# 05_580_hw_1 - Physics 580 Quantum Mechanics I Prof P M...

This preview shows pages 1–2. Sign up to view the full content.

Handout 5 3 September 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Homework 1 Prof. P. M. Goldbart University of Illinois This homework is due to be handed in to the PHYS 580 Homework Box by 2:30 p.m. on Thursday September 9. 1) Lagrangian for a compound plane pendulum : Consider a pendulum comprising an upper segment, of length L 1 and bob mass M 1 , and a lower segment that is freely hinged to the bob of the upper segment, of length L 2 and bob mass M 2 . The segments move in a common vertical plane in the presence of gravity (acceleration g ). Let the angular displacements of the upper and lower segments from the vertical direction be, respectively, θ 1 and θ 2 . a) Construct a Lagrangian for the compound plane pendulum in terms of θ 1 and θ 2 . b) Hence, obtain the equations of motion for the compound plane pendulum. c) By linearizing about the position of mechanical equilibrium, determine the frequen- cies of the normal modes of oscillation of the compound pendulum in terms of its parameters, 2) Stephenson-Kapitza pendulum : This type of pendulum is a simple pendulum of length L and natural frequency ω , except that the point of support is forced to vibrate. In the simplest case, which we consider here, the point of support undergoes vertical harmonic oscillations of frequency Ω ω and amplitude qL . a) Construct the Lagrangian for the Stephenson-Kapitza pendulum, choosing as the co- ordinate the angular displacement θ away from the downward vertical. b) Hence, obtain the equation of motion for the Stephenson-Kapitza pendulum. What is striking about the Stephenson-Kapitza pendulum is that if the support is vibrated suﬃciently rapidly (i.e., for q > 2), the pendulum has a stable conﬁguration when it is inverted (i.e., when θ = π ), about which it can undergo stable, small oscillations at a frequency given by ω 1 + 1 2 q 2 2 . For further details, see E.I. Butikov, On the dy- namic stabilization of an inverted pendulum , Am. J. Phys. 69 (2001), 755-768; and/or http://www.fam.web.mek.dtu.dk/FVP/Jon/Theory1.html . The method of multiple-scale analysis provides a mathematically precise approach to this problem; for this and many other powerful techniques, see, e.g., C.M. Bender and S.A. Orszag,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

05_580_hw_1 - Physics 580 Quantum Mechanics I Prof P M...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online