PHY 711 -- Assignment #3

Finish reading Chapter 1 in Fetter & Walecka.

• Work Problem #1.16 at the end of Chapter 1 in Fetter and Walecka. Note that you might want to use the equation in FW #1.15 or the equivalent equation derivated in class.

PHY 711 -- Assignment #5

Read Chapter 2 in Fetter & Walecka.

• Suppose that you would like to install a Foucault Pendulum at a location of your choice. Find the latitude of your location and determine the period of the pendulum to make a complete circle of the direction of its swing.

PHY 711 -- Assignment #6

Start reading Chapter 3, especially Section 17, in Fetter & Walecka.

• Using calculus of variations, find the equation y(x) of the shortest length "curve" which passes through the points (x=0, y=0) and (x=2, y=8).

1. The figure above shows a box of mass m sliding on the frictionless surface of an inclined plane (angle θ). The inclined plane itself has a mass M and is supported on a horizontal frictionless surface. Write down the Lagrangian for this system in terms of the generalized coordinates X and s and the fixed constants of the system (θ, m, M, etc.) and solve for the equations of motion, assuming that the system is initially at rest. (Note that X and s represent components of vectors whose directions are related by the angle θ.)

PHY 711 -- Assignment #12

Continue reading Chapter 4 in Fetter & Walecka.

1. Consider the the mass and spring system described by Eq. 24.1 and Fig. 24.1 in Fetter & Walecka. Explicitly consider the case of N=6 and find the 6 coupled equations of motion. Compare the normal mode eigenvalues for this case (obtained with the help of Maple or Mathematica) with the equivalent analysis given by Eq. 24.38.

PHY 711 -- Assignment #13

Finish reading Chapter 4 in Fetter & Walecka.

1. Consider the system of 3 masses (m₁=m₂=m₃=m) shown attached by elastic forces in the right triangular configuration (with angles 45, 90, 45 deg) shown above with spring constants k and k'. Find the normal modes of small oscillations for this system. For numerical evaluation, you may assume that k=k'.

PHY 711 -- Assignment #15

Continue reading Chapter 7 in Fetter & Walecka.

Consider the example presented in Lecture 21, slide 23, where a one-dimensional Poisson equation was solved using a Green's function constructed from the corresponding homogenious solutions. Verify the results on this slide and check that the resultant potential Φ(x) satisfies the particular Poisson equation for x ≤ -a, -a ≤ x ≤ a, and for x ≥ a.

PHY 711 -- Assignment #17

Start reading Chapter 5 in Fetter & Walecka.

1. The figure above shows a rigid 3 atom molecule placed in the x-y plane as shown. Assume that the rigid bonds are massless.
   1. Find the moment of inertia tensor in the given coordinate system placed of mass M in terms of the atom masses, bond lengths d, and angle α.
   2. Find the principal moments moments of inertia I₁, I₂,I₃ and the corresponding principal axes.
   3. (Extra credit.) Find the principal moments and axes for a coordinate system centered at the ceter of mass of the molecule.

PHY 711 -- Assignment #18

Continue reading Chapter 5 in Fetter & Walecka.

1. Work problem 5.9, parts (a) and (b) at the end of the chapter.

PHY 711 -- Assignment #19

Continue reading Chapter 9 in Fetter & Walecka.

1. Consider the example discussed in class on slides > 11, concerning the flow of an incompressible fluid in the z direction in the presence of a stationary cylindrical log oriented in the y direction. For this problem, consider the case where the log is replaced by a stationary sphere. Find the velocity potential for this case, using the center of the sphere as the origin of the coordinate system and spherical polar coordinates.

PHY 711 -- Assignment #20

Continue reading Chapter 9 in Fetter & Walecka.

1. Using the analysis covered in class, estimate the speed of sound in the fluid of He gas at 1 atmosphere of pressure and at 300K temperature.

PHY 711 -- Assignment #21

Continue reading Chapter 9 in Fetter & Walecka.

1. Consider a cylindrical pipe of length 0.5 m and radius 0.05 m, open at both ends. For air at 300 K and atmospheric pressure in this pipe, find several of the lowest frequency resonances.

PHY 711 -- Assignment #22

Continue reading Chapter 9 in Fetter & Walecka.

1. Equation 51.69 of F & W gives the expansion of a plane wave in cylindrical coordinates in terms of an infinite summation of Bessel functions of order m. Using the asymptotic form of the Bessel function (given in the notes and in the appendix D3.28), show the validity of this identity in the limit kr → ∞.