4.2: Chapter Checklist
- Page ID
- 34367
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You should now be able to:
- Apply symmetry arguments to find the normal modes of systems of coupled oscillators by finding the eigenvalues and eigenvectors of the symmetry matrix.
Problems
4.1. Show explicitly that (4.7) is true for the \(K\) matrix, (4.43), of system of Figure \( 4.3\) by finding \(SK\) and \(KS\).
4.2. Consider a system of six identical masses that are free to slide without friction on a circular ring of radius \(R\) and each of which is connected to both its nearest neighbors by identical springs, shown below in equilibrium:
- Analyze the possible motions of this system in the region in which it is linear (note that this is not quite just small oscillations). To do this, define appropriate displacement variables (so that you can use a symmetry argument), find the form of the \(K\) matrix and then follow the analysis in (4.37)-(4.55). If you have done this properly, you should find that one of the modes has zero frequency. Explain the physical significance of this mode. Hint: Do not attempt to find the form of the \(K\) matrix directly from the spring constants of the spring and the geometry. This is a mess. Instead, figure out what it has to look like on the basis of symmetry arguments. You may want to look at appendix c.
- If at \(t = 0\), the masses are evenly distributed around the circle, but every other mass is moving with (counterclockwise) velocity \(v\) while the remaining masses are at rest, find and describe in words the subsequent motion of the system.
4.3.
- Prove (4.56).
- Prove that if \(A\) and \(A^{\prime}\) are normal modes corresponding to different angular frequencies, \(\omega\) and \(\omega^{\prime}\) respectively, where \(\omega^{2}\) \neq \omega^{\prime 2}\), then \(b A+c A^{\prime}\) is not a normal mode unless \(b\) or \(c\) is zero. Hint: You will need to use the fact that both \(A\) and \(A^{\prime}\) are nonzero vectors.
4.4. Show that (4.43) is the most general symmetric \(6 \times 6\) matrix satisfying (4.44).