PHYS 350 - 2005W [Week 11]

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You can find the lecture notes at this link or at the Wiki.

Week 11 - Hamilton's Equations [10] [12]

Summary

We end our tour of classical mechanics with yet another formulation that places the momenta and coordinates on equal footing. You can find the notes for this week and next on the Wiki.

Reading List

  • ``Lagrangian Dynamics''
    REF: Wells, D. A. 1967, Chapter 16. Top

  • ``Classical Mechanics''
    REF: Goldstein, H. 1980, Chapter 8. Top

Problem Set - 2 December 2005 at 1pm

Here are the rules. If you do better on this assignment than the midterm the scaled mark for this assignment will replace that of the midterm. If in the opinion of the instructor you have collaborated, your assignment will be shredded before it is graded. This determination will be final and without appeal (we have a crosscut shredder). You have nothing to lose by doing the assignment and nothing to gain if you collaborate.

The assignment will cover material from the entire course up to 30 November 2005. The questions will be revealed at 1pm on 30 November 2005. Although the instructor may find queries on the assignment submitted after 1:10pm on 30 November 2005 entertaining, he will not entertain them.

Problem 1 - Question The First Top

Calculate the moment of inertia of a solid, regular hexagonal prism about its symmetry axis. Express your answer in terms of the mass of the prism and the length of each side.

Problem 2 - Question The Second Top

A uniform solid cylindrical drum of mass M and radius a is free to rotate about its axis, which is horizontal. A cable of negligible mass and equilibrium length l0 is wound on the drum, and carries on its free end a mass m. Write down the Lagrangian function in terms of appropriate generalized coordinates. You may assume that the cable does not slip on the drum and that the cable is elastic with a potential energy 1/2 k x2. Find and solve the equations of motion assuming that the mass is released from rest with the cable unextended.

Problem 3 - Question The Third Top

Find the normal modes of oscillation of two pendulums of different masses M and m but the same length l. Both pendulums are attached to the same horizontal bar; the points of attachment are separated by a distance s0. The pendulums are connected by a spring with spring constant k. When both masses are at the bottom of their arcs, the spring has its equilibrium length.

Problem 4 - Question The Fourth Top

Write down the kinetic energy of a particle in cylindrical polar coordinates in a frame rotating with angular velocity ω about the ''z''-axis. Show the the terms proportional to ω and ω2 reproduce the Coriolis force and the centrifugal force respectively.
Last modified: Wednesday, 30 November 2005 12:14:25