Physics 263: Problem Set #6

1. (BTM 3.1.7.) The trick of approximating a sum by an integral (rather than the other way around) was reviewed in class on Friday (just write the Riemann sum for the integral of ln x from 1 to n). [Ok, so we didn't get to this. My bad.] If you're having trouble with the constraints, just imagine concrete numbers. Suppose N=10 and there are three levels, i=1,2,3. Then the number of particles in level 1 + level 2 + level 3 must be N, because every particle is somewhere. So N = n1 + n2 + n3. Now what is the analogous result for the energy E? Once you have the constraints, you also have the function to maximize (namely S), so just apply our Lagrange multiplier procedure for n1, n2, ... instead of for x,y,z,...
2. (BTM 3.2.4.) Not much to say here: just carry out equation (3.2.35).
3. (BTM 3.2.5.) In the first case, the density (mass per unit area here, instead of mass per unit volume) is a constant and in the second case the density is another constant times r. How can you determine these constants? (Hint: it involves M.) Given those constants, you just carry out equation (3.2.49). What are the limits of integration for r and theta in all of the integral? The answers are in the back of the book.
4. (BTM 3.2.8.) Draw a picture and then pick a coordinate system to use. Just because the problem describes the "y-axis" as the axis of rotation doesn't mean you can't identify it with another axis in your coordinate system. You choose a coordinate system in order to simplify the problem, taking advantage of symmetries. What are the symmetries here? The answer is in the back of the book.
5. (BONUS: BTM 3.2.7.) Nothing tricky here, just follow the instructions. Consider x and y each as functions of u and v.