# 4237

I. Georg’ 43 2. Georgi 5.1, but connect mass 4 to a wall with another spring of spring constant k, so it looks like mass 1 3. Consider 3 equal masses of mass In connected by equal springs with spring constant k. with masses 1 & 3 also connected to the walls by springs with constant k (diagram (c) from last week’s homework, or Georgi 3.3 with all masses, springs the same). a) Write down the M-1 K matrix (you did this last week). Write down S, the 3×3 reflection symmetry matrix. Show that WI KS= SA1-I K. b) What are the eigenvalues of 5? Do not directly compute the eignevalues using the determinant for-mula, but instead use the fact that two mirror reflections is the same as the original. c) Find 3 eigenvectors of S. Show that the eigenvector with a unique eigenvalue is also an eigenvector of WI K. d) Unless you are very lucky in your initial guess, the two eigenvectors of S you found that have the same eigenvalue will not be eigenvectors of M-1 K. Show that the two remaining eigenvectors of WI K you found last week are eigenvectors of S and show that these can be written as a linear combination of the eigenvectors you found already. 4. You will use this in the next problem. Prove that if = .E„N_ocos(n0) = 0 if tn is odd, and E„N.ocos(nO) = I if in is even and nonzero. One approach is to use these identities: Ecos(nO) = &(Eta() N aN+1 Ean n.0 1-a

If you evaluate the complex sum and multiply by C:a-7„ you should find that If m Is odd, you get fcot(0/2) and if in is even you get I. 5. Because Pt I K is symmetric for our model system of coupled pendula, normal modes with different fre-quencies should be orthogonal. Let’s check this for the example system of N-1 masses on pendula, con-nected by springs to each other and to the walls.

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