PHY 711 Classical Mechanics

MWF 11-11:50 AM OPL 107 http://www.wfu.edu/~natalie/f07phy711

Instructor: Natalie Holzwarth Phone:758-5510Office:300 OPL e-mail:natalie@wfu.edu

Homework Assignments

PHY 711 -- Assignment #1

August 29, 2007

Read Chapter 1 in Fetter & Walecka and make note of useful appendices.

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PHY 711 -- Assignment #2

August 31, 2007

Continue reading Chapter 1 in Fetter & Walecka.

  • In the last lecture, we derived the differential cross section for the elastic scattering of two hard spheres -- dσ/dΩ|CM( θ) = D2/4, where D is the sum of the radii of the two spheres. Now suppose that in lab frame of reference, the incident mass (m1) has an initial velocity v1 and the target mass (m2) is at rest.
    • Find the relationship between the center of mass scattering angle θ to the lab frame scattering angle χ.
    • Find the relationship between the lab and CM differential cross sections.
    • Evaluate these expressions for the case that m1=m2. Check your results to make sure that the total scattering cross section is the same in the two frames of reference.

PHY 711 -- Assignment #3

September 3, 2007

Continue reading Chapter 1 in Fetter & Walecka.

PHY 711 -- Assignment #4

September 5, 2007

Start reading Chapters 3 and 6 in Fetter & Walecka.

This problem is one more example of scattering theory from Chap. 1.

  • Work problem 1.16 in Fetter & Walecka. Assume γ>0. (Extra credit if you consider the complete problem.)

PHY 711 -- Assignment #5

September 7, 2007

Continue reading Chapters 3 and 6 in Fetter & Walecka.

  • Find the function y(x) with the end points y(0)=0 and y(1)=1, which minimizes the integral

    I({y(x),dy(x)/dx}) = ∫10 ( (dy/dx)2 + 2 y(x) ) dx.

PHY 711 -- Assignment #6

September 10, 2007

Continue reading Chapters 3 and 6 in Fetter & Walecka.

  • Find the function r(θ) with the end points r(0)=r(2π)=a, which maximizes the enclosed area:

    A({r(θ)}) = (1/2) ∫02π (r(θ))2 dθ
    while keeping the length fixed at the value 2πa:

    ∫02π r(θ) dθ = 2 &pi a.

PHY 711 -- Assignment #7

September 11, 2007

Continue reading Chapters 3 and 6 in Fetter & Walecka.

  • Work problem 3.3 in Fetter & Walecka.

PHY 711 -- Assignment #8

September 17, 2007

Continue reading Chapters 3 & 6 in Fetter & Walecka.

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PHY 711 -- Assignment #9

September 19, 2007

Continue reading Chapters 3 & 6 in Fetter & Walecka.

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PHY 711 -- Assignment #10

September 21, 2007

Continue reading Chapters 3 & 6 in Fetter & Walecka.

  1. Work problem #6.18 in Fetter & Walecka.

PHY 711 -- Assignment #11

September 25, 2007

Finish reading Chapter 6 in Fetter & Walecka.

  1. Consider the Hamilton-Jacobi equations presented in Section 35 of your text. Find the trajectory q(t) and S(q,α,t) for the example:
    V(q) = mg q,

    where g is the constant gravitational acceleration and the potential represents that of a constant gravitational potential, with q(t) representing the vertical displacement.

PHY 711 -- Assignment #12

October 8, 2007

Continue reading Chapters 5 in Fetter & Walecka.

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PHY 711 -- Assignment #13

October 10, 2007

Continue reading Chapters 5 in Fetter & Walecka.

  1. Work problem #5.10 in Fetter & Walecka.

PHY 711 -- Assignment #14

October 12, 2007

Start reading Chap. 7 in Fetter & Walecka.

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PHY 711 -- Assignment #15

October 15, 2007

Continue reading Chapters 5 in Fetter & Walecka.

  1. Consider a function h(x)=x(1-x) within the interval 0 ≤ x ≤ 1. Find the Fourier expansion of this function and plot the error in the expansion when including 1, 2, and 3 non-trivial terms.

PHY 711 -- Assignment #16

October 22, 2007

Continue reading Chap. 7 in Fetter & Walecka.

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PHY 711 -- Assignment #17

October 24, 2007

Read Appendix A in Fetter & Walecka.

  • Work problem A.7 (pg. 509) in Fetter & Walecka.

PHY 711 -- Assignment #18

October 26, 2007

Read Chap. 8 in Fetter & Walecka.

  • Work problem 8.5 in Fetter & Walecka.

PHY 711 -- Assignment #19

October 29, 2007

Start reading Chapter 9 in Fetter & Walecka.

  1. Consider Eq. 48.14 in Fetter & Walecka. Write out the expressions in cartesian coordinates and convince yourself that the identity is correct.

PHY 711 -- Assignment #20

October 31, 2007

Continue reading Chapter 9 in Fetter & Walecka.

  1. The above photo shows a syringe with a liquid (density ρ=1000 kg/m3 ) inside the barrel (cross sectional area 8 x 10-5 m2). Suppose a force of F = 1 N is applied to the barrel. What is the velocity of the liquid coming out of the needle (cross sectional area 8 x 10-7 m2)?

PHY 711 -- Assignment #21

November 14, 2007

Start reading Chap. 10 in Fetter & Walecka. The following problem uses material from the end of Chap. 9.

Suppose that a shock wave in an adiabatic ideal gas, having γ=1.5 is created with a pressure ratio p2/p1 = 2, using the notation used in class and used in Fetter & Walecka. Find the corresponding ratios and differences for the following other properties of the gas ahead (2) and behind (1) the shock front:

  1. density -- n2/n1
  2. temperature -- T2/T1
  3. entropy difference -- s2-s1 (express your answer as a multiple of cv (heat capacity at constant volume)
  4. Mach number -- M1

PHY 711 -- Assignment #22

November 26, 2007

Start reading Chap. 12 in Fetter & Walecka. The following problem uses material from the end of Chap. 10 and the lecturenotes on solitary waves.

In class and in the lecture notes, we derived the soliton form of the surface displacement given by ζ(x,t)=η(x-ct) expressed in Eq. 25 of the notes. From this result find the following quantities to lowest order in &eta0/h, the ratio of the wave amplitude to the water depth.

  1. Velocity potential function φ(x,t)=χ(x-ct).
  2. v_x(x,z=h+ζ,t).

PHY 711 -- Assignment #23

November 26, 2007

Continue reading Chapters 12 in Fetter & Walecka.

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PHY 711 -- Assignment #24

November 30, 2007

Finish reading Chap. 12 in Fetter & Walecka.

  1. Work problem 12.13 in Fetter & Walecka, parts a and b or c (extra credit for considering both b and c).
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