Homework #2, 2013

Due on Day 5 at 5pm

Q1: Physics Journals

Read the journal club assignment. Find a second appropriate article in one of the listed journals (you should have found your first article as part of HW#1). Read the article. Write down the first author, article title, journal name, journal volume, page number and year of the article. Write a one paragraph summary of the article.

Note: This question will be repeated in HW#3 so that you become familiar with at least three different articles.

Q2: Five coupled mechanical oscillators

The boxes in the diagram below represent five oscillators of identical mass m constrained to move in the horizontal direction without twisting. (Imagine the masses are on frictionless tracks and you are looking down from the top). Gravity plays no role in this problem. The lines joining the masses represent springs. Spring constants are labeled. The equilibrium distance between neighboring blocks is a. The sides of the system are immovable walls.

Figure 1.

Find the natural frequencies of the system.

Note: This question is designed to give you practice solving a set of 5 coupled differential equations. It is also designed to show that mechanical systems with linear restoring forces are always solvable with the matrix algebra approach. The math is not as bad as it might look, however, most mortal beings will need the help of a computer to find the eigenvalues of a 5×5 matrix.

Q3: A chain of coupled pendulums

Consider an infinite periodic system of coupled pendulums. The length of each pendulum is L. The moving masses have mass m. A portion of the system is shown below. The springs between the masses are identical and have spring constant κ. At equilibrium the masses lie on the x-axis with a spacing a. Assume that motion is restricted to the x-axis.

(a) Find the dispersion relation for small oscillations of this system.

(b) Explore the dispersion relation.

Part (b) is deliberately open ended to encourage you to ask questions yourself. Such questions could include: what are the interesting features (max freq, min freq, edge of Brillioun zone)? What is different about this system compared to others you have studied? Are there limiting cases as you change κ? How quantitative can you be?