Physics 410

Advanced Mechanics

 

Fall Term, 2006

1:50 – 3:00 MWF,  Youngchild Hall Room 115

Professor Matthew Stoneking

 

Catalog course description: Treats various topics selected from: mechanics of rigid bodies, Lagrangian and Hamiltonian formulations, variational principles, fluids, classical scattering, relativistic mechanics, and theory of small vibrations.

 

Text: Classical Dynamics of Particles and Systems, 5th Edition, by Stephen T. Thornton and Jerry B. Marion, (Thomson Brooks/Cole Publishing, 2004).

 

Other mechanics texts on reserve in the library:

Mechanics, by Keith R. Symon

Classical Mechanics, by Herbert Goldstein

An Introduction to Mechanics, by Daniel Kleppner and Robert J. Kolenkow

Classical Mechanics, a Modern Perspective, by Vernon Barger and Martin Olsson

Classical Mechanics, by A. Douglas Davis (not on reserve)

 

Office hours: Wednesday 9:30 – 11:00 AM, Thursday 1:30 – 3:00 PM. Appointments can also be made to meet with instructor.

 

Grading policy:

Grades will be assigned based on the following elements, weighted as indicated:

·        Midterm Exam: 20%

·        Final Exam: 25%

·        Problem sets: 30%

·        Problem Presentations: 15%

·        Attendance / Participation: 10%

 

Exams:

The midterm exam will be a take home exam.  The final exam will be cumulative and will take place during the regularly scheduled final exam time (Friday, 8 December, 1:30 PM).

 

Problem sets:

Problem sets will be assigned almost every week (probably ~8 total), and will usually be due in class on Friday of each week.  Students are encouraged to collaborate on problem solutions, but each student must submit his own write-up of solutions and each student must participate in the solution of each problem.

 

Problem Set #1 (due: Fri. 9/29): T&M, Chap. 7, problems 3, 6, 7, 9, 10, 14, 15, 21

Problem Set #2 (due: Mon. 10/9): T&M, Chap. 7, problems 11, 20*, 23, 24, 28, 30*, 31*, 37*

Problem Set #3 (due: Wed. 10/18): T&M, Chap. 10, problems 8*, 9*, 15*, 19, 22*

Problem Set #4 (due: Fri. 11/3): T&M, Chap. 12, problems 1*, 2, 7*, 8*, 11, 13*, 17, 26, 28

Problem Set #5 (due: Mon. 11/13): T&M, Chap. 13, problems 8, 14, 17, 18, 19, 22

Problem Set #6 (due: Fri. 12/1): David, Chap. 12, problems 10, 12*, 13*, 16*, 18*, 19

*problems presented in class

 

Problem Presentations:

Each student will make three problem solution presentations in class.  Grades will be based on the effectiveness of the presentation style as well as the quality and completeness of the solution.  Presentations should therefore be rehearsed and solutions should be checked by the instructor prior to class.

 

Outline of Topics Covered:

I.                                Lagrangian and Hamiltonian Dynamics (T&M chapters 6 and 7)

II.                             Motion in Noninertial Reference Frames (M&T chapter 10)

III.                           Dynamics of Rigid Bodies (M&T chapter 11)

IV.                          Coupled Oscillations (M&T chapter 12)

V.                             Vibrating Systems and the Wave Equation (M&T chapters 13)

VI.                          Fluid Mechanics (supplemental reading)

VII.                        Special Relativity (M&T chapter 14)

 

Schedule: The following is a tentative schedule of topics to be covered in this course.

1. Wed. 20 September

Review of Newtonian mechanics and derivation of Lagrange’s Equations from Newtons’ 2nd Law (see sections 7.6, 7.7)

2. Fri. 22 September

Example problems using Lagrange’s Equations, Hamilton’s Principle (of Least Action)

Read T&M Chapter 6 and Chapter 7 (sections 7.1 – 7.4)

3 Mon. 25 September

Generalized momenta, ignorable coordinates and constants of the motion,

Velocity-dependent potential energy and the Lagrangian for a charged particle in an electromagnetic field

Read T&M, Chapter 7 (sections 7.6 – 7.9)

4. Wed. 27 September

Lagrange Undetermined Multipliers and Constraint Forces

Read T&M Chapter 7 (section 7.5 )

5. Fri. 29 September

Hamiltonian dynamics and the Virial Theorem

Read T&M Chapter 7 (sections 7.10 - 7.13)

Problem set #1 due

6. Mon. 2 October

Noninertial reference frames and fictitious forces

Read T&M Chapter 10 (sections 10.1 – 10.3)

7. Wed. 4 October

Problems involving centrifugal and Coriolis forces

Read T&M Chapter 10 (section 10.4)

8. Fri. 6 October

Problem session on Lagrangian and Hamiltonian dynamics

9. Mon. 9 October

Coriolis force and the Larmor Theorem in electrodynamics

Rigid body rotational dynamics: angular momentum and the inertia tensor

Read T&M Chapter 11 (sections 11.1 – 11.4)

Problem Set #2 due

10. Wed. 11 October

Rotational kinetic energy, principal axes, the parallel-axis theorem, and Euler’s equations

Read T&M Chapter 11 (sections 11.5 – 11.7, 11.9)

11. Fri. 13 October

Stability of rotations about principal axes (the tennis racket theorem)

Read T&M Chapter 11 (section11.12)

12. Mon. 16 October

The Force-free (torque-free) symmetric top, Euler angles

Read T&M Chapter 11 (sections11.8, 11.10)

13. Wed. 18 October

Problem session on noninertial reference frames

Problem Set #3 due

14. Fri. 20 October

Euler angles and the symmetric top with one point fixed

Read T&M Chapter 11 (section 11.8, 11.11)

Midterm exam handed out

15. Mon. 23 October

Two coupled harmonic oscillators, normal modes

Read T&M Chapter 12 (sections 12.1 – 12.3, 12.6 – 12.8)

16. Wed. 25 October

The Loaded String

Read T&M Chapter 12 (section 12.9)

 

Fri. 27 October Midterm Reading Period … no class

Midterm Exam Due

 

17. Mon. 30 October

This class session will be rescheduled because the instructor will be out of town at a physics conference.

18. Wed. 1 November

Problem session on coupled oscillations

There will be a substitute instructor for this class session because the instructor will be out of town at a physics conference.

19. Fri. 3 November

Waves on a string under tension.  The classical wave equation.  Phase velocity. Fourier analysis.

Read T&M Chapter 13 (sections 1-4, 6-8)

Problem set #4 due

20. Mon. 6 November

Wave packets and group velocity. Fluid statics

Read T&M Chapter 13 (section 9)

21. Wed. 8 November

Fluid statics, the continuity equation.

Read Davis Chapter 12 (sections 1-2)

22. Fri. 10 November

Fluid momentum equation, Bernoulli’s theorem.

Read David Chapter 12 (sections 3-4)

23. Mon. 13 November

Applications of Bernoulli’s equations, sound waves in a gas.

24. Wed. 15 November

Fourier transforming differential equations, sound waves and electromagnetic waves revisited, plasma waves.

Problem Set #5 due

25. Fri. 17 November

Water waves.  Viscosity.  The Navier-Stokes equation.

26. Mon. 20 November

Poiseuille flow (viscous flow through a cylindrical pipe).

Special relativity: the postulates and the Lorentz transformation equations, velocity addition

Read T&M Chapter 14 (sections 14.1 – 14.4)

 

Thanksgiving Break

 

27. Mon. 27 November

Invariant spacetime interval, spacetime diagrams, time dilation and length contraction, line-of-sight Doppler effect

Read T&M Chapter 14 (sections 14.5)

17. Tues. 28 November (makeup class)

Doppler effect for general orientation and relative motion of source and receiver.  Four-vectors.   Relativistic momentum and energy.

Read T&M Chapter 14 (sections 14.7 - 14.9)

28. Wed. 29 November

Problem session on fluids

29. Fri. 1 December

Understanding the Twin Paradox using spacetime diagrams.  Examples using relativistic energy and momentum conservation.

Read T&M Chapter 14 (sections 14.6, 14.10 – 14.11)

Problem set #6 due

 

FINAL EXAM: Friday 8 December 1:30 PM