PH 2202, Intermediate Mechanics II

This course is a continuation of the treatment of mechanics started in PH 2201. Topics covered include: rigid-body dynamics, rotating coordinate systems, driven harmonic oscillator and normal modes, and the Lagrangian and Hamiltonian formulations of mechanics. Recommended background:  PH 1110/11, PH 1120/21, PH 1130, PH 1140, PH 2201, MA 1021, MA 1022, MA 1023, MA 1024, MA 2051 and concurrent registration in or completion of MA 2071. (The more important courses are in bold.)  In contrast to PH 2201, this material  is more theoretically oriented and much of it will be new to you. It requires a higher level of mathematical background.

The text is "Classical Mechanics", by J.R. Taylor, available in the bookstore. The physics library, OH 118, has a copy on reserve. A good supplementary text, with a lot of solved problems, is "Introduction to Classical Mechanics" by D. Morin. We meet in OH 223 on MTWThF at 2:00 pm. Instructor: Professor NA Burnham,

Course objectives

Final grade determination

The twelve homework assignments. No late work accepted. [200 pts]
The two exams. 10% penalty for make-ups. [200 pts]

Your class attendance is expected, although not required. If your numerical grade lies on the border line between two letter grades, then your class participation will determine which letter grade you shall receive. For your final grade in the course, nominally A > 320/400, B > 280/400, and C > 240/400. Although the borders between letter grades could change slightly depending on the difficulty of the exams, note that there is an allowance of forty free points, that is, you may miss approximately two homework assignments without penalty. After Thanksgiving, I'll give you an indication of how you are doing. Because I do not curve, there is no harm in helping each other...if you all earn As, you will all get an A. Similarly, if you all earn fewer than half of the points in the course, you will certainly all receive NRs. Please respect my decision not to discuss grades by email.


I will make at least one problem per exam similar to a problem from the supplementary problem sets. The individual exams are closed book, closed notes, although you may bring a formula sheet, limited to one side of a Letter page. There is a 10% penalty on make-ups. Each exam is worth 25% of your final grade. No calculators or other electronic devices are allowed.

Grading guidelines

I will use the following scheme for grading. You'll notice a big emphasis on effective communication, an aspect of your education that corporations examine during the hiring process.
Points For each problem (out of five possible points):
-5 No symbolic solutions
Symbolic solution has wrong dimensions
-1 to -5  Write-up hard to read or understand
No commentary
-1 Vectors confused with scalars or vice versa
Missing or incorrect units on numerical answers
-1 No boxes around symbolic and numerical answers
  In general, for any given problem:
 5 =  Very good -- write-up clear and correct
 4 = Good -- write-up clear and mostly correct, or understandable and correct
 3 = Acceptable -- write-up understandable and mostly correct, or poor write-up and correct, or clear write-up and incorrect

For the summaries (out of five possible points):
Equations in summary
-1 to -5
Symbols in summary
-1 to -5 Not enough detail
Shorter or longer than specified length, in increments of 50 words
No word count
  And for an entire assignment:
-5 Electronic submission
-1 No name
-1 Ragged edges
-1 No staple

Approach to assignments

I want to give you the opportunity to examine the material from a variety of perspectives. Thus, in addition to the traditional lectures and homework, you will also be reading and writing about mechanics for reading summaries. The reading summaries will help you understand the fundamental principles of the material. The supplementary and homework problems will help you with your problem-solving abilities, and the emphasis on clear problem solutions is good preparation for your career.

Reading summaries

Reading summaries provide you with a chance to preview upcoming material. Because the quality of hand writing varies so much, please use a word processor to typeset them. The reading summaries should be 300 to 600 words in length, with the word count listed at the end. If your first draft is too long, keep a copy for yourself to study from, and submit a revised version. Use no equations or symbols. Here is an example summary of Taylor Sections 1.1 to 1.5.
Mechanics is the study of motion. Its history dates from Galileo and Newton and was developed further by Lagrange and Hamilton. About a hundred years ago, scientists started to develop relativistic mechanics and quantum mechanics, and to distinguish among them, the traditional form of mechanics has become known as "Classical Mechanics."

Newton's famed three laws of motion are based upon ideas that concern space, time, mass, and force. Space can be described by coordinate systems, the simplest one being the cartesian coordinate system, which has three orthogonal axes. The position of an object can be determined by a position vector, and the next few pages of the text reviews vector algebra. After further reminders of time, reference frames, mass, and force, Newton's laws are introduced.

The First Law says that a particle moves with a constant velocity unless acted upon by an external force. The Second Law states the relationship between mass, a scalar, and acceleration, a vector. Their product equals the vector sum of all the forces acting on a particle. Since acceleration is the second derivative of position with respect to time, the Second Law is a differential equation. Newton's Laws hold in what are called inertial reference frames. They do not always hold for relativistic or quantum systems. Nonetheless, they are valid over a wide range of scales in size and speed and are thus worthy of study.

The Third Law states the relationship between "action" forces and "reaction" forces. They are equal in magnitude and opposite in direction, but act on two different bodies. From the Third Law, which considers only one particle, one can construct a theory for many particles. Internal forces in a multiparticle system have no influence on the total momentum. In other words, if the external forces are constant, then momentum is conserved. Taylor gives an example where, apparently, the Third Law is violated, but then explains that mechanical momentum is not the only form of momentum. It can, for example, be electromagnetic. However, Taylor assures us that for the rest of his text, we will consider situations only in which the Third Law holds.

353 words

Supplementary problems

The solutions to the supplementary problems are already online, giving you the opportunity to practice the material before the homework is due. The "due dates" indicated on the calendar below are my suggested deadlines for understanding the supplementary material.

Homework assignments

There are twelve homework sets of one to three problems each, plus a reading summary (ten to twenty-five possible points per assignment). If you can not or choose not to attend class, I expect to see your work in my mailbox (near the Physics Department office) at 2:00 pm the day that it is due. If you are not able to perform your homework on time, I still recommend that you do it, as exam problems will be similar. Far fewer problems are assigned than should be, due to limitations on grading time. You should work through the supplementary problems before attempting the homework. You are encouraged to collaborate on the homework problems, but you must each write up your own solutions. I will give the grader the grading guidelines above. A link at the course myWPI site takes you to the problems labeled "NAB."

Supplementary proglems
Homework problems
Supplementary problems Homework problems
SP0: P.6.1
HW0: Solve Problem 6.2; summarize Sections 6.0-6.3. The calculus of variations is our lead-in to Lagrangian mechanics. [10 pts]
SP1: P.7.1, 7.15. HW1: Solve Problems 7.3, 7.18; summarize Sections 7.0-7.4. After you learn to use Lagrangians well, you will wonder why you were taught anything else. [15 pts] SP2: P.7.31, 7.37. 
HW2: Solve Problems 7.27, 7.40; summarize Sections 7.5-7.8. Section 7.8 is our introduction to the Hamiltonian. [15 pts]
SP3: P.13.1, 13.13. HW3: Solve Problems 13.6, 13.23; summarize Sections 13.0-13.6. The Hamiltonian is commonly used in quantum mechanics, a subject that many of you will study next year. [15 pts] SP4: P.9.2, 9.3.
HW4: Solve Problems NAB1, NAB2; summarize Sections 9.0-9.5. The topic is noninertial frames, notably the Earth and the origin of the tides, and centrifugal force. [15 pts]
SP5: P.9.8, 9.15.
HW5: Solve Problems NAB3, NAB4. Summarize Sections 9.6-9.10. More on noninertial frames, including the Coriolis force and its affect on the motion of projectiles. [15 pts] SP6: P.9.16, 9.27.  
HW6: Solve Problems 9.18, 9.30; summarize Sections 10.0-10.3. How the distribution of mass within a rotating body affects its motion. [15 pts]
SP7: P.10.4, 10.11, 10.23. HW7: Solve Problems NAB5, NAB6, 10.18; summarize Sections 10.4-10.8. The precession of tops. [20 pts]
SP8: P.10.29, 10.35, 10.40. NOT 10.43. HW8: Solve Problems 10.30, 10.36, 10.47; summarize Sections 11.0-11.4. Many real-life systems can be modeled as a set of interacting springs. [20 pts]
SP9: P.11.1, 11.5. HW9: Solve Problems NAB7, NAB8, NAB9; summarize Sections 11.5-11.7. The coordinated movements of interacting springs and masses. [20 pts] SP10: P.11.15, 11.20, 11.24, 11.32. HW10: Solve Problems NAB10, NAB11, NAB12. Summarize Sections 12.0-12.4. The characteristics of and approach to chaos. [20 pts]
SP11: P.12.12, 12.13, NOT 12.27
HW11: Solve Problems 12.11, 12.16, 12.19; summarize Sections 12.5-12.8. Methods of describing chaos. [20 pts] --


The syllabus is embodied by this calendar. SP = supplementary problems (not to be turned in), HW = homework, EX = exam, e = easy, h = hard. The blue and brown shading of the text is to help you distinguish between the two halves of the course.
Week of Monday Tuesday Wednesday Thursday Friday
25. October 2009
What's due 
Today's material
Unconstrained motion
1. November  SP1
Conservation laws
Hamilton 1D
Hamilton nD
Rotating frames
8. November SP3
Submit to Nick Hoar
Submit to Nick Hoar
15. November SP5
Centrifugal force
Coriolis force
Chap. 9
22. November --
Chaps. 6,7,13,9

Chaps. 6,7,13,9
Do the rest of the reading summaries over Thanksgiving.

29. November SP7
Inertia tensor
Principal axes
Chap. 10
Normal modes
6. December
Helpful Lagrangians
The general case
Chap. 11
Approach to chaos
13. December
Chap. 10
Chaps. 11-12

Chaps. 10-12


Communication and office hours

My office is OH 219. I am sometimes in my lab, OH 009, in the Physics Library, OH 118, or in the department office, OH 119. My mailbox is between the doors of OH 118 and 119. Email (checked once daily), web, office phone with voice mail (508) 831-5365; fax (508) 831-5886; my basic weekly schedule is posted at Please put "PH 2202" in the subject line of your emails for a faster response. Drop in for quick homework hints any time I am free; otherwise for longer discussions please come to an office hour or make an appointment. The web address for this page is My research is described at and at links therein.


There is often a physics graduate student in the physics library, OH 118. He or she sits near the sign labeled "Physics Help". The principal reason for Physics Help is actually introductory physics, not our class. Still, the Helpers should be able to assist you. There should be a schedule posted on the library door. The PLA for this course is Mr Nicholas L Hoar,

If you need course adaptations or accommodations because of a disability or if you have medical information to share with me, please see me. Students with disabilities are encouraged to contact the Disability Services Office (DSO) as soon as possible to ensure that such accommodations are implemented in a timely fashion. The DSO is located in Daniels Hall, (508) 831-5235.

Academic dishonesty

Individual integrity is vital to the academic environment because education involves the search for and acquisition of knowledge and understanding, which are, in themselves, intangible. Evaluation of each student’s level of knowledge and understanding is a vital part of the teaching process, and requires tangible measures such as reports, examinations, and homework. Any act that interferes with the process of evaluation by misrepresentation of the relation between the work being evaluated (or the resulting evaluation) and the student’s actual state of knowledge is an act of academic dishonesty. The moral equivalent of academic dishonesty in larger society is treason.

In Intermediate Mechanics, you are encouraged to collaborate on the homework, although you must prepare the homework for submission yourself. You may not submit homework after the answers have been made public. You may bring to the classroom exams a formula sheet; it is limited to one side of a standard Letter-sized page. During an exam, you may have only the exam, your formula sheet, and writing implements on your desk. (No calculators, telephones, or other electronic devices.) You may not give or receive information during exams, except to ask the instructor to clarify a question.

Educational research has shown that:

  1. The most learning occurs in an environment characterized by high expectations and respect and care for individual students, and where the value of collaboration is stressed over competition.
  2. The most learning occurs in an active classroom environment where students take responsibility for learning rather than being passive receptors of the professor’s knowledge.
  3. Students can learn as effectively or more effectively from peers than from a professor.
  4. Facilitating development of students’ communication, teamwork, and interpersonal skills is as important as helping them learn physics.
  5. Professors and students are equals in the learning process. I have as much to learn about teaching and people as they have to learn about physics.
Adapted from Prof. Chrys Demetry, WPI Materials Science

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N.A. Burnham, October 2009