PHY 711 -- Assignment #2

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=4). What is the length of this "curve"?

PHY 711 -- Assignment #4

Continue reading Chapter 3, in Fetter & Walecka.

Consider a point particle of mass m moving (only) along the x axis according to a force F_x=-K x, where K is a positive constant. The particle trajectory as a function of time, x(t), has the initial value x(t=0)=C, where C denotes a given length, and its initial velocity is 0.

• Write down and solve Newton's second law for this system, finding the form of the trajectory x(t).
• Now write down the Lagrangian for this system and solve the Euler-Lagrange equations for the trajectory x(t). How does your answer compare with (a)?

PHY 711 -- Assignment #6

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, and g) and determine the equations of motion. (Note that X and s represent the lengths of vectors whose directions are related by the angle θ.)
2. Assume that initially positions X(t=0)=0, s(t=0)=0 and that their corresponding velocities are also 0. Find the trajectories X(t) and s(t) for t > 0.

PHY 711 -- Assignment #7

Continue reading Chapter 3 and 6 in Fetter & Walecka.

This problem considers the same system as Problem #5.

1. Starting from the Lagrangian used to analyze Problem #5, determine the corresponding Hamiltonian in canonical form.
2. Determine and solve the canonical equations of motion, comparing the resulting trajectories with those obtained by working Problem #5.

PHY 711 -- Assignment #9

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 with its origin placed at the center of mass of the molecule.

PHY 711 -- Assignment #11

Finish reading Chapter 5 in Fetter & Walecka.

Consider problem 5.10 at the end of chapter 5 in Fetter & Walecka

1. Work part (a).
2. (Extra credit) Work part (b).

PHY 711 -- Assignment #14

Continue 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 derived in class.

PHY 711 -- Assignment #16

Continue reading Chapter 7 in Fetter & Walecka.

1. Consider the function f(x) = x (1-x²) 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 for example.

PHY 711 -- Assignment #19

Read Chapter 8 in Fetter & Walecka.

1. In the last few slides of Lecture 26, we considered an annular membrane stretched between the circular radii a ≤ r ≤ b. Find a wave form f(r) for the case that a=0.5 and b=1, assuming angular uniformity (m=0) and that f(a)=f(b)=0.

PHY 711 -- Assignment #20

Read Chapter 9 in Fetter & Walecka.

1. A tank having an area of 100 m² is open to the atmosphere and contains 1000 m³ of water. It has a spigot at it bottom which has a height of 1 m above a drain in the floor. A hose having a diameter of 1 cm is used to empty the water from the tank when the spigot is opened. Using Bernoulli's analysis, estimate the time it takes to empty the tank via the floor drain.

PHY 711 -- Assignment #21

Continue reading Chapter 9 in Fetter & Walecka.

1. Consider the example discussed in Lecture 27 & 28, 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 #22

Continue reading Chapter 9 in Fetter & Walecka.

1. Estimate the speed of sound for the following ideal gas materials at a pressure of p = 101325 Pa and temperature T= 274 K:
2. ρ = 1.29 kg/m³ (approximating dry air)
3. ρ = 0.18 kg/m³ (approximating He gas)

PHY 711 -- Assignment #23

Continue reading Chapter 9 in Fetter & Walecka.

1. Consider a cylindrical pipe of length 0.1 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, including at least one that has non-trivial radial dependence.

PHY 711 -- Assignment #24

Finish reading Chapter 9 in Fetter & Walecka.

1. In class, we discussed how to visualize the non-linear behavior of an adiabatic ideal gas with parameter γ. Using Maple or Mathematica or other software and using a parametric plot formalism, create an animated gif file to show the traveling waveform s(w), where s is a shape of your choice and w=x-u(s(w))t. You will also need to choose the value of γ as well.

PHY 711 -- Assignment #25

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 the ideas are discussed in Lecture 32.

PHY 711 -- Assignment #28

Start reading Chapter 12 in Fetter & Walecka.

1. Work problem 12.4 at the end of Chapter 12 in Fetter & Walecka.

PHY 711 -- Assignment #29

Continue reading Chapter 12 in Fetter & Walecka.

1. Work problem 12.13 at the end of Chapter 12 in Fetter & Walecka.

PHY 711 -- Assignment #30

Start reading Chapter 13 in Fetter & Walecka.

1. Work problem 13.8 at the end of Chapter 13 in Fetter & Walecka. Note that this problem is non-trivial and homework credit will be earned for significant if only partial solutions.