Homework #9 — PHYS 601 — Fall 2014
Due on Friday, November 21, 2014 online
Professor Victor Yakovenko
Office: 2115 Physics
Web page: http://physics.umd.edu/~yakovenk/teaching/
GPS: Goldstein, Poole, Safko, Classical Mechanics, 3rd edition, 2002, ISBN 0-201-65702-3
LL: Landau and Lifshitz, Mechanics, 3rd edition, 1976, ISBN 978-0-7506-2896-9
Total score is 36 points.
Small Oscillations, LL Ch. 5, GPS Ch. 6
1. Foucault Pendulum, 12 points. Foucault pendulum is modeled as an object of
mass m moving in the horizontal (x, y) plane subject to a potential V = γ(x2 + y 2 )/2.
Use the non-inertial reference frame attached to the rotating Earth at a geographic
location characterized by the polar angle θ measured from the North Pole. The body
experiences the Coriolis force 2mv × Ωz , where v = (ẋ, ẏ) and Ωz = Ω cos θ is the
vertical projection of the angular velocity Ω = 2π/T of the Earth with T = 24 hours.
(a) Write two coupled equations of motion for (x, y) and find the eigenfrequencies
and normal modes of the system. Sketch and physically interpret the motion in
the normal modes.
(b) Suppose the object initially oscillates along the x direction (1, 0). Calculate subsequent motion of the object and find precession frequency of the line of oscillations
in the (x, y) plane.
(c) Calculate the total angle of precession in 24 hours. Verify your answer in the
limiting cases θ = 0 (North Pole), θ = π (South Pole), and θ = π/2 (the Equator).
2. Problem I.1 from the physics qualifier exam in January 2013, 12 points.
Consider a system consisting of three particles of masses m1 = m, m2 = M , and
m3 = m. At the start, m1 moves with velocity v0 , collides with the spring extending
on the left side of m2 , and sticks to it, thus creating a system of three masses connected
by two massless springs of force constant γ and equal lengths.
m1 = m
γ
v0
-
A A A A
A A A A
γ
m2 = M
A A A A
A A A A
m3 = m
All questions below refer to the system after the collision.
(a) Write down the Lagrangian of the system and the Lagrange equations of motion.
(b) Determine the eigenfrequencies and eigenvectors (normal modes) of the system.
Discuss symmetry of the normal modes.
2
Homework #9, Phys601, Fall 2014, Prof. Yakovenko
(c) What is the center-of-mass velocity of the system?
What is the maximal displacement of the mass m2 away from the center of mass
during the subsequent motion of the system after the collision?
(d) Calculate the energies allocated to each of the three normal modes after the
collision. Verify that their sum is equal to the initial kinetic energy.
3. Problems 6.8 and 6.9, 12 points. Triangular molecule with the right angle. Answer
questions from the textbook and additional questions.
Additional question: Discuss symmetries (parities) of the normal modes of vibration
and sketch their patterns of displacement.
Directions: You can find normal modes of vibration from a 6 × 6 matrix by eliminating
zero modes. Alternatively, you can introduce degrees of freedom that eliminate solid
rotation and center-of-mass displacement from the beginning and work with a 3 × 3
matrix. See discussion and solved problems in Ch. 5.24 of LL.
November 16, 2014