Ps10 - connecting them are constrained to move in a circle so that each atom has just one(angular degree of freedom Show that the 0-frequency mode

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PHY 504 Problem Set #10 due 10 November 2009 1. The Lennard-Jones potential models the interaction between a pair of neutral atoms: V ( r ) = A r 12 B r 6 where the first term represents a van der Waals attraction at large distances and the second term approximates the hard-core Fermi repulsion. a) For a diatomic molecule express the frequency of oscillations about equilibrium in terms of A , B , and the masses of the two atoms. b) Consider a triatomic molecule consisting of 3 identical atoms bound by Lennard-Jones two-body potentials. Argue that an arrangement with the 3 atoms at the corners of an equilateral triangle will always be an equilibrium configuration. c) Obtain the normal mode frequencies for oscillations about equilibrium. To simplify the problem, you may assume that the atoms and the springs
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Unformatted text preview: connecting them are constrained to move in a circle, so that each atom has just one (angular) degree of freedom. Show that the 0-frequency mode is a rotation about the center of mass. 2. A double pendulum (Fig. 1.4), consisting of masses m 1 and m 2 at the ends of rods of lengths l 1 and l 2 which are hinged at the location of m 1 , swings in the xz-plane. a) Write down the full Lagrangian in terms of the coordinates in Fig. 1.4, and the Lagrangian for small displacements about equilibrium. b) Find a transformation of coordinates, as done in class, which diagonalizes both the kinetic and potential energy terms. c) Describe the normal modes of vibration and the general solution for θ 1 and 2 , in terms of the initial angular displacements and velocities....
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This note was uploaded on 11/16/2010 for the course PHY 504 taught by Professor Staff during the Fall '08 term at Kentucky.

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