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# hw6 - Florida State University Department of Physics PHY...

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Unformatted text preview: Florida State University Department of Physics PHY 5246 Assignment # 6 (Due Friday, 20th November, 2009) (1) (15 points) Fetter and Walecka 4.9. If you show that the secular equation has three degenerate modes at ω 2 = 0 using Mathematica, Maple, or similar software, it will be worth 3 points. If you do it by rearranging the rows or columns of the matrix, it will be worth 8 points. (2) (15 points) Fetter and Walecka 4.3. (3) (10 points) A particle slides on the inside surface of a frictionless cone. The cone is ﬁxed with its tip on the ground and its axis vertical. the half-angle at the tip is α (see the ﬁgure). Let r be the distance from the particle to the axis, and let θ be the angle around the cone. Find the equations of motion. r0 α If the particle moves in a circle of radius r0 , what is the frequency, ω of this motion? If the particle is then perturbed slightly from this circular motion, what is the frequency, Ω, of the oscillations about the radius r0 ? Under what conditions does Ω = ω ? (4) (a) (8 points) A mass m is ﬁxed to a point on the rim of a wheel of radius R that rolls without slipping on the horizontal ground. The wheel is massless, except for a mass M located at its center. Find the equation of motion for the angle through which the wheel rolls. For the case where the wheel undergoes small oscillations, ﬁnd the frequency. (b) (12 points) The mass m is now free to slide without friction along the rim of the wheel. Choose two suitable generalized coordinates, construct the Lagrangian, and obtain the equqations of motion. Find the normal mode frequencies and normalized normal modes for small oscillations. (5) (10 points) A mass M is free to slide along a frictionless rail. A pendulum of length l amd mass m hangs from M (see the ﬁgure). Find the equations of motion. For small oscillations, ﬁnd the normal modes and their frequencies. 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 M 111111 000000 111111 000000 1111111 0000000 111111 000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 l 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 m ...
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