The Essential Exponential! Mathematical Modeling IN Beginning Physics. Robert G. Fuller and Vicki L. Plano Clark University of Nebraska Lincoln

Transcription

1 The Essential Exponential! Mathematical Modeling IN Beginning Physics Robert G. Fuller and Vicki L. Plano Clark University of Nebraska Lincoln

2 Poster Abstract: Mathematical Modeling in Beginning Physics - The Essential Exponential! Robert G. Fuller, Univ. of Nebraska - Lincoln, 314 Ferguson Hall, Lincoln, NE ; ; fax ; rfuller2@unl.edu Vicki L. Plano Clark, Univ. of Nebraska - Lincoln In our attempt to humanize physics 1 and take seriously the warning of Albert A. Bartlett, "Humanity s greatest weakness is its inability to understand the exponential function" 2, we have developed a variety exponential modeling activities for lectures, recitations, laboratories and examinations. We use these in our beginning physics courses for students who intend careers based on the life sciences. In this poster we will present a variety of those activities and encourage all other physics teachers to include such activities in their courses. When was the last time you taught your students the Rule of 70? Come and see our poster and find out about it! 3 1 Partially supported by NSF DUE Grants # and # A. A. Bartlett, Arithmetic, Population, and Energy, AAPT Announcer, 32(2) 154, and Forgotten Fundamentals of the Energy Crisis, Am. J. Phys. 46(9), , see

3 If you have any one- Exponential expressions: decay: -bx y(x)=a e growth: +bx y(x)=a e Halving value i constant (decay) Doubling value is constant (growth) Rate of change of y proportional to y Constant percent change Linear when plotted on a Semi-log graph you have them all!

4 The greatest shortcoming of the human race is our inability to understand the exponential function. AAB, 1978 For example: If the rate of consumption grows at a mere 7% per year, the consumption in the next decade EXCEEDS the total of all the previous consumption!

5 Rule of 70 T 2 = 70/P where T 2 = Doubling Time P = percent growth per unit time How does this come about? (Hint: ln 2 = 0.693) So for physicists it could be the rule of 69.3

6 Gall Bladder Surgery (from AAB) (5 points for each correct answer and its explanation) As a part of your job as a health care provider, you have been asked to come up with a plan to help Americans pay for their health care costs. You have been presented with the following data: The cost of gall bladder surgery in the United States is given below: Year Cost 1950 $ $ $ $ $ $ 803* The Medicare Program went into effect in $ $ $ $ $ $ 4900 (a) Draw an appropriate graph of these data, label your axes. Explain. From these data, or graph, estimate, and explain the reasoning you use to make your estimate (b) When did gall bladder surgery cost $100? Explain. (c) When will gall bladder surgery cost twice as much as it did in 1980? Explain. (d) The present annual percentage change in the cost of gall bladder surgery. Explain. (e) The cost of gall bladder surgery near the end of your lifetime say, in the year Explain.

7 Optical Density (Student ID numbers for initial values) You may earn up to 10 points based on showing your work and explaining it. Pretend you are a successful research scientist who has invented a thin plastic filter that can protect human skin from ultraviolet radiation. In fact, the optical density of a thin sheet of this plastic, onetenth of a millimeter in thickness, has an optical density that is exactly the same, to three significant figures, as three digits of your student ID number. To determine what value of optical density this is put a decimal in front of the first non-zero digit in the last four digits of your student ID number and then round this decimal to three significant figures, e.g. if the last four digits of your student ID number are 5678 your film would have an OD of 0.568, or if the last four digits of your student ID number are 0234 your film would have an OD of (2 points) Determine the optical density of your filter to three significant figures: Optical Density (OD) = (4 points) What is the fraction (3 sig. fig.) of ultraviolet radiation that is transmitted by one sheet of your filter? 3. (4 points) To adequately protect human skin from ultraviolet radiation, the UV intensity must be reduced to 1% of the incident radiation, i. e. you must build a filter that has an optical density of How thick (3 sig. fig.) must this filter be in millimeters?

8 Chinese student enrollment (30 points) The total student enrollment at a large Midwest university was 18,800 students in Assume the total student population there is growing at a constant rate of 2.2% per year. 27. (5 pts) Estimate the total student enrollment at that university in (10 pts) The numbers of Chinese students enrolled in that university are give in the chart below. Assume the number of Chinese students continues to change at the same rate in the future. Derive a mathematical expression for the number of Chinese students at that university as a function of the number of years since (5 pts) Use your expression to predict the number of Chinese students at that university in (5 pts) Using the same procedures you used to answer parts (a) and (c), predict when all of the students at that university will be Chinese. 31. (5 pts) Examine what you did to answer part (d) and explain the flaws in your procedures or assumptions.

9 Nerve Membrane (24 points ) A typical nerve membrane can be treated as a parallel plate capacitor and a resistor. The sodium and chloride ions can be treated as an ion pump that keeps the nerve membrane at a rest potential of 100 mv. A nerve pulse from the axion can be modeled by a switch, S, that shorts out the ion pump for a short time. See the circuit diagram. Assume the capacitance of a unit area of the membrane is 34.6 nf. The resistance of a unit area of the membrane is 2.0 M Ω. (a) (6 points) What is the total charge on a unit area of the membrane when it is at its rest potential ( i.e. completely polarized)? Show your work. (b) (6 points) If the nerve pulse is suddenly applied to the polarized membrane at time t = 0.0 s {i.e. the switch is closed}, and remains applied for 144 ms{i.e. the switch is opened again}, at what time will the nerve potential first be 50 mv, i.e. half of its original value? Show your work. (c) (12 points) Using the conditions stated in part (b), draw a graph of the voltage across the membrane as a function of time for 288 ms. Label your axes. Show the units. Indicate on your graph the voltage on the membrane at 288 ms.

10 Physics 142 Lecture Quiz 6 October 6, 2000 Name Reci. Section 8:30 11:00 12:30 Please do your own work. Open book and notes. Please do not talk to your neighbors. You may earn up to 10 points based on showing your work and explaining it. The membrane that surrounds one of your nerve cells can be assumed to behave like a small parallel plate capacitor. To send a message the cell is fully charged(v = 100 millivolts = 100 mv) and then suddenly discharged. The discharge can be modeled, to first approximation, as an exponential decay. The time is takes the electric potential across the cell to drop to one half of its original value,t 1/2, is given, in milliseconds, by putting a decimal in front of the first non-zero digit in the last four digits of your student ID number and then round this decimal to three significant figures, e.g. if the last four digits of your student ID number are 5678 your film would have an OD of 0.568, then multiply that decimal by 10, so you get a number between 1 and 10 milliseconds, with three significant figures. Drop to one-half time = T 1/2 =. ms 1. (6 points). Plot the electric potential across the cell as a function of time. LABEL your axes with appropriate numbers and units! 2. (2 points) How long does it take the potential across the cell to drop to 6.25 mv? 3. (2 points) If your diet is high in complex carbohydrates, then the T 1/2 is increased to three times its original value. With this new value of T 1/2 how long would it take the potential across the cell to drop from 100 mv to 6.25 mv?

11 Sample Exponential Lab Activities -1 Thermal Cooling Graph Metal can with water Temperature sensor Mechanical Damping

12 Sample Exponential Lab Activities -2 Electrical Damping (LRC Circuit Voltage) Graph RC Circuit Voltage A + + MBL V 0 C R V - - sensor B

13 Sample Exponential Lab Activities 3 Absorption of White Light by Color Filters 100 Absorption by Color Filters Filter Thickness (# of Filters) Absorption of Beta Radiation by Aluminum Absorption of Radiation G-M Sensor Radiation Source y = 24.79ē x R 2 = Absorber Thickness (cm)

14 Sample Exponential Lab Activities -4 Optical Density vs. Liquid Concentration Optical Density = log 10 I 0 I Optical Density as a Function of Sample Concentration y = x R 2 = OD Concentration (g/ml)

15 A proposed Book The Essential Exponential! The greatest shortcoming of the human race is our inability to understand the exponential function by Albert A. Bartlett, University of Colorado at Boulder with Robert G. Fuller and Vicki L. Plano Clark University of Nebraska Lincoln A Guide to this Mathematical Model for Teachers Introduction: Why the Exponential is Essential by AAB Organizing Learning Experiences for Exponential Modeling by RGF & VLPC Classroom Activities on Exponential Modeling by RGF and VLPC Mechanics Fluids Waves and Sound Heat and Thermodynamics Electricity and Magnetism Light and Optics Twentieth Century Physics Background Readings Collection of AAB articles from the American Journal of Physics The Forgotten Fundamentals of the Energy Crisis 46(9) , 1978 Sustained availability: A Management Program, 54(5), , 1986 and The Physics Teacher The Exponential Function, published in 11 parts from and articles from other journals. Other related articles about Exponential Modeling from physics journals Perhaps include economics and other content, e.g. L. David Roper s work, M. King Hubbert s work, etc. Bibliography Index (cross listing by content and type of activity, i.e. lecture, discussion, examination question, homework assignment, or laboratory.) (clip and mail to M. McDowell, Physics, UNL, Lincoln, NE, Yes, I want to keep informed about this book! Name: Mailing address: address: Or send your name and mailing address to mmcdowell1@unl.edu. Use The Essential Exponential as the subject. Zip

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