Physics 110A: Hints on Chap 11 probs



17. Start out with the result of the mass and "spring" matrix from the development in class (or on p. 434). It cleans things up to divide the (K −ω²M) matrix by mL² and then call g/L ω0². By "find normal modes," it means establishing the relationship between amplitudes of the two angles for the two normal mode frequencies.

19. Watch out for the cross term in the kinetic energy. Best write the position vector of the pendulum bob, differentiate to get velocity, then square it to get kinetic energy. Make the small angle approximations to the Lagrangian before developing the equations of motion.

28. Try writing down the kinetic energy and potential and coming straight out with the mass matrix and "spring" matrix directly, as laid out in Section 11.5. The K matrix is just second derivatives of U. For the mass matrix, don't forget the ½ already taken account of. Also, off-diagonal elements contribute to the sum twice, so don't double them on account of the ½.

32. See comments for #28. Despite being a three-star near the end of the chapter, this one is not really harder than the others you've done, aside from producing 3×3 matrices.


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