Physics 311 - Fall 97

Problem set 2, solutions

1. Gravity waves. Landau and Lifshitz, Fluids give the complete solution as w² = g/k tanh(kh), where tanh denotes the special function "tangent hyperbolic" that may be familiar to some of you (if not, here is a graph of tanh(x) for x between 0 and 5:
   dimensional analysis we did to find the dispersion relation, neglecting surface tension but including the effect of finite water depth h.

2. Order of magnitude estimates. Estimate the following to within a factor of ten; make sure to clearly spell out your assumptions. There aren't any wrong answers here (although some might be ridiculous). We aren't looking for exact numbers, just quick estimates -- spend at most 10-15 minutes on each. You'll find that the PQRG is handy.
   
   1. Estimate the pressure at the center of the earth (dimensional analysis is useful).
   2. Estimate the energy stored in the earth's magnetic field.
   3. Estimate the annual energy consumption by automobiles in the United States. Express your result in exajoules (EJ); 1 exajoule = 10¹⁸ J and also, conveniently, 1 exajoule = 1 quad = 10¹⁵ Btu. Check your result against available data.

3. Roasting a turkey. According to my Betty Crocker Cookbook, an 8 lb. stuffed turkey will attain an internal temperature of 185° F in a 350° F oven in 3.5 hours. Approximately how much time is required to roast a stuffed 16 lb. bird? How about a goat? An ox? [Hint: use dimensional analysis to determine how the roasting time scales with the weight. You'll need to know something about thermal diffusion.]

4. The diffusion equation. Let c(x,t) be the concentration (number of particles per unit volume) of dye molecules in a tube of water. x t Assume that the concentration only varies in one direction, x.
   
   1. Following the development in class for the heat equation, derive the continuity equation for the concentration and the particle flux j(x,t) (number of particles passing through a unit area per unit time), of the form

       

   2. Next, combine this with Fick's Law of diffusion,

       

      where D is the diffusion constant, to arrive at a diffusion equation for c(x,t). What is the physical meaning of Fick's Law? What are the dimensions of D?

   3. Suppose at an initial time t = 0 you put N₀ dye molecules at the center of the tube. What is c(x,t) at subsequent times? (You needn't repeat the derivation from class; just translate the results when necessary).
   4. Optional: generate an animation that shows the changing color of the liquid as the dye diffuses. Same conditions apply as for the next problem.

5. Fun with computers (you can skip one of the other problems if you do the optional part of this or of the previous problem; I hope that good animations can be used for the joy of future students):
   
   1. Write a program (in MAPLE, or MathCad, or whatever) that will reproduce, more or less, the animation of diffusion found on the web notes, or at least several frames of it. Bring the program on disk for real-time testing (or generate the animation on your web page if you have one).
   2. Optional: Generate an animation of two-dimensional diffusion. The relevant expression is t(x,t) t(y,t), where the functional form of t is given by eq. (1.38) of the notes.