Program of Study Advanced Placement Physics C D. Forbes, M. Hannum TJHSST Page 1
Course Selection Guide Description for 2011/2012: TJHSST: FCPS: Students study a mathematically substantial formulation of Newtonian mechanics (first semester) and electricity and magnetism (second semester), including vector and calculus-based treatment of particle kinematics (motion), Newton s interaction model, energy, linear momentum, angular momentum, systems of particles, oscillators, and Newtonian gravity in the first semester. Topics covered in the second semester include electromagnetic fields, superposition, electrostatics, magnetostatics, induction, electric currents and elementary circuits, Maxwell s equations in integral form and the Lorentz force law. Students should allot time to complete at least seven hours of quantitative problem solving homework per week to attain mastery. Students are thoroughly prepared to take both the Mechanics and Electricity and Magnetism sections of the Advanced Placement Physics C examination. The purpose of this course is to prepare students to take the Advanced Placement Physics C examination, for which college credit and/or placement may be given if a qualifying score is achieved. Advanced Placement Physics is a second-level course which surveys a broad selection of physics topics at a level above Physics 1. It is designed for students who have completed a core science curriculum and are now ready to pursue more advanced and specialized studies in mechanics and electricity and magnetism. AP Physics C serves as the foundation in physics for students who wish to pursue physical science or engineering degrees. All students are required to take the Advanced Placement exam. Page 2
Course Title: Advanced Placement Physics C Grade Level(s): 10, 11, 12 [sophomores by permission only] Unit of Credit: 1.0 Prerequisite: Chemistry 1 Co-requisite: BC Calculus Course Description: The AP Physics C courses at TJ are intended to be the equivalent of the first two semesters of the university sequence in physics for scientists engineers. The three-semester or two-year sequence of lower-division physics courses at a major university always includes classical mechanics as one semester and classical electricity & magnetism as another semester. Both courses are strongly calculus-based, and are explicitly aimed at those intending to major in engineering or the physical sciences. There is a significant laboratory component to both courses. After the AP exams in early May, [There are two exams, one for mechanics and one for E&M.] Additional topics such as wave motion and Special Relativity will be explored as time allows. Page 3
Course Syllabus AP Physics C Course Policies 2012/2013 [Please see the detailed syllabi for Mechanics and E & M for information on the course curriculum. Those documents were written to comply with the College Board s AP certification policy. If they should conflict with this document in any way, the policies outlined in this document will be considered correct and current.] SHOULD YOU CHOOSE TO DROP AP PHYSICS If, after reading this document, doing the summer reading and assignments, and attending the first several weeks of the course, you decide that this is not the year for you to take a university course in physics then you must understand and agree to the following: Juniors All juniors at TJ are required to be enrolled in physics. If you choose to leave AP Physics for whatever reason, then you MUST be placed into a section of Physics 1. While Student Services and the Physics Department will make a good effort to accommodate your request with a simple straight-across switch of classes there are limits. Due to the increasing number of AP Physics sections, and the corresponding decrease in the number of Physics 1 sections, the rest of your schedule may have to be adjusted, and you may have to sacrifice another course. In any event, if you wish to start fresh (no AP grade to follow you), your choice to switch into Physics 1 must be officially made and approved by the date of the first interim report. If you decide to drop after that date, your AP grade will be transferred to your Physics 1 teacher. [NOTE: after the official course change date of September 16 th, a withdraw from AP Physics will be placed on your transcript.] No transfer to Physics 1 requests will be considered after the end of the 1 st quarter. Seniors Your cases will be decided on an individual basis by Student Services. However, just as with juniors, your choice to switch into another course must be officially made and approved by the date of the first interim report. No promise is made or implied that there will be an opening in an appropriate course. Page 4
Sophomores There is essentially no place for you to go. So, your cases will again be decided on an individual basis by Student Services. Due to the time constraints, your choice to switch into another course must be officially made and approved by the end of the second week of classes (September 16 th ), which is the official course change date. No promise is made or implied that there will be an opening in an appropriate course. The protocol outlined above will be strictly adhered to by all concerned; students, parents, counselors, and teachers. Primary Text: Physics for Scientists and Engineers (Extended Version), Tipler, Paul A., 4 th ed. 1999. New York: W. H. Freeman. Volumes I & II, or, the Standard, or, the Extended edition. [There are several possible ISBN numbers for this text due to the various formats for the text.] Replacement Cost: The 4 th edition is out of print. If lost or damaged beyond the ability to reissue, then you are responsible for procuring a used copy from Amazon.com or another vendor. Otherwise, the assessment is $100.00. Grading Details The following general outline will be followed for the year and each quarter: For the Year: 2 Major Topic Exams [given near the end of each semester]: Mechanics 10% Electricity & Magnetism 10% 4 Quarters (@ 20% each) 80% Total 100% Page 5
Each Quarter: Percentage Unit Tests (2 to 3) 50% ±5% Homework Quizzes [HWQs] (10 to 12) 27% ±5% Laboratory Work 15 % ±5% Homework [WebAssign] 8% ±2% TOTAL 100% [NOTE: due to AP exam testing, the 4 th quarter may deviate from this weighting.] Also, there exist regular opportunities for bonus points. These Student Learning Objectives [SLOBs] are single problems given during lunch. Problems earn between 1 and 2 points each. [Note: each quarter s grade will be truncated to 100% if excess points are earned. In other words, you may not store up bonus points to use 4 th quarter. Also, new FCPS regulations state that extra credit may not raise a student s grade more than one step; say, from B to B+, or A- to A, but not B+ to A.] The lowest quiz and the lowest homework grade will be dropped each quarter for all students. Do not blow off a quiz or a homework assignment you will undoubtedly need to drop one that you actually attempted with sincere effort! There are NO re-tests for credit in this course. There are NO test corrections for credit in this course. Late Assignments: All assignments are due at the beginning of the class period on the assigned date. A significant penalty will be assessed for work submitted after its due date. Page 6
MAKE-UP WORK: Work missed due to an absence must be made up. You will receive full credit when making up work due to an excused absence; make-up due to an unexcused absence will receive no credit. Make-up work must be completed within the time limits set by the TJHSST Science & Tech Department: Quizzes missed due to excused absences should be made up the next class day unless some accommodation has been agreed upon with the teacher. Major tests missed due to excused absences should be made up within 2 days unless the absence is an extended one. Again, accommodations can be made in special situations. Class/lab work should be made up within 2 block days. Labs not made up within 1 week will count as a zero except in the most extenuating of circumstances (e.g., hospitalization, etc.). Make-up work will not be done during class time. You may complete the work before school, during lunch or 8 th period, or (by arrangement) after school. Failure to complete make-up work will result in a zero for the assignment. [Note: long-term absence due to illness, injury, or a family crisis will need to be addressed on an individual basis.] IT IS YOUR RESPONSIBILITY TO INITIATE THE MAKEUP PROCESS AND COMPLETE THE WORK. WORKING IN GROUPS Progress in science and technology relies upon cooperation, collaboration, and effective communication. Learning the necessary skills and content of science in particular, physics also can benefit from group learning and communication. I have yet to meet a successful university physics student that did not attend at least occasionally a study group. Virtually no one understands everything right off the bat. However, while you are encouraged to assist each other, you are not helping if you do someone else s work for them! This next statement cannot be overemphasized: IF YOU DO NOT SPEND SIGNIFICANT MENTAL ENERGY ENGAGING THE COURSE MATERIAL, YOU WILL DO POORLY ON THE TESTS AND HOMEWORK QUIZZES. In other words, if you don t make a significant effort to complete the problem sets on your own, your grade will suffer. Page 7
HONOR CODE Workers in science and technology have a long and proud heritage of individual responsibility and respect for the work of others; plagiarism, altering or fabricating data, and providing false information are serious violations of the integrity of the enterprise. Students in physics are expected to follow this model. Many activities will involve teamwork or cooperative effort. Open discussion of lab methods within your lab group, problem solving strategies, and reading assignments is expected and encouraged. This does not, however, constitute permission to submit identical copies of reports, or to divide problem sets among individuals, or to copy another student s solution to a problem. If you need clarification regarding what may constitute misconduct, please talk to me. This course subscribes to, and supports, the TJ Honor Code. In other words, if you are caught violating the honor code, or if two or more papers exhibit statistically improbable linguistic or mathematical coincidences (i.e., you copied), you will receive a zero for that assignment, the appropriate grade-level administrator will be informed, your counselor will be notified, and your parent(s) or guardian(s) will be contacted. This applies to ALL involved parties, both the cheater(s) and the cheatee(s). Those who facilitate a violation of the honor code (i.e., give assistance or supply answers) are equally guilty of ethical misconduct. Page 8
COMPUTOR USE: [Please see the NETWORK USER GUIDELINES for general computer usage rules.] Students will find themselves engaged in lab and computer work in the physics classroom on a regular basis. In order to maintain the computers in functional condition, we must abide by certain rules. In particular, students are NOT ALLOWED TO INSTALL ANY FILES ON THE HARD DRIVES OF THE COMPUTERS IN THIS CLASSROOM (without the explicit permission of the instructor). No portable media may be connected to the machines without the explicit permission of the instructor. Care must be taken with all physics equipment including computers. Most importantly: NO FOOD OR DRINK IS ALLOWED WHEN WORKING ON A COMPUTER in the classroom. You may not use any other student s computer code, or spreadsheet template, or laboratory data/procedure/analysis without proper attribution. If you do not cite all sources you used in a document or assignment, then you are guilty of plagiarism, and are in violation of the Honor Code. LABORATORY LAB SAFETY: Safety is always a concern when working in a science lab. This is especially true when the class size taxes the physical space of the room. It is expected that TJ students will never exhibit inappropriate disruptive or dangerous behavior at any time. However, having noted the standard to which TJ students are held, any student who does act in a disruptive or dangerous manner will receive a grade of zero for the lab. Continued disruptive or dangerous behavior will result in grades of zero and non-participation in the class pending a parent conference. LAB EQUIPMENT: Treat the equipment carefully and with respect. I know things wear out and break. Also, accidents happen. I break things regularly. If equipment breaks while you are using it, please let me know, so that I may order a replacement. If the equipment was broken through careless use, YOU are responsible for its replacement. EQUIPMENT AND SUPPLIES Each student is expected to provide the equipment and supplies listed below, and bring them to class on a daily basis. Graphing calculator (TI-83 Plus, TI-83 Plus Silver, TI 84, or TI 89 preferred) 1 bound composition book filled with quad-ruled graph paper. This is your problem set notebook. #2 pencil(s) and black or blue pen(s) Page 9
In addition, you must have a computer with internet access in your home. The computer needs to have the following software/capabilities: 1. A spreadsheet program compatible with Excel, 2. A word processor compatible with MS Word, 3. An internet browser, 4. Continuous email access. AP Physics C students are expected to maintain up to date email addresses in the Blackboard and WebAssign systems. TIME COMMITMENT Experience indicates that students in a quality university calculus-based physics course must be able to dedicate AL LEAST 8 hours a week to reading and homework practice on a regular basis as well as more time on occasion. This is NOT a maximum but a minimum. The amount of time a person needs to master the material to the level of expectations in the course is very much dependent on student aptitude and prior preparation and can easily exceed TWICE the minimum mentioned above. Page 10
Preface to Program of Study This document is intended to outline both the depth and breadth of the AP Physics C course at Thomas Jefferson High School for Science and Technology (TJHSST) as well as to draw lines of distinction between the recommended topics provided by the College Board and the extensions that we provide as a department. Before we address this side-by-side comparison of topics, there are several important details that should be noted about the structure and goals of AP Physics C and the academic backgrounds of our students at TJHSST that distinguish our course from other other high schools across the country(and FCPS), and the recommendations of the College Board. The AP Physics C curriculum is based on the standard introductory physics courses offered at colleges and universities around the United States. As such, the course is designed for students who are planning on pursuing degrees in the physical sciences or engineering. It is meant to build on the conceptual understandings built during a prior course in physics. The emphasis of the course is placed on solving a variety of challenging problems requiring calculus and deep mathematical understanding Category C courses also build on the conceptual understanding attained in a first course in physics, such as the Category A course described above. These courses normally form the college sequence that serves as the foundation in physics for students majoring in the physical sciences or engineering. The sequence is parallel to or preceded by mathematics courses that include calculus. 1 Most high schools in the nation (and FCPS) follow the College Board guidance and consider AP Physics C to be a second year course. However, at TJHSST more than three quarters of the enrolled students will receive their first exposure to physics via AP Physics C. This constitutes an important distinction. It is strongly recommended that both Physics B and Physics C be taught as second-year physics courses. A first-year physics course aimed at developing a thorough understanding of important physical principles and that permits students to explore concepts in the laboratory provides a richer experience in the process of science and better prepares them for the more analytical approaches taken in AP courses 2. As a first year course we must take extra time to develop basic conceptual understandings in students as well as provide them multiple opportunities to explore deep mathematical connections. Taking time to develop both, while under the time constraints of the FCPS calendar equate to our AP Physics C course covering the required material in about half of the recommended time. 1 AP Physics Course Description, The College Board 2012, pg. 6 2 AP Physics Course Description, The College Board 2012, pg 7 Page 11
If AP Physics is taught as a second-year course, it is recommended that the course meet for at least 250 minutes per week (the equivalent of a 50-minute period every day). However, if it is to be taught as a first-year course, approximately 90 minutes per day (450 minutes per week) is recommended in order to devote sufficient time to study the material to an appropriate depth and allow time for labs 3. These collegiate courses are typically broken into a three semester sequence, and it is very important to note that the AP Physics C course represents TWO distinct semester courses out of this sequence Mechanics, followed by Electricity and Magnetism. Each of these courses is tested by the College Board separately. There are two AP Physics C courses Physics C: Mechanics and Physics C: Electricity and Magnetism, each corresponding to approximately a semester of college work. Mechanics is typically taught first, and some AP teachers may choose to teach this course only. If both courses are taught over the course of a year, approximately equal time should be given to each. 4 Another very important distinction between the AP program at TJHSST and other schools is that the majority of the high schools in the United States (and FCPS specifically) only cover the first semester of the material. This translates into approximately half the material that we cover. Our students have been exposed to a full two semesters of calculus based physics by the end of the school year. Aside from the deep coverage of physics topics, it should be noted that our AP Physics C courses spend significant time developing the tools of calculus. Traditionally nearly two thirds of our enrolled students are co-enrolled in AP Calculus BC, with the rest having completed BC prior to starting AP Physics. The initial topics of the course require a calculus treatment; therefore we must develop the required mathematical tools during the first weeks of school. The development of these core calculus topics precedes the more formal treatment the students receive in their calculus class. This results in our students being asked to understand and use new mathematics while at the same time learning fundamental concepts in physics. The very quick incorporation of calculus into the course is accomplished with a multi part summer assignment that covers vector algebra and the conceptual basis of both differential and integral calculus. With the assumption that students have mastered these topics, we are able to begin the course with three-dimensional kinematics. Of final note is that upon completion of the AP Tests in May we continue to cover more advanced topics beyond the scope of the AP Physics Course Description. We return to Mechanics and discuss the differential equations that define mechanical waves. With this as a basis, we then explore electromagnetic waves and special relativity in a comprehensive manner. 3 AP Physics Course Description, The College Board 2012, pg. 8 4 AP Physics Course Description, The College Board 2012, pg. 11 Page 12
Mechanics Course Semester 1 Comparison of College Board Requirements to Thomas Jefferson Program of Study College Board Requirements Thomas Jefferson Program Notes (A) Kinematics (including vectors, vector algebra, components of vectors, coordinate systems, displacement, velocity, and acceleration) 1. Motion in one dimension Our course starts with three dimensional motion and all kinematic variables are first defined in 3D. Then, 1D motion is treated as a special case of this more general 3D motion. (a) Students should understand the general relationships among position, velocity, and acceleration for the motion of a particle along a straight line, so that: 1. Given a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, they can recognize in what time intervals the other two are positive, negative, or zero and can identify or sketch a graph of each function of time. 2. Given an expression for one of the kinematic quantities, position, velocity or acceleration, as a function of time, they can determine the other two as a function of time, and find when these quantities are zero or achieve their maximum and minimum values. (b) Students should understand the special case of motion with constant acceleration so they can: Level of calculus that is expected is much higher at TJ including the use of differentiation and integration during the first weeks of school. Advanced methods including the chain rule and non-polynomial functions are used frequently. Page 13
1. Write down expression for the velocity and position as functions of time, and identify or sketch graphs of these quantities. 2. Use the equations v v0 at, 2 x x 1 0 v0t at, and 2 2 2 v v0 2a x x0 to solve problems involving one-dimensional motion with constant acceleration. Reaction Time Lab (c) Students should know how to deal with situations in which acceleration is a specified function of velocity and time so they can write an appropriate differential equation and solve it for v(t) by separation of variables, incorporating correctly a given initial value of v. 2. Motion in two dimensions, including projectile motion (a) Students should be able to add, subtract, and resolve displacement and velocity vectors, so they can: 1. Determine components of a vector along two specified, mutually perpendicular axes. 2. Determine the net displacement of a particle on the location of a particle relative to another. 3. Determine the change in velocity of a particle or the velocity of one particle relative to another. The incorporation of Ordinary, Separable, Differential Equations is a major theme of the TJ course. We begin discussion of ODE s in the context of kinematics, but quickly exceed the expectations of the College Board through the use of more complex calculus. For example students are required to understand how to solve second order ODE given multiple initial conditions Introduce the use of the dot product to find projections of one vector onto another. Page 14
(b) Students should understand the general motion of a particle in two dimensions so that, given functions x(t) and y(t) which describe this motion, they can determine the components, magnitude, and the direction of the particles velocity and acceleration as functions of time. (c) Students should understand the motion of projectiles in a uniform gravitational field, so they can: 1. Write down expression for the horizontal and vertical components of velocity and position as functions of time, and sketch or identify graphs of these components. 2. Use these expressions in analyzing the motion of a projectile that is projected with an arbitrary initial velocity. 3.Vector operations (B) Newton s Laws of Motion 1. Static equilibrium (first law) students should be able to analyze situations in which a particle remains at rest, or moves with constant velocity, under the influence of several forces. TJ far exceeds s the College Boards guidelines when it comes to basic vector operations. Students use rotational vector quantities, multiple coordinate systems, and make heavy use of unit vectors and complex vector manipulations and vector identities (example. BAC-CAB) We address relative motion between two points in rotating systems as well as make use of more advanced calculus methods in combination with relative motion Students are required to write down and use the vector forms for the position, displacement, and acceleration as functions of time for projectiles in uniform gravitational fields. Students analysis includes mature use of calculus to find rate of change of vectors. Students are expected to analyze projectile motion in the context of relative motion Can-it Lab Page 15
2. Dynamics of a single particle (second law) (a) Students should understand the relation between the forces that acts on an object and the resulting changes in the objects velocity, so they can: 1. Calculate, for an object moving in one dimension, the velocity change that results when a constant force F acts over a specified time interval. 2. Calculate, for an object moving in one dimension, the velocity change that results when a force F(t) acts over a specified time interval. 3. Determine, for an object moving in a plane whose velocity vector undergoes a specified change over a specified time interval, the average force that acted on the object. (b) Student should understand how Newton s Second Law applies to an object subject to forces such as gravity, the pull of strings, or contact forces, so they can: 1. Draw a well-labeled, free body diagram showing all real forces that act on the object. 2. Write down the vector equation that results from applying Newton s Second Law to the object, and take components of this equation along appropriate axis. (c) Students should be able to analyze situations in which an object moves with specified acceleration under the influence of one or more forces so they can determine the magnitude and direction of the net force, or one of the forces that make up the net force, such as motion up or down with constant acceleration. F = m a Demo TJ students are expected to perform this type of analysis both analytically as well as graphically. Page 16
(d) Students should understand the significance of the coefficient of friction, so they can: 1. Write down the relationship between the normal and frictional forces on a surface. 2. Analyze situations in which an object moves along a rough inclined plane or horizontal surface. 3. Analyze under what circumstances an object will start to slip, or to calculate the magnitude of the force of static friction. (e) Students should understand the effect of drag forces on the motion of an object, so they can: 1. Find the terminal velocity of an object moving vertically under the influence of a retarding force depending on velocity. 2. Describe qualitatively, with the aid of graphs, the acceleration, velocity, and displacement of such a particle when it is released from rest or is projected vertically with specified initial velocity. 3. Use Newton s Second Law to write a differential equation for the velocity of the object as a function of time. 4. Use the method of separation of variables to derive the equation for the velocity as a function of time from the differential equation that follows from Newton s Second Law. TJ far exceeds the College Board guidance on the origins of friction. Students will understand the molecular and atomic origins of friction and be able to relate this physical property back to core chemistry concepts Friction Lab TJ students must describe these kinematic variables quantitatively. Air Drag Lab TJ students must solve the second order differential equation to determine the position of a falling object under the influence of drag force as a function of Page 17
time. 5. Derive an expression for the acceleration as a function of time for an object falling under the influence of drag forces. TJ far exceeds College Board expectations with Differential Equations. TJ students must develop models of drag forces that are proportional to velocity raised to different integer and fractional powers. The College Board only requires drag forces to be directly proportional to velocity. 3. Systems of two or more objects (third law) (a) Students should understand Newton s Third Law so that, for a given system, they can identify the forces pairs and the objects on which they act, and state the magnitude and direction of each force. (b) Students should be able to apply Newton s Third law in analyzing the force of contact between two objects that accelerate together along a horizontal or vertical line, or between two surfaces that slide across one another. (c) Students should know that the tension is constant in a light string that passes over a massless pulley and should be able to use this fact in analyzing the motion of a system of two objects joined by a string. (d) Students should be able to solve problems in which application of Newton s Laws lead to two or three simultaneous linear equations involving unknown forces or accelerations. (C) Work, Energy, Power 1. Work and the work-energy theorem Students must apply Newton s Third Law and analyze contact forces between two objects that are moving in circular paths. Students must analyze the forces involved with massive strings and massive pulley. Atwood Machine Demo Students must analyze systems that lead to more than three simultaneous linear equations. Page 18
(a) Students should understand the definition of work, including when it is positive, negative, or zero, so they can: 1. Calculate the work done by a specified constant force on an object that undergoes a specific displacement. 2. Relate the work done by a force to the area under a graph of force as a function of position, and calculate this work in the case where the force is a linear function of position. 3. Use integration to calculate the work performed by a force F(x) on an object that undergoes a specified displacement in one dimension. 4. Use the scalar product operation to calculate the work performed by a specified constant force F on an object that undergoes a displacement in a plane. (b) Students should understand and be able to apply the work-energy theorem, so they can: 1. Calculate the change in kinetic energy or speed that results from performing a specified amount of work on an object. 2. Calculate the work performed by the net force, or by each of the forces that make up the net force, on an object that undergoes a specified change in speed of kinetic energy. 3. Apply the theorem to determine the change in an object s kinetic energy and speed that results from the application of specified forces, or to determine the force that is required to bring an Students relate the work done by a force to the area under a graph in cases where force is non-linear. Students use methods of integration that exceed the College Board expectations like integration by substitution and integration by parts. Students use the scalar product operation to calculate the work performed by a specified variable force F(x, y) on an object that undergoes a displacement. Page 19
object to rest in a specified distance. 2. Forces and potential energy (a) Students should understand the concept of a conservative force, so they can: 1. State alternative definitions of conservative force and explain why these definitions are equivalent. 2. Describe examples of conservative forces and non-conservative forces (b) Students should understand the concepts of potential energy, so they can: 3. State the general relation between force and potential energy, and explain why potential energy can be associated only with conservative forces. 4. Calculate a potential energy function associated with a specified one-dimensional force. 5. Calculate the magnitude and direction of a onedimensional force when given the potential energy function U(x) for the force. 6. Write an expression for the force exerted by an ideal spring and for the potential energy of a stretched or compressed spring. 7. Calculate the potential energy of one of more objects in a uniform gravitational field. 3. Conservation of energy Students calculate the turning points of objects trapped in potential wells via the magnitude and direction of the associated force. Students write expressions for the force and potential energy for non-linear springs. Page 20
(a) Students should understand the concepts of mechanical energy and of total energy so they can: 1. State and apply the relation between the work performed on an object by non-conservative forces and the change in an object s mechanical energy. 2. Describe and identify situations in which mechanical energy is converted to other forms of energy. 3. Analyze situations in which an object s mechanical energy is changed by friction or by a specified externally applied force. (b) Students should understand conservation of energy, so they can: 1. Identify situation in which mechanical energy is or is not conserved. 2. Apply conservation of energy in analyzing the motion of systems of connected objects, such as an Atwood machine. 3. Apply conservation of energy in analyzing the motion of objects that move under the influence of springs. 4. Apply conservation of energy in analyzing the motion of objects that move under the influence of other non-constant one-dimensional forces. TJ objectives far exceed that of the College Board because TJ students are expected to be able to solve complex problems that incorporate combinations of all of the above scenarios. Page 21
(c) Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton s Laws. 4. Power Students should understand the definition of Power, so they can: (a) Calculate the power required to maintain the motion of an object with constant acceleration (e.g., to move on object along a level surface, to raise and object at a constant rate, or to overcome friction for an object that is moving at constant speed). (b) Calculate the work performed by a force that supplies constant power, or the average power supplied by a force that performs a specified amount of work. TJ objectives far exceed the College Board concerning the set up, and solving of Differential Equations relating both constant Power and as function of time P(t) and velocity as a function of time v(t) (D) Systems of particles, linear momentum 1. Center of mass (a) Students should understand the technique for finding center of mass, so they can: 1. Identify by inspection the center of mass of a symmetrical object. 2. Locate the center of mass of a system consisting of two such objects. 3. Use integration to find the center of mass of a thin rod of non-uniform density. Students use integration to find center of mass of more complicated, geometries including non-uniform densities. They must also make use of Page 22
non-cartesian coordinate systems. (b) Students should be able to understand and apply the relation between center-of-mass velocity and linear momentum, and between center-of-mass acceleration and net external force for a system of particles. (c) Students should be able to define center of gravity and to use this concept to express the gravitational potential energy of a rigid object in terms of the position of its center of mass. 2. Impulse and momentum Students should understand impulse and linear momentum, so they can: (a) Relate mass, velocity, and linear momentum for a moving object, and calculate the total linear momentum of a system of objects. (b) Relate impulse to the change in linear momentum and the average force acting on an object. (c) State and apply the relations between linear momentum and center-of-mass motion for a system of particles (d) Calculate the area under a force verses time graph and relate it to the change in momentum of an object. (e) Calculate the change in momentum of an object given a function F(t) for the net force action on the object. 3. Conservation of linear momentum, collisions (a) Students should understand linear momentum conservation, so they can: Students must use integration to find center of gravity/ gravitational potential energy of extended bodies with nonuniform densities. Page 23
1. Explain how linear momentum conservation follows as a consequence of Newton s Third Law for an isolated system. 2. Identify situations in which linear momentum, or component of the linear momentum vector, is conserved. 3. Apply linear momentum conservation to onedimensional elastic and inelastic collisions and two-dimensional completely inelastic collisions. Collisions Video Analysis Lab 4. Apply linear momentum conservation to twodimensional elastic and inelastic collisions. 5. Analyze situations in which two or more objects are pushed apart by a spring or other agency, and calculate how much energy is released in such a process. (b) Students should understand frames of reference, so they can: 1. Analyze the uniform motion of an object relative to a moving medium such as a flowing stream. 2. Analyze the motion of particles relative to a frame of reference that is acceleration horizontally or vertically at a uniform rate. TJ objectives exceed that of the College Board as TJ students must model and analyze one-dimensional collisions in the center of mass reference frame. TJ objectives exceed that of the College Board as TJ students must set up, and solve differential equations that describe the motion of a rocket using the concepts of impulse and conservation of momentum. Page 24
(E) Circular motion and rotation 1. Uniform Circular Motion Students should understand the uniform circular motion of a particle, so they can: (a) Relate the radius of the circle and the speed or rate of revolution of the particle to the magnitude of the centripetal acceleration. (b) Describe the direction f the particle s velocity and acceleration at any instant during the motion. (c) Determine the components of the velocity and acceleration vectors at any instant and sketch or identify graphs of these quantities. (d) Analyze situations in which an object moves with specified acceleration under the influence of one or more forces so they can determine the magnitude and direction of the net force, or of one of the forces that make up the net force, in situation such as the following: 1. Motion in a horizontal circle (e.g. mass on a rotating merry-go-round, or car rounding a banked curve) 2. Motion in a vertical circle (e.g. mass swinging on the end of string, cart rolling down a curved track, rider on a Ferris wheel) 2. Torque and rotational statics (a) Students should understand the concept of torque so they can: 1. Calculate the magnitude and direction of the torque associated with a given force. Students can prove the relationship between radius of the circle, and the speed of revolution of a particle to the centripetal acceleration. Page 25
2. Calculate the torque on a rigid object due to gravity. TJ objectives far exceed the College Board standards when it comes to vector operations of torque. TJ students analyze and calculate the torque on rotating objects using the vector cross product. (b) Students should be able to analyze problems in statics, so they can: 1. State the conditions for translational and rotational equilibrium of a rigid object. 2. Apply these conditions in analyzing the equilibrium of a rigid object under the combined influence of a number of coplanar forces applied at different locations. (c) Students should develop a qualitative understanding of rotational inertia, so they can: 1. Determine by inspection which set of symmetrical objects of equal mass has the greatest rotational inertia. 2. Determine by what factor an object s rotational inertia changes if all its dimensions are increased by the same factor. (d) Students should develop skill in computing rotational inertia so they can find the rotational inertia of: 1. A collection of point masses lying in a plane about an axis perpendicular to the plan. 2. A think rod of uniform density, about an arbitrary axis perpendicular to the plane. Moment of Inertia Demo Students calculate moments of inertia of a thin uniform rod with non-uniform density. Page 26
3. A thin cylindrical shell about its axis, or an object that may be viewed as being made up of coaxial shells. (e) Students should be able to state and apply the parallel-axis theorem. 3. Rotational kinematics and dynamics (a) Students should understand the analogy between translational and rotational kinematics so they can write and apply relations among the angular acceleration, angular velocity, and angular displacement of an object that rotates about a fixed axis with constant angular acceleration. (b) Students should be able to use the right-hand rule to associate an angular velocity vector with a rotating object. (c) Students should understand the dynamics of fixedaxis rotation, so they can: 1. Describe in detail the analogy between fixedaxis rotation and straight-line translation 2. Determine the angular acceleration with which a rigid object is accelerated about a fixed axis when subjected to a specified external torque. 3. Determine the radial and tangential acceleration of a point on a rigid object. 4. Apply conservation of energy to problems of fixed-axis rotation. TJ objectives far exceed that of the College Board with the different geometries students are expected to find the moment of inertia of. Students must calculate the moment of inertia of solid, 3 dimensional objects, with varying densities. Students can prove the parallel axis theorem. Students make use of the vector cross product to calculate the direction of the angular velocity vector. Bouncing Ball Video Analysis Lab Page 27
5. Analyze problems involving string and massive pulleys. 4. Angular momentum and its conservation (a) Students should be able to use the vector product and the right-hand rule, so they can: 1. Calculate the torque of a specified force about an arbitrary origin. 2. Calculate the angular momentum vector for a moving particle. 3. Calculate the angular momentum vector for a rotating rigid object in simple cases where this vector lies parallel to the angular velocity vector. (b) Students should understand angular momentum conservation, so they can: 1. Recognize the conditions under which the law of conservation is applicable and relate this law to one- and two- particle systems such as satellite orbits. TJ objectives far exceed the College Board with the application of differential equations involving torque and other angular quantities. TJ students write and solve differential equations that relate torque to angular momentum and angular acceleration to determine angular velocity and position. TJ objectives far exceed the College Board in the area of the calculation and understanding of angular momentum in the context of tops and precession. TJ students can calculate and draw angular momentum vectors on a top during precession and calculate the precession rate. Real Atwood Machine Demo Page 28
2. State the relations between net external torque and angular momentum, and identify situations in which angular momentum is conserved. 3. Analyze problems in which the moment of inertia of an object is changed as it rotates freely about a fixed axis. 4. Analyze a collision between a moving particle and a rigid object that can rotate about a fixed axis or about its center of mass. (F) Oscillations and Gravitation 1. Simple harmonic motion [SHM] (dynamics and energy relationships) Students should understand simple harmonic motion, so they can: (a) Sketch or identify a graph of displacement as a function of time, and determine from such a graph the amplitude, period and frequency of the motion. (b) Write down an appropriate expression for displacement of the form sin or cos to describe motion. Students can analyze a collision (elastic and inelastic) between a moving particle and a rigid body that results in both rotational and translational momentum. (c) Find an expression for velocity as a function of time. (d) State the relations between acceleration, velocity, and displacement, and identify point in the motion where these quantities are zero or achieve their greatest positive and negative values. (e) State and apply the relation between frequency and period. Page 29
(f) Recognize that a system that obeys a differential equation of the form must execute simple harmonic motion, and determine the frequency and period of such motion. (g) State how the total energy of an oscillating system depends on the amplitude of the motion, sketch, or identify a graph of kinetic or potential energy as a function of time, and identify points in the motion where this energy is all potential or all kinetic. (h) Calculate the kinetic and potential energies of an oscillating system as functions of time, sketch or identify graphs of these functions, and prove that the sum of kinetic and potential energy is constant. (i) Calculate the maximum displacement or velocity of a particle that moves in simple harmonic motion with specified initial position and velocity. (j) Develop a qualitative understanding of resonance so they can identify situation in which a system will resonate in response to a sinusoidal external force. 2. Mass on a spring Students should e able to apply their knowledge of simple harmonic motion to the case of a mass on a spring so they can: (a) Derive the expression for the period of oscillation of a mass on a spring. (b) Apply the expression for the period of oscillation of a mass on a spring. (c) Analyze problems in which a mass hangs from a spring and oscillates vertically. (d) Analyze problems in which a mass attached to a spring oscillates horizontally. Mass on a Spring / Pendulum Labs Page 30
(e) Determine the period of oscillation for systems involving series or parallel combination of identical springs, or springs of differing lengths. 3. Pendulum and other oscillations students should be able to apply their knowledge of simple harmonic motion to the case of pendulum so they can: (a) Derive the expression for the period of a simple pendulum. (b) Apply the expression for the period of a simple pendulum. (c) State what approximation must be made in deriving the period. (d) Analyze the motion of a torsional pendulum or physical pendulum in order to determine the period of small oscillations. 4. Newton s law of gravity Students should know Newton s Law of Universal Gravitation, so they can: (a) Determine the force that one spherically symmetrical mass exerts on another. (b) Determine the strength of the gravitational field at a specified point outside a spherically symmetrical mass. TJ objectives far exceed the College Board requirements as TJ students must calculate the force that other (nonspherically symmetrical) objects exert on each other. Objects often have non-uniform density distributions. TJ objectives far exceed the College Board requirements as TJ students must calculate the gravitational field around other (non spherically symmetrical). Objects often have nonuniform density distributions. Students apply the general methodology of Maclaurin series expansions. Cavendish Video Lab Page 31
(c) Describe the gravitation force inside and outside a uniform sphere, and calculate how the field at the surface depends on the radius and density of the sphere. 5. Orbits of planets and satellites Students should understand the motion of an object in orbit under the influence of gravitational forces, so they can: (a) For a circular orbit: 1. Recognize that the motion does not depend on the object s mass; describe qualitatively how the velocity, period of revolution, and centripetal acceleration depend upon the radius of the orbit; and derive expressions for the velocity and period of revolution in such an orbit. 2. Derive Kepler s Third Law for this case of circular orbits. 3. Derive and apply the relations among kinetic energy, potential energy, and total energy for such an orbit. (b) For a general orbit: 1. State Kepler s three laws of planetary motion and use them to describe in qualitative terms the motion of an object in an elliptical orbit. 2. Apply conservation of angular momentum to determine the velocity and radial distance at any point in the orbit. Students have an in-depth understanding of and can apply the Shell Theorem to both uniform and non-uniform spheres to calculate gravitation field/force inside or outside the sphere. Students recognize that the motion does not depend on the object s mass; describe quantitatively how the velocity, period of revolution, and centripetal acceleration depend upon the radius of the orbit; and derive expressions for the velocity and period of revolution in such an orbit. Students derive Kepler s second law and use all Laws to quantitatively describe the motion of objects in elliptical orbits. Page 32
3. Apply angular momentum conservation and energy conservation to relate the speeds of an object at the two extremes of an elliptical orbit. 4. Apply energy conservation in analyzing the motion of an object that is projected straight up from a planet s surface or that is projected directly towards the planet from far above the surface. Page 33
Electricity and Magnetism Course Semester 2 Comparison of College Board Requirements to Thomas Jefferson Program of Study A. Electrostatics College Board Requirements Thomas Jefferson Program Notes a) Students should understand the concept of electric charge, so they can: (1) Describe the types of charge and the attraction and repulsion of charges. Lab & demonstrations (2) Describe polarization and induced charges. b) Students should understand Coulomb s Law and the principle of superposition, so they can: Polarity of water molecule demonstration The principle of superposition is applied throughout the course (1) Calculate the magnitude and direction of the force on a positive or negative charge due to other specified point charges. (2) Analyze the motion of a particle of specified charge and mass under the influence of an electrostatic force. 2. Electric field and electric potential (including point charges) a) Students should understand the concept of electric field, so they can: Coulomb lab determine the charge on repelling metal spheres. (1) Define it in terms of the force on a test charge. (2) Describe and calculate the electric field of a Page 34