PHY 711 -- Assignment #3

Continue reading Chapter 1 in Fetter & Walecka.

• Work Problem #1.15a at the end of Chapter 1 in Fetter and Walecka or derive for yourself the equivalent equation discussed 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=1). What is the length of the "curve"?

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 #11

Finish reading Chapters 3 and 6 in Fetter & Walecka.

• Consider the example of the mass m initially at height h falling from rest with constant acceleration g to verify the analysis of the action S at time t>0 discussed in the lecture notes.

PHY 711 -- Assignment #13

Continue reading Chapter 4. in Fetter & Walecka.

• Consider the example of of an infinite one-dimensional chain of masses m and M connected with identical springs having constant k..
  • Verify the normal frequencies obtained in the class notes for this system.
  • Now assume that k/m ≡4 ω₀² and k/M ≡ ω₀², where ω₀ is a given constant. Plot the normal mode frequencies for this case as a function of qa.

PHY 711 -- Assignment #14

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 #16

Continue reading Chapter 7 in Fetter & Walecka.

Consider the example presented in the last two slides of Lecture 22, where a one-dimensional Poisson equation was solved using a Green's function constructed from the corresponding homogeneous 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

Continue reading Chapter 7 in Fetter & Walecka.

1. Consider the function f(x) = x² (1-x) in the interval 0 ≤ x ≤ 1. Find the coefficients A_n of the Fourier series based on the terms sin( n π x). Extra credit: Plot f(x) and the Fourier series including 3 terms.

PHY 711 -- Assignment #19

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 #20

Finish reading Chapter 5 in Fetter & Walecka.

1. Consider a Cartesian coordinate system in which a vector V has components (V_x,V_y,V_z). Now suppose that the coordinate system is rotated by the 3 Euler angles so that in the new orientation, the vector has components (V'_x,V'_y,V'_z). Find the rotation matrix and the new vector components for the case that α=90 deg, β=90 deg, and γ=0 deg.

PHY 711 -- Assignment #21

Start reading Chapter 9 in Fetter & Walecka.

1. Approximate the ocean as an incompressible fluid and ignore effects of fluid motion to estimate the pressure difference at a height of 100 meters below the sea relative to the pressure at the sea surface. Please mention the density of sea water you assume for your estimate.

PHY 711 -- Assignment #22

Continue reading Chapter 9 in Fetter & Walecka.

1. Consider the example discussed in Lecture 29, 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 homework problem, 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 #23

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 #24

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 #25

Finish reading Chapter 9 in Fetter & Walecka.

1. Assume the ideal gas law and adiabatic conditions for He gas, having an initial pressure of p₀= 101325 Pa (1 atm) and initial temperature of T₀=300K. Calculate the following when the pressure is changed p₁=2p₀.
   1. T₁.
   2. The change in the internal energy per unit mass Δ ε.
   3. The change in the entropy per unit mass Δ s

PHY 711 -- Assignment #26

Start reading Chapter 10 in Fetter & Walecka.

1. Work Problem 10.3 at the end of Chapter 10 in Fetter and Walecka. Note that some of ideas are discussed in Lecture 33.