General Considerations: The numerical computations can be done in C, MATLAB, or Mathematica. The choice is yours. For graphic plotting, you can use MATLAB/Mathematica, or xmgr. Normally, the current week's lab is required to be submitted by the next lab in two weeks time. The labs problem set may not specify precisely and exactly what is expected. This is deliberately so, to have open-ended questions. In this sense, you can treat the lab problems as a small scale project. Marks will be gauged according to your efforts and ingenuity. Most of the problems will be based on (but not necessarily follow exactly) the textbook problems. You must discuss/interpret/explain your method/motivation/results in your report. Feel free to ask the lecturer/TA questions, and discuss among students. But computational work and report must be done independently.
Q1: (question 7. on page 29). Table 1.1 below gives orbital data for six planets. Fit this data to a power law of the form y = C xp (x to the power p). Plot the data points as well as the fitted curve. The largest asteroid, Ceres, has a period of about 4.5 earth years; about how far from the sun would you place it? (Hint: Kepler used data similar to this to formulate his third law. What was his basic model of planetary motion? Was it pictorial, analogical, or mathematical?)
Table 1.1
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Planet Semimajor Sidereal period
axis (106 km) (days)
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Mercury 57.9 87.97
Venus 108.2 224.7
Earth 149.6 365.26
Mars 227.9 686.98
Jupiter 778 4332.4
Saturn 1427 10759
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Q2: (pictures on page 27). On page 27 of Figure 1.12 in the textbook, the author drew (apparently by hand) some pictures of a function f(t) and its fourier (integral) transform. In this lab, we want to make a set of pictures (preferably in one page) that resemble the figure 1.12, but more quantitatively. For each picture on the left, give a mathematical expression for it, make a plot of it, and find its numerical fourier transform of it, and plot on the right. [There is the issue of using sine, or cosine, or exponential for the transform, your choice should be such (if you can) so that the fourier transform is real and easily plotted]. It's totally upto you to give f(t) that resemble the plots on the left. However, in your report, your f(t) should be precisely specified and your plots should be carefully labelled. [If you need help on the concept of Fourier transform, and particularly discretized numerical implementation, read "Numerical Recipes", chap.12, by Press et al, available in CZ3 lab]. To get a quick introduction on Fourier transform, look at the webpage http://aurora.phys.utk.edu/~forrest/papers/fourier/. Be careful about the slightly different definition of Fourier integrals.