Problem 8

# Problem 8 - 4. Now write down the two equations to 1 st...

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Problem 8 In this problem, we wish to obtain, by using an expansion in large k, the lower frequency “g-mode” normal mode in the coupled oscillator problem for the case where k/m >> g/l. This will show you the technique to extract low frequency information from a system which includes both high and low frequencies. Proceed as follows: 1. Write down the 2 coupled oscillator equations for angles x(t) and y(t). k is the spring constant, m is the mass. 2. We are interested only in low frequencies, ie, omega << (k/m) 1/2 . Begin by scaling both equations. Use this inequality and identify the largest (dominant) terms. 3. Expanding x and y as x = x 0 + x 1 + …. ., etc, write down the 2 equations in zeroth order, involving x 0 and y 0 . Thus, find any relationship between x 0 (t) and y 0 (t). Does this make sense for the low frequency response?
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Unformatted text preview: 4. Now write down the two equations to 1 st order, involving x 1 (t) and y 1 (t). In doing so, use optimal ordering. This means that if there are 2 or more small terms and there is no way you can figure out their relative sizes, assume they have the same scaling, ie, dont discard one in favor of the other. 5. Important step: Find an annihilator that will kill the dominant term. In this case, the annihilator is really simple but its easy to miss (note: any operation that kills is ok). Using this, you should get more info on the lowest order x (t) and y (t). Thus, find equations for these variables and find the general solutions. Thus, this is the lowest order low frequency response. Does it make sense given what you know about the coupled oscillator problem?...
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## This document was uploaded on 12/29/2011.

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