Mathematics-Physics High School SMILE Meeting
29 February 2000
Notes take by Porter Johnson

Porter Johnson (IIT Physics)
told us of Birthdays on Today's Date (Leap Day!): Gioacchino Rossini [http://www.naxos.com/person/Gioachino_Rossini/26313.htm] in 1792 and Herman Hollerith http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hollerith.html in 1860. The first being the composer (who wrote William Tell), and the latter the inventor of punched card computers. Interesting stuff!

placed two tuning forks on the table, each mounted on its own sounding box. They appeared to be identical except one was shorter than the other by about 1 cm.

Larry asked us to predict how their frequencies (pitches) would compare. And then he tapped one, damped it, and then the other. The longer fork had a lower pitch, which is what one would expect, because its mass would be greater; the stiffness of each would be comparable, being of same material and cross-section. But then Larry attached two identical small masses, one to each of the tines of the shorter fork, and positioned them carefully to a premarked location. Once again he sounded the forks, and now they appeared to have the same pitch! The added mass on shorter fork had lowered its pitch to match that of the longer fork. Neat!

But Larry had more. He placed the forks so the open ends of their sounding boxes faced each other, and then struck one fork. We heard it sound, and then Larry damped it; but we then could hear the faint sound coming from the other fork. Resonance! But the sound was weak, and to make it more obvious that the second fork was set into resonance by sound from the first, he suspended a ping pong ball from a light string so that the ball very lightly touched the upper end of one of the tines of the second fork. Then he repeated the experiment. The ping pong ball bounced away from the tine of the second fork as it vibrated from the resonant transfer of sound energy from the first fork. Beautiful!

Were the forks at exactly the same frequency? Someone suggested sounding them both at once; we would hear beats if the frequencies were slightly different. And we did! By adjusting the location of the small masses, Larry controlled the beat frequency that we heard. He set up a frequency meter with microphone, and displayed the frequency of each fork. The beat frequency would be the difference of the two (262 - 256) Hz. Very nice!

Walter McDonald (CPS substitute/Medical Technician) showed us about inequalities. He showed us both algebraic expressions and their geometric graphs to visualize those inequalities. Eg. A parabola

y = x2 - 1

was plotted, and then a region within the parabola representing

y > x2 - 1.

(handout) Good math connections!

Fred Schaal (Lane Tech HS) showed us classroom use of plotting calculators, and he used one to project to the screen so we could see what was going on. Betty Roombos followed Fred's directions to do this, and it worked well. Fred had her (and others with calculators he had passed out) plot the equations of straight lines and find their intercepts with the x and y axes. He used number pairs to determine a set of lines:

 x-intercept y-intercept line 10 1 y = -(10/1)x+10 9 2 y = -(9/2)x+9 8 3 y = -(8/3)x+8 7 4 y = -(7/4)x+7 6 5 y = -(6/5)x+6 5 6 y = -(5/6)x+5 4 7 y = -(4/7)x+4 3 8 y = -(3/8)x+3 2 9 y = -(2/9)x+2 1 10 y = -(1/10)x+1

The intersections of successive pairs of lines were determined by using the plotting computers, and one could even zoom up close to the intersections to see more detail and better determine the actual number pair locating each intersection. He then used the intersection points as data, and did a linear, quadratic, and cubic fit to those points on a calculator. He compared the results with the "tangent curve" described below by Porter Johnson. A fine phenomenological math lesson!

Bill Shanks (Joliet Junior College)
raised the question, What makes things fly? Following the usual textbook explanation, he sketched a wing cross-section, said that the air over the top of the wing had to travel farther than that on the bottom, and so had to move faster, resulting in lower pressure at the top wing surface than on the bottom - and so providing lift on the wing a la Bernoulli. But then Bill sketched a flat-looking wing (supersonic aircraft?) and pointed out there would be very little difference in path over top and bottom. Same for a kite. No asymmetry - no difference - no lift! But then Bill pointed out that - assuming a wing inclined upward relative to the air stream - air striking the bottom surface would be deflected downward. Then the change in its momentum would be downward, corresponding (Newton's 2nd Law) to a downward force on the air by the wing. But the air must be exerting an equal and opposite force (Newton's 3rd Law) on the wing, which would be upward - resulting in lift! So where does that leave Bernoulli? (or those who invoke his "law"?) Refreshing, Bill!

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

x/a + y/(b-a) = 1

Let us regard the parameter b as being fixed, while a varies continuously between the values 0 and b. A solid region in the first quadrant is filled by these lines. To determine the boundary of that region, we solve the above relation for y, to obtain

y(a) = (b - a) (1 - x/a) = b + x - a - (bx)/a

In this relation, keep x fixed, and vary a. The maximum value of y under such variation can be calculated by setting the derivative dy/da to zero:

dy/da = - 1 + (bx)/a2 = 0 ,
so that
a = Ö(bx)

We substitute this value of a into the expression for y(a) to obtain

ymax = b + x - 2 Öbx = (Ö b - Ö x)2.

To show that that this is indeed the maximum value of y(a), calculate the quantity ymax - y(a):

ymax - y(a) = a + bx/a - 2 Ö(bx) = ( a - Ö(bx) )2 /a .

Clearly, the right side is non-negative, and it is zero when a is chosen as above. The formula for ymax gives the greatest value that y can have for a given value of x, and thus lies at the top of the region traced out by the straight lines described above. Thus, the top of the region is given by

y = (Ö b - Ö x )2.
or
Ö x + Ö y = Öb.

This curve, which represents the envelope of all the straight lines, can also be written as

(x - y)2 + b2 = 2 b (x + y)

The curve is a parabola with the line of symmetry [axis] lying along the line x=y. It is symmetric under interchange of the variables x and y. The point closest to the origin has coordinates

x = b/4 = y .

This is the "symmetry point of the parabola", and is the "tangent point" for the curve with a = b/2, namely

2x/b + 2y/b = 1 .

Every other point on this curve is a "tangent point" for one and only one of the straight lines described above, which have intercepts a and b-a, respectively.

Here is an Excel-generated image of the lines and the asymptotic curve:

Incidentally, these "envelope curves" occur frequently in geometrical optics, in which light rays move in straight lines in a uniform medium. Clearly, the "bundle" formed by all light rays can have a nontrivial structure. The boundary of that "bundle" is called a caustic in geometrical optics. As an example, see http://www.math.harvard.edu/archive/21a_spring_06/exhibits/coffeecup/index.html The Coffeecup Caustic. Here is a brief description of the effect, taken from that reference.

You are drinking form a cylindrical cup in the sunshine. Sometimes, when the sun shines into the cup, you can see a crescent of light as the sunshine reflects from the inside of the cup onto the surface of the drink. A picture of a real cup is shown, and you can do your own on-line computer simulations of the effect. Check it out!

See You on 14 March!