Course Goals, Physics 2210, Spring 2011
Contents:
Physics 2210 is your second course in classical mechanics (following Physics 1110). We will cover roughly the first five chapters of Taylor's text (plus, perhaps, Ch 9. Phys 3210 continues with more advanced topics from Taylor), and related portions of Boas, largely Chapters 2, 4, 7 and 8.
Classical mechanics is a general framework rather than a description of a particular physical system. It describes and explains motion of objects (and groups of objects), subject to forces. It forms the basis for essentially all of modern physics - of course, Quantum Mechanics ultimately is required too, to extend the domain of applicability. But, Quantum mechanics nevertheless builds on classical mechanics. Classical mechanics lies at the heart of a huge variety of technology and natural phenomena. The mathematics we learn in this course is a natural extension of the calculus and algebra you already know, and will be used in some form in almost every formal scientific application you will likely encounter. The tools and topics we cover in this course set the stage for applications in such diverse fields as engineering, planetary and atmospheric sciences, fluid mechanics, and thermodynamics. Classical mechanics is an extraordinarily practical branch of physics - until the advent of computers it was hard to apply to most realistic systems, but today it is commonplace to calculate nonlinear phenomena, and extraordinarily complex models, building on the fundamentals we will cover this term. Classical mechanics is one of the oldest branches of physics, but it is still very much alive today, particularly in the domains of mechanical and civil engineering, fluid mechanics, statistical mechanics, and chaos theory. And, in the words of John Taylor, it provides "a wonderful opportunity" to learn a variety of mathematical techniques used throughout physics!
COURSE SCALE LEARNING GOALS
1. Math/physics connection: Students should be able to translate a physical description of a sophomore-level classical mechanics problem to a mathematical equation necessary to solve it. Students should be able to explain the physical meaning of the formal and/or mathematical formulation of and/or solution to a sophomore-level physics problem. Students should be able to achieve physical insight through the mathematics of a problem.
2.
Visualize the
problem: Students should be able to sketch the
physical parameters of a problem including sketching the physical situation and
the coordinates (e.g. , equipotential lines, a resonance curve, a pendulum with its angle as the coordinate,) as
appropriate for a particular problem.
3. Expecting and checking solution: When appropriate for a given problem, students should be able to articulate their expectations for the solution to a problem, such as direction of a force, dependence on coordinate variables, and behavior at large distances or long times. For all problems, students should be able to justify the reasonableness of a solution they have reached, by methods such as checking the symmetry of the solution, looking at limiting or special cases, relating to cases with known solutions, checking units, dimensional analysis, and/or checking the scale/order of magnitude of the answer.
4. Organized knowledge: Students should be able to articulate the big ideas from each chapter, section, and/or lecture, thus indicating that they have organized their content knowledge. They should be able to filter this knowledge to access the information that they need to apply to a particular physical problem, and make connections/links between different concepts.
5.
Communication. Students should be able to justify and
explain their thinking and/or approach to a problem or physical situation, in
either written or oral form. Students should be able to write up problem solutions that
are well-organized, clear, and easy to read.
6. Build on Earlier Material. Students should deepen their understanding of Phys 1110 material. I.e., the course should build on earlier material.
7. Problem-solving techniques: Students should continue to develop their skills in choosing and applying the problem-solving technique that is appropriate to a particular problem. This indicates that they have learned the essential features of different problem-solving techniques (eg.,solving differential equations with constant coefficients, using fourier series methods to solve PDEs with appropriate boundary conditions, etc). They should be able to apply these problem-solving approaches to novel contexts (i.e., to solve problems which do not map directly to those in the book), indicating that they understand the essential features of the technique rather than just the mechanics of its application. Students should move away from using templates. They should be able to justify their approach for solving a particular problem.
…7a. Vectors and coordinate systems: Students should be able to compute dot and cross products and solve vector equations without reference to books or external materials, and they should demonstrate comfort with these mathematical tools. Students should recognize whether variables are scalars or vectors, and vector and scalar variables should be clearly distinguishable in students’ written work. Students should be able to project a given vector into components in multiple coordinate systems, and to choose the most appropriate coordinate system in order to solve a given problem. Students should be able compute surface and volume integrals in Cartesian, cylindrical, and spherical coordinate systems (i.e., know the expressons for dV in these coordinate systems and how to apply them in a particular situation).
…7b. Approximations: Students should be able to recognize when approximations are
useful, and use them effectively (eg., recognize when air resistance is a small effect, Students should be able to indicate how
many terms of a series solution must be retained to obtain a solution of a
given order, and should be able to identify when a Taylor expansion is
appropriate and what the variable of expansion is in a given problem.
…7c. Series expansions: Students should be able
to recognize when a series expansion is appropriate to approximate a solution,
and expand a Taylor Series beyond zeroth order.
…7d. Orthogonality: Students should recognize that both vectors and functions can be
orthogonal and that any function can be built from a complete orthonormal basis. Students should be able to expand functions in an orthonormal basis (e.g. find the coefficients for a Fourier series) and interpret the
coefficients physically. Students
should be able to determine from the even or odd symmetry of a function which
terms in the expansion are zero. Students should be able to define the terms complete and orthonormal in the context of an orthonormal basis.
…7e. Differential equations: Given a physical situation, students should be able to write
down the required ordinary differential equation, identify the method of
solution, and correctly calculate the answer. Students should be able to identify
the type of differential equation (homogeneous, linear vs. nonlinear, constant
vs. variable coefficients, 1st, 2nd, or higher order, etc.) and choose the
correct method to solve that type of ODE. Students should be able to explain how the type of differential equation
helps determine which methods of solution will be applicable.
…7f. Superposition: Students should recognize that – in a linear system
– the solutions may be formed by superposition
of components.
8.
Problem-solving strategy: Students
should be able to draw upon an organized set of content knowledge (LG#3), and
apply problem-solving techniques (LG#4) to that knowledge in order to organize
and carry out long analyses of physical problems. They should be able to connect the pieces of a problem to
reach the final solution. They
should recognize that wrong turns are valuable in learning the material, be
able to recover from their mistakes, and persist in working to the solution
even though they don’t necessarily see the path to the solution when they begin
the problem. Students should be able to articulate what it is that needs
to be solved in a particular problem and know when they have solved it.
9. Intellectual maturity: Students should accept responsibility for their own learning. They should be aware of what they do and don’t understand about physical phenomena and classes of problem. This is evidenced by asking sophisticated, specific questions; being able to articulate where in a problem they experienced difficulty; and take action to move beyond that difficulty.
Important comment on preparation:
Physics 2210 is a challenging, upper-division physics course. Unlike earlier courses, you are fully responsible for your own learning. Physics 2210 covers much material you have not seen before, at a higher level of conceptual and mathematical sophistication than you may have encountered in a physics class so far.
Therefore you should expect:
YOU control the pace of the course by asking questions in class. We tend to speak quickly, and questions are important to slow down the lecture. This means that if you don’t understand something, it is your responsibility to ask questions. Attending class and the homework help sessions gives you an opportunity to ask questions. We are here to help you as much as possible, but we need your questions to know what you don’t understand.
Physics 2210 covers some of the most fundamental physics and mathematical methods in the field. Your reward for the hard work and effort will be learning important and elegant material that you will use over and over as a physics major (and beyond!) Here is what we have experienced:
How to succeed in this course: The topics in Phys 2210 are among the greatest intellectual achievements of humans. Don’t be surprised if you have to think and work hard to master this! You can perform very well in this class if you follow this time-tested system:
1. Read the text section before lecture. If you read it first, it’ll sink in faster during lecture.
2. Take detailed notes on your reading and write down questions so you can ask them in class.
3. Come to class and stay involved. Come to office hours with questions.
4. Start the homework early. Give yourself time to work and understand. No one is smart enough to do the homework in the last hour before class, and no one is smart enough to learn the material without working problems.
5. Work together. Do your own thinking, but talking to others is a great way to get unstuck.
6. Don’t get behind. It’s very hard to catch up.