1. 1 Test #2 of 4 Thurs. 10/17/02 – in class Bring an index card 3”x5”. Use both sides. Write anything you want that will help. You may bring your last index card as well. Lagrangian method (most of exam) Line integrals / curls Generalized forces / Lagrange multipliers / constraint forces Tuesday 10/15 – Real-time review / problem session :08

2. 2 Tides :05 Why are there bulges on BOTH sides of the earth?

3. 3 Tides :55 Why are there bulges on BOTH sides of the earth?

4. 4 Class #22 of 30 Review of rotating reference frames Coriolis force Non-axi-symmetric rotations The inertia tensor Torques for off-axis rotations Diagonalization and eigenvalues Principal axes :02

5. 5 Vector velocity – last time :60 Latitudes – Socorro 34N – (.38 km/s=880 mph) Nome 64N – (.20 km/s=473 mph) Hilo 19N – (.44 km/s=1021 mph) Hobart 43S – (.34 km/s=790 mph) Velocities are all relative to INERTIAL frame.

6. 6 :60 Derive Newton in rotating frame

7. 7 Newton in a rotating frame :60 Coriolis Term Centrifugal Term Newtonian Term

8. 8 :60 Coriolis force for a rocket going up

9. 9 Problem L22-1 :30 A 100 kg weather rocket is launched straight up from Langmuir Lab. At t=10 sec, it’s velocity is 100 m/s. At t=100 sec, it’s velocity is 1000 m/s. What is the magnitude and direction of the Coriolis force at t=10 and 100 sec? An airplane flies due North from Albuquerque. Because of Coriolis force, the airplane will be deflected from its true North path A) Which direction will the deflection be in? B) How many miles off course will the airplane be after one hour? Assume v=300 m/s.

10. 10 You thought you knew all about angular momentum…  P=mv … what could be more familiar?  Let’s just relax  Cross product  …and again :02

11. 11 Inertia Tensor  Inertia tensor -- single particle :02

12. 12 Charles “Chuck” Hermite (1822-1901) :02 French mathematician who did brilliant work in many branches of mathematics, but was plagued by poor performance in exams as a student. However, on his own, he mastered Lagrange's memoir on the solution of numerical equations and Gauss's Disquisitiones Arithmeticae. He was forced to leave Ecole Polytechnique after one year when it was decided that his congenitally deformed right leg would not allow him to take a commission in the military, making him not worth their time.Lagrange'sGauss's In 1869, he became a professor at École Normale, and in 1870 at Sorbonne. All during his career, was generous in his help of young mathematicians. He showed that e was a transcendental number Hermite also discovered some of the properties of Hermitian matrices. etranscendental number Hermitian matrices

13. 13 Inertia Tensor is Hermitian Inertia tensor is symmetric Inertia tensor is Hermitian :02 Hermitian matrices have real eigenvalues If the eigenvalues differ, the eigenvectors are orthogonal

14. 14 Inertia Tensor :02  Derived all this for a single particle Below, for multiple particles

15. 15 Angular momentum of non-axi-symmetric object  Inclined bar-bell Bar length 2L :02 x y z

16. 16 Non-axisym. object II :60  Let

17. 17 Non-axisym. object III :60

18. 18 But that’s not all!!  Principal Axes  The ones about which rotations do not produce torques  These are the eigenvectors of the inertia tensor  How do we find them? :02 x y z

19. 19 Diagonalization :60  Set phi=0  If is an eigenvector

20. 20 Diagonalization :60

21. 21 Problem L22-2 :30 x y z Given all the above, what are the principal axes (the eigenvectors) for this problem?

22. 22 Class #22 Windup Tuesday bring a tennis racket or Halliday & Resnick and a rubber band. Exam 11/19 Central force / rotating reference frames Office hours Th/Fri 11/7-8. HW 11 due 11/12 in class. HW 12 due 11/14 as usual. :60