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 halfangle 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
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Full Document
 Fall '09
 ROBERTS
 Physics, mechanics, SMALL OSCILLATIONS, Fetter

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