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Mechanics}Normal Mode

One can work through the formalism of the usual normal mode analysis or learn how to deal with normal mode frequencies the easy way:

The highest normal mode frequency is due to the two masses oscillating out of phase. The \omega^2 contribution from the pendulum is just g/l. The \omega^2 contribution from each mass due to the spring is k_i/m_i. This is choice (D).

(If one had to guess, one can immediately eliminate choice (A), since that is the lowest normal mode frequency. In normal modes, there's always an in-phase frequency, which tends to be the lowest frequency.)

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Comments
fjornada
2009-10-18 19:56:52
take l \rightarrow \infty and m_1 \rightarrow \infty.
that's just the case of a mass (m_2) oscillating around a fixed point. only (D) has such a limit.
his dudeness
2010-07-30 05:30:04
Perhaps once you've got it down to choices (D) and (E), you can argue that if you let one of the masses go to infinity so that it's effectively fixed, the normal mode frequency should have some dependence on K, and a larger K will lead to a larger frequency. In (E), the K dependence drops out for this case, so that leaves choice (D).
NEC
hungrychemist
2007-10-08 22:47:17
take limit as K-> 0, you should get the lowest normal mode frequencyNEC
grep
2006-10-27 14:20:32
If normal mode oscillation isn't your thing, you can use dimensional analysis to eliminate B and E. Then just remember that in general for pendulum systems the frequency decreases when l increases. This leaves only choice D
welshmj
2007-08-02 20:10:18
I don't think dimensional analysis can be used here. It certainly does not eliminate B or E.
wizang
2009-10-02 00:53:46
I checked the units, and they work out on all of the possible answers. Keep in mind that the spring constant is measured in terms of force per unit length. Taking this into consideration, all the answers end up at \frac{1}{s}
NEC

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