure to glue doubled sticks all along their lengths, and not just at the ends.

He closed with the following observations:

• Buttresses are good for bridges that permit support below the roadway, as is not often allowed in contests.
• Every bridge begins in the mind of  an engineer.
• In earthquake engineering, the idea is to save the people, even if the structure collapses.  It is often said they don't make cars like they used to, and it is true.  Cars were once designed to survive a serious collision, even though the occupants might be killed.  Today the occupants are more likely to be saved, although the vehicle may be destroyed.

Notes taken by Porter Johnson