HW Chapter 11

HW Chapter 11 - 3. Since we are dealing with small...

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PHYS4102 HOMEWORK CHAPTER 11 Assigned Thursday, March 10; Due Thursday, March 24. Problem 1. A particle of mass m ﬁnds itself in a region of space where the potential is given by V ( x,y,z ) = V 0 a 2 (4 x 2 + 5 y 2 + 6 z 2 - 2 ax - 5 ay ) where V 0 and a are positive constants. (a) Determine the location and value of V at its minimum. (b) Evaluate the mass matrix and the potential matrix. (c) Determine the normal mode oscillation frequencies about the equilibrium point. Problem 2. Taylor, Problem 11.11 Problem 3. Three springs of equal length and spring constants are placed in a frictionless circular trough of radius a . At the junctions of the springs are masses m, m, and 2 m . Determine the normal modes and their frequencies of oscillation. Hint: Label the three masses m 1 = m , m 2 = m , m 3 = 2 m . By symmetry, at equilibrium the masses will be separated by 120 degrees of arc. Deﬁne the angle zero to be at the equilibrium position of m 1 . Then m 2 is at 2 π/ 3 and m 3 is at 4 π/
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Unformatted text preview: 3. Since we are dealing with small oscillations, the angu-lar position of m 1 is 1 , that of m 2 is at 2 / 3 + 2 and that of m 3 is at 4 / 3 + 3 . Use the i as your generalized coordinates, and solve the prob-lem using the Lagrangian approach of Taylor section 11.6 to nd the mass and spring-constant matrices. Having those in hand, you can easy nd the eigenfrequencies, perhaps making use of Mathematica to solve any unwieldy systems of linear equations you may encounter. With the eigenfrequencies in hand, the normal modes can easily found, making use of the ansatz of Equation (11.10). 6102 Problem. Use a computer to nd the normal mode frequencies of a 1-dimensional system of 10 masses connected by 11 springs, with rigid walls on both ends. All masses 0.1 kg and all springs have k = 5N/m. 1...
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This note was uploaded on 12/10/2011 for the course PHYS 4102 taught by Professor Fertig during the Spring '11 term at University of Georgia Athens.

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