By week:
|
You can find the lecture notes at this link
or at the Wiki.
Week 11 - Hamilton's Equations
Problem Set
Problem 1 - Question The First 
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
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
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
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 12:14:23
|
