A Method For Teaching Algegra Based Advanced Placement Physics And Calculus Based Advanced Placement Physics, In The Same Clas
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Teresa A. Walker 1532 Blackbird RD NE Rio Rancho, NM 87144 (505) 892-8041 [email protected]
April 17, 2008
Dave Westpfahl, Ph. D W. D. Stone, Ph. D Julie Ford, Ph. D. New Mexico Tech Master’s of Science Teaching Program 801 Leroy Place Socorro, NM 87801
Dear Drs. Westpfahl, Stone, and Ford:
You will find attached my final Independent Study Project entitled “A Model for Teaching Algebra-based Advanced Placement® Physics and Calculus-based Advanced Placement® Physics in the Same Classroom”. This will complete my Master’s of Science Teaching degree.
Hours of research and writing have gone into producing this project to support AP Physics teachers and students. In an effort to provide improved education for high school students, Advanced Placement ® (AP) Physics provides students with advanced physics concepts, prepares them for college, and makes them more attractive to competitive colleges. Providing this report to physics educators will provide them support in teaching both algebra-based AP Physics and calculus-based AP Physics in the same classroom. Providing this program at their school will allow physics instructors with a smaller population of students to offer the class by increasing enrollment.
I would like to thank you for your time in reading and reviewing this project. Please contact me if you have additional concerns or questions.
Sincerely,
Teresa A. Walker A MODEL FOR TEACHING ALGEBRA-BASED ADVANCED PLACEMENT® PHYSICS AND CALCULUS-BASED ADVANCED PLACEMENT® PHYSICS IN THE SAME CLASSROOM
Prepared for Physics Educators And New Mexico Tech Masters of Science Teaching (MST) Program
Particularly
David J. Westpfahl W. D. Stone Julie Ford Professor, Astrophysics Professor, Mathematics Professor, Technical Communication
Prepared by Teresa Walker [email protected] http://www.orgsites.com/nm/walker/ Instructor Rio Rancho High School, Rio Rancho Public Schools Rio Rancho, New Mexico
Submitted April 17, 2008 ABSTRACT
This project provides a plan for teaching algebra-based Advanced Placement® Physics (AP Physics B) and calculus-based Advanced Placement® Physics (AP Physics C) in the same classroom. This project provides a general timeline for presenting all the material required by both classes and in particular provides self-guided lesson plans for students when the material of each of the courses is entirely different. The material covered in the self-guided lesson plans for AP Physics B includes waves and optics, thermal physics and fluid mechanics, and nuclear physics. The material covered in the self-guided lesson plans for AP Physic C includes rotational motional, calculus-based electricity including Gauss’s Law, and calculus-based magnetism including Ampere’s Law.
The purpose of this project is to provide high school physics educators with support and to provide physics students with quality lessons on the specific topics described. The goal for teaching these two classes concurrently in the same classroom is to improve enrollment in AP Physics. Literature research was done on the specific topics using college physics textbooks and the internet. The author’s experience teaching the subjects this way for six years also contributed to the lesson plans presented.
iii TABLE OF CONTENTS
ABSTRACT ……………………………………………………………………………………. iii LIST OF TABLES AND FIGURES …………………………………………………………..... v INTRODUCTION ……………………………………………………………………………… 1 OVERVIEW OF COURSE TOPICS AND ORDER………………………………………….... 3 PRACTICAL THOUGHTS ON MANAGING THE COURSES TOGETHER ……………….. 6 RECOMMENDED BOOKS AND INTERNET RESOURCES ………………………………... 7
UNIT 3, WAVES AND OPTICS (PHYSICS B) AND ROTATIONAL MOTION (PHYSICS C) ……………………………………………………………………………….…… 9 Physics B Unit 3, Lesson 1, Wave Beginnings (B31Wave) …………………………………….. 9 Physics B Unit 3, Lesson 2, Reflection and Refraction (B32Reflect) …………………………. 14 Physics B Unit 3, Lesson 3, Lenses and Mirrors (B33Lenses) …………………………...……. 15 Physics B Unit 3, Lesson 4, Diffraction and Thin Films (B34Diffract) ……………………….. 20 Physics C Unit 3, Lesson 1, Calculus-based Rotation (C31Rotate) …………………………… 23 Physics C Unit 3, Lesson 2, Rotational Inertia (C32Inertia)………………………...…………. 25 Physics C Unit 3, Lesson 3, Incline and Pulley (C33Incline) ………………………….…….… 27 Physics C Unit 3, Lesson 4, Other Rotational Situations (C34Other) ……………………….… 28
UNIT 5, FLUID MECHANICS AND THERMAL PHYSICS (PHYSICS B) AND ELECTRICITY IN DEPTH (PHYSICS C) ……………………………………………………. 30 Physics B, Unit 5, Lesson 1, Pressure and Buoyant Force (B51Pressure) …………………..… 30 Physics B, Unit 5, Lesson 2, Bernoulli and Torricelli (B52Bernoulli) …………………...……. 33 Physics B, Unit 5, Lesson 3, Gas Laws (B53Gas) ……………………………………………... 34 Physics B, Unit 5, Lesson 4, Thermodynamics (B54Thermo) ………………………………… 36 Physics C Unit 5, Lesson 1, Gauss’s Law for Spheres (C51Gauss) …………………………… 39 Physics C Unit 5, Lesson 2, Continuous Charge Distributions (C52Continuous) …………….. 41 Physics C Unit 5, Lesson 3, RC Circuits (C53RC) ……………………………………………. 43 Physics C Unit 5, Lesson 4, Electric Potential from Electric Field (C53Potential) ……………. 45 Physics C Unit 5, Lesson 5, Other Field, Potential and Capacitance Problems (C54Other) ...… 48
UNIT 7, PHYSICS B NUCLEAR AND QUANTUM PHYSICS AND PHYSICS C MAGNETISM IN DEPTH ………………………………………………………………..…… 50 Physics B Unit 7, Lesson 1, Nuclear Physics and Radioactivity (B71Nuclear) ……………….. 50 Physics B Unit 7, Lesson 2, Quantum Physics and Wave Particle Duality (B72Quantum) …… 52 Physics B Unit 7, Lesson 3, Bohr and Energy Level Diagrams (B73Bohr) …………………… 53 Physics C Unit 7, Lesson 1, Ampere’s Law (C71Ampere) ……………………………………. 55 Physics C Unit 7, Lesson 2, LR Circuits (C72LR) …………………………………………….. 56 Physics C Unit 7, Lesson 3, Other Magnetism Problems (C73Other) …………………………. 58 CONCLUSION…………………………………………………………………………………. 59 BIBLIOGRAPHY AND WORKS CITED…………………………………………………...… 61
iv LIST OF TABLES AND FIGURES
Table 1. Side by Side Comparison of Order and Topics Covered for Physic B and Physic C ………………………………………………………………………………. 3 to 5 Table 2. Additional Book and Internet Resources for Physics B and Physics C ………………………………………………………………………...…….. 7 to 8 Table 3. Unit 3 Overview, Physics B Waves & Optics and Physics C Rotational Motion ………………………………………………………………………………….... 9 Figure 1. An ambulance experiencing the Doppler Effect…………………………………….. 10
Table 4. Table of harmonics and overtones……………………………………………………. 13
Table 5. Synopsis of mirrors and lenses………………………………………………………. 16
Table 6. Answer chart for the practice problems for ray tracing…………………..…….. 19 to 20
Figure 2. Free body diagrams for an Atwood Machine ……………………………………….. 26
Table 7. Unit 5 Overview, Physics B Fluid Mechanics & Thermal Physics and Physics C Electricity in Depth ………………………………………………………………………….…. 30
Figure 3. Schematic of energy input and output for a heat engine…………………………….. 38
Table 8. Unit 7 Overview, Physics B Nuclear and Quantum Physics and Physics C Magnetism in Depth ……………………………………………………………………………………….... 50
v INTRODUCTION
The purpose of my project is to provide support materials and guidance for physics teachers who wish to teach both types of Advanced Placement® (AP) Physics concurrently in the same classroom setting. The two courses are algebra-based Physics B and calculus-based Physics C. Physics C consists of two parts, (1) Mechanics and (2) Electricity and Magnetism.
This year, and for the past 6 years, I have taught the courses concurrently in the same classroom. Approximately 30 students each year at my high school took the courses, 60% taking the single course of Physics B and 40% taking the two parts of the Physics C courses. The single course of Physics B or the two courses of Physics C are essentially equal to two semesters of freshman level college physics. I am often asked, “Why do you teach two AP classes at the same time?” This introduction answers the “Why” question. It also provides information to answer the other important question, “How?”
What is the value of AP classes to students? College professors, high school teachers, and students may all have different views of these courses, both pro and con. Research also gives a variety of findings, also both pro and con. College Board administers all AP exams and for the school year beginning in the fall semester of 2007 has audited the content of every course that carries the designation AP, which is a registered trademark of the College Board. Two studies conducted at the University of Texas and funded by College Board show that students who take AP classes and pass the tests have higher college grade point averages and receive a bachelor’s degree in fewer years (Chute, 2007). College Board itself reports two different studies, one from University of California at Berkeley and one from the National Center for Educational Accountability, which find that an AP exam grade, and particularly an exam grade of 3 (which is considered passing), are strong predictors of ability to complete a college degree (College Board, 2007). However, there have been reports of the erosion of the value of AP classes and exams (Marklein, 2006). Other independent research studies showed little or no relationship between AP classes and college success (Chute, 2007). Selective schools such as Harvard, Yale, and MIT require students to take additional courses even if those students have passed the AP exam in that subject (Marklein, 2006). College Board president Gaston Caperton has stated that College Board has accelerated its drive to reach underserved students of differing ethnic, racial and economic backgrounds (Caperton, 2005). His hope is that AP classes can help to increase rigor in schools (Marklein, 2006).
Regarding the value of AP classes to students, my educational philosophy is to give all students, the best chance at success in their lives. I teach students of differing academic levels, some who are absolute geniuses. I also teach those who are learning English for the first time or have significant learning disabilities. Not everyone can be a brain surgeon, the epitome of all career choices according to my grandmother. Some students can be brain surgeons or engineers or computer systems analysts. They just don’t know it. College physics is required for many math, science, and technology-related degrees. I don’t want students’ possibilities limited by an inability to pass college physics classes. One of the best ways I can prepare students for college physics and open the most avenues for success is to teach AP Physics. Whether they scrape by or perform brilliantly, they have gained greater ability. They will be better able to deal with the
1 difficulty, disappointment and failure they will encounter because they experienced these things in AP Physics.
What value does teaching this class have for me? I hear a lot of talk about teacher burnout. Colleagues complain about the stress of our job. Young teachers leave the profession within a year or two. Even I get tired of explaining the solution to a common misunderstanding for the 100th or the 1000th time. AP Physics has its common misunderstandings, but it also has new and exciting ones every single year. I am fond of telling my students that I have completed 30 years worth of AP Physics problems and yet, every year, there is at least one new one that makes me think. The problem encourages me to be flexible, inventive, and to look at things in a new way. Also, I get to teach wonderful students in AP Physics. They are not all at the top of their class, but they are willing to take risks, try something new and difficult, and to work hard. They have their own unique personalities, and they are never dull. The variety inherent in the class and the students helps me avoid teacher burnout. This variety keeps me loving my job, which benefits me and all my students, not just one classroom.
Job satisfaction isn’t the only factor to consider when discussing concurrent physics courses. Money is a significant issue. It is the subject of a question on one of my final exams. “What question do I always ask you? What makes the world go round?” The answer is money. Most teachers know that administrators require a minimum number of students in an elective class for the course to be offered. In my school they talk about money and numbers of teaching positions in terms of full-time equivalents (FTE’s). I can’t tell you everything about FTE’s, but I do know that in public education, just like in the corporate world, money is important. To keep within my school district’s budget, an elective class must have at least 20 students. AP Physics is an elective class and I want to teach it so I will do all that is reasonable to get 20 students enrolled in the class. One of those reasonable things is to offer Physics B and Physics C in the same classroom.
I have provided you with the reasons why I teach AP Physics as I do. Perhaps you are just teaching one of the AP Physics classes. Or maybe you already teach concurrent courses or are considering doing so. I have developed a step-by-step plan for teaching this material. This plan includes self-guided lesson plans for students taking either calculus or algebra-based physics when the studied topics don’t overlap. This provides students with the greatest benefit when time is limited. If you would like to contact me or need additional information, please visit my website at http://www.orgsites.com/nm/walker/.
2 OVERVIEW OF COURSE TOPICS AND ORDER
At this point, if you are reading this I am going to assume that you are actually considering teaching some course of AP Physics, either Physics B, Physics C, both courses separately, or both concurrently in the same classroom. There are a few items of general information that I think would be helpful and I am going to go over those in this section.
College Board oversees all AP classes and administers AP exams. The courses that would actually be taught simultaneously are algebra-based Physics B and calculus-based Physics C. Physics C consists of 2 parts, (1) Mechanics and (2) Electricity and Magnetism. Physics B is one course designed to cover the first 2 semesters of introductory college-level algebra-based physics. Physics C, Mechanics and Physics C, Electricity and Magnetism are designed to cover 1 semester each of introductory college-level calculus-based physics. I teach 2 AP Physics courses at Rio Rancho High School (RRHS). They are titled AP Physics B and AP Physics C. The curriculum of both C courses is covered in AP Physics C at RRHS. Topics overlap in the 2 different courses about half of the time. The other half of the time, totally different topics are covered. A comparison of the topics covered can be found on the College Board web site at: http://apcentral.collegeboard.com/apc/members/repository/ap05_phys_objectives_45859.pdf. The goal of teaching either of the classes is for a student to understand the concepts of 2 semesters of introductory college-level physics, comparable with the first 2 semesters of physics offered at a college or university. In order to do this I have chosen to structure the scheduling of the classes as shown in the table below.
Table 1. Side by Side Comparison of Order and Topics Covered for Physic B and Physic C.
Physics B Physics C, Mechanics AND Electricity and Magnetism
Course Unit Outlines Course Unit Outlines 1) Newtonian Mechanics 1) Newtonian Mechanics a) Kinematics including vectors, a) Kinematics including vectors, displacement, velocity, and acceleration displacement, velocity, and acceleration for 1 dimensional motion and 2 for 1 dimensional motion and 2 dimensional/projectile motion dimensional/projectile motion b) Newton’s Laws including static b) Newton’s Laws including static equilibrium, dynamic equilibrium of a equilibrium, dynamic equilibrium of a single particle and systems of two or single particle and systems of two or more objects more objects c) Work, energy and power including the c) Work, energy and power including the Work-Energy Theorem, conservative Work-Energy Theorem, conservative forces, potential energy and forces, potential energy and conservation of energy conservation of energy d) Conservation of linear momentum for a d) Conservation of linear momentum for a system of particles including impulse, system of particles including center of collisions, and explosions mass, impulse, collisions, and explosions
3 2) Centripetal Motion, Gravitation, and 2) Centripetal Motion, Gravitation, and Oscillations Basic Rotational Dynamics Oscillations, Basic Rotational Dynamics and Torque and Torque a) Centripetal motion and gravitation a) Centripetal motion and gravitation including uniform centripetal motion, including uniform centripetal motion, Newton’s Law of Universal Newton’s Law of Universal Gravitation, and circular orbits of Gravitation, and circular and general satellites and planets orbits of satellites and planets b) Oscillations including pendulums, mass b) Oscillations including pendulums, mass on a spring, and simple harmonic on a spring, and simple harmonic motion motion c) Basic Rotational Dynamics and Torque c) Basic Rotational Dynamics and Torque 3) Waves and Optics 3) Rotational Motion a) Properties of traveling waves a) Torque and rotational statics b) Wave motion including properties of b) Rotational kinematics and dynamics standing waves, Doppler effect, and c) Angular momentum and its superposition conservation for point particles and for c) Physical optics including interference extended bodies including rotational and diffraction, dispersion of light and inertia the electromagnetic spectrum d) Geometric optics including reflection and refraction, mirrors and lenses 4) Electricity 4) Electricity Basics a) Electrostatics including charges, field, a) Electrostatics including charges, field, potential, force from Coulomb’s Law, potential, force from Coulomb’s Law, field and potential of point charges, and field and potential of point charges, and field and potential of a planar system of field and potential of a planar system of charges charges b) Electrostatics of conductors and parallel b) Electrostatics of conductors and parallel plate capacitors plate capacitors including dielectrics c) Electric circuits including current, c) Electric circuits including current, resistance, and power, steady state resistance, and power, steady state direct current circuits with battery and direct current circuits with battery and resistors only and steady state resistors only and steady state capacitors in circuits capacitors in circuits 5) Fluid Mechanics and Thermal Physics 5) Electricity in Depth a) Fluid mechanics including hydrostatic a) Gauss’s Law pressure, buoyancy, fluid flow b) Electric field and potential due to continuity, and Bernoulli’s equation charge distributions of b) Temperature and heat including the spherical/cylindrical symmetry mechanical equivalent of heat, heat c) Spherical and cylindrical capacitors transfer, and thermal expansion d) Capacitors in circuits specifically c) Kinetic theory and thermodynamics transients in RC circuits including the kinetic theory of ideal gases, the ideal gas law, the first of law of thermodynamics and PV
4 relationships, and the second law of thermodynamics and heat engines 6) Magnetism 6) Magnetism Basics a) Magnetostatics including forces on a) Magnetostatics including forces on charged particles moving in magnetic charged particles moving in magnetic fields, forces on current-carrying wires fields, forces on current-carrying wires in magnetic fields, and magnetic fields in magnetic fields, and magnetic fields due to long current-carrying wires due to long current-carrying wires b) Electromagnetism including b) Electromagnetism including electromagnetic induction, Faraday’s electromagnetic induction, Faraday’s Law, and Lenz’s Law Law, and Lenz’s Law 7) Atomic and Nuclear Physics 7) Magnetism in Depth a) Atomic physics and quantum effects a) Biot-Savart and Ampere’s Law including photons and the photoelectric b) Inductance including LR and LC effect, atomic energy levels, and wave- circuits particle duality c) Maxwell’s equations b) Nuclear physics including nuclear reactions, conservation of mass number and charge, and mass-energy equivalence
Out of the seven units that I teach for each class, four of the units are essentially the same and three units are completely different. Daily handouts for each of the units that are completely different are the purpose of this project.
5 PRACTICAL THOUGHTS ON DAILY MANAGEMENT OF THE COURSES
I would like to provide you with some practical ideas for day-to-day management when teaching Physics B and Physics C in the same classroom. I think it breaks down to a few useful suggestions. My most important suggestion is to keep a firm grip when scheduling time, both throughout the semester and the class period. Secondly, have high expectations for time management and initiative from your students and encourage and motivate them. Finally, I think you should find the best resources and support system. The goal of these suggestions is to allow you to provide the most beneficial experience for your students in the most efficient way.
Managing time is absolutely the most important aspect of teaching AP Physics and teaching Physics B and Physics C in the same classroom requires even more dedicated time management. Do as much as you can together with both classes for your own convenience and the students’ ease of understanding. In the Overview of Course Topics and Order, I tried to give you my best thoughts on how I schedule topics throughout the semester. Of course, it is important to make sure there is time to cover all topics. When you are covering topics that are different, time management through the class period is important. I typically divide the class time in half and teach each separate topic for half the class. I switch which part of the class is taught first every day so that no group claims favoritism. At the times when the topics are different, it is so important for the students to be responsible, both for their time and their learning which leads me to the next thing I’d like to discuss.
If you are teaching Physics B and Physics C concurrently in the same classroom, you are probably doing this to provide as many students the opportunity to take AP Physics as possible. Make sure students are aware of this from the beginning of the class and that because of this opportunity that they need to flex their college muscles, be independent, and take on more responsibility for their own learning. Students may need to be reminded of this over the semester, particularly later in the semester as the exams loom closer and stress is high. I try to keep motivation as high as possible by reminding students of what they stand to gain from the class. I particularly emphasize the college experience they are gaining before even going to college. I also remind them of the benefit of having a less strenuous freshman year of college. I will also say that the occasional granola bar or slice of pizza as positive reinforcement (feel free to read this as “bribe”) is very useful.
Resources and support are so important. There are numerous books, websites, and people that can be resources for teaching AP Physics. You will not be able to use them all so find and choose the ones that will be most helpful to you. In the next section, I have provided a section on the books and websites that have been most helpful to me. I also suggest that all teachers teaching any class of AP Physics seek out the AP Summer Institute class nearest to them if they have never taken it and even consider taking a refresher if it has been a while. You will receive valuable materials and practical advice from the instructor as well as a chance to meet and network with other AP Physics teachers in your geographic area. I also suggest joining the AP Electronic Discussion Group. When you join this group, be prepared to receive a flood of mail and figure out how to put it into its own folder. You will not be able to read all e-mails, but it will allow you to know what teachers across the country are discussing and ask for their help if needed. This has been very helpful to me on several occasions.
6 RECOMMENDED BOOK AND INTERNET RESOURCES
Sometimes I have students complain that they don’t understand the textbook. I understand that. It has happened to me many times. However, I will not accept that as an excuse for why they don’t understand the material. My resource mantra chanted over and over again to my students and to myself is “If you can’t figure something out from your book, then find another resource and keep doing that until you find one that is best for you.” I believe that this is an absolutely necessary skill. Other resources include additional textbooks, paperback review books, the internet, and other students and instructors. Below you will find some possible resources listed. I have used three different textbooks in teaching Physics B over the years and two different textbooks teaching Physics C. There are also other books I use for reference and several internet websites that I check first for further information. This not a comprehensive list of helpful materials, just a list of the ones I use.
Table 2. Additional Book and Internet Resources for Physics B and Physics C.
Type of List of Resources Resource Physics B College Physics by Raymond A. Serway and Jerry S. Faughn. 7th Edition. Textbooks Published by Thomson Brooks Cole. Physics by John D. Cutnell and Kenneth W. Johnson. 6th Edition. Published by Wiley & Sons, Inc. Physics: Principles with Applications by Douglas C. Giancoli. 6th Edition. Published by Prentice Hall. Physics C Fundamentals of Physics by David Halliday, Robert Resnick, and Jearl Walker. Textbooks 7th Edition. Published by John Wiley & Sons, Inc. Physics for Scientist and Engineers by Paul A Tipler and Gene Mosca. 5th Edition. Published by W. H. Freeman and Company. Principles of Physics A Calculuus-Based Text by Raymond A. Serway and John W. Jewett, Jr. 4th Edition. Published by Thomson Brooks Cole. Other Book Conceptual Physics by Paul G. Hewitt. 10th Edition. Published by Addison Resources Wesley. Cracking the AP Physics B & C Exams by Stephen A. Leduc. 2006-2007 Edition. Published by Random House. Advanced AP Physics B Course Home Page. Placement® http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2262.html. Physics AP Physics C: Electricity and Magnetism Course Home Page. Websites http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2263.html AP Physics C: Mechanics Course Home Page. http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2264.html Table of Information and Equations Tables for AP® Physics Exams. http://apcentral.collegeboard.com/apc/public/repository/physics_equation_tables_ 2008_09.pdf
7 Table 2. (continued) Additional Book and Internet Resources for Physics B and Physics C.
Other AP Physics B by Dolores Gende. http://apphysicsb.homestead.com/. Internet How Stuff Works. http://www.howstuffworks.com/. Resources HyperPhysics by Carl R. Nave. http://hyperphysics.phy- astr.gsu.edu/hbase/hph.html. The Physics Classroom Tutorial by Tom Henderson. http://www.glenbrook.k12.il.us/gbssci/phys/Class/BBoard.html. The Physics Factbook by Glenn Elert. http://hypertextbook.com/facts/index- topics.shtml. The Physics Hypertextbook by Glenn Elert. http://hypertextbook.com/physics/. Wikipedia. http://www.wikipedia.org/.
8 UNIT 3, PHYSICS B WAVES & OPTICS AND PHYSICS C ROTATIONAL MOTION
Table 3. Unit 3 Overview, Physics B Waves & Optics and Physics C Rotational Motion.
Physics B Waves & Optics Physics C Rotational Motion 1) Wave Beginnings - Understand and 1) Rotation Beginnings – Contrast and make calculations for waves including compare linear quantities and rotational the wave equation, superposition, the quantities and make calculations for Doppler Effect, beats, standing waves, equilibrium requiring τnet =0. waves on a stretched string, 2) Moment of Inertia and Pulleys – Gain a fundamental frequencies and deeper conceptual understanding of moment resonance. of inertia, use calculus to determine the 2) Reflection and Refraction- Make moment of inertia of a uniform rod and be calculations for and have a conceptual able to take moment of inertia of a pulley understanding of Snell’s Law, ray into account. tracing for interfaces such as air-water 3) Rotational Motion for an Inclined Plane and air-prism, and apparent depth. – Compare and contrast the use of 3) Lenses and Mirrors - Understand Newton’s 2nd law and conservation of mathematically and conceptually the energy to solve problems for objects of lens/mirror equation for diverging and varying moments of inertia rolling down an converging lenses and concave and inclined plane. convex mirrors and ray trace for any of 4) Other Rotational Situations – Understand these and describe the image. conceptually and be able to make 4) Interference, Diffraction and Thin calculations for the rotational analogies to Films- Understand and make the Work-Energy Theorem, power, calculations for double slit diffraction, Newton’s 2nd Law, angular momentum, and single slit diffraction, diffraction precessional motion. gratings and thin films.
Physics B Unit 3, Lesson 1, Wave Beginnings (B31Wave)
Wave Beginnings - Understand and make calculations for waves including the wave equation, superposition, the Doppler Effect, beats, standing waves, waves on a stretched string, fundamental frequencies and resonance.
Start your homework today by labeling the top of each paper with the date and the exciting code: B31Wave and the page number. Make sure to label any additional pages with your exciting code and the page number so neither one of us gets confused. Feel free to add anything else at the top you need to help you remember what topics you are studying. Today you are going to make sure you know some wave basics and specific information on several topics. Wave basics and their calculations are a significant part of every AP Physics B test. I am covering a lot of topics in this lesson, but it is most important to know the basics. Topics such as the Doppler Effect, beats, standing waves, waves on a stretched string, and fundamental frequencies and resonance are part of this class, of course, but you will have to choose to what depth you want to study this material. According to the prerequisites for this class, you should have done some study of waves before this time. I expect that you understand the essentials about the following terms and concepts:
9 crest, trough, wavelength, transverse, longitudinal, period, frequency, superposition, nodes, antinodes, standing waves, constructive and destructive interference, compression and rarefactions of longitudinal waves, in phase waves, out of phase waves, decibels, nanometers, the electromagnetic (EM) spectrum including radio waves, infrared, visible light, ultraviolet, x-rays and -rays (gamma rays), the trend of wavelengths and frequencies for EM waves, the visible spectrum of light along with its trends in wavelengths and frequencies and the ballpark limits for the wavelengths at both ends of the visible spectrum, the speed of light, speed of sound, interference, diffraction, polarization and the Doppler effect. Now that was a very long list. 1. Go back and read the long list again and write down any term you need to double check or if you know everything about all these wave terms, write down as the answer to this homework question #1 as “I know all this stuff”. 2. If you need some help on these terms two great places to look are your textbook and the internet. Usually there are separate chapters or sections in your textbook for (A) waves, (B) sound waves, and (C) light waves. Write down the chapters and page numbers or internet resources for each of these 3 topics. 3. What is the wave equation? Tell what each variable is and its most common unit. 4. What is the speed of sound? 5. What is the speed of light? 6. Explain in some detail what the drawing below has to do with the Doppler Effect. Be sure to describe the effect on wavelength, frequency and pitch.
v
Figure 1. An ambulance experiencing the Doppler Effect.
7. How are frequency and pitch related to each other? 8. If you are standing still and a car/ambulance is coming toward you, what happens to the pitch of the sound you observe? 9. If you are standing still and a car/ambulance passes you and moves way from you, what happens to the pitch of the sound you observe? This is the Doppler Effect. **It is important to know that the actual frequency/pitch of the sound does not change, just the frequency/pitch observed!** Think about that vroom noise a car
10 makes as it passes you now in terms of frequency and pitch. Make the noise. See if your mimicry follows the physics. Now there are calculations that can be made to find the specific frequency for the observer. This can be done with a single equation if there is a plus/minus and minus/plus symbol is used in the numerator and denominator of the equation ( ,∓). Or there can be multiple equations for different situations. There are three different velocities in the equation, the velocity of the sound, the velocity of the source, and the velocity of the observer. Haul your textbook out or nimble up those internet searching fingers and find this equation in a reference. 10. Write down a single equation or a multiple equations if needed to calculate the observed frequency. 11. Look at the equation(s). Could you use it/them if you had to; if the equation and all the information you needed to solve it were given to you? Be honest. Now for the good news…students have been very, very rarely asked to find an actual frequency. However, questions on the concept of the Doppler Effect abound. Make sure you can answer them. My students always ask me what this has to do with Doppler Weather Radar, the most common time they hear the word “Doppler”. If you want to know the answer to this question, you might try this website: http://premiuma.accuweather.com/phoenix2/help/adc/pr_radar.htm#how_doppler from accuweather.com or a more basic one on radar from How Stuff Works. http://www.howstuffworks.com/radar.htm. Beat Frequency - Beat frequency or just “beats” is a specific sound and frequency pattern created due to two objects that are making a sound at nearly the same frequency. There is alternating constructive and destructive interference. My students in the band always seem to know exactly what I am talking about when we discuss this, better than I do! Wait until the lab and you will get the chance to simulate this. Woo hoo! There is a very characteristic frequency pattern that is associated with beat frequency. This is typically found in all textbooks. 12. Using a book or the internet, find the picture and make a sketch. Here is a hint on a possible web site: http://en.wikipedia.org/wiki/Beat_%28acoustics%29 Standing Waves and Waves on a Stretched String - Standing waves are commonly produced on stretched strings and most easily understood if we discuss both at the same time. Those stretched strings can be found on guitars, violins, violas, cellos, and basses! A standing wave on a string vibrates so quickly that the trough and crest can be seen at the same time. Check out this picture on the internet or find one in your textbook http://www1.union.edu/newmanj/lasers/Light%20as %20a%20Wave/standing%20wave2.JPG. Standing waves exhibit nodes, places where the string doesn’t really vibrate up and down at all, where you supposedly could touch the string and not disturb the waves. Notice that standing waves on a string are classic places to be able to see and understand the terms node, antinode, trough, crest, wavelength, and amplitude. **Please note that the simultaneously seen crest and trough are not part of the same wave. The crest goes with the next trough down the line not the one directly underneath it and vice versa. You have to be very careful when counting the number of standing waves on a string.** On the internet picture I referenced above, there is actually only one entire wavelength and a small part of another. If you don’t understand this, you need to discuss this with me in greater detail later. 13. Now make a quick sketch of several standing waves on a string. Trace one full wavelength and label it correctly with the following terms: node, antinode, trough, crest, wavelength, and amplitude.
11 The speed of a wave along a string is related to wavelength and frequency using the wave equation (v =f ) , but it is also related to the tension in the string and the mass per unit length of the string according to the following equation. F v T
In this equation v is the speed (m/s), FT is the force of tension (N), and is the mass per unit length (kg/m). I am sorry that they are using again. It is not my fault that the letters are used over and over again. Don’t get mad at me. Also the symbol is sometimes written as the quantity m/L or mass divided by length which is actually just the definition of mass per unit length or length density. So it is plausible that there could be a free response problem where there is a mass hanging over a pulley connected to a string that there is a standing wave on. You could be given a picture of these waves and the amount of the mass and the mass density per unit length. Then you could calculate the weight of the mass, the tension in the string, the velocity of the waves on the string, the number of waves from the picture, and the frequency of the waves. This would be a great free response problem! I have not seen one exactly like it, but I have seen all these elements in other problems. Also please note that the equation above for velocity as a function of tension and length density is not found on your equation sheet. This does not mean it has never been used on the AP exam. It has. They just provided the equation for you. I also wonder if it isn’t possible to make up a multiple-choice question or two out of this equation considering the following trends. 14. If the tension in the string increases what happens to the velocity, wavelength, and frequency? 15. If the string is thicker or in other words has a greater length density, what happens to the velocity, wavelength and frequency? Fundamental Frequencies and Resonance - A resonant frequency is a natural frequency related to the physical properties of the object that is vibrating and causing the sound. The lowest resonant frequency is the fundamental frequency, also called the first harmonic and usually abbreviated f1. Both standing waves on strings and columns of air create standing waves and have these fundamental frequencies/harmonics. This part of sound wave discussion is significant to musical instruments since a significant number of these make music due to strings (guitar, violin, etc.) or columns of air (flute, trumpet, etc.) Luckily for us the standing waves for strings and air columns are related. The following discussion will be much more understandable if you have some pictures to look at. So please find the information about fundamental frequencies or harmonics in your textbook or find a couple of good sites on the internet. You should have separate pictures for strings and for columns of air. 16. Write the page numbers from your book or give the URL from the websites that you are using and have the picture in front of you before reading further. First let’s talk about strings. If you had trouble finding something about fundamental frequencies/harmonics of standing waves, check out Hyperphysics, Vibrating Strings, http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html. Scroll down to harmonics. Let’s consider a string that has only two nodes, one at each end. This means that there is just one crest and trough or one big “bump”. This corresponds to one half of a wavelength. Remember what I said about counting standing waves carefully. So the length of the string, L = λ/2. Or we can rewrite this as = 2 L. We are going to keep the tension in the string and also it’s mass per unit
12 length the same so that speed of the wave remains the same. With only 2 nodes we have the largest wavelength we can get and therefore the lowest frequency also known as the fundamental frequency, f1. Also recall that v = f, so v = 2L f1 and solving for this fundamental frequency f1=v/2L. However, the velocity doesn’t change so if we want to consider different frequencies, there will different wavelengths also. If you didn’t understand all that logic, read the previous paragraph again, check out your book or internet resource, talk to your classmates and then go on. If you still have questions, you will have an opportunity to discuss this with me later. This lowest frequency is the fundamental frequency. It is also called the first harmonic.
Next let’s consider a standing wave on a string that has three nodes, two big “bumps”, or one full wavelength. These are all the same. This is the second largest wavelength possible which leads us to the second harmonic and the first overtone (f2). Please note that overtones can be remembered to be over harmonics so that the first overtone is the second harmonic and the second overtone is the third harmonic. Notice that the number of the harmonic and the number of the frequency are the same and the number of the overtone is one less. Some books use f0 for the fundamental frequency, but I have found that to be used less often. Remember I didn’t make up the lettering system. You don’t get to shoot the messenger, ha ha! For this second harmonic
(f2) we know that = L and using the wave equation we the equation we get v = L f2and the second harmonic’s frequency, f2= v/L note that 2f1 = f2. Let’s see if we can see some patterns here and understand the general equation found in the book. 17. Copy the table below into your notebook and fill in the table in a similar fashion to the table
was filled out for f1 and f2 .
Table 4. Table of harmonics and overtones.
Frequency Harmonic Overtone Nodes Bumps Wavelength Frequency Equation First Not 2 1 L / 2 v f f = 1 Applicable = 2L 1 2L Second First 3 2 L 2 / 2 v f f = 2 f1 2 = L 1 L
f 3
f 4 nth f n
Remember that columns of air also create standing waves and share some equations and properties with standing waves on strings. Make sure to have the pictures you found earlier of the waves in columns of air readily available. There are two possibilities with columns of air, a column of air open at both ends and a column of air that is only open at one end. If you have a pipe that is open at both ends, it uses the same equation as the standing waves on a string.
13 v f n nf n 1,2,3... n 2L 1 If the column of air is closed at one of the ends, this restricts air movement at that end and we get only odd multiples of the fundamental harmonic. The equation for this is v f n nf n 1,3,5... n 4L 1 Be sure to study the pictures that you have found to help make this clearer. With the two equations directly above, you should be able to handle any standing waves on strings or in columns of air that come your way. 18. Write down the equation for the frequency for the 7th harmonic for a column of air open at both ends.
19. Write down the equation for f3 for a column of air open at only one end. Notice that these equations are not given on your equation sheet. However, these types of questions have appeared on the exam so at least a cursory knowledge of the properties of standing waves is necessary. 20. Yahoo! We are done with basic waves. Just a note, we will not be studying speed as a function of bulk modulus or temperature, intensity of waves or intensity level in decibels. Now please write down the two most important things you learned from this lesson.
Physics B Unit 3, Lesson 2, Reflection and Refraction (B32Reflect)
Reflection and Refraction- Make calculations for and have a conceptual understanding of Snell’s Law, ray tracing for interfaces such as air-water and air-prism, and apparent depth.
Label each page of today’s homework with today’s code B32Reflect, the page number, and the date. You are going to learn all the concepts of this unit in terms of light although these concepts are true of other types of waves. You are going to gain some understanding of reflection and refraction. 1. Please find the sections of your book or websites that discuss reflection, refraction, Snell’s Law and thin films and write down the references for this information. 2. Show your understanding of the following terms in the most concise way possible, either a few words, an equation, or a picture. You may want to complete this as you do the rest of the work for this section. A) reflection B) refraction C) Snell’s Law D) thin films E) light ray F) incident angle G) reflected angle H) normal I) diffuse reflection J) index of refraction K) critical angle L) prism M) total internal reflection N) prism O) apparent depth P) fiber optics Reflection and Refraction - Please answer the following questions. Write down any additional questions you have so you can ask me later. 3. What is the shape of the path that light follows as it moves? 4. How are the reflections from smooth surfaces different from bumpy surfaces?
14 5. How does the measurement of angles in this section differ from the way we have most commonly measured angles in the past. 6. What can you tell me about the relationship between the angle of incidence and the angle of reflection? Snell’s Law of Refraction and Geometric Optics - Answer the following questions 7. Does the speed of light ever change? How does the speed of light in a vacuum compare to its speed in water? 8. I like to create a picture in my mind about index of refraction. Tell me how at least a couple of these words relate to index of refraction: thick, thin, clear, opaque, sheer, or dense? How do would these terms you have chosen relate to water and air? 9. Write down a mathematical definition of index of refraction. 10. As a light wave refracts which of these change and which do not: speed, frequency, and wavelength? Write each word down with the word “yes” (it changes) or “no” (it doesn’t). 11. Draw two pictures, one showing light traveling from air to water and the other showing light traveling from water to air. On both pictures be sure to include the normal, the incident ray, the reflected ray, and the refracted ray. Below each picture write a “rule” that helps you remember how the refracted ray bends based on the refractive index of the incident medium and the refractive index of the refracting medium. The “rule” must include the word “normal”. 12. Write down Snell’s Law and the name of each variable in it and how they are related to the subscripts. Feel free to illustrate or use previous illustrations if that will help. 13. What is dispersion and what does it have to do with rainbows? 14. Make a drawing of a prism that is an equilateral triangle and show an incident beam of white light that enter the first plane of the prism at an angle that is different than 90°. Then show the ray tracing of how the prism refracts red, green, and blue light through the prism and on the other side. Which bends more red or green? Green or blue? 15. Why couldn’t the incident beam of light in the last question enter at 90°? 16. What is total internal reflection and what does it have to do with fiber optics? 17. Give a mathematical equation for critical angle. 18. Write down a rule for total internal reflection and its relation to the index of refraction for the two media involved that starts “Total internal reflection can only occur……” Apparent Depth - Here is an interesting web site that shows a virtual lab on using apparent depth to measure index of refraction. http://www.mathsphysics.com/Physics/RandA.htm 19. Write an equation for apparent depth. 20. Okay we made it through the very exciting work on reflection and refraction! Good job! Write down the two most significant fundamentals that you learned in this lesson.
Physics B Unit 3, Lesson 3, Lenses and Mirrors (B33Lenses)
Lenses and Mirrors - Understand mathematically and conceptually the lens/mirror equation for diverging and converging lenses and concave and convex mirrors and ray trace for any of these and describe the image.
Today’s lesson will cover mirrors, lenses, ray tracing, magnification, and the lens/mirror equation. Please label all homework papers with the exciting code B33Lenses and the date and
15 the page number. Please also locate references in either your textbook or on the internet that discuss ray tracing for lenses and mirrors and write these references down.
Equations for this lesson are found below. The subscript o as in so refers to the object and the subscript i as in si refers to the image. 1 1 1 Lens/Mirror equation f so si s h R m= i m = i f = so ho 2
m is magnification (a unitless number) so is length the object is from the mirror or the lens si is distance the image is from the mirror or lens f is focal length R is the radius of curvature hi is the image height ho is the object height.
All variables except m are measured in length units. They are often given in centimeters. This is just fine as long as all other numbers in the equation are in centimeters also. Any unit will work as long as all units are the same for these equations. So you usually do not need to convert the units. R the radius of curvature can be given for a mirror. The focal length of the mirror is half this radius of curvature. Generally the rays of light for lenses and mirrors move from left to right like you normally read. However, problems have been known to appear where the light moves from right to left. After some practice you will already know exactly what to do and this will not be a problem for you. Below are some illustrations of how to classify the mirrors and lenses. This is very important because once you can classify the mirror or lens, it leads you into specific ways that problem is done.
Table 5. Synopsis of mirrors and lenses.
Concave mirror Convex mirror Converging lens Diverging lens Skinny woman Pregnant woman Fattest in the middle Skinniest in the Skinniest at the ends middle The ends converge Fattest at the ends At least one convex The ends diverge from face the center This is a double At least one concave convex lens face This is a double concave lens
16 General information for mirrors includes: concave ↔ f is positive (on left side of the mirror) Usually the left side is the positive side – but positive is always the object side convex ↔f is negative (Usually on the right side of mirror) so is always positive If si is positive then the image is real. If si is negative then the image is virtual. (on the wrong side of the mirror from where we expect it – opposite object side) If m is positive then the image is upright. (right side up) If m is negative then the image is inverted. (upside down)
General information for lenses includes: converging ↔ f is positive–Usually left side –Always object side diverging↔ f is negative (Usually right side) so is always positive If si is positive then the image is real. If si is negative then the image is virtual. (on the wrong side of the lens from where we expect it – object side, not other side) If m is positive then the image is upright. (right side up) If m is negative then the image is inverted. (upside down)
The following information gives the “Big Picture” of Ray Tracing. You will not be able to understand this unless you practice! After you have done several different types of ray tracing diagrams I hope you realize you are making the same rays over and over again using some general rules. You must know where the focus is for each type of lens and mirror. Concave mirrors and converging lenses are extremely similar (BASICALLY THE SAME), AND so are convex mirrors and diverging lenses (BASICALLY THE SAME). YOU MAY WANT TO MEMORIZE THAT CONVEX AND DIVERGING ARE THE SAME BY REMEMBERING THAT YOU WANT TO GET AWAY (DIVERGE) FROM SOMEONE WHO VEXES YOU (CONVEX) Concave Mirrors and Converging Lenses create real objects can when the object distance is greater than the focal length. This is the only time real objects are created. Convex mirrors and diverging lenses ALWAYS create virtual objects AND objects located between the focal point and a converging lens or concave mirror ALWAYS create virtual objects. General information on the 3 principal rays – The AP Physics test usually only requires 2 out of the 3. Did I mention that you will not be able to understand this unless you practice. Draw a ray parallel to the principal axis that touches the top of the object and then goes through the focus Draw a ray that goes through the focus and the top of the object and then goes parallel to the principal axis. For mirrors a line through the center of curvature (C or 2f) and the top of the object, the reflection is along the same line. For lenses a line through the top of the object and the horizontal and vertical center of the lens (optical center). Note: If one of the basic rays passes through the object, then the image will be virtual and the rays must be put on the opposite side of the lens or mirror than where they usually are. To locate the object, draw an arrow from the principal axis to the place where the 3 principal rays meet. If
17 the object is virtual, it is located where the 3 dashed lines intersect. PLEASE NOTE, if the curvature of the mirror isn’t just right, all 3 rays may not intersect perfectly. This is not a problem. Remember this is a graphical method and those are never as perfect as the actual mathematical calculations. You can make a best guess (I draw a really big dot) or just take a point where 2 of the principal rays intersect. WEIRD HINTS: We expect to see an object through a lens! The expected side for the image is the opposite side from the object! We expect an object to be reflected from a mirror! The expected side for the image is the same side as the object! In order to understand ray tracing, you must practice it several times!
Specific ray tracing directions for specific situations are below. Please skim through them and then use then to complete the practice problems assigned on the next page.
RAY TRACING -CONCAVE MIRRORS FOR OBJECTS LOCATED MORE THAN 1 FOCAL LENGTH FROM THE MIRROR Draw a ray parallel to the principal axis that touches the top of the object and then reflects at the surface of the mirror through the focus. Draw a ray that goes through the focus and the top of the object and then reflects at the surface of the mirror parallel to the principal axis. Draw a ray that goes through the center of curvature (C = 2f) and reflects along the same line RAY TRACING - CONVERGING LENSES FOR OBJECTS LOCATED MORE THAN 1 FOCAL LENGTH FROM THE LE NS Draw a ray parallel to the principal axis that touches the top of the object and then bends at the center of the lens to pass through the focus on the other side of the object Draw a ray that goes through the focus on the object side of the lens and the top of the object and then bends at the center of the lens to go parallel to the principal axis Draw a ray from the top of the object and passes through the center of the lens (the point at which the vertical center of the lens intersects the principal axis) RAY TRACING- CONCAVE MIRRORS FOR OBJECTS LOCATED BETWEEN THE FOCAL POINT AND THE MIRROR Draw a ray parallel to the principal axis that touches the top of the object and then reflects at the surface of the mirror through the focus, HOWEVER, since this ray passes through the object the image is virtual. Draw a dashed line on the opposite side of the mirror that is an extension of the reflection line. Draw a ray that goes through the focus and the top of the object and then reflects at the surface of the mirror parallel to the principal axis, then extend the reflected line on the other side of the mirror with a dashed line. Draw a ray that goes through the center of curvature and the top of the object (C = 2f) and then extend this line on the other side of the mirror with a dashed line. RAY TRACING -CONVERGING LENSES FOR OBJECTS LOCATED BETWEEN THE FOCAL POINT AND THE LENS Draw a ray parallel to the principal axis that touches the top of the object and then bends at the center of the mirror to pass through the focus on the other side of the object. HOWEVER, because the rays drawn will never meet, the image is virtual. Now draw a dashed line on the same side of the lens that is an extension of the line after the bend at the center of the lens. Draw a ray that goes through the focus on the object side of the lens and the top of the object and then bends at the center of vertical plane of the lens to go parallel to the principal axis, THEN extend the line (dashed) on the same side of the lens
18 Draw a ray from the top of the object and passes through the center of the lens (the point at which the vertical center of the lens intersects the principal axis), THEN extend this line with a dashed line backward from the object RAY TRACING FOR CONVEX MIRRORS FOR ALL OBJECTS – Remember this mirror only results in virtual objects Draw a ray parallel to the principal axis that touches the top of the object and then bends as it goes through the mirror to go through the focus on the other side of the mirror (remember f is negative for this mirror). The line on the other side of the mirror should be dashed. Draw a ray that would go through the focus on the other side of the mirror and bend it at the center of the mirror to reflect parallel to the principal axis. Extend the bent line on the other side of the mirror with a dashed line Draw a ray that goes through the center of curvature as it would appear on the other side of the mirror and the top of the object (C = 2f). This line should be dashed on the other side of the mirror. RAY TRACING –DIVERGING LENSES FOR ALL OBJECTS – Remember this lens only results in virtual objects Draw a ray parallel to the principal axis that touches the top of the object and then reflects back from the lens to go through the focus on the same side of the lens that the object is on (remember f is negative for this lens). The line that reflects from the lens should be dashed Draw a ray from the top of the object that would go through the focus on the other side of the lens and reflect it at the center of the lens parallel to the principal axis. Extend the line parallel to the axis on the left side of the mirror with a dashed line Draw a ray that goes through the vertical and horizontal center (optical center) of the lens and the top of the object. This line does not need to be dashed. Practice: Make ray-tracing diagrams for problems 1 through 6. This is done most easily by using graph paper! 1. Concave mirror, object distance 20 cm, object height 5 cm, radius of curvature 30 cm 2. Concave mirror, object distance 10 cm, object height 5 cm, radius of curvature 30 cm 3. Convex mirror, object distance 10 cm, radius of curvature 40 cm, object height 5 cm 4. Converging lens, object distance 10 cm, focal length 15 cm, object height 5 cm 5. Converging lens, object distance 20 cm, focal length 15 cm, object height 5 cm 6. Diverging lens, object distance 10 cm, focal length 20 cm, object height 5 cm 7. Fill in the chart shown below so, si, f, R (only for mirrors), ho, hi and m for problems 1 through 6 from your ray diagrams (always the first row you come to). Then solve for these quantities mathematically using the equations you were given in this lesson and fill in those rows on your chart. How did your graphical method (ray tracing) compare to the math? What are the strengths and weaknesses of ray tracing? Table 6. Answer chart for the practice problems for ray tracing.
Problem # and type S0 Si F R H0 Hi m (mirrors only) 1 concave mirror diagram >f 1 concave mirror math >f
19 Table 6. (continued) Answer chart for the practice problems for ray tracing.
2 concave mirror Diagram 3 convex mirror diagram 3 convex mirror math 4 converge lens diagram < f 4 converge lens math 5 converge lens diagram >f 5 converge lens math >f 6 diverge lens diagram 6 diverge lens math Physics B Unit 3, Lesson 4, Interference, Diffraction, and Thin Films (B34Interfere) Interference, Diffraction and Thin Films- Understand and make calculations for double slit diffraction, single slit diffraction, diffraction gratings and thin films. Label each page of today’s homework with today’s code B34Interfere, the page number, and the date. Again, you are going to learn all the concepts of this lesson in terms of light although these concepts are true of other types of waves. I have specifically seen problems that involve sound from stereo speakers. 1. Find the pages in your book or websites on the internet that discuss Young’s double slit experiment, single slit diffraction, diffraction gratings and thin films and write down these references. Let’s start with interference. We have talked before about constructive and destructive interference. 2. How are amplitude and intensity affected for two waves that undergo constructive interference? 3. How are amplitude and intensity affected for two waves that undergo destructive interference? 4. Constructive and destructive interference are a significant part of Young’s Double-Slit Experiment. Skim through one of your resources and then in two sentences tell me what constructive and destructive interference have to do with Young’s experiment. 20 Look on page 2 of your Table of Equations for AP Physics Exams and find all the equations that are given for the topics we have studied in this unit in the section labeled WAVES AND OPTICS. 5. There is a classic illustration that goes with Young’s experiment that illustrates the variables for this situation in the two equations found on our equation sheet, d sin θ = m λ and xm ~ mλL/d. Find the classic illustration and make a quick, but understandable sketch of it. Make sure you have labeled xm, d, θ, and L on the illustration. The letter y might be shown on the illustration instead of xm. They represent the same thing. Now I want you to gain a better understanding of the illustration. Young’s experiment yields bright fringes (maxima) and dark fringes (minima). Make sure you know what these terms mean. When a bright fringe occurs the light that travels there has to arrive exactly in sync. When this happens, Ta Da!, there is constructive interference or light of greater intensity/amplitude. The light from each slit has traveled one wavelength or an integer multiple of a wavelength further or shorter in the path it takes. The only exception to this is the central maximum where light has traveled the same distance from both slits. The difference in the length the light traveled is typically referred to as “path difference” and this is often asked about on the test. 6. If we aren’t considering the central maxima, what is the minimum path difference for the first bright fringe? For dark fringes, the light has to arrive completely out of sync. That means that a crest from the light from one slit has to arrive exactly when a trough from the light from the other slit arrives. Then, Aha!, there is destructive interference or darkness. 7. How does the path difference for dark fringes differ from that of bright fringes? Okay now go back and look at that lovely illustration you drew for question 5. Now answer the following questions. 8. What is L? 9. What is θ? 10. What is xm? 11. What is d? 12. Now even more irritating, what is m? Books and resources typically give different equations for bright fringes than for dark fringes. I am crazy and think you can use the same two equations for both. Or maybe I am not so crazy. 13. Give three examples of a numerical value for m for bright fringes using the two equations given on the AP equation sheet. 14. Give three examples of a numerical value for m for dark fringes using the two equations given on the AP equation sheet. 15. Now consider some other things. One of our equations has an equal sign, =, and the other has an approximation sign, ~. So of course this leads me to a question. Why? The answer to this question is a question. What is the small-angle approximation and how does that approximation get us from the equation with the equal sign to the equation with the approximation sign? 16. In 1 to 2 sentences tell how Young’s experiment could determine the wavelength of an unknown source of light. 17. What does Young’s experiment have to say about the wave nature of light? Now have a quick read in your resource about single slit diffraction and diffraction gratings. Notice that the typical book or resource gives new equations for single slit diffraction and 21 another one for diffraction gratings. Now again your teacher is totally living the la vida loca and thinks you can use the same two equations on the AP equation sheet for these two situations. 18. What is d for single slit diffraction? 19. How is there a path difference for single slit diffraction? 20. What is d for a diffraction grating? 21. Make a drawing of waves showing the intensity of light on the screen for (a) double slit diffraction, (b) single slit diffraction and (c) a diffraction grating. Okay, I think that is it for double slits, single slits, and diffraction gratings. But wait… There is another significantly important type of interference that happens with thin films. Locate this topic in one of your references because now I want you to think about thin films. I used to say that I hated thin films. Truthfully I still don’t love them, but at least now I can do problems involving them and get the problems right every time! That is because I have created Walker’s Sandwich method of thin films. Woo hoo! You know what a sandwich is, don’t you? I am sure you do, but I have to define the “Walker” thin film sandwich. A sandwich is a filling of something placed between two pieces of bread. The two pieces of bread are the same, but the sandwich filling is totally different. A piece of bread with a filling and then another filling without that second piece of bread is not a “Walker” thin film sandwich. Wait until I give you my totally exciting discussion of little old ladies and East European women body builders for Lenz’s law. That will be even more fun than my sandwich discussion! And that leads us to my thin film discussion. Okay, you are familiar with thin films without ever having discussed them mathematically. You have seen water puddles with a slightly oily surface on them where you can see a rainbow haven’t you? That rainbow is caused by thin film interference. You can see this same rainbow near the surface of a soap bubble can’t you? Thin films are also used on eyeglasses and other types of glass as anti-glare coatings. They are also used in fiber optics. So first let’s contemplate this whole stupid sandwich thing. Think of the soap bubble. The soapy water that forms the bubble creates a thin film! And light traveling from air through the soapy water into the air inside the bubble creates an interference pattern that allows us to see the rainbow! It is the indices of refraction of the air-soapy water-air interfaces that create the sandwich! So if you have a ray of light (think arrow) that goes from a medium with a low refractive index (air) into a medium with a higher refractive index (soapy water) and then back into a medium with lower refractive index (air), you have a thin film sandwich! If you are looking at a film of water on a piece of glass, then you are going from a low index of refraction (air 1) to a higher index (water 1.33) to an even higher one (glass 1.5). This is NOT A SANDWICH. Thin film problems on the AP Exam involve calculating the minimum thickness of the thin film and doing ray tracing and saying where phase changes occur. It is important to note that it is not required that the two layers on either side of the thin film have the exact same indices of refraction, they just need to follow the same trend, either larger than the thin film or smaller than the thin film. If this is true, then it is still a sandwich. Okay I know this isn’t easy, but read the thin film notes below, then answer these questions. 22 Thin Film Notes A. When going from thinner to thicker there is a ½ phase change at the REFLECTION SURFACE. B. When going from thicker to thinner there is no phase change. C. Recall that the wavelength of the light changes in the second medium. Remember the velocity slows and since the FREQUENCY NEVER CHANGES then the wavelength must also be smaller so divide the vacuum wavelength by the refractive index to get the new wavelength. D. So in going from thinner to thicker there is a /2 path difference, but the wave also has to travel an additional distance of 2t where t is the thickness of the film. So if 2t = /2 then the waves recombine in phase and there is constructive interference. A general equation for constructive interference is: 2nt = (m + ½ ) OR two times the thickness times the refractive index of the thin film = a multiple of /2 E. If the extra distance 2t traveled by the light is a multiple of , then the two waves recombine out of phase then 2t = and you get destructive interference. The general equation for destructive interference is: 2nt = m OR Two times the thickness times the refractive index of the thin film = a multiple of F. The above equations are valid when there is only one phase reversal, i.e. if it is a thin film sandwich where the thin film is contained between two media that have a smaller index of refraction or a larger index of refraction than the film itself (i.e. soap bubble which is an air-soapy water-air sandwich!). If it is NOT a sandwich, i.e. you go thin, thick, thicker or vice versa, THEN THE TWO EQUATIONS ARE REVERSED! 22. Let’s say that you have a soap bubble with a refractive index of 1.35. The refractive index of air is 1. What is the minimum thickness of the bubble for constructive interference to occur? At which interfaces is there a 180° phase change? 23. There is a puddle of water (n=1.33) with a thin film of vegetable oil on top of it (n=1.47) and then air (n=1) on top of that. What is the minimum thickness of the oil for constructive interference to occur? At which interfaces is there a 180° phase change? 24. Let’s consider a biology slide with a glass bottom (n=1.55) then water (n=1.33) and then a plastic slide cover (n=1.46). Is this is sandwich or not? 25. That’s it for this lesson. Write the two most important things you learned in this lesson. Physics C Unit 3, Lesson 1, Rotation Beginnings (C31Rotate) Rotation Beginnings – Contrast and compare linear quantities and rotational quantities and make calculations for equilibrium requiring τnet =0. Start your homework today by labeling the top of each paper with the date and the exciting code, C31Rotate, and the page number. Make sure to label any additional pages with your exciting code and the page number so neither one of us gets confused. Feel free to add anything else at the top you need to help you remember what topics you are studying. 23 1. Find references in your textbook or on the internet that describe rotational kinematics and situations for equilibrium where the net torque is zero. Write these references down. 2. Remember that rotational quantities are analogous to mechanics quantities you already know. And they use the same equations. In most textbooks, there is a chart with the two sets of kinematics equations (i.e. v = v0 + at and = 0 + t) found side by side. Find this chart and write the page number as the answer to this question. Also some books use the word “angular” to refer to these quantities. I have also seen them referred to as “rotational” and “radial”. I will use all three terms interchangeably AND you should be familiar with all three terms. So let’s look at some basics. Read the ~ sign as “is analogous to”. a (rotational acceleration) ~ a, acceleration w (rotational velocity) ~ v, velocity q (rotational displacement) ~ x , displacement t (torque) ~ F, force I (moment of inertia) ~ m, mass L (angular momentum) ~ p, momentum K (angular kinetic energy) ~ K, linear kinetic energy -- Wait a minute. Those are both the same! Rest assured they are supposed to be. There are a few additional things to know. All angular components are equal to their linear components divided by the radius. So q =x/r and w = v/r and a=a/r. This makes sense to get rid of the unit of length contained in linear quantities. So if you divide the linear quantity acceleration, a, by length you really get 1/s2 or radians/s2. Because of this, I think of radians as not really being units, but as filling in a space. Also remember 360° = 2p radians=1 revolution. 3. You also need to be aware of how centripetal acceleration relates to rotational quantities. Derive or find an equation for centripetal acceleration in terms of r and ω. 4. What is the difference between tangential acceleration and total translational acceleration? 5. Write the equation for rotational kinetic energy. 6. Write down an equation that does not contain an integral for moment of inertia. 7. Write down the integral equation for moment of inertia. There will be more on this soon. 8. Write down the units for moment inertia. 9. (a) Write down the page for the Table with Moments of Inertia. (b) What is the moment of inertia for a thin cylindrical shell? (c) What is the moment of inertia for a solid disk? (d)What is the moment of inertia for a solid sphere? 10. The units for rotational kinetic energy are the exact same units as our linear ones. But wait, the equations are totally different. For this part of the homework, prove that both equations give the exact same units. 11. There are other analogous equations. Write the equation for torque that is not r x F. You don’t have to look this up. It is analogous to FNET =ma. 12. What are the units for torque? 13. We have learned that the requirements for equilibrium in terms of force are that Fnet = 0 along both the x and y axes. Now with inclusion of torque, what are the requirements for equilibrium? 24 14. This is the end of today’s lesson. Now I suggest you read carefully through examples in your book about basic rotational kinetic energy, calculations of moment of inertia that do not require calculus, and equilibrium problems that include torque. Also please write as the answer for this question the two most scintillating things you learned in this lesson. Physics C Unit 3, Lesson 2, Moment of Inertia and Pulleys (C32Inertia) Moment of Inertia and Pulleys – Gain a deeper conceptual understanding of moment of inertia, use calculus to determine the moment of inertia of a uniform rod and be able to take moment of inertia of a pulley into account. Okay, okay, you need to do that labeling thing again. Today’s label is C32Inertia. Please put this code, the date, and the page number as usual at the top of each page as you do the homework. That is the first exciting thing. The next exciting thing is that we are actually going to use calculus today! Yesterday you should have written down the integral form of the equation for moment of inertia. We will start with that. First I want you to find either in your book or on the internet or both a place where an example is done to find the moment of inertia of uniform solid cylinder. 1. For your first homework problem, copy down the process step by step, to find the moment of inertia of a uniform solid cylinder. Write the process down like you had to explain to someone else how to do it. This means not just writing down mathematical steps, but short phrases explaining the process. This has not ever been asked on the AP exam that I can find a history of, but similar questions have been and it is important to be able to understand this process. I apologize for the fact that both lower case “r” and upper case “R” are used, but the basic rule is that lower case r is a variable and upper case R is a constant, in this case one of the limits of integration. 2. Now see if you can achieve an analogous process to find the inertia of a uniform rod that is pivoted about one end. Use a linear density for the rod instead of a volumetric one like you did for the cylinder. By the way we will be using these densities over and over again for this APC course, so let’s get the typical nomenclature down. ●Volumetric density, like the one you studied in chemistry where density equals mass divided by volume, is given the Greek letter lower case rho, ρ. The typical unit of ρ is kg/m3. ●Surface area density, the only example I can think of is number of pixels on a computer screen, is given the Greek letter lower case sigma, σ, typical unit kg/m2. ●Linear density, mass per unit length is given the Greek letter lower case lambda, λ, typical unit kg/m. You will be using λ for this problem. We will use these same Greek letters again when we discuss charge density in association with Gauss’ law and then again with Ampere’s law …you should be hearing an evil witch- like laugh in your head right now. Typically this problem, the one about the moment of inertia for a uniform rod, is done using an infinitesimally small sliver of the rod of width dx which then has a mass of λ dx and your limits of integration are from 0 to L. Try this integration and then see if your answer matches the one in the table that you found yesterday. I have seen this very integration on the AP exam more than once. 25 Let’s consider a pulley with significant rotational inertia. An object rolling down an incline and object(s) attached to a pulley where rotational inertia has to be taken into account are probably the two most classic rotational physics problems. Since they are on the exam so often, you will go over them in some significant detail. Please note just because you are learning about an object rolling down an incline and a pulley where moment of inertia is important, this does not mean that you will not have to deal with the easier situations of an object just sliding down a ramp or a pulley where moment of inertia is not important. You will be asked to do both on the test and you must read carefully to see which kind of a problem you are being asked to do. Remember, this is as much a reading test as it is a physics test. Reading is Fundamental! Let’s consider the classic problem of an Atwood machine. This is a pulley with significant moment of inertia. The pulley is considered to have a radius R and a rotational inertia I. There are two masses looped over it, m1 and m2. Let’s say that m2 is heavier than m1 and that m2 is accelerating down and m1 is accelerating upwards. You are typically asked to find the acceleration of the masses and the tension in each side of the rope. So let’s look at this problem. The free body diagram for each object is shown below. F F T1 T2 m m 1 2 I F F T1 T2 F = m g F = m g g1 1 g2 2 Figure 2. Free body diagrams for an Atwood Machine. The first thing I would like you to notice is that unlike previous problems where we have ignored rotational inertia, the tension on each side of the rope is not the same. So I have labeled them as FT1 and FT2. In a previous lesson you learned the definition of torque () = I. Torque is also a cross product equal to r x F. I will take an arbitrary sign convention of clockwise is positive and counterclockwise is negative. In other words, any force that acts to make the pulley rotate clockwise we will consider as being positive while any force that acts to make the pulley rotate counterclockwise we will consider as negative. Remember rotational mechanics have significant analogies in linear mechanics. So we know that FNET = ma and we can also say that NET = I and NET = r x F so that I = r x F. Notice that there are three unknowns, a, FT1, and FT2 and you will need three equations. Here are the equations. First let’s sum FNET around m1. The equation is FT1 – m1g = m1a. Now let’s sum FNET around m2. The equation is FT2 – m2g = - m2a. Remember my sign convention is always that down is negative and m2 is accelerating down. 26 The third equation is provided by rotational motion and that is FT2R – FT1R = I. Recall also that a = a/R. Substituting this in we get F R – F R = I . Here are our three equations in a nice T2 T1 R neat list: FT1 – m1g = m1a FT2 – m2g = - m2a a FT2R – FT1R = I R Notice that the acceleration (a) in all three equations is exactly the same acceleration. This will always be the case for the types of problems that we are going to consider. You will get the excitement of considering other types of problems where this is not true once you are in higher- level classes in college! This problem works out nicely if we consider the pulley to be a solid cylinder (seems like we talked about this before) with a moment of inertia of ½ MR2. 3. Substitute the moment of inertia of the pulley into the third equation and solve the three simultaneous equations for the acceleration in terms of the three masses, m1, m2, and M the mass of the pulley. 4. Well, that seems like plenty to learn today to me. Write down the two stimulating and essential things you learned from this lesson. Physics C Unit 3, Lesson 3, Rotational Motion for an Inclined Plane (C33Incline) Rotational Motion for an Inclined Plane – Compare and contrast the use of Newton’s 2nd law and conservation of energy to solve problems for objects of varying moments of inertia rolling down an inclined plane. Please label all pages of this homework assignment with its code, C33Incline, the date and the page number. I will just be boring and tell you to find the section in your book or an internet website that describes the rolling motion of a rigid object. 1. Write down the pages in your book or the URL of a website that can be used as a resource for rotational motion on an inclined plane. Most problems consider an object rolling down an incline. So you will start with that. Let’s start by considering a sphere. Please start by making a free body diagram; you know I always think that is valuable. If you are to create the free body diagram of a sphere on an incline, the forces you need to consider are the gravitational force, the normal force, and the frictional force. If there is no frictional force, the object won’t roll; it will just slide down the incline! Wow! I have seen a big deal made of these free body diagrams. Now ask yourself exactly where each force would originate on the object? In a previous lesson we know that it does matter where the force of gravity and other forces actually are due to torque. In this free body diagram, the force of gravity should start at the center of the sphere (the center of mass for a uniform object) and point down. The normal force should start at the exact point of contact between the sphere and the incline and point perpendicularly upward from the incline. The frictional force should also start at the point of contact of the sphere and the incline and point uphill and parallel to the ramp. 2. Draw a correct free body diagram of a sphere on an incline. Let’s consider an incline of a given height, h, and a given angle, θ. You can solve for the velocity of the sphere at the 27 bottom of the incline by using conservation of energy. And you can solve for the acceleration using the same principle. However, like other inclined plane problems you have solved you can also use Fnet = ma. You need to be able to do both. Let’s start with conservation of energy. There are two types of energy that you are considering as the ball rolls, its translational kinetic energy (1/2 mv2) and its rotational kinetic energy (1/2 I2). In this case the velocity of the in the ½ mv2 equation is the velocity of the center of mass of the sphere. Added together these are the sphere’s total energy and at the bottom of the ramp should equal its gravitational potential energy at the top of the ramp according to the law of conservation of energy. 3. Look up the moment of inertia of a solid uniform sphere of mass M and radius R. 4. Set the sum of both types of kinetic energy equal to the gravitational potential energy. Then solve for v at the bottom of the incline assuming the sphere starts from rest. Remember that = v/R and substitute that in and magically (not really) all the R’s should cancel out. 5. Now that you know the v at the bottom and v0 = 0 and knowing θ and h you can find the length of the ramp and find the acceleration of the sphere. Oh, by the way, you can probably find an example on the internet that will show you this problem exactly for a sphere or a cylinder. 6. Now let’s consider doing the same thing using Fnet= ma and of course our new and exciting friend τ which equals both I and F x r! You will consider that the sphere has a coefficient of friction . Go back to that free body diagram you drew and prove that the acceleration of the object is a = g sin θ - g cos θ using Fnet = ma. 7. Now you can use the two equations for torque, τ = I and FR and set them equal to each other. You already looked up moment of inertia for the sphere. The force F that cases the rolling is friction and so τ = the force of friction times the radius. Or in other words I = mg cos θ. Substitute the correct moment of inertia into this equation and solve for . Remember that = a/R. Substitute that in too or it won’t be right. 8. Put the you solved for in part 7 into the equation that you solved for in part 6 and solve for the acceleration. It should be the same answer you got for question 5. Say Aha! 9. Now use kinematics and solve for the velocity at the bottom of the incline. This answer should be the same as question 4. Say Yahoo! 10. Okay that should give you the basics for dealing with these kinds of problems. There however, is one more classic problem you need to know the answer to. A solid sphere, a solid cylinder and a box are having a race down similar inclines. They all have the same mass and the cylinder and sphere have the same radius. The box slides down a frictionless incline and the sphere and cylinder roll down the same incline without slipping. Who wins the race? 11. What if we compare two spheres rolling down an incline where one has some slipping and the other has no slipping? Which one will win then? 12. What causes one sphere to have some slipping and the other to have no slipping? 13. Check out a website provided by Hyperphysics: http://hyperphysics.phy- astr.gsu.edu/hbase/hoocyl.html#hc1 or check out any website that talks about spheres, cylinders, and boxes racing. Physics C Unit 3, Lesson 4, Other Rotational Situations (C34Other) 28 Other Rotational Situations – Understand conceptually and be able to make calculations for the rotational analogies to the Work-Energy Theorem, power, Newton’s 2nd Law, angular momentum, and precessional motion. Okay, we have finally made it. This is the last new material for rotational situations. Yahoo! Label this lesson C34Other, the date and the page number. There are a couple of other rotational analogues we haven’t discussed. So let’s get these figured out. 1. There is a rotational analogue for the Work-Energy theorem. Find that equation and write it down. Could you use it? 2. I have always told you that the centripetal force can do no work. Did I lie? Is there work done during rotational motion? 3. What is the rotational analogue for power? 4. This has nothing to do with angular quantities, but I know it is shown in this chapter because it is in one of those cute tables that show the angular quantities on one side and the rotational quantities on the other side. There is a differential equation for force that uses translational quantities of momentum and time and is Newton’s second law! Write it. Remember it is a differential equation! Prove to me mathematically how this is equivalent to Fnet = ma. 5. Write the analogous rotational equation for the one you gave the answer to in number 4. 6. What is the equation for angular momentum that is a cross product? 7. Give simplified version of the previous equation that is in terms of the magnitudes of the vectors only. 8. Write the equation for angular momentum in terms of rotational quantities only that is analogous to our old friend p = mv. 9. Is angular momentum conserved? Is there an analogous equation for my poo = p equation? 10. Now let’s talk about the ice skaters. They are very important and often asked about. Think about the last time you saw an ice skater doing a spin. When he wanted to spin faster and faster he pulled his arms in. How does that change his moment of inertia? Write down a couple of sentences that describe what is happening in terms of physics during the spin. 11. Read/skim something about precessional motion. Write down 1 to 2 sentences about what precessional motion in terms of observation and physics. 12. Read through any example problems you can find about conservation of angular momentum where there is a net torque. 13. You have completed the last rotational motion assignment! Yahoo! Make me happy by writing down the two most important notions from today’s lesson. 29 UNIT 5, PHYSICS B FLUID MECHANICS & THERMAL PHYSICS AND PHYSICS C ELECTRICITY IN DEPTH Table 7. Unit 5 Overview, Physics B Fluid Mechanics & Thermal Physics and Physics C Electricity in Depth Physics B Fluid Mechanics & Thermal Physics C Electricity in Depth Physics 1) Pressure and Buoyant Force – 1) Gauss’s Law for Spheres - Comprehend Understand both conceptually and and be able to analyze Gauss’s law for mathematically the variance of pressure spherical distributions including with fluid depth, pressure as a function of conductors, nonconductors of uniform force and area, and buoyant force. charge density and nonconductors with 2) Bernoulli and Torricelli – Understand densities as a function of r. basic flow rate through pipes and be able to 2) Continuous Charge Distributions - Be describe and utilize Bernoulli’s equation able to determine and recognize appropriate and related equations including Torricelli’s problem solving techniques for finding the Law, particularly understanding how these electric field and the potential from concepts relate to the conservation of mass continuous charge distributions particularly and energy. rings or arcs of charge. 3) Gas Laws – Be able to interpret and make 3) RC Circuits - Grasp the numerical and calculations related to the following topics: theoretical concepts associated with RC thermal expansion, the ideal gas law, the circuits. kinetic theory of gases and rate of energy 4) Electric Potential from Electric Field - transfer during conduction. Using the integral relating electric field to 4) Thermodynamics – Be able to interpret potential and the derivative relating and make calculations related to the potential to electric field, be able to make following topics: thermodynamic work and determinations both mathematically and PV diagrams, the laws of thermodynamics, conceptually. and heat engines. 5) Other Field, Potential and Capacitance Problems - Gain insight and awareness of other Gaussian surfaces, more difficult continuous charge distributions, and capacitance for geometries different from rectangular parallel plates. Physics B, Unit 5, Lesson 1, Pressure and Buoyant Force (B51Pressure) Pressure and Buoyant Force – Understand both conceptually and mathematically the variance of pressure with fluid depth, pressure as a function of force and area, and buoyant force. Let’s do first things first. First haul out your Table of Equations for AP Physics Exams. There is a section on it labeled Fluid Mechanics and Thermal Physics. Look at the equations. The topics relating to these equations are what we will be covering before the next test. Pressure is the one of the unifying concepts of this unit so let’s start with a discussion of pressure and then move on 30 to a discussion of buoyant force. Please label all of today’s homework with B51 Pressure, the date, and the page number. 1. Let me be blatantly obvious, please write down these 3 equations you will be using in this lesson, (1) P = P0 + ρgh, (2) Fb = ρVg and (3) P = F/A. Write down your best guess of what each letter represents. 2. Look in your book or on the internet and find a discussion of the following topics and write down a reference or two for the study of these topics: fluids, density, specific gravity, pressure, Pascal’s Principle, buoyancy, and Archimedes’ Principle. You might see Bernoulli in there, but we will be studying Bernoulli tomorrow. We will not be studying the topics related to deformation of solids. Don’t forget to write down some page numbers or URL’s. 3. I am pretty sure that you already know the equation for density, but just to be sure, write down the formula for density. 4. This ain’t your chemistry teacher’s density. Or, in other words, there are no units of grams per centimeter cubed or grams per milliliter here in physics. Write down the units used in physics for density. 5. Write down the density of water at 4° C in correct physics units. 6. Write down the page number of the table in your textbook which lists the densities of common substances. You may need to refer to this in order to do homework. 7. Let’s figure out what the heck specific gravity is. It really is an easy concept. What is the difference between the density of glycerin which is 1260 kg/m3 and its specific gravity of 1.26? Why is specific gravity unitless? What is the specific gravity of copper which has a density of 8920 kg/m3? What is the density of gold which has a specific gravity of 19.3? Why is specific gravity useful? 8. Referring back to the last equation I asked you to write down, P = F/A. Write down the name of each variable and its units. 9. Define the Pascal in terms of Newtons and meters. Define the Pascal in terms of the units of kg, m and s and only. 10. Write a sentence that explains the concept of why a person can lay on a bed of nails. 11. Write a sentence that explains why a woman of the same weight will poke holes in the football field when she is wearing high-heeled shoes when she doesn’t make any holes when she is wearing soccer cleats. 12. Find that equation I gave at the beginning with an h in it which gives pressure as a function of height. It is up next. If I have a bathtub with 3 inches of water in it and a milk carton with 3 inches of water in it and a glass with three inches of water in it, which one will have the greatest pressure at the bottom? Why? Would the answer be different if there were milk in the milk carton? 13. Explain in less than 3 sentences why there is atmospheric pressure on earth in terms of air, fluid mechanics concepts, and height. 14. What is the difference between gauge pressure and absolute pressure? 15. Okay now a new equation the one for buoyant force. Write down the name of each variable and tell me what it relates to, either the object or the fluid the object is in. I assume you know the units of each variable, so you should make sure you do. 16. It is amazing that you can also write an equation for the force of gravity in terms of density and volume. Please write this equation. 31 17. Now pretty please draw a pretty picture of a rectangular piece of wood floating in tub of water. Make sure to show a part of the wood below the water and part of it above the water. Draw a free body diagram showing the two most significant forces acting on the wood. Make sure both forces are expressed in terms of density and volume and that each density and volume have subscripts on them indicating which density and volume they are! Using this information derive an equation that will give you a fraction or percent of the wood that is submerged. Give an equation that will give the fraction or percent of the wood that is above the surface. This is a very, very classical question and is often found on the AP exam. Make sure you read carefully to see whether the question asks for the volume submerged or volume above the surface. 18. Why didn’t you include the buoyant force of the air in the previous problem? 19. Now for the very exciting and absolutely necessary Archimedes’ crown problem. I always want to say that jewelers use this method for determining if an object is really gold or not, but actually jewelers use chemical tests (Is It Real Gold, 2007). This problem is classic and is shown in almost every physics textbook. A crown that is supposedly gold is weighed both in air and in water. Let’s pretend that the weight in air is 10 N and the weight in water is 8 N and go through the problem. First things first of course, please draw two free body diagrams: (1) the crown hanging on a spring scale in air and (2) the crown under the same conditions except that it is in water. 20. You should have drawn three forces for the each of the situations to be technically correct. There should be the force of the spring scale up and a buoyant force up and the force of gravity down. Yes, it is true even the crown in the air is experiencing a buoyant force because it is immersed in a buoyant fluid, air. Yes, in all those free body diagrams I have shown of books and tables and lawnmowers and boxes there should always have been a buoyant force due to air, but I never showed it. Why? Because the buoyant force of air is so small it can generally be neglected. So if you forgot the buoyant force of air, I will forgive you. However, the buoyant force of the water on the crown is not negligible. Now go back to your free body diagram and write the equations for buoyant force and gravity in terms of density. Make sure you use a subscript that distinguishes the two kinds of density for each. They are very different. 21. Okay now I would like you to use Fnet = ma for each FBD and prove what you have is equivalent to the equations I have listed below. (1) Fs in air = ρcrownVcrowng = Fg (2) Fs in water + ρfluidVcrowng = ρcrownVcrowng = Fg Please note from the example I have started with Fs in air is 10 N and Fs in water is 8 N. 22. Notice that the volume of the fluid replaced is the same as the volume of the crown. Also I hope that you can see that the difference between the two weights is equal to the buoyant force which also equals ρfluidVcrowng. Now you should be able to solve for the volume of the crown, the mass of the crown and the density of the crown. Solve for these things and since we discussed the density of gold previously, discuss whether the crown is gold or not. 23. Write down the two most striking things that you learned about fluid mechanics in this homework assignment. 32 Physics B, Unit 5, Lesson 2, Bernoulli and Torricelli (B52Bernoulli) Bernoulli and Torricelli – Understand basic flow rate through pipes and be able to describe and utilize Bernoulli’s equation and related equations including Torricelli’s Law, particularly understanding how these concepts relate to the conservation of mass and energy. Today’s homework code is B52Bernoulli. Please make sure to put this on each page of your homework along with the page number of the page you are writing on. Okay, let’s get started. 1. Find and write down some resources in your textbook or on the internet that discuss Bernoulli’s equation and Torricelli’s equation and how they relate to the flow of water. 2. There are two main forms of Bernoulli’s equation. One is found on your AP equation sheet. It is a little different in that it has an entire word in it, “constant”. Find and write this equation down. 3. The other form of Bernoulli’s equation you can usually find on the internet or in your textbook and it is very similar to an equation that you learned about while discussing conservation of energy. Find this equation and write it down. 4. Write in a couple of sentences describing how Bernoulli’s equation relates to conservation of energy. 5. Now before you start using Bernoulli’s equation, let’s do some thinking about flow rates. Knowing what you do about basic physics units, give me two possible units for flow rate, one for mass flow rate and one for volumetric flow rate. Now consider how these two flow rate units are related…okay that’s enough of that. Hmm, aren’t kg and m3 the units of something else? Hmm. Okay I am hoping with all this hmming that you see that you either multiply or divide by density to change one unit into the other. 6. Now how else can you calculate volume? Last time I looked volume for a cylinder (and what is a pipe, but a really long cylinder) it was the cross sectional area of the end multiplied by the height of the cylinder. But we really want to talk about volumetric flow rate. How can we get that time unit in there? Don’t we already have this very exciting quantity with units of length per time that you learned about on practically the first day of physics? I think it is called velocity. Yes that’s it. So you can also get volumetric flow rate by taking the cross sectional area of the pipe times the velocity of the fluid. Now in terms of variables and knowing that information, write an equation for mass flow rate in a pipe. 7. Okay now you need to find out what happens if you change the cross sectional area of the pipe. Write an equation for our first (1) mass flow rate in a pipe of given cross sectional area. Write another equation for a second (2) mass flow rate in a pipe of with a different cross sectional area. What is the same in both places? The first thing is density. The “assumption” of using these equations is that our fluid is incompressible. You can’t squish it and change its density. What is also true of this situation is that mass is conserved. What flows past one point in a certain time in the pipe also flows past any other point in the same time. So mass flow rate is a constant and your two equations are equal to each other. Set them equal to each and show that you get this equation A1v1 = A2v2. 8. Now for a very, very important concept. Using the equation you derived in the problem before and the Bernoulli equation, tell me what happens to velocity of flow and pressure in a pipe if you decrease the cross sectional area of a pipe while keeping the average height of the pipe the same. 33 9. Let’s derive Torricelli’s Law. Using Bernoulli’s equation, consider a situation where you have a large tank of water. The top of the tank is open to the atmosphere and there is a hole near the bottom a distance h down from the top. Water is spouting from the hole near the bottom of the tank, but because the tank is so large we can consider the velocity of fluid flow at the top of the tank to be zero. (a) Derive an equation for the velocity of the water exiting the hole in terms of h. (b) What equation does it look like? (c) Knowing the hole at the bottom of the tank has a radius of r, derive an equation for volumetric flow rate from this hole. (d) Knowing that the tank is filled with a liquid of density ρ, derive an equation for mass flow rate from this hole. 10. Let’s consider one more thing. There is something that can be used to measure pressure in a pipe called a Venturi tube. Essentially it is a vertical tube set into a horizontally flowing pipe. The height of fluid in the tube gives the pressure in the pipe related to the equation P = P0 + ρgh. Draw a quick sketch of a pipe that has been “crimped” to a smaller cross sectional area in the middle with the same, but larger cross sectional area at both ends. Add 3 Venturi tubes, one each for beginning, middle, and end and sketch qualitatively the height you would expect in each of the 3 tubes. 11. That’s it for Bernoulli. Tomorrow, thermodynamics. Write down the two most striking ideas from this lesson. Physics B, Unit 5, Lesson 3, Gas Laws (B53Gas) Gas Laws – Be able to interpret and make calculations related to the following topics: thermal expansion, the ideal gas law, the kinetic theory of gases and rate of energy transfer during conduction. Today you will be learning about the behavior of gases and along with that I will throw in a little bit of thermal expansion and a small discussion of conduction. The code for this assignment is B53Gas. Please label each page of your assignment with that code, the date and the page number. 1. First of all I would like you to find some references. You will need to write down book page numbers or an internet URL for each of the following topics: (a) thermal expansion, (b) ideal gas law, (c) kinetic theory of gases, and (d) heat transfer by conduction. 2. Let’s start with thermal expansion. Find an equation for thermal expansion in your reference and write it down. It should contain an α and I prefer the one with the Δ’s. The α is not rotational acceleration, but is the coefficient of linear expansion and of course the Δ means “change in”. Make sure the equation you have written down is either the same as the one on your Table of Equations for AP Physics Exams or that you can understand how they are the same. Also write down a reference for a table of coefficients of expansion for different substances. You may need these values for homework. If you are a given a problem of this nature on the AP exam, you will be given α or you will be given data and asked to solve for α. The topic of thermal expansion is often one of the first thermal physics topics covered in your textbook, usually right after a discussion of temperature. Since the equation requires temperature in the calculations, you need to understand temperature also. During thermal physics we will use two temperature scales, °C and K, the Celsius and Kelvin temperature scales. The Kelvin temperature scale is an absolute temperature scale which means that it is 34 referenced to absolute zero, the temperature at which all molecular motion stops. Please note that temperatures in Kelvin don’t use the degree sign. Degrees of the K and °C scales are the same size and temperatures can be converted from °C to K and vice versa with little trouble. You should add 273.15 to temperatures in °C to get K and subtract 273.15 from temperatures in K to get °C. Truthfully for our purposes, you really only need 3 significant figures and so 273 will be good enough. We will not use °F scale and so you don’t need to know how to convert from or to this scale. Now after saying all this very important stuff there are two rules that you need to remember for every single calculation involving temperature in thermal physics: (a) When you use a ΔT in calculations you may use either °C or K in your calculations; (b) When you use an absolute T in calculations (it is not ΔT or a subtraction, T – T0), you must use temperature in K! If you mess up rule (b), you will have done all your calculations incorrectly. Now write down in a sentence or two why it is okay to use either scale when you are using ΔT or T-T0 in a calculation. 3. Okay after that long discussion of temperature, let’s get back to thermal expansion. You are lucky enough in New Mexico to see the evidence of this process everywhere in buckling sidewalks and cracks in asphalt. You are only given an equation for the coefficient of linear expansion on your homework sheet. There are equations for area expansion and volumetric expansion. These equations are analogous and use α also, but it the constant is replaced by β where β=2α in the area equation. The constant is γ in the volumetric equation where γ = 3α in the volumetric equation. Write down why all the constants are multiples of α. 4. Now on to the ideal gas law. Write it down. Write down the constant R from your AP Physics equation sheet. Write down what the unit of pressure multiplied by volume is, or in other words 1 Pa times 1 m3 equals what? Write down what the temperature units you absolutely must put into this equation. 5. I love PV = nRT and you will definitely use it. This section will feel a bit “chemistry-like”, but we will use our excellent physics skills. You need to know Avogadro’s number. Write it down with a full sentence surrounding it so that I can tell you know what it means. 6. Chemistry books often define Boyle’s Law or Gay-Lussac’s Law, but not me. You only need one equation for all these laws. Assume that you have a constant amount and therefore number of moles, n, of a gas and remember that R is a constant, and then derive an equation for an ideal gas that starts at one pressure, volume and temperature and then changes to a different pressure, volume and/or temperature. Put some kind of subscripts on the variables so you can tell them apart. You can derive any kind of constant pressure, constant volume, or constant temperature equation directly from this one equation. Of course this is a very useful equation. It is not found on your equation sheet. I expect you to be able to derive it at a moment’s notice. You will probably have it memorized quite quickly. 7. By the way just for funsies define these terms: (a) isothermal, (b) isobaric, (c) isochoric, and (d) adiabatic. I have seen multiple-choice questions that are just as easy as knowing the definitions of these words! 8. Now on to the kinetic theory of gases. References usually give a list of postulates or assumptions of this theory. I have seen anywhere from 4 to 12 assumptions/postulates of this theory. At the risk of being helpful (you know how I hate that, ha ha!), here are two websites where I looked it up (a) http://www.grc.nasa.gov/WWW/K-12/airplane/kinth.html and (b) http://en.wikipedia.org/wiki/Kinetic_theory. You may hate my choice of internet resources, but the point is to show how very differently the assumptions of this law can look. Make sure you check out your textbook too and I actually love the way it is presented in the 35 Chemistry in the Community textbook. You do not have to memorize the scientific language in these postulates, but you should know the basics of this theory. So please -jot down, -in bulleted form, -in just a few words each, the basics of this theory. 9. There are several equations on your equation sheet related to the kinetic theory of gases. There is an equation for the ideal gas law in terms of Boltzmann’s constant (kB). There is an equation for average kinetic energy in terms of kB. There are two equations for vrms which is the root-mean-square velocity of the molecules of the gas. One of the vrms equations is for a mole of an ideal gas and one is for a molecule of an ideal gas. Also please recall that we use kilograms for mass and the periodic table of elements uses grams, so there are ~2 E -3 kg of H2 in a mole as well as 2 grams in a mole. Now finally here are the points I want to make! (a) Double the temperature in Kelvin (OF COURSE) and what happens to the average kinetic energy? (b) Double the temperature in Kelvin and what happens to the root-mean-square velocity of the molecules of an ideal gas? (c) The average energy of an ideal gas is related to what variable? (d) How does this relate to the unit of pressure multiplied by volume? 10. Conduction, convection, and radiation! Define them and give an example of each in as few of your own words as you can. There is no calorimetry on the APB exam any more. Did I mention my years of research using high pressure, high temperature flow calorimeters? Okay, I will stop whining and get over it. 11. Okay now back to conduction. There is an equation for the rate of energy transfer through a slab of area A and length L due to conduction. Write it down from one of your resources. Compare it to the one given on your Table of Equations for AP Physics Exams. Know what the variables are and be prepared to use the equation. 12. That is it for our first pass through thermal physics. Please write down the most sensational perceptions that you have gained from this lesson. Physics B, Unit 5, Lesson 4, Thermodynamics (B54Thermo) Thermodynamics – Be able to interpret and make calculations related to the following topics: thermodynamic work and PV diagrams, the laws of thermodynamics, and heat engines. Today’s lesson is about thermodynamics specifically PV diagrams and PV work, the laws of thermodynamics and heat engines. For today’s assignment, please label each page with the code B54Thermo, the date and the page number. 1. Check out references on today’s topics on the internet or in your textbook and write down how you can find these references again. 2. Okay in the previous assignment I had you define the terms isothermal, isobaric, isochoric and adiabatic. Make sure you remember these definitions. You can often visualize the topics below in terms of an ideal gas being compressed in a syringe and then you can relate this to engines. So having said these very exciting things, lets consider the definition of work W = -PΔV. This is an equation for the work done on a gas. What was that underline for? You will soon see (evil cackling wicked witch type laughter is heard in the background) my pretties. So let’s visualize that syringe. Let’s consider taking the syringe full of ideal gas and immersing it in a cold bath while leaving the plunger of the syringe free to move. We can understand from the ideal gas law as the gas cools the volume will decrease. Leaving the plunger free keeps the pressure at atmospheric pressure and constant. The change in volume 36 will have a negative sign, so the work done on the gas will be positive. If the gas expands the change in volume will be positive and the work done on the gas will be positive. Please realize that there is work done on the gas by the environment and work done by the gas on the environment. These are equal in magnitude, but opposite in sign and you must, absolutely must keep the sign convention straight in your head. The equation W = -PΔV is for the work done on a gas! Find a picture or mnemonic device or something to help you keep this straight. For now, just write down a couple of phrases that will summarize the signs of W for this very long paragraph. Also please note that if there is no change in volume there ain’t no work done on the gas no how. Remember I only used bad grammar to make really important points! How does this underlined quadruple negative sentence relate to those vocabulary words at the beginning of this paragraph? 3. Let’s see if you actually understand what you just wrote down. I have a syringe of ideal gas at 1 E 5 Pa. I add heat to the syringe and its volume changes from 1 m3 to 2 m3 (okay, okay, it is a really, really, really big syringe…golly gee it is just an example). What is the work done on the gas? What is the work done by the gas? 4. State the first law of thermodynamics in words only the fewest number of words possible. 5. Write an equation for the first law of thermodynamics found in a physics resource. If your equation has a minus (-) sign in it, change it to a plus. Your equation should now match the equation found on your AP equation sheet. Other resources have different ideas about the sign convention and this has been discussed and debated by scientists, but I suggest that you stick with the currently accepted system. I am not going to debate the merits of sign conventions. Please also look at the very front page of your Table of Equations for AP Physics Exams, the one with all the constants on it. At the bottom of the page there are a couple of asterisks. Look at the asterisk next to IV. Read the sentence next to it. This tells you that the current accepted sign convention is that W represents the work done on the gas. Back to the syringe, this means if V gets smaller that ΔV is negative and W (W = -PΔV) done on the gas is positive and W done by the gas is negative. If V gets larger, the ΔV is positive and W (W = -PΔV) done on the gas is negative and W done by the gas is positive. I must think all this sign convention stuff is important if I keep repeating it over and over again, huh? 6. Now what is this Q in the equation form of the first law of thermodynamics? Write down what it is. Please remember that Q is a type of energy so that you never forget that Q and W have the same units. 7. What is the U in the equation form of the first law of thermodynamics? It is not potential energy like that of a spring or gravity. What are the units of U? 8. Remember the end of the previous homework assignment where we discussed kinetic molecular theory and Kavg and the fact that Pressure multiplied by Volume gives units of energy. I am hoping that you can see by those equations that K = f(T) and PV = f(T) and that energy of a gas is a function of temperature which leads me to my second quadruple negative of this assignment. If there is no change in temperature there ain’t no change in the internal energy of the gas no how. How does this relate to those vocabulary words I reminded you of in question 2 of this assignment? 9. These problems typically are associated with PV diagrams. Let’s contemplate what certain types of processes look like on this diagram. What is the shape of the line/curve that represents an isobaric process on a PV diagram? 10. What is the shape of the line/curve that represents an isochoric process on a PV diagram? 37 11. What is the shape of the line/curve that represents an isothermal process on a PV diagram? What is the shape for an adiabatic process? If an isothermal line/curve and an adiabatic line/curve were drawn on a PV diagram starting from a common point, how could you tell one from the other? 12. If you have triangular shaped process (or indeed any closed process of any shape) drawn on a PV diagram, it starts and ends at the same pressure and volume. What can you tell me about the beginning internal energy and the ending internal energy of the gas? 13. How would you find the work for an isobaric process on a PV diagram? 14. How would you find the work for a triangular shaped process on a PV diagram? 15. Okay, please find a PV diagram for a Carnot cycle either in your textbook or on the internet. Make a quick sketch of it. Mark on your sketch which parts of the 4 step process are isothermal. Mark on your sketch which parts of the 4 step process are adiabatic. Put an arrow to show where in the process heat is added and another arrow to show where in the process heat is removed. 16. Write down the equation for the efficiency of a Carnot engine. Make sure it matches the one on your AP equation sheet. If it doesn’t use the one on your equation sheet or manipulate the equation you have until it matches. You will use the one on your equation sheet the most so it is easiest to work with this equation all the time. No I am not just being irritating by not just telling you to write down the one from your equation sheet in the first place. You have to be able to see where these equations actually come from! What is TH? What is TC? What do the units of these two quantities absolutely have to be in? 17. You can see my all time favorite picture to describe overall efficiency of a heat engine on the below. The Carnot efficiency of a heat engine is considered the “ideal efficiency”. There is another equation for efficiency on your equation sheet. Write it down. Yes, these two equations are often set equal to each other so be prepared for that. Tell me in a few words each what QH, QC and W are. Using the very attractive picture below that I drew all by myself with the help of Microsoft word, write a conservation of energy equation for QH, QC and W. Write in a sentence or two how the regular old efficiency equation (not the Carnot one) relates to grades on homework or tests. Q H W Q C Figure 3. Schematic of energy input and output for a heat engine. 38 18. Okay that is it for our second pass of thermodynamics. I know this is a lot of material. It will become clearer as you practice what you learn. Good luck! Now please write down the two most fantastic understandings you took from this lesson. Physics C Unit 5, Lesson 1, Gauss’s Law for Spheres (C51Gauss) Gauss’s Law for Spheres - Comprehend and be able to analyze Gauss’s law for spherical distributions including conductors, nonconductors of uniform charge density and nonconductors with densities as a function of r. This most exciting day of days has arrived. Today’s lesson is about Gauss’s Law. Today you will be learning about spherical Gaussian surfaces. There actually is some nice calculus associated with Gauss’s law, but it probably won’t show up where you think it will. Please label each page of your assignment with the code C51Gauss, the date and the page number. 1. Find references for Gauss’s law in your textbook and on the internet and write them down. It is worth it to do some looking on the internet. These problems are classic problems and getting information about them from more than one independent source will help you understand this material better. An internet site that I think has pretty comprehensive coverage of Gauss’s law is Hyperphysics. This is their link for a basic statement of Gauss’s law, http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html#c1 and here is a link for some of the types of spherical distributions you will be looking at, http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elesph.html#c3. 2. In one of your references find an integral version of Gauss’s law. Write it down. Compare it to the equation for Gauss’s law on your Table of Equations for AP Physics Exams. Let’s make sure you understand the basics of this equation. So what does the circle on the integral sign mean? 3. Which charge does Q refer to? 4. Name epsilon naught in words and give its value. 5. Find the symbol ФE somewhere in one of your resources. It is not on your equation sheet, but it relates to Gauss’s law in a very significant way. Describe the meaning of this symbol in a phrase or so. 6. There are a couple of common Gaussian surfaces. One of these is a closed cylinder. This can be a short and fat cylinder (tuna fish can) that can be used for a continuous sheet of charge. The cylinder can also be long and skinny (tennis ball can) that can be used for a long line of charge. The most common Gaussian surface you will encounter is a sphere. You will practice with this surface extensively, but you will need to know the cylindrical ones also and I will assign those in the future. 7. What is the surface area of a sphere with a radius of r? 8. Now let’s look at the integral side of the equation of Gauss’s law. You might think you will get to do a lot of calculus with this lovely integral sign, however, when you integrate E·dA over the closed surface of a sphere, the limits of integration are essentially zero and the area of the sphere, 4πr2, and the closed integral of E·dA turns out to be very simple for spheres is always E(4πr2). So knowing what the integral side of the equation of Gauss’s law equals, find an equation for the electric field a distance r from a point charge using a spherical 39 Gaussian surface and knowing that our k of 9E9 is actually equal to 1 divided by the quantity 4 times pi times epsilon naught. Have you ever seen this equation before? 9. Let’s consider a metallic (AKA conducting) sphere, either a solid sphere or a hollow spherical shell of a given radius with a charge Q. The answer is the same for both the solid sphere and the hollow shell. Find out what the electric field is inside this shell a distance r from the center of the shell, but still with r being less than the actual radius of the shell. Please also recall that for conducting surfaces all the charge resides on the outside of the shell spread as far apart as possible since like charges repel. What is Qenclosed? So then what is the electric field inside a conductor? 10. Now a conducting sphere (solid or hollow – they give the same result) of a given radius and charge Q. Using Gauss’s law, derive an equation for the electric field a distance r from the center of the sphere, but this time r is larger than the actual radius of the sphere. Remember that it doesn’t matter if you already know what the answer is; you still must show the work. Have you seen this equation before? Doesn’t it look just like Coulomb’s law for a point charge? And it doesn’t matter what the radius of the conducting sphere is, as long as the equation is for the electric field outside the conducting sphere. As a matter of fact it reminds me of when we find the acceleration due to gravity using the law of universal gravitation. The distance used is r even though all the mass is not a point at the center of the earth. 11. So you can now handle showing that the electric field inside a conductor is zero and you can derive Coulomb’s law using Gauss’s law for a distance r from a point charge or the center of a conducting sphere. These have all actually been a part of AP exam in the past. Now let’s look at a nonconducting (plastic) sphere of radius R with a uniform charge distribution and total charge Q. Typically you are asked to find the electric field inside the sphere and outside the sphere. Or in other words E for r 40 nonuniform charge density. Let’s say that the charges varies radially outward with a volumetric charge density of ρ = Br where r is a variable radial distance from the center of our sphere and B is a constant. Now we have to find Qenc in this sphere and this is where you get to do the nice calculus. Remember while talking about moment of inertia, I told you that you would be using the symbols ρ, σ and λ for density again when we talked about Gauss’s law? Well here is ρ again. Density is mass divided by volume, right? It is the same for charge density so in this case ρ = charge divided by volume or ρ = Q/V or looking at a differential form of this equation and rearranging a little bit, dq = ρ dV or a small amount of charge equals charge density times a small amount of volume. However, you need everything in terms of r which is actually what you will be integrating. So what is dV in terms of r? Well, V = (4/3) πr3 and taking the derivative with respect to r, dV = (4/3) πr2 (1/3) dr or 4πr2 dr. Hey that’s just the area of a sphere times dr! You are, of course, totally correct. So substituting our function for dV and our function for ρ into dq = ρ dV, you 2 should get dq = (Br) (4πr ) dr. Now you can integrate this to find Qenc both inside the sphere and for the entire sphere. If you need to find the electric field and therefore the charge inside the sphere your limits of integration are from 0 to some variable distance r. If you need to find the electric field outside the sphere, then you need to find the total charge on the sphere and your limits should be 0 to the radius of the sphere R. Knowing all this very exciting information, find equations for the electric field both inside (r Physics C Unit 5, Lesson 2, Continuous Charge Distributions (C52Continuous) Continuous Charge Distributions - Be able to determine and recognize appropriate problem solving techniques for finding the electric field and the potential from continuous charge distributions particularly rings or arcs of charge. Good morning. Another day, another assignment. Today you will be learning about finding electric potential (V measured in volts) and electric field for some other continuous charge distributions. No Gauss’s law today, but you will be using some of the same skills you used in the previous assignment. And there is a little of calculus today, so this is exciting! Please label each page this assignment C52Continuous along with the date and the page number. 1. You are going to be looking at finding electric field and electric potential from continuous 41 charge distributions. Typical possibilities include rings of charge or lines of charge. Find references on these types of problems either in your textbook or on the internet. Usually electric field and electric potential are found in different chapters, but the concepts are similar and you are often asked to solve for both of these in the same problem. So I am combining them! Write down the references. 2. You will also be considering calculus forms of equations you have already worked with. I am going to write the calculus form and I want you to write down an analogous equation we have seen before that is almost the same and tell me when you use this previous equation you have seen. (a) E = k ∫ dq /r2; (b) V = k ∫dq/r; (c) E = -(dV/dr) and ΔV = -∫E dr. Notice that these equations are not on your equation sheet, but they are differential forms of long used equations and you will be expected to use them. 3. Remember again I told you those symbols for types of density would come up again. You saw ρ yesterday. Today we have λ. Let me just remind you of what those symbols are because σ is coming up again soon. ●Volumetric density is given the Greek letter lower case rho, ρ. ●Surface area density is given the Greek letter lower case sigma, σ. ●Linear density is given the Greek letter lower case lambda, λ. Sorry there is no answer to this number. It is purely informational. I know you are devastated. Try to hang on. 4. The most common type of problem to consider is a ring of charge with a linear charge density λ and a radius R. Let’s find the total charge of the ring. Let’s consider a small arc of the ring ds, the length of the arc (using rotational variables) is r dθ, so dq = λ r dθ and integrating Q = ∫ λ r dθ with the limits of integration being from 0 to 2π. Notice that r doesn’t vary with θ. Show the work to prove Q = 2πRλ. Or we could have skipped all that calculus stuff and said the total charge is just λ times length. The length of the ring is 2πR (circumference of a circle) and so Q = 2πRλ. 5. Now let’s find the electric field at the center of the ring. REMEMBER that electric field is a vector and since it is a complete ring every tiny arc of the circle ds has another tiny arc of circle that is exactly opposite it and it doesn’t matter if the ring is positively charged or negatively charged, every single small piece if electric field is cancelled out by a piece of equal magnitude and opposite direction on the other side of the ring. So what is the electric field at the center of the ring? I know it is an easy question. 6. Now to find the electric potential at the center of the ring. We have that equation V = ∫dq/r and from number 4) we know that dq = λ r dθ so V =k ∫ λ (r dθ) / r using the limits from 0 to 2π, then what does V equal? You didn’t really need calculus for this either! DO NOT fall into the trap of thinking well, E = kQ/r2 or E = k ∫ dq /r2and all the charge is an equal distance from the center and the total charge of the ring is 2πRλ and telling me the electric field is 2πRλ/R2 or some such nonsense. You must keep in mind the vector summative nature of E and the scalar summative nature of V. Those things haven’t changed since we first discussed them for point charges and this is another very nice way to see if you understand vector and scalar quantities for electricity. 7. Now doing 4, 5, & 6 might have been the beginning to a very nice free response problem. What happens next? Let’s consider not a full ring of charge, but an arc of half a ring and let’s orient it so that it looks like a bowl sitting on a table. Make a sketch of this arc. You will definitely need it soon. The arc still has a radius of R and a charge density λ and it is positively charged. What is the total charge of this arc? 8. What is the electric potential of this arc at a point that would be the center of the entire ring, if there were an entire ring? Notice that when there was a whole ring I asked for E first and it 42 was definitely easier since it didn’t require any math really, only a knowledge of the nature of vectors. Now I am asking for electric potential first. Hmmm. 9. Now let’s find the electric field of this arc at a point that would be the center of the entire ring if there were an entire ring. Let’s consider the two top edges of the bowl. Mark off a tiny part of the ring at the two top edges and let’s make two small arrows that show the direction of the electric field and the center of the arc. One goes left and the other goes right and they cancel out so this is going to be easy…Wait a minute, not so fast. Let’s now consider two small pieces of the arc each about half way down the bowl. Mark these two pieces at a symmetrical place on both sides. Now draw arrows for the direction of the electric field at the center of the arc. The arrows are equal in magnitude and opposite in the x direction, but not in the y direction. What is the direction of the electric field at the center for this ENTIRE positively charged arc? Why? Now you will need to sum up (integrate) the electric field due to all those small arcs. This is not as easy as the previous considerations. 10. Now let’s actually solve for the electric field. Consider one of the two small arcs at the top of the bowl. The charge of that arc is dq = λ r dθ. You also know that E = k ∫ dq /r2 so in this case E = k ∫λ r dθ /r2 and we need to consider only the y components of the field because all the x components cancel out. So E = k ∫ (λ dθ /r) (sin θ). Since λ and r are both constants, they can come outside of the integral and E = (kλ/r) ∫sin θ dθ with the limits of integration being from π to 2π. Solve for the electric field at the center of the arc. 11. Try finding the electric field at the origin for a continuous line of charge of length L and a distance A away from the origin. The line of charge has a continuous charge density of λ also. 12. I think that is it for continuous charge distributions. Be sure to bring any questions you have to the next time we meet. Please write down the two most remarkable ideas from this lesson. Physics C Unit 5, Lesson 3, RC Circuits (C53RC) RC Circuits - Grasp the numerical and theoretical concepts associated with RC circuits. Please label each page of this assignment with the code C53RC, the date and the page number. 1. Find some resources either in your textbook or online dealing with circuits containing resistors and capacitors, also called RC circuits. Write down these references. 2. The first thing I want to do is remind you, believe it or not, of something we learned in mechanics when we talked about air resistance. I showed you these types of equations that represented (1) exponential increase up to an asymptote and (2) exponential decrease. The (-t/τ) equations can be represented by these general forms (1) K = K0 (1 – e ) for exponential (-t/τ) increase up to an asymptote and (2) K = K0 e for exponential decrease. You saw the equation for velocity of an object that experienced air resistance where the resistive force was equal to some constant times velocity, FAR = bv. Go back and remind yourself of that concept and write down for me the equation for the velocity of that object that took the form of one of the two equations above. 3. What is τ for the equation you just wrote? 4. Now let’s consider a resistor of resistance R connected in series with a capacitor of capacitance C and a battery with an EMF or voltage of ε. Please draw this circuit and label each piece with its letter and throw in a switch too! 43 5. Please recall what I kept saying about capacitors when we talked about circuits before. At the instant the switch is closed (t=0) the capacitor acts like a ______and after a long time (as t → ∞) the capacitor acts like a______which is exactly what it looks like. Please fill in the blanks. You should already know the calculations for these limiting conditions. 6. Now you want to consider the time in between t = 0 and t → ∞. During this time the current of the capacitor, the voltage over the capacitor, the charge of the capacitor and the voltage over the resistor all change from min to max values or vice versa. Write down Ic, Vc, Qc, and VR and tell me whether they are a min or max at both t = 0 and t → ∞. 7. In order to do these problems you will be using Kirchoff’s loop rule, but you need to be reminded of or find out some basics first. What is the differential equation to find current in terms of Q and t? 8. Write down the equation for the voltage drop over a capacitor. You already know this equation from our previous discussion of capacitors. 9. Now go around the circuit starting at the battery and show that the Kirchoff’s loop rule for this simple RC circuit is ε – IR – Q/C = 0. 10. However, current I is also equal to dQ/dt. Substitute this into the previous equation. Divide the entire equation by R and rearrange it so that you get dQ/dt by itself on the left hand side of the equation and show that you get dQ/dt = (ε/R) – (Q/RC). 11. Then get a common denominator for the equation of RC and write down the equation you get now. 12. Rearrange the equation again to get all the terms involving Q on the left and everything else on the right and show that you get dQ/(Q – Cε) = (-1/RC) dt. Integrate this equation with the limits from 0 to Q on the left and 0 to t on the right. Then get rid of the logarithm. Show that (-t/RC) (-t/RC) you get the form Q = Cε (1 – e ) or Q = Qmax (1 – e ) since the capacitance times the voltage (Cε ) is the maximum charge on the capacitor. Sketch a graph of Q versus t. 13. Now differentiate the equation you just found with respect to t and show that you get I = (-t/RC) (-t/RC) (ε/R) e or I = Imax e . Sketch a graph of I versus t. 14. Now consider the limiting conditions of t = 0 and t → ∞ and evaluate the last two equations in terms of these limiting conditions. Do you get the values that you expect for Q and I? 15. Also consider VR and VC in terms of the equations you just got. Figure out analogous equations for VR and VC using reasoning and sketch graphs of both VR and VC versus time. 16. What is the time constant τ for this equation? 17. Knowing that you cannot exponentiate a number with units, 1 Farad times 1 Ohm equals what unit? 18. Those are the basics that you need to know for charging a capacitor. Now let’s consider discharging the same capacitor, C, using the same resistor, R. Consider that the switch is closed on the preceding circuit for a good long time and the capacitor completely charges and current stops flowing in the circuit. What is the energy stored in the capacitor? 19. Then you are able to magically open the switch and remove the battery from the circuit without discharging the capacitor and now you have a circuit with just a fully charged capacitor and a resistor and an open switch. Draw this circuit. 20. What will happen when you close the switch now? To the current through the circuit? To the charge of the capacitor? To the voltage over the capacitor? To the voltage over the resistor? Write a phrase or two to answer each of these questions. 21. Write the Kirchoff’s loop rule for the voltage drops around this new circuit. 22. Substitute in dQ/dt for I and rearrange the equation getting all the Q terms on the left and 44 everything else on the right. Integrate both sides of the equation and get an equation for Q in terms of t for a discharging capacitor. Is it what you expect? 23. Differentiate the equation that you got in the step before this to get current I as a function of t. Is it what you expect? 24. Make quick graphs of QC, IC, VR and VC versus time. 25. What happened to the energy stored in the capacitor? Did it disappear? 26. How would the time constant change if you had two resistors in series with the capacitor? If there were two resistors in parallel with each other and then in series with the capacitor? If there were two capacitors in series with 1 resistor? Two capacitors in parallel with each other and then in series with 1 resistor? What values should you use for C and R for these situations? 27. You should now understand the basics of RC circuits. Please write down the two most notable impressions from this lesson. Physics C Unit 5, Lesson 4, Electric Potential from Electric Field (C54Potential) Electric Potential from Electric Field - Using the integral relating electric field to potential and the derivative relating potential to electric field, be able to make determinations both mathematically and conceptually. Today we will be discussing the relationship between electric potential and electric field. You will use some of the equations I gave you in the previous assignment and you will look at the same types of charged spheres that were discussed in the first Gauss’s Law assignment. Please label this assignment C53Potential along with the date and page number. 1. You will need to find references about how electric potential and electric field are related either in your textbook or on the internet. If you are looking on the internet, the best resources I have found are college websites. Write down your references. 2. Let’s recall an equation I gave yesterday, ΔV = -∫E dr. I didn’t really use it, but it went along with the other equations. I hope that you see that it is related to an equation we have for the electric field inside a capacitor, E = -V/d. Doing a little rearranging E = -ΔV/Δd, -ΔV = E Δd, and replacing d with r and moving the negative sign I get ΔV = -E Δr. This last one looks more like the integral equation that I gave. Now recall that potential is not generally an absolute value, it is a difference between two levels like gravitational potential energy. Perhaps it is measured between 120 V and ground or it can be measured between any two other points. Perhaps it also helps to visualize measuring voltage in a circuit where you have to measure the voltage over (or between) two points. In more detail the equation ΔV = -∫E dr is given as VB – VA = -∫E dr where the upper limit of integration is rB and the lower limit is rA and rB and rA are only the radial distances away. Let’s look at this for a simple situation for a 2 point charge and substitute in the equation for E for an electric field VB – VA = -∫(k q/r ) dr. Using the limits of integration rB and rA as being the radial distance from the point charge, 1 1 show that you get VB – VA = kq [ ]. rB rA 3. Now it is the usual convention to define the potential a long way away as zero. In other words define rA = ∞ and VA= 0 and then simplify the above equation and take off the subscripts. Write down what you get and write down where you have seen it before. Please 45 note that the equation you got for this homework question is the potential for a point charge. If you are asked to find the potential difference between two different points in an electric field, you will get something more like your answer in number 2. 4. Now let’s find the potential inside and outside a conducting sphere. Gee it seems like I am asking all the questions I did for Gauss’s Law for spheres…and I am! And now I am going to tell you the answer (I have lost it, gone totally round the bend giving out answers like this). The potential inside a charged conducting sphere of total charge Q and radius R is kQ/R the same as at a point on the surface of the sphere. The potential does not change inside a conducting sphere, it is constant, always kQ/R. There are nice written explanations of why this is in references such as your textbook and on the internet. Feel free to read about what it says in one of my favorite internet references, HyperPhysics: http://hyperphysics.phy- astr.gsu.edu/hbase/electric/potsph.html. Let me see if I can present you with a picture in your mind to hold this there and then you will go through an example. Let’s find the electric potential at the center of a charged sphere of radius R and total charge Q. Recall the previous assignment about ring of charge. When we found the potential at the center of the ring, it was equal to the total charge of the ring divided by the radius multiplied by k, of course. This was because all the charge was equally distant from the center of the ring. This is the same with the conducting sphere. Recall that for conducting spheres all the charge resides on the outside of the sphere distributed as evenly as possible so that the individual charges can get as far away from each other as possible because they are repulsed! So now I hope it is okay with you that the potential at the center of a conducting sphere is kQ/R. But let’s contemplate a point say halfway between the center and the surface. Now all the charge is not equally distant. That is true, but I would like you to recall two things. The electric field inside a conductor in equilibrium has to be zero and E = -dV/dr. So if the electric field is zero inside a conductor and electric field is equal to the rate of change of electric potential, then the electric potential inside the conductor cannot change. Consider my explanation, read your book’s explanation and check the internet too, but let’s go on to a mathematical example. I want to get the potential of a conducting sphere of total charge Q and radius R at its surface and also its center. I am going to use the convention that potential is zero at r = ∞. 2 For the potential at the surface V∞ - VR = - Edr and E outside the sphere equals kQ/r so V∞ - R Qdr V = -k . Evaluate this integral and write down your answer. Don’t forget the limiting R 2 R r condition about r and V at infinity. 5. I hope you just wrote down, taking care of the multiple changes of sign, the answer that VR = kQ/R, just the same thing we get for a point charge. Don’t just hand wave at the sign changes. You will have to do problems that are more complex than this and if you can’t figure out the sign changes now, you certainly won’t be able to then and it will matter more. Now let’s consider the math showing us the potential inside the conducting sphere. A method of solving these problems that I have used over and over is to start at infinity and move in towards the point on the sphere that I want to find. And yes I do imagine myself in a tiny spaceship traveling in from infinity with wonderful goggles on that allow me to view how the potential changes as I get closer to and then inside the conducting sphere. The equation is constantly flashed up on the viewing screen in my spaceship! Okay now 46 R Edr considering this situation, in general I would have V∞ - VR = - Edr and VR – V0 = - . If R 0 I add the left sides of both equations (V∞ - VR) + (VR – V0), I get V∞ - V0. And then if I add R the two right hand sides of the equation I get - Edr + (- Edr ) and I just evaluated the first R 0 integral like 2 minutes ago and the electric field inside the conductor is 0 so the second integral is just zero. So solve all this and show me that at for V0 you get kQ/R! 6. That was a lot of work just to prove what the potential inside a conducting sphere is just kQ/R. However, the process will help us as we go along. Let’s now consider a nonconducting sphere of radius R with a total charge Q and a uniform charge distribution. We can consider the potential outside the sphere, but I hope that you have seen so many situations where that reduces down to just what it is for a point charge that you will see that the answer is slightly boring, V = kQ/r. Remember that big R is the constant radius and little r is the variable radial distance! You will get asked this easy question on the AP exam, but they know it is easy and it will only be worth 1 or 2 points. Just say it acts like a point charge and write down the equation. The interesting question is to find the electric potential inside the sphere! On number 11 of the C51Gauss assignment you should have found that the equation for the electric field inside the sphere was E = kQr/R3. So lets find V for some arbitrary distance r from the center of the sphere and of course r kQ 2 2 kQ 2 2 = - (r – R ) or Vr - VR = (R – r ) . Now don’t forget that VR= kQ/R. Substitute 2R 3 2R 3 kQ r 2 this in and do the algebraic manipulation to show that Vr = (3 ) which is the typical 2R R 2 form found in most resources. 7. Now as if I haven’t already put you through enough pain, let’s go for one more. Recall in the Gauss’s law assignment, I asked you to find the electric field for a nonconducting sphere of nonuniform charge with a charge density ρ = Br and a radius R. I asked you to find the electric field both inside and outside the sphere. This was number 13 on that assignment. Here are some intermediate calculations I made and my final answers. Qenc for r from 0 to R (the actual radius of the sphere) I got Qenc for r>R (outside the sphere) to be 4 2 πBR . This led me to find that E inside of the sphere was Br /(4є0). E outside the sphere is just good old kQ/r2. Substituting in E = kπBR4/r2 or if I want to get rid of k then E = BR4/ 2 (4є0r ). Now we went through all that just so that we can look at the potential of the sphere. What is the potential at r > R? You can just treat it as a point charge and V = kQ/r using the entire charge of the sphere πBR4 and V = kπBR4/r. What is the potential at r=R? It is also 47 just kQ/r using the r = R so V = kπBR4/R = kπBR3. Again, these questions about outside the sphere will be asked, but their solutions are rather simple. Again the difficult question is to find V inside the sphere. However, it is analogous to what was done for the nonconducting r sphere of uniform charge. Again Vr - VR = - Edr . Substituting in the correct electric field R r 2 2 equation (E = Br /(4є0) inside the sphere) I get Vr - VR = -B/(4є0) r dr . Solve this for Vr. R 8. Change the limits of integration on the last problem and show the potential at the center of 3 the sphere, V0, equals BR /(3є0). 9. Now just to drive you absolutely crazy, there is another way to do this. Let’s solve for the potential at the center of the sphere since it reduces to such a cozy fraction. Do you recall the equation V = k ∫dq/r? If not, this is a good time to recall it. It is found in the lesson R dq C52Continuous in number 2. Substituting = k . We have an equation for Q inside the 0 r sphere as πBr4. I can either differentiate q in terms with respect to r and get dq = 4πBr3dr which is exactly the same as dq = ρ dV or (Br) 4πr2 dr. I can substitute this equation into the R 4Br 3dr integral and get V = k . Please simplify and evaluate this equation for the 0 r potential at the center of the sphere. 10. I think that is enough for finding electric potential for now. There will be more excitement soon! Please write down the two most sensational concepts you learned from this lesson. Physics C Unit 5, Lesson 5, Other Field, Potential and Capacitance Problems (C55Other) Other Field, Potential and Capacitance Problems - Gain insight and awareness of other Gaussian surfaces, more difficult continuous charge distributions, and capacitance for geometries different from rectangular parallel plates. Now let’s just clear up some loose ends about Gauss’s law and some other ideas related to symmetry of spheres. Label each page of your homework with the code C54Other, the date and the page number. 1. Let’s start with references. Today we will be specifically discussing the electric field due to an infinite line of charge of uniform density, the electric field due to an infinite nonconducting sheet of charge, finding electric field from potential, finding the electric field and potential at a distance x away from the center of a ring of charge along the x axis going through its center and finding the capacitance of two conducting plates that are spherical shells. These are typical examples found in textbooks and on the internet. It is worth your while to find an example and follow along with these. Please find references for the types of problems I have listed and write those references down. 2. Derive an equation for the electric field a distance r away from an infinite line of charge of uniform charge density λ. This requires the use of a symmetrical cylindrical Gaussian surface. This is usually a textbook example and can be found on the internet if you need some help. There are a few questions below related to this problem. 48 3. Considering problem number 2, what is the direction of the area vector? E dA 4. In the Gauss’s law equation, = Q/є0, what does the symbol, , between E and dA mean? 5. Why don’t the end caps of the Gaussian cylinder in number 2 affect the answer? 6. Derive an equation for the electric field a distance r away from an infinite nonconducting sheet with a uniform surface area charge density σ. This also requires the use of a cylindrical Gaussian surface. 7. In number 6, only the end caps of the Gaussian surface matter and the outer cylindrical shell doesn’t. Why is this so? 8. Consider a ring of charge of radius R with a total charge +Q and uniform linear charge density. Derive an equation for the electric field a distance x away from the center of the ring on the x-axis that runs through the center of the ring. Again this is a common textbook example or internet example. 9. Consider a ring of charge of radius R with a total charge +Q and uniform linear charge density. Derive an equation for the electric potential a distance x away from the center of the ring on the x-axis that runs through the center of the ring. You should be able to find this example in your textbook or on the internet. 10. In the last lesson, you considered multiple problems in which you found the electric potential using an equation for electric field. Now you need to consider how a problem might progress if you are given the potential and need to find the electric field. Let’s consider a conducting sphere of radius R that is charged up to a potential V. Determine the charge Q on the shell in terms of R and V. 11. Let’s consider that conducting sphere of radius R and potential V from the previous problem. Two halves of a conducting spherical shell that have a radius larger than the inner conducting sphere are carefully placed around the inner conducting sphere without touching it or disturbing it. The charge on the outer shell is zero, but there is definite movement of charge. Assuming the charge on the inner sphere is positive, sketch a picture showing the charge distribution on the inner sphere and the outer shell. 12. If the outer shell is then connected to ground, what will happen? Again sketch a picture showing the charge distribution on the inner sphere and the outer shell. 13. These two metal spheres actually form a capacitor! There are two metal surfaces a given distance apart and both have a net charge on them, equal in magnitude and opposite in sign. How do you find the capacitance of this kind of capacitor? First you will find the potential between the outer shell and the inner sphere, then we can use the equation Q = C ΔV. You will go over this in a practice problem. 14. That’s it for other capacitance, electric field and potential problems. Write down the two most significant perceptions from this lesson, please. 49 UNIT 7, PHYSICS B NUCLEAR AND QUANTUM PHYSICS AND PHYSICS C MAGNETISM IN DEPTH Table 8. Unit 7 Overview, Physics B Nuclear and Quantum Physics and Physics C Magnetism in Depth Physics B Nuclear and Quantum Physics Physics C Magnetism in Depth 1) Nuclear Physics and Radioactivity – 1) Ampere’s Law – Gain an awareness of Understand the three types of radioactive how Ampere’s law is used to calculate decay and the byproducts of each, be able magnetic field and be able to apply this to to balance nuclear equations and be able to both short answer questions and problems. use the most famous physics equation 2) LR Circuits – Understand the ideas and E=mc2 making calculations in both Joules calculations associated with LR circuits. (J) and electron volts (eV). 3) Other Magnetism Problems – Consider 2) Quantum Physics and Wave Particle and evaluate additional calculus-based Duality – Gain an appreciation and magnetism problems such as the current comprehension of the photoelectric effect, induced in a rectangular loop of wire a de Broglie wavelength, Heisenberg distance away from a wire that carries Uncertainty Principle and wave particle changing current and therefore a changing duality. magnetic field. 3) Bohr and Energy Level Diagrams – Achieve appropriate comprehension of the Bohr model of the atom and how this relates to energy level diagrams, particularly understanding how to calculate energy, frequency and wavelength of emitted photons. Physics B Unit 7, Lesson 1, Nuclear Physics and Radioactivity (B71Nuclear) Nuclear Physics and Radioactivity – Understand the three types of radioactive decay and the byproducts of each, be able to balance nuclear equations and be able to use the most famous physics equation E=mc2 making calculations in both Joules (J) and electron volts (eV). Today’s exciting homework code is B71Nuclear. Don’t forget to put this code, the date, and page number at the top of each page of homework. Please find the pages in your book that discuss nuclear physics. Some of the parts of this discussion might include these key words: binding energy, alpha, beta, gamma, or nuclear reactions. 1. Write down the pages numbers in your book where information is found about radioactive decay. As usual there are always great references on the Internet. Here is the URL for the discussion of this information in Hyperphysics: http://hyperphysics.phy- astr.gsu.edu/hbase/nuccon.html#c1. 2. Now let’s start with the basics. What are the three different types of radioactive decay? 3. What particle is associated with each type? 4. Which type is the most dangerous and why? 50 5. You also need to know how to balance nuclear equations, so let’s do a little “chemistry” learning. If I talk to you about carbon-12, carbon-13, and carbon-14, what do the 12, 13, and 14 mean? 12 6. If I write the symbol for carbon-12 in this fashion, 6 C , what do the 12 and the 6 mean? 7. Write similar symbols for carbon-13 and carbon-14. 8. Write similar symbols for the neutron, proton and electron. 9. Write similar symbols for the beta particle and the alpha particle. 10. Why didn’t I have you write a similar symbol for the gamma particle? 11. Write down any interesting thoughts you have about the way the symbols were written for carbon-12, carbon-13, and carbon-14 and the way the symbols are written for the neutron, proton, and electron. 12. Now we are going to do a little bit of nuclear equation balancing. Consider this equation, 238 234 A 92 U 90 Th Z X , what are the values for A, Z, and X? You need to look at a periodic table. What is the name of the particle? 14 14 A 13. Next equation, 6 C 7 N Z X . What are the values for A, Z, and X? What is the name of the particle? 12 12 A 14. Next equation, 7 N 6 C Z X . What are the values for A, Z, and X? What is the name of the particle? 15. Write a chemical equation for gamma decay. 16. What is the difference between fission and fusion? 17. Now for the true excitement, during both fission and fusion there is tiny loss of mass and this entire mass is converted to energy according to Einstein’s famous equation, E=(Δm)c2. Give c for this equation with 5 significant figures. The delta is important, as you will see in a minute. 18. Please notice that quantities of the same units as K = ½ mv2 are found in the equation E = mc2. If m is given in kilograms and c in m/s, what is the unit of E? 19. I know that last one was a dorky question, but I just wanted to make sure. It used to bother me that you lost energy both when small molecules were put together to make larger molecules (fusion) or when large molecules are split into smaller molecules (fission), but I have a nice car analogy. If I had a car and I took it completely apart separating every single piece that could be, I would certainly lose a tiny piece of mass, no matter how careful I was, even if it was only a tiny piece. And if I had all the parts to make a car starting from scratch and putting each and every piece together, I would certainly have a tiny piece of mass left over when I was done. So now it makes more sense to me. 20. Let’s consider the mass of just a single proton. Find the energy released in Joules by the annihilation of just one single proton. The mass of the proton in kilograms is on the constants sheet as is the value of c in m/s. 21. Wow, that is a lot of energy! Physicists like to be as efficient as the next person and using the mass in kilograms of something as small as a proton is really just inconvenient to write, so there is another mass unit, the atomic mass unit or amu. The unit itself is just abbreviated u. What is the mass of a proton in atomic mass units? Give at least 7 significant figures. 22. Why do I want so many significant figures? Why is it possible to have so many significant figures? 23. What is the conversion between the unit u and kg? 51 24. What is mass defect? What is mass deficit? What is binding energy? What is disintegration energy? 25. Of course since there is a more convenient unit of mass, there is also one for energy. The Joule even though I consider it to be small unit, is just too large. The new energy unit is the electron volt, abbreviated eV. What is the conversion between electron volts and Joules? 26. On the Table of Information for AP Physics, the following line is found: 1 u = 1.66 X 10-27 kg = 931 MeV/c2. Explain the 931 MeV/c2. 27. Let’s consider a typical problem now. A fusion reaction of deuterium (Hydrogen-2) plus tritium (Hydrogen-3) yields Helium-4 and one neutron. What energy is released by this reaction considering only one molecule of deuterium and one molecule of tritium? I am giving the following values for masses: mass of deuterium is 2.014102 u, mass of tritium is 3.016049 u, mass of helium 4 is 4.002603 u and the mass of a neutron is 1.008665 u. Give the energy released by the reaction both in Joules and mega-electron volts. 28. How many molecules of deuterium are needed to provide the energy needs of the entire United States for one year if the U. S. uses 6.25 E 34 MeV per year? 29. How many kilograms of ocean water would be needed to provide the U. S. energy use for one year? Assume one drop of water is 0.1 grams and there is 1 deuterium molecule for every 6000 H2O molecules. 30. I think that’s it for beginning nuclear physics! Woohoo! Please write down the two most valuable ideas from this lesson. Physics B Unit 7, Lesson 2, Quantum Physics and Wave Particle Duality (B72Quantum) Quantum Physics and Wave Particle Duality – Gain an appreciation and comprehension of the photoelectric effect, de Broglie wavelength, Heisenberg Uncertainty Principle and wave particle duality. Today’s exciting homework code is B72Quantum. Don’t forget to put this code, the date, and page number at the top of each page of homework. 1. Please find references in your book or on the Internet that discuss today’s topics including Planck, Einstein and the photoelectric effect, photons, de Broglie wavelength, Heisenberg Uncertainty Principle and wave particle duality. Please write these references down. 2. Give a sentence or two in synopsis of Max Planck and his contributions to this equation found on your AP Equation Sheet, E = hf. 3. Give the name of each variable in the equation, E = hf, its units and two values if it is a constant. 4. What is the photoelectric effect and who discovered it? 5. What did Einstein win the Nobel Prize in Physics for? 6. What is a photoelectron? 7. What is stopping potential? 8. What is a photon? 9. What is the work function? Which has a larger work function sodium or iron? Does this make sense to you 10. Write the equation from your Table of Equations for AP Physics Exams that includes the work function. 52 11. Sketch the graph of Kmax versus frequency. Give the work interpretation of the x-axis intercept on this graph. 12. True or False and Why? Increasing the intensity of a light with a frequency below the threshold frequency will cause photoelectrons to be ejected from a metal. 13. Why the heck is the word “quantum” so important in the phrase quantum physics? 14. What is wave particle duality? Describe an experiment that illustrates the wave nature of light. Describe an experiment that demonstrates that wave is a particle. 15. What is the equation for the de Broglie wavelength? Summarize de Broglie’s discovery in a sentence or two. 16. Consider a physics teacher of mass 80 kg moving at her top speed walking speed of 19 minutes per mile. What is her wavelength? 17. What is the mass of a photon? 18. There is another equal sign on your Table of Equations for AP Physics Exams right after E = hf. Write the rest of the equation. How is this related to the de Broglie equation? 19. Can a photon have momentum? 20. Do the answers to 17 and 19 bother you? Answer yes or no and then write one additional sentence of explanation. 21. Give 1 to 3 sentences of synopsis for the Heisenberg Uncertainty Principle. 22. Give 1 to 3 sentences of synopsis for the Compton Effect and draw a quick sketch that represents Compton scattering, a combination of nuclear physics and the conservation of momentum. 23. I guess that is it for quantum physics for today. Please write down the two most outstanding perceptions from today’s lesson. Physics B Unit 7, Lesson 3, Bohr and Energy Level Diagrams (B73Bohr) Bohr and Energy Level Diagrams – Achieve appropriate comprehension of the Bohr model of the atom and how this relates to energy level diagrams, particularly understanding how to calculate energy, frequency and wavelength of emitted photons. Do not freak out or anything, but this is the last typed lesson you will get in this class! The homework code for today is B73Bohr. Please write this code, the date, and page number at the top of each page of homework. 1. Find references in your textbook or on the Internet relating to atomic physics particularly the Bohr model of the hydrogen atom and energy level diagrams. 2. Make a quick sketch of the Rutherford Gold Foil Experiment and describe it in 1 to 3 sentences. 3. Read through the page or two of narration and equations concerning the Bohr model of the hydrogen atom. There are associations between this and several other topics you have studied. Write one sentence for each explaining how the Bohr model of the hydrogen atom relates to the following topics: (a) centripetal motion; (b) kinetic energy; (c) Coulomb’s law; (d) electric potential energy; (e) electric potential. Just a note that you do not need to know every iota of narrative and mathematical information of Bohr’s model of the hydrogen atom. However, having some knowledge of this makes it so much easier to know and do what truly is required. 53 1 1 1 4. A nice equation when considering atomic spectroscopic data is this one, RH ( 2 2 ) , n n f ni although I don’t think you will use this equation mathematically very often in this unit. It gives a nice reference point for evaluating the specific wavelengths produced in the hydrogen spectrum. It was known that only very specific wavelengths were emitted for the hydrogen spectrum 50 years before it was known why. You get to evaluate the why pretty darn soon! 1 Lucky you. Other classic equations you should consider are En E1 2 for hydrogen and a n f more general form of this equation that can be used for one-electron atoms such as ionized Z 2 helium or doubly ionized lithium, En (13.6eV ) . These two equations lead us to a n 2 discussion of energy level diagrams. Find a copy of the energy level diagram for hydrogen. Make a quick sketch of it. Where is zero on this diagram? Where is the largest magnitude of E? Is it a positive or negative value? What are the units of E? 5. Okay, if an electron absorbs a certain amount of energy, it is excited to a higher orbit. Then after the electron hangs around at this orbit for a while it returns to its lower orbit and emits a photon of energy. The energy level of the photon emitted is the difference between the energy of the two levels. You can find the energy of the photon emitted by subtracting the energy of the two levels. Since you know the exciting equation E = hf, then you can find the frequency of the photon and then you can find the wavelength of the photon using the equation c = f. Okay now derive an equation for specifically in terms of E and the known constant c. If you know the wavelength you should have some idea whether or not the emitted photon wavelength is in the visible light spectrum or higher than that or lower than that. And this is a nice way to tie in a question involving light waves with nuclear physics! 6. Arrows can be used to show the transition from one energy level to another. On your sketch from #3, draw an arrow in a new color that you didn’t use on the diagram that shows the transition from energy level 1 to 3. Draw another arrow that shows a transition from level 2 to level 3. Draw another arrow that shows a transition from the ground state to the being completely removed from the atom. Which way do the arrows point? 7. Okay now there are supposed to be a couple of big Aha!’s here. Let me be totally obvious and list them out. (a) Wavelengths of emitted photons are only possible if they coordinate to the change in energy from one level to another. The absolute values of the energy levels aren’t as important to the problems that you will be doing as the difference in the energy levels. (b) Given an energy level diagram you should be able to draw an arrow that corresponds to a change in energy level and draw it in the correct direction. (c) Given the diagram you should be able to find the frequency, wavelength and momentum (remember E = pc) of the emitted photon. (d) Given numbers for the absolute values for the energy levels for a real or fictitious atom, you should be able to construct an energy level diagram and then do all of the above. 8. Just for clean up, describe the Michelson-Morley Experiment in 2 to 4 sentences. 9. Same as the last question, describe the Davisson-Germer Experiment in 2 to 4 sentences. 10. That is it for nuclear physics. Make my day and yours by listing the 2 most significant insights from today’s lesson. 54 Physics C Unit 7, Lesson 1, Ampere’s Law (C71Ampere) Ampere’s Law – Gain an awareness of how Ampere’s law is used to calculate magnetic field and be able to apply this to both short answer questions and problems. Please label the pages of this assignment with the code C71Ampere, the date and a page number. 1. The beauty of Ampere’s Law is that once you have already covered Gauss’s Law, Ampere’s Law is not that big of a deal. Find references that deal with Ampere’s Law in your textbook or on the Internet. Write down those references. 2. Write down Ampere’s Law from one of your references. Check it against the equation that appears on your Table of Equations for AP Physics Exams. Make sure they match or that you understand any differences between them. 3. What is the meaning of the circle on the left hand side of the integral? It is not the same as for Gauss’s Law. 4. Define the current that I refers to. How is this analogous to the Q in Gauss’s Law? 5. Name mu naught in words and give its value. 6. What is the most common shape of Amperian Loop’s you will be using? 7. Analogously from what you know of the exciting calculus of the left hand side of Gauss’s law, solve the integration for the left hand side of Ampere’s Law. 8. Let’s solve Ampere’s law for the magnetic field a distance r away and outside a wire carrying a current of I. Where have you seen this equation before? Very exciting isn’t it? 9. Now consider the magnetic field inside a wire of radius R carrying current I where the current is uniformly distributed over the cross sectional area of the wire. I hope that it is obvious to you that the magnetic field outside the wire is still that same old equation we have always used. This is why I am only asking you to consider the more interesting question of the magnetic field inside the current carrying wire. What is Ienc in this situation? The magnetic field that I want you to consider is inside the wire a distance r from the center of the wire, with r 55 field is for a coaxial cable. Find out the basics of the coaxial cable. If you drew an Amperian Loop that went outside the coaxial cable, what would Ienc be? Knowing this, give a reason or two to use coaxial cables. 15. Let’s consider that the coaxial cable is made from an inner conducting wire and an outer conducting sleeve and that the space between the inner conducting wire and the outer conducting sleeve is filled with silicon oil that has a dielectric constant = 2.5. Derive an expression for the capacitance of this cylindrical capacitor. 16. One final thing, there is a lovely law called the Biot-Savart Law. Find this law from a reference source and write it down. Tell me whether or not it appears on your equation sheet. Using your textbook, other books or the internet as aides, use the Biot-Savart Law to derive the expression for the magnetic field outside a wire carrying a current I, a distance r from the center of the wire. Tell me which you like better, Ampere’s Law or Biot-Savart. Give supporting reasons. 17. That’s all folks, for Ampere’s Law. Please write down the two most noteworthy insights from today’s lesson. Physics C Unit 7, Lesson 2, LR Circuits (C72LR) LR Circuits – Understand the ideas and calculations associated with LR circuits. Please label each page of this assignment with the code C72LR, the date and the page number. 1. Find some resources either in your textbook or online dealing with circuits containing resistors and inductors, also called LR circuits. Write down these references. 2. What is an inductor? What does it look like in a circuit diagram? 3. What is the symbol for inductance? What is the unit of inductance? 4. Please recall what I kept saying about capacitors when we talked about circuits before. At the instant the switch is closed (t=0) the capacitor acts like a wire of zero resistance and after a long time (as t → ∞) the capacitor acts like a broken wire because this is what it is really and what it really looks like in a circuit! Of course I have a similar sentence for inductors. Remember how the laws for capacitors and resistors in regular circuits just seemed to be switched around? For resistors in series, they add normally, but for capacitors in series, they add upside down? Well in these RC and LR circuits there is an analogous situation and the laws seem to be switched around again. So here is the sentence for inductors. At the instant the switch is closed (t=0) the inductor acts like ______and after a long time (as t → ∞) the inductor acts like a ______because this is what it is really and what it really looks like in the circuit diagram! Please fill in the blanks. I also have a very important sentence for inductors. They hate change like an old grandma or grandpa person. More about that later. 5. Again you will end up with the classic “look” equations for (1) exponential increase up to an asymptote and (2) exponential decrease. Again, these equations can be represented by (1) K (-t/τ) (-t/τ) = K0 (1 – e ) for exponential increase up to an asymptote and (2) K = K0 e for exponential decrease. If you have forgotten the basics of these equations, please remind yourself. 6. There is an equation for the voltage drop across an inductor. It is a differential equation. Find it and write it down. 7. Now let’s consider a resistor of resistance R connected in series with an inductor of 56 inductance L and a battery of with an EMF of ε. Please draw this circuit and label each part of the circuit with its variable letter and throw in a switch too! 8. Now we want to consider the time in between t = 0 when you will shut the switch and t → ∞. During this time the current through the inductor, the voltage over the inductor, and the voltage over the resistor all change from min to max values or vice versa. Write down IL, VL, and VR and tell me whether they are a min or max at both t = 0 and t → ∞. 9. In order to do these problems we will be using Kirchoff’s loop rule. Let’s go around the circuit starting at the battery and show that the Kirchoff’s loop rule for this simple LR circuit is ε – IR – L(dI/dt) = 0. 10. Before you integrate this equation, first divide the entire equation by R, (ε/R) – I – (L/R) (dI/dt) In integrating this equation, it is easiest to let (ε/R) – I equal U and then dU = -dI. Make these substitutions into the differential equation. Rearrange the equation to get dU and U on one side and dt and everything else on the other side. Then integrate both sides, using the limits of integration from U0 to U on the dU side and from 0 to t on the dt side. Then substitute back in for U. Rearrange until you get a nice equation with our classic form! Of course you should be showing all steps. 11. Does current exponentially increase to an asymptote or exponentially decrease? How does this make sense with previous thoughts about how an inductor acts at t=0 and as t → ∞? And what about all that stuff about old grandma and grandpa people, how does that make sense? 12. Make a sketch of I versus t for the circuit and for VL and VR versus t also. 13. What is the time constant τ for this equation in terms of L and R? 14. Knowing that you cannot exponentiate a number with units, 1 Henry divided by 1 Ohm equals what unit? 15. The equation you found in number 10 is for the current when “charging” an inductor. But there is no stored charge. However, there is definitely stored energy. Write the equation for the energy stored in an inductor. Consider that I is always changing and you get a changing amount of energy stored in the inductor until the switch has been closed for a long time. 16. Those are the basics that you need to know for storing energy in an inductor. Consider that the switch is closed on the preceding circuit for a good long time and the inductor stores all the energy it can. Then you are able to open the switch and magically remove the battery from the circuit without disturbing the inductor and now you have a circuit with just a fully energized inductor and a resistor and an open switch. Draw this circuit. 17. What will happen when you close the switch now? To the current through the circuit? To the voltage over the inductor? To the voltage over the resistor? Write a phrase or two to answer each of these questions. 18. Now I would really like to have you write the Kirchoff’s loop rule for the voltage drops around this new circuit without a battery and then make you solve the differential equation and get an equation for the current I as a function of time, but I am going to let you just write down the equation without any calculus! Oh…the guilt. The equation should be familiar! 19. Make quick graphs of I, VR and VL versus time. 20. What happened to the energy stored in the inductor? Did it just disappear? 21. Now consider the following. How would the time constant change if you had two resistors in series with the inductor? If there were two resistors in parallel with each other and then in series with the inductor? Etc. etc.? 22. You should now understand the basics of LR circuits. Please put pen to paper and note the two most remarkable ideas from this lesson. 57 Physics C Unit 7, Lesson 3, Other Magnetism Problems (C73Other) Other Magnetism Problems – Consider and evaluate additional calculus-based magnetism problems such as the current induced in a rectangular loop of wire a distance away from a wire that carries changing current and therefore a changing magnetic field. I do not want you to be too relieved or happy, but I do want you to know that this is the final lesson of the semester. Please label today’s homework with the code C73Other, along with the date and the page number. 1. Find references in your textbook for the chapters on magnetism skim the chapters checking for example problems that you have not covered yet in this class. Also find the section in your book or a reference from the Internet about Maxwell’s equations. Write these references down. 2. There are several classic calculus-based magnetism problems that do not use Ampere’s law. Let’s consider a few of these. First, find the integral equation for F = BIL and write it down. 3. Consider a closed 2-dimensional semicircle of wire carrying a current I where there is a magnetic field that is in the same plane as the semicircle and is perpendicular to the straight side of the wire. Find the total magnetic force acting on the wire. Try it on your own, see if it is an example in your textbook or try the Internet. Here is a possible URL to give you a hint: http://www.phys.uri.edu/~gerhard/PHY204/tsl185.pdf. 4. Consider a 2-dimensional closed rectangular loop of wire where there is a magnetic field that is in the same place as the rectangle and perpendicular to two sides of the rectangle. Sketch this situation. What are the forces on the two sides of the rectangle that are perpendicular to the magnetic field? What does it cause the rectangle to do? Give me your best mathematical treatment of this problem. Feel free to use these resources: you, your peers, others sentient beings knowledgeable about physics, books, and the Internet. 5. What about a circular loop of wire with a radius of R carrying a current of I. What is the magnetic field at a point a distance a away from the loop along the axis that goes through center of the loop? Feel free consult good resources that have been mentioned before. 6. Consider a rectangular loop of wire with a resistance R and a length L and a width W that is a distance c away from a current carrying wire where the current is a function of time according to the equation I = mt +b. What is the current induced in the wire? Again, you should consult good resources. 7. Find a reference for this equation, I = NevdA, and tell what all letters in the equation are named and what the units of each letter are and what the equation is most often used for. 8. List the 4 Maxwell equations in derivative form (not the gradient form which uses a symbol like an upside down Δ). Give a sentence that gives your best conceptual interpretation of each law. 9. That’s it. Please write down the final two most salient thoughts related to this lesson. 58 CONCLUSION Why do students take Advanced Placement Physics? Why do instructors teach the class? Should an instructor teach the algebra-based Physics B course or the calculus-based Physics C courses? What is the difference between Physics B and Physics C? How can enough students be recruited so that the class can be taught? How can you guide students to know and understand some difficult concepts on their own? These are the questions that I have answered with this project. Some of them I deliberately set out to answer. One of these questions appeared of its own volition as the project progressed. The purpose of this project is to provide support materials for high school physics instructors and students. These materials were specifically written to provide physics instructors with self-guided study for students when they were teaching algebra-based Physics B and calculus-based Physic C concurrently in the same classroom. The topics covered are those where the two courses are completely different. This project is my guide of “How” to accomplish teaching these two courses in the same classroom. What is the benefit to students taking Advanced Placement Physics? There are mixed feelings about this among educators, students, and reports in the media. Some universities do not give college credit for AP classes. Some studies show that students are greatly benefited in college from AP classes. Other studies show that taking these classes in high school makes no significant difference in their secondary education. I think willing students benefit from these courses. I do not think it matters whether or not the college of their choice gives them credit for their passing AP score or not. These students have been exposed to college-level material and have coped with the difficulties that are inherent in this kind of material. They are more prepared for all college-level material. I always tell my students that if they take the same course again in college, they will be smiling and getting high grades while other students in the class are crying! What is the benefit to instructors teaching the class and what is the benefit to offering the two classes concurrently in the same classroom? Teachers get to interact with some of the brightest and most motivated students. They get to teach material that still challenges them. This helps keep instructor morale high. Offering both classes at the same time allows for a greater pool of students to take the class. In smaller schools or larger schools where competition for top students is fierce, this will enable more students to take the class at a mathematical level that they are prepared for. Schools, students and educators benefit from having AP Physics as part of the curriculum. My original vision for these lessons for my students to complete independently was to provide them with detailed information during the time what they were learning was totally different from some of their classmates. I started out creating detailed notes for the topics. Then I realized the students would gain a greater understanding if I didn’t just provide detailed notes, but if I asked them good questions that would lead them to the information they need to know. Then I could discuss the results with them. At this point in my project I changed the emphasis from just setting down information to a combination of giving some information and asking students questions that would lead them to the information they need to know and then discussing that information with them. 59 The purpose of this project is to provide high school physics instructors and students with support material for AP Physics B and AP Physics C when the material of the courses is totally different, particularly if the two courses are taught concurrently in the same classroom. A timeline overview of when the topics are taught was given. Specific self-guided student lessons for these topics were provided. More rigorous high school physics courses lead to improved college physics performance. Being successful in physics in college opens the door to more career choices for students. Teaching more rigorous high school physics courses benefits all physics students and the educators who teach them. 60 BIBLIOGRAPHY AND WORKS CITED "Advanced Placement Program." Wikipedia. 2008. Wikipedia. 22 Jan 2008 "Advanced Placement Report to the Nation 2007." 2007. CollegeBoard. 22 Jan 2008 "AP Physics B Course Home Page." AP Central. 2008. CollegeBoard. 15 Apr 2008 "AP Physics C: Electricity and Magnetism Course Home Page." AP Central. 2008. CollegeBoard. 15 Apr 2008 "AP Physics C: Mechanics Course Home Page." AP Central. 2008. CollegeBoard. 15 Apr 2008 American Chemical Society, Chemistry in the Community: Chemcom. 5th Ed. USA: W. H. Freeman, 2006. "Beat (Acoustics)." Wikipedia. 30 Jan 2008. Wikipedia. 10 July 2007 Brain, Marshall. "How Radar Works." 2007. How Stuff Works. 10 July 2007 Caperton, Gaston. "Foreword to Access, Equity, and Achievement." 10 Nov 2005. CollegeBoard. 29 Jan 2008 Celis, William. "Advanced Placement Exams Suffering an Erosion in Value." New York Times 28 Dec 1994 22 Jan 2008 Chute, Eleanor. "Muddle on How to Value Advanced Placement Tests." Pittsburgh Post-Gazette 06 May 2007 22 Jan 2008 Cutnell, John D., and Kenneth W. Johnson. Physics. Sixth Ed. USA: Wiley & Sons Inc., 2005. Elert, Glenn. "The Physics Hypertextbook." 2008. 13 Apr 2008 Elert, Glenn. "Topic Index." The Physics Factbook. 2008. 13 Apr 2008 61 Elert, Glenn. "Waves & Optics." The Physics Hypertextbook. 2007. 10 July 2007 "Fastest Growing Occupations." Employment Projections. 2007. U S Department Of Labor. 20 Jun 2007 Foley, John. "Measurement of the refractive index of a liquid by the real and apparent depth method." Leaving Certificate Physics Experiment. 2003. 17 July 2007 Gende, Dolores. "AP Physics B." 13 Apr 2008 Giancoli, Douglas. Physics: Principles with Applications. Sixth. Upper Saddle River, NJ: Pearson Prentice Hall, 2005. Halliday, David, Robert Resnick, and Jearl Walker. Fundamentals of Physics. 7th Ed. USA: John Wiley & Sons Inc., 2005. Henderson, Tom. "The Physics Classroom Tutorial." 2007. Glenbrook South High School. 13 Apr 2008 "How Does Doppler Radar Work?." Frequently Asked Questions About Radar. 2007. Accuweather.com. 10 July 2007 "How Stuff Works." 2008. How Stuff Works, Inc. 13 Apr 2008 "Is it Real Gold?." Jewelry-Design-Gemstone.com. 2006-2007. Leslie Leirness Jewelry Designs. 24 Feb 2008 "Kinetic Theory of Gases." 5 Oct 2006. NASA. 5 Mar 2008 "Kinetic Theory." Wikipedia. 2008. Wikipedia. 5 Mar 2008 "Learning Objectives for AP® Physics." College Board AP. CollegeBoard. 20 June 2007 Leduc, Stephen A.. Cracking the AP Physics B & C Exams. 2006-2007 Edition. New York: Random House, 2006. "Light as a Wave." 10 July 2007 62 Marklein, Mary Beth. "Advanced Placement: A detour for college fast track?." USA Today 20 Mar 2006 22 Jan 2008 Morsch, Laura. "What Some Fastest Growing Jobs Pay." CareerBuilder.com 27 Jan 2006 20 Jun 2007 Muller, Gerhard. "Magnetic Force on Semicircular Current." 25 April 2007. Department of Physics University of Rhode Island. 19 Mar 2008 Nave, Carl R. "Air Column Resonance." HyperPhysics. 2008. Department of Physics and Astronomy Georgia State University. 12 July 2007 Nave, Carl R. "Electric Field of a Conducting Sphere." HyperPhysics. 2008. Department of Physics and Astronomy Georgia State University. 8 Mar 2008 Nave, Carl R. "Gauss's Law." HyperPhysics. 2008. Department of Physics and Astronomy Georgia State University. 8 Mar 2008 Nave, Carl R. "Harmonics." HyperPhysics. 2008. Department of Physics and Astronomy Georgia State University. 12 July 2007 Nave, Carl R. "HyperPhysics." 2008. Department of Physics and Astronomy Georgia State University. 13 Apr 2008 Nave, Carl R. "Nuclear." HyperPhysics. 1997. Department of Physics and Astronomy Georgia State University. 8 Jan 2008 Nave, Carl R. "Potential: Charged Conducting Sphere." HyperPhysics. 2008. Department of Physics and Astronomy Georgia State University. 9 Mar 2008 Nave, Carl R. "Quantum Physics." HyperPhysics. 2000. Department of Physics and Astronomy Georgia State University. 8 Jan 2008 Nave, Carl R. "Resonance." HyperPhysics. 2008. Department of Physics and Astronomy Georgia State University. 12 July 2007 63 Nave, Carl R. "Sound and Hearing.” HyperPhysics. 2007. Department of Physics and Astronomy Georgia State University. 10 July 2007 < http://hyperphysics.phy- astr.gsu.edu/hbase/sound/soucon.html#soucon>. Nave, Carl R. "Vibrating String." HyperPhysics. 2008. Department of Physics and Astronomy Georgia State University. 10 July 2007 Nave, Carl R. "Waves and Vision." HyperPhysics. 2007. Department of Physics and Astronomy Georgia State University. 10 July 2007 "Proton." Wikipedia. 2008. Wikipedia. 20 March 2008 Sadler, Philip M. and Robert H. Tai. "Success in Introductory College." 7 Jan 2000. John Wiley & Sons, Inc. 20 Jun 2007 Serway, Raymond A, and Jerry S Faughn. College Physics. 7th Ed. USA: Thomson Brooks Cole, 2006. Serway, Raymond A, and John W Jewett, Jr. Principles of Physics A Calculus-Based Text. 4th Ed. USA: Thomson Brooks Cole, 2006. "Table of Information and Equation Tables for AP® Physics Exams." College Board AP. CollegeBoard. 15 April 2008. "The College Board Announces States' Results in the 2007 Advanced Placement Report to the Nation: A Larger Percentage of High School Graduates Achieve High AP® Standards." Press Releases. 2 Feb 2007. CollegeBoard. 22 Jan 2008 "The Value of Advanced Placement and Pre-AP Classes." Superintendent's Blog. 23 Aug 2006. McPherson Unified School District 418. 22 Jan 2008 Tipler, Paul A, and Gene Mosca. Physics for Scientists and Engineers. 5th Ed. New York: W.H. Freeman and Company, 2004. "Wikipedia." 2008. Wikipedia. 13 Apr 2008 64 Zobel, Leonard A. . "Parts of a Wave." Zona Land. 2007. 25 Jun 2007 65