PHY 711 Classical Mechanics

MWF 11-11:50 AM

OPL 107

http://www.wfu.edu/~natalie/f08phy711

Instructor: Natalie Holzwarth

Phone:758-5510

Office:300 OPL

e-mail:natalie@wfu.edu

Homework Assignments

• Problem Set #1 (8/27/08
• Problem Set #2 (8/29/08)
• Problem Set #3 (9/01/08)
• Problem Set #4 (9/03/08)
• Problem Set #5 (9/05/08)
• Problem Set #6 (9/08/08)
• Problem Set #7 (9/10/08)
• Problem Set #8 (9/12/08)
• Problem Set #9 (9/15/08)
• Problem Set #10 (9/17/08)
• Problem Set #11 (9/19/08)
• Problem Set #12 (9/22/08)
• Problem Set #13 (10/03/08)
• Problem Set #14 (10/06/08)
• Problem Set #15 (10/08/08)
• Problem Set #16 (10/10/08)
• Problem Set #17 (10/13/08)
• Problem Set #18 (10/20/08)
• Problem Set #19 (10/22/08)
• Problem Set #20 (10/24/08)
• Problem Set #21 (10/27/08)
• Problem Set #22 (10/29/08)
• Problem Set #23 (10/31/08)
• Problem Set #24 (11/03/08)
• Problem Set #25 (11/17/08)
• Problem Set #26 (11/21/08)
• Problem Set #27 (12/01/08)

PHY 711 - Assignment #1

August 26, 2008

1. In class, we showed that the relationship between the impact parameter b and the scattering angle $\chi$ for elastic scattering between two hard spheres has the form:
   
   $$b = D \cos \left( \displaystyle{\frac{\chi}{2}} \right).$$
   
   
   Using the above diagram which shows the geometry of two hard spheres at the moment of impact, derive this formula.

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   PHY 711 -- Assignment #2

   August 29, 2008

   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 1, 2008

   Continue reading Chapter 1 in Fetter & Walecka.

   • This is an exercise to help you become familiar with the Maple software. You may wish to take advantage of the "Maple Tour" and with the example files on our web page -- General Maple Examples or Rutherford scattering integrals. This is an open-ended assignment depending on your time and interest. Minimally, please turn in a Maple worksheet output with the following:
     • An integral.
     • A derivative.
     • A plot.

   PHY 711 -- Assignment #4

   September 3, 2008

   Skim Chap. 2 and start reading about the calculus of variation in Chap. 3 in Fetter & Walecka.

   1. Consider the "centrifugal force" - ω × (ω × r) described in Eq. 11.5b of Chap. 2, where ω describes the Earth's rotation about the North pole. Using the latitude of Winston-Salem (approximiately 35^o) and the estimated magnitude of ω, estimate the magnitude and direction of the centrifugal acceleration in Olin 107.
   PHY 711 - Assignment #5
   September 4, 2008
   1. Consider the following integral which involves a continuous function $y(x)$:
      

      $$I = \int_0^1 \left[ \left( \frac{dy}{dx} \right)^2 + \frac{y}{a} \right],$$ (1)

      
      
      where a is a given constant. Assuming the boundary conditions $y(0)=0$ and $y(1)=0$, find the function $y(x)$ that extremizes the integral $I$.
   
   
   PHY 711 - Assignment #6
   September 6, 2008
   1. In class, we considered the famous Brachistochrone problem in which a mass slides down a frictionless track with shape $y(x)$, starting at $y(0)=2a$ and ending at $y(a \pi) = 0$. The integral that calculates the travel time $T$ between these two points is:
      

      $$\sqrt{2 g} T = \int_0^{a \pi} \sqrt{\frac{1+(dy/dx)^2}{2a-y}} dx.$$ (1)

      
      
      Here $g$ denotes the gravitational acceleration. Evaluate this integral for the follow two shapes $y(x)$. Numerically compare these two results (as factors of $\sqrt{a}$) to determine which is larger. Explain.
      1. In parametric form with $0 \le \theta \le \pi$:
         $x = a(\theta - \sin \theta)$
         $y = a(1 + \cos \theta)$
      2. In conventional form with $0 \le x \le a \pi$:
         $y(x) = 2a - \frac{2}{\pi} x$

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   PHY 711 - Assignment #7
   September 10, 2008
   1. Consider a particle of mass $m$ moving in 3-dimensions in a force field of the form (as expressed in Cartesian coordinates):
      
      $${\bf {F}}(x,y,z) = K \frac{x \hat{\bf {x}}+ y \hat{\bf {y}}+ z \hat{\bf {z}}}{\sqrt{x^2 + y^2 + z^2}},$$
      
      
      where $K$ is a constant.
      1. Write down the Lagrangian and Lagrange's equations of motion for this particle in Cartesian coordinates - $L(x,y,z,\dot{x},\dot{y},\dot{z})$
      2. Now, transform the coordinate system to cylindrical coordinates:
         
         $$x = \rho \cos \phi$$
         
         
         
         
         $$y = \rho \sin \phi$$
         
         
         
         
         $$z = z$$
         
         
         so that you can write the Lagrangian and Lagrange's equations of motion for the particle in cylindrical coordinates - $L(\rho, \phi, z, \dot{\rho}, \dot{\phi}, \dot{z})$.
      3. Identify any ``constants of motion'' in either coordinate system.
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   PHY 711 - Assignment #8
   September 12, 2008

   This exercise is designed to illustrate the differences between partial and total derivatives.

   1. Consider an arbitrary function of the form $f=f(q,\dot{q},t)$, where it is assumed that $q=q(t)$ and $\dot{q}\equiv dq/dt$.
      1. Evaluate
         
         $$\frac{\partial}{\partial q} \frac{df}{dt} - \frac{d}{dt} \frac{\partial f} {\partial q}.$$
         
         

      2. Evaluate
         
         $$\frac{\partial}{\partial \dot{q}} \frac{df}{dt} - \frac{d}{dt} \frac{\partial f} {\partial \dot{q}}.$$
         
         

      3. Evaluate
         
         $$\frac{df}{dt}$$
         
         
         .
      4. Now suppose that
         
         $$f(q,\dot{q},t) = q \dot{q} t, \;\;\; {\rm {where}} \;\;\ q(t)={\rm {e}}^{-t/\tau}.$$
         
         
         Here $\tau$ is a constant. Evaluate $df/dt$ using the expression you just derived. Now find $f(t)$ and take its time derivative directly to check your previous results.
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   September 13, 2008
   PHY 711 - Problem Set # 9

   Continue reading Chapter 3 in Fetter and Walecka.

   Consider the following Lagrangians. For each, determine the equations and constants of motion. Assume that $A$, $B$, $M$, $g$, $h$, $q$, and $c$ are constant parameters.

   1. 
      
      $$L(\theta,\phi,\psi,\dot{\theta},\dot{\phi},\dot{\psi}) = \... ...\phi} \cos(\theta) + \dot{\psi} \right)^2 -Mgh \cos(\theta).$$
      
      

   2. 
      
      $$L(x,y,z,\dot{x},\dot{y},\dot{z}) = \frac{1}{2}M (\dot{x}^2 ... ...{z}^2 ) + \frac{1}{2} \frac{q}{c} B ( x \dot{y} - y \dot{x}).$$
      
      

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   PHY 711 -- Assignment #10

   September 17, 2008

   Start reading Chapter 6 in Fetter & Walecka.

   • Consider the two Lagrangians given in HW 9. For each, find the corresponding Hamiltonian in canonical form and write the form of the canonical equations of motion.

   PHY 711 -- Assignment #11

   September 19, 2008

   Continue reading Chapters 6 in Fetter & Walecka.

   1. Work problem #6.18 in Fetter & Walecka.

   PHY 711 -- Assignment #12

   September 22, 2008

   Finish reading Chapter 6 in Fetter & Walecka.

   Read parts of the paper by Hans C. Andersen "Molecular dynamics simulations at constant pressure and/or temperature" in which he constructs a Lagrangian function to represent a system of particles held at constant pressure α.

   1. Starting with the Lagrangian function (3.2), derive the Hamiltonian function (3.6).
   2. Derive the equations of motion (3.7).
   October 3, 2008
   PHY 711 - Problem Set # 13

   Finish reading Chapter 5 in Fetter and Walecka.

   In most Classsical Mechanics texts (besides Fetter and Walecka), the Euler angles are defined with a different convention as shown below. (This figure was slightly modified from one available on the website http://en.wikipedia.org/wiki/Euler_angles.)

   

   In this case, the first rotation is about the original $\hat{\bf {z}}$ axis by $\phi$ corresponding to the rotation matrix
   

   $${\cal{R}}_{\phi} = \left( \begin{array}{ccc} \cos \phi &-\s... ...\sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{array} \right).$$ (1)

   
   
   The second rotation is about the new $\hat{\bf {x}}$ axis by $\theta$ corresponding to the rotation matrix
   

   $${\cal{R}}_{\theta} = \left( \begin{array}{ccc} 1 & 0 & 0 \\... ...theta \\ 0 & \sin \theta & \cos \theta \end{array} \right).$$ (2)

   
   
   In this case, the last rotation is about the new $\hat{\bf {z}}$ axis by $\psi$ corresponding to the rotation matrix
   

   $${\cal{R}}_{\psi} = \left( \begin{array}{ccc} \cos \psi &-\s... ...\sin \psi & \cos \psi & 0 \\ 0 & 0 & 1 \end{array} \right).$$ (3)

   
   
   For this convention, write a general expression for the angular velocity vector $\bf {\omega}$ in terms of the time rate of change of these Euler angles - $\dot{\phi}$, $\dot{\theta}$, and $\dot{\psi}$ corresponding to the 29.7 of your text.

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   October 6, 2008
   PHY 711 - Problem Set # 14

   Start reading Chapter 7 in Fetter and Walecka.

   Suppose that an infinite continuous string satisfies the wave equation:
   

   $$\frac{\partial^2 u}{\partial^2 t} = c^2 \frac{\partial^2 u}{\partial^2 x}.$$ (1)

   
   
   Find $u(x,t)$ for the following initial conditions:
   1. 
      

      $$u(x,0) = 0 \;\;\;\;\; \frac{ \partial u}{\partial t}(x,0) = \frac{2 A x}{(x^2 + a^2)^4},$$ (2)

      
      
      where $A$ and $a$ are positive constants.
   2. 
      

      $$u(x,0) = \frac{ A }{(x^2 + a^2)^2}\;\;\;\;\; \frac{ \partial u}{\partial t}(x,0) = 0,$$ (3)

      
      
      where $A$ and $a$ are positive constants.

   You can visualize your results using the animate feature of Maple. For example, to animate the result we obtained in class today, you can use the following syntax:

   > with(plots);
   > animate(plot, [exp(-(x+t)^2)+exp(-(x-t)^2), x = -30 .. 30], t = 0 .. 20);
   

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   PHY 711 -- Assignment #15

   October 8, 2008

   Continue reading Chapter 7 in Fetter & Walecka.

   1. Consider the initial value wave equation problem posed in problem #2 of homework set 14. Solve that problem using the Fourier transform method. Discuss the convergence of that solution as a function of the Fourier coefficients.

   PHY 711 -- Assignment #16

   October 10, 2008

   Continue reading Chapter 7 and also Appendix A in Fetter & Walecka.

   1. Work Problem A.9 in Appendix A.
   October 13, 2008
   PHY 711 - Problem Set # 17

   Continue reading Chapter 7 in Fetter and Walecka.

   Consider the differential equation
   

   $$\left( - \frac{d^2}{dx^2} - \lambda \right) \phi(x) = F_0 \sin \left( \frac{\pi x} {L} \right),$$ (1)

   
   
   where $\phi(x=0)=0$ and $${\frac {d \phi}{dx}(0)=0}$$ and where $\lambda$, $F_0$, and $L$ are constants.
   1. Show that the solution takes the form
      

      $$\phi(x) = \frac{F_0}{ \frac{\pi^2}{L^2} - \lambda} \left( \si... ...- \frac{\pi}{\sqrt{\lambda} L} \sin(\sqrt{\lambda} x) \right).$$ (2)

      
      

   2. Use the method of Laplace transforms to verify this solution.
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   PHY 711 -- Assignment #18

   October 20, 2008

   Start reading Chapter 9 in Fetter & Walecka.

   1. Convince yourself of the validity of Eq. 48.14 by expanding the left and right hand sides in cartesian coordinates.

   PHY 711 -- Assignment #19

   October 22, 2008

   Continue reading Chapter 9 in Fetter & Walecka.

   1. Suppose you have a cylindrical pipe of length L=1.0 m open at both ends with a radius a=0.1 m. The speed of sound is 330 m/s. Find the first 10 or so resonant frequencies of sound within the pipe in units of Hz (cycles per second; 2 π rad/sec corresponds to 1 Hz). What is the lowest frequency mode that has a non-trivial radial variation? Note that the first few zeros of the derivatives of Bessel functions z'_mn are given by
      1. z'₀₁ = 0.00000
      2. z'₁₁ = 1.84118
      3. z'₂₁ = 3.05424
      4. z'₀₂ = 3.83170
      5. z'₃₁ = 4.20119

   PHY 711 -- Assignment #20

   October 24, 2008

   Continue reading Chapter 9 in Fetter & Walecka.

   1. Consider an ideal gas characterized by a given γ parameter under adiabatic conditions. Find expressions for the following variable of the gas as a function of its density ρ:
      1. Pressure p(ρ)
      2. Internal energy per unit mass ε(ρ)
      3. Temperature T(ρ)
      4. Sound velocity c(ρ)
      5. Fluid velocity v(ρ)

   PHY 711 -- Assignment #21

   October 27, 2008

   Continue reading Chapter 9 in Fetter & Walecka.

   1. Consider an ideal gas characterized by a given γ parameter under adiabatic conditions on either side of a shock front as discussed in Sec. 52 of your text. Find an expression for the Mach number M₁ = (v₁-u)/c₁ in terms of the density and pressure ratios on either side of the shock. Express Eqs. 52.53 and 52.54 as functions of M₁ and plot ρ₂/ρ₁ and T₂/T₁ vs M₁ for the case γ=1.4.

   PHY 711 -- Assignment #22

   October 29, 2008

   Finish reading Chapter 9 and start 10 in Fetter & Walecka.

   1. Consider an ideal gas characterized by a given γ parameter under adiabatic conditions on either side of a shock front as discussed in Sec. 52 of your text.
      1. Solve the appropriate equations to find the density ratio ρ₂/ρ₁ as a function of the pressure ratio p₂/p₁.
      2. Find the range of physical values of the density ratio ρ₂/ρ₁
      3. Derive Eq. 52.58 for the entropy difference on either side of the shock wave.
      4. Evaluate the entropy difference s₂-s₁ as a function of the density ratio ρ₂/ρ₁.
      5. Show that s₂-s₁ > 0 in the physical range of values by evaluating the expressions for γ=1.4.

   PHY 711 -- Assignment #23

   October 31, 2008

   Continue reading Chapter 10 in Fetter & Walecka.

   1. Work problem 10.3 in Fetter & Walecka. (Hint: Think about how you might be able to apply a version of Eq. 53.15 for this case.)

   PHY 711 -- Assignment #24

   November 3, 2008

   Continue reading Chap. 10 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,t).

   PHY 711 -- Assignment #25

   November 17, 2008

   Finish reading Chap. 12 in Fetter & Walecka.

   1. Work problem 12.13(a). (Extra credit for working parts (b) and/or (c).)

   PHY 711 -- Assignment #26

   November 21, 2008

   Finish reading Chap. 11 in Fetter & Walecka.

   1. Work problem 11.2 in Fetter & Walecka.

   PHY 711 -- Assignment #27

   December 1, 2008

   Finish reading Chap. 13 in Fetter & Walecka.

   1. Work problem 13.8 in Fetter & Walecka. Note that the T_zz=0 and T_xz=0 conditions apply only for z=0. Additional hints available in class.
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