Cantilevers, physical science, and rational functions

As an experiment, I opted to conclude the week on forces not with a quiz but rather with demonstration of a non-linear system in which a meter stick cantilever arm balances an increasing amount of mass on the other of the suspension line. This was a modification from earlier terms.
The final mass was a whopping 3050 grams suspended by a single string

In 056 Center of gravity and static force, a single 200 gram meter stick is suspended from an overhead track using a string. The exercise begins with no mass hung from one end of the meter stick and the string tied at the 50 cm mark. The meter stick hangs balanced.

A 50 gram mass is added as close to the end as possible, and the string is moved towards the mass in order to rebalance the system. The length of the balancing arm of the meter stick was recorded in a table. Then additional masses were added. The data gathered is contained in the following table.

Cantilever length (cm) mass (gmf) Theory (gmf)
50 0 -0
59.5 50 47
61.5 60 60
63 70 70
65.5 90 90
67.7 110 110
69.6 130 129
71.3 150 148
72.8 170 168
74 190 185
77.8 250 250
81.5 350 341
92 1050 1050
95.7 2050 2126
97 3050 3133

When graphed, the result is as seen below.

The small blue squares are the actual data, the green diamonds are the theoretic values based on a rational function. Note that the power function is a poor fit to the data at high mass values.
where rho is the density per linear centimeter for the meter stick. For the system above the value was two grams per linear centimeter. In a spreadsheet with the data arranged as seen above, the formula becomes =2*(5000-100*A2)/(A2-100).

I stumbled onto the system a couple of years ago without realizing that the mathematical relationship involved a rational function. The relation can be derived by setting the torques to be equal. Rational function are fairly rare in the lives of most people, and finding a simply physical system that is modeled by one was a real plus.

I had intended to stop near 300 grams, the limit on my smaller hanging mass set. At around 300 grams I asked whether the meter stick could balance an even higher mass, such as a person. The class was in deep doubt. Some of the students have absorbed that experimental facts on the ground are what count, not asserted hypotheses.

I realized I had to go farther. I found a hanger that would handle the kilogram slot masses. One kilo. Two kilos. Three kilos. "Take a picture!" shouted an amazed student. I then used a second meter stick to remind the students that the meter stick had a mass of only 200 grams, yet was balancing 3050 grams (three kilograms plus a 50 gram hanger).

Few in the class have completed college algebra. In the present text, rational functions are in chapter four - some sections do not reach chapter four. Thus this is a mathematical functional form few students have seen. Terra incognita.

For homework the students are to graph the data using a spreadsheet and add a trend line for the regression type that best fits the data. No spreadsheet offers a fit to an arbitrary function such as a rational function, I am uncertain that such a capacity exists in anything easily and openly available. A WolframAlpha inquiry on fitting to a rational function has no response.

On the next Wednesday class I plan to try to "explain" how the function "works" at cantilever lengths (Lc) of 50 and 100 centimeters. The function produces physically real results only for 50 ≤ x < 100, x ≠ 100. Here the asymptote has a very physical meaning: the mass of the cantilever, or more accurately the torque, is irrelevant to the mass being suspended as the mass is directly under the suspension point. Mathematics made visual and concrete.

This summer I will be teaching a course in exponential and trigonometric equations. My plan to use physically real systems, spreadsheets, and WolframAlpha to bring a deeper level of insight and understanding to the students. A narrower but deeper river. Chapter four and four point two will definitely be on my syllabus. Along with the decay rate of the subsequent bounces of a super ball and the trigonometric sine function of a RipStik, and low mass chains. I may also play with the path function in SVG if I have a fair number of CIS majors in the class - using spreadsheets to generate the data points for an SVG chart. I have used SVG in MS 100 College Algebra in the past. Hard work and good fun.

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