Curriculum Map. Discipline: Science Course: AP Physics C

Transcription

1 Curriculum Map Discipline: Science Course: AP Physics C Unit 1: c What is the role of the calculus in physics? What physical meaning can be assigned to the various time derivatives of position and their graphical representations, vector dot products, and vector cross products? How are these quantities computed? Review of differentiation and integration of polynomials and some trig functions Derivation of kinematics formulae via calculus Kinematics problems in one dimension using calculus Review of basic vector calculus as applied to space curves Derivation of dot and cross product theorems; applications (scroll down) Vector problems Given r(t), students will compute v(t), v(t), and a(t), create their graphs for 1-D motion, state the physical interpretations of increasing/decreasing functions, relative extrema, concavity, and inflection points. Students will use determinants to compute vector cross products, apply vector theorems to compute the angle to two vectors, make use of unit vector notation, and perform other vector computations readily. Unit 2: Motion detector computer lab c

2 How can the work done by a force be computed if the force is continually changing in magnitude or direction? What useful information does this work give a scientist or engineer? Why is it important to determine a frame of reference in motion problems? How do Newton's law of motion apply to real-world problems? Tangential and centripetal acceleration, unit normal and tangent vectors, parametizations and curvature of a space curves, vector calculus, integration, and differential equations involved in solving 3-D motion problems. Relative velocities and projectile motion problems with geometric complexities. Friction, Newton's laws, free body diagrams, tension, incline planes, drag problems (with and without first order, linear, separable differential equations), banked track problems, accelerometers, and terminal velocity. (scroll down) Work-energy theorem, gravitational and elastic potential energy, kinetic energy, nonuniform force fields, nonconservative forces, work done by a force that varies as a function of 3-D position coordinates that acts on a body moving along some space curve (using path integrals), power, graphical interpretations of work and force graphs, work done by forces that vary with time rather than position. Find parametric equations that describe the motion of a particle moving along a parabolic path at constant and nonconstant speeds. Set up and solve integrals to find work done by nonconstant forces. Solve static and dynamic problems using Newton's Laws and kinematics and, alternatively, using energy methods. Work/force lab using force sensor, Pasco software, and Excel: students gather force sensor data and use technology to perform numerical integration to find work done by a nonconstant force. Unit 3: (scroll down) c What is the relationship between potential energy and work done by a conservative force?

3 How can this relationship be exploited to solve work, motion, and energy problems involving gravity, springs, friction, inclines, etc. When and why is momentum conserved? gradient vectors; writing the vector components of a force as partial derivatives of a potential energy function, finding work done by a force field that varies of a coordinate plane via partial derivatives and integration; complex work-energy problems involving gravitational and elastic potential energy, friction, and inclines; work/lifting problems involving nonconstant forces; elastic/gravitational potential energy problems involving projectile motion center of mass for discrete and continuous systems; conservation of momentum in different reference frames; center of mass of a uniform pyramid; motion (scroll) of center of mass in a system with moving components; center of mass & optimization; rocket problems (conserving momentum where mass changes) elastic collisions; inelastic collisions (complete & incomplete); impulse & change in momentum; 2-D collisions (vector method & system of scalar equations) Students will compute partial derivatives of functions and find gradient vectors Students will set up and solve conservation of energy equations involving gravitational & elastic potential energy, work done by friction, and kinetic energy. Students will set up and compute integrals to find the set of mass of various uniform bodies. (scroll) Students will set up and solve systems of equations in elastic and inelastic collision problems. Students will use Interactive Physics software to simulate collisions. Interactive Physics computer simulation Unit 4: Collision lab What is rotational inertia? How do the concepts of rotational inertia, torque, and angular momentum help in solving motion problems?

4 Under what circumstances are angular momentum and mechanical energy conserved? Rotational mechanics: kinematics, kinetic energy, inertia, momentum, and the rotational analogs to Newton's 2nd law of motion & the work-energy theorem; Atwood pulley problems with pulleys with non-negligible mass; rolling with and without slipping. Torque & angular momentum as vector cross products; conservation of angular momentum and mechanical energy problems; static equilibrium; rotational dynamics. Students will calculate complex moments of inertia using integration and the parallel axis theorem. Students will explain the precession of a gyroscope in terms of torque, angular momentum, and vector cross products. Students will write net force and net torque equations for problems in dynamics and static equilibrium. Interactive Physics computer simulations (scroll) Unit 5: How is the center of gravity related to the center of mass? What are Newton's shell theorems and how are they useful? What is the relationship between simple pendulums, simple harmonic motion, and differential equations? Center of gravity vs. center of mass; stress/strain (tension/compression, shearing, hydrolic). Universal gravitation; Newton's shell theorems; gravitional potential energy; escape speed; Kepler's laws of planetary motion; orbital mechanics. Oscillations; simple harmonic motion (simple pendulums, torsions pendulums, masses on springs, physical pendulums; 2nd order, linear, homogeneous differential equations with constant coefficients. Students will set up and solve integrals to find the net graviational force between two non-point-like objects. Students will calculate the elevation necessary for a satellite to achieve geosynchronous orbit. Students will write equations of motion for a periodic system in the form of (scroll)

5 of differential equations and then solve them and find their periods. Interactive Physics computer simulations Unit 6: What is an electric field and how is it useful in solving problems? What do some common electric fields look like? How does changing the magnitude of a charge, or the configuration of a group of charges, affect the associated magnetic field? Intro to electricity and magnetism: charge; elementary charge; conductors vs. insulators; SI unit-- Coulomb; permittivity constant/coulomb's constant Electric fields: definition; field lines around various charge configurations; dipole moments; vector calculus electric field problems; simple harmonic motion and potential energy of a dipole in a uniform electric field (derivation). Intro to Gauss' law Students will calculate the magnitudes and directions of electric field vectors at a given point due to a uniform charge distribution with geometric symmetry. Students will articulate the similarities and differences between gravitational and electric fields. Given a charge configuration, students will choose, based on the geometry of the situation, the appropriate Gaussian surface. Demo: charging via friction, conduction, and induction Unit 7:

6 What is Guass' law and how is it useful in physics? How does the concept of electric potential related to gravitational potential? What is the relationship between electric field and potential? Guass' Law: equivalence to Coulomb's law; electric flux; integrating a dot product over a surface; properties of charged conductors and insulators; lightning; shell theorems for electric charge. Electric Potential: definition and SI units; comparison to gravitational potential; equipotential surfaces; integrating dot product along a path; path independence; finding potentials for discrete systems; using calculus to set up and solve integrals to find the potentials for continuous systems; induced dipole moments. (scroll) Capacitance: definition and units; parallel plate, spherical, and cylindrical capacitors; capacitors in parallel and in series; complex capacitor circuit configurations; potential energy and energy density derivations; dielectrics. Students will set up and solve integrals to compute electrics field strengths and potentials for a variety of problems, with and without Guass' law. Students will explain the process of a lightning strike, how the shell theorems apply in different situations, and the reasoning behind defining the potential of a charge to be zero at infinity. (scroll) Students will simplify complex capacitor circuit problems, find the equivalent capacitance, and find the charge stored on each capacitor. Interactive Physics computer simulations April: How are Kirchoffs laws useful in solving circuit problems? How can complex circuit diagrams be simplified, and how can one find current and voltage drops throughout the circuit? What is an RC circuit and what mathematical techniques can be employed to solve RC circuit problems?

7 Partial derivatives, gradients, writing potential as the gradient of the electric field; potential energy for a system of charges; change in potential along and through a conductor. Current density, electron drift speed, resistivity and conductivity, definition of resistance vs. Ohm's law, electrical power, real and ideal emf devices, Kirchoff's laws, resistors in series and parallel, complex resistor circuits with multiple batteries (accompanied with a quick review a linear algebra techniques), ammeters and voltmeters, RC circuit differential equations and solutions. Given an electric field in 3 dimensions, students use partial derivatives to form a gradient vector representing electric potential as a function of position. Students will set up and solve linear systems using Kirchoff's laws for complex circuits. Students will set up and solve differential equations for RC (resistor-capacitor) circuits with and without and emf source. Presentation of Engineering Open House observations Unit 9: Interactive Physics computer simulations, magnetism lab What is the difference between an electric and a magnetic field and how are they related? In what way are charges affected by magnetic fields? How can one use Ampere's law to calculate magnetic field strengths in various circuit configurations? Relationship between magnetic force, charge, speed, and magnetic field; properties of magnetism and magnetic fields; Hall effect; circular and helical motion of a charged particles in magnetic fields; auroras; cyclotrons & synchrotrons; mag. force on a current carrying wire; torque on a current-carrying wire loop in a mag. field. Magnetic fields produced by current-carrying wires; Biot-Savart law; integral derivation for mag field due to infinitely long wire; forces between parallel and antiparallel current-carrying wires; Ampere's law (along with review of line intergrals and comparison with Gauss's law); mag fields inside a wire; solenoids and toroids. (scroll) Electric motors and generators; magnetic flux, Faraday's law of induction, and Lenz's law; induced electric fields; inductance and inductors; RL circuits (with associated differential equations and solutions)

8 Students will set up and solve differential equations for RL (resistor-inductor) circuits with and without and emf source. Students will describe the magnetic fields associated with current-carrying wires in various geometric configurations and explain their descriptions in terms of Ampere's law. Students will set up and solve integrals involving the Biot-Savart law. nteractive Physics computer simulations & AP practice tests Unit 10: What are mathematical commonalities between mechanical and electrical oscillators? How are RLC circuits mathematically identical to damped harmonic oscillators? How does AC differ from DC and what are the advantages and disadvantages of each? How are phasor diagrams helpful in analyzing AC circuit problems? Damped electrical and mechanical oscillators and associated differential equations (derived from an energy perspective and via Kirchoff's laws), RLC circuits, alternating current, phasors, reactance, impedance, resonance, Vrms, AC power, transformers. AP prep Various topics of students interest such as relativity, quantum mechanics, etc. Students will present to the rest of the class what they learned about civil engineering (after meeting with Urbana civil engineers). Students will draw AC circuit diagrams and phasor diagrams, find inductance and reactive capacitances, calculate impedance, phase angle, and power output. Interactive Physics computer simulations AP practice tests