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(C) For the first system (series combination),the effective spring constant is $k_{eff1} = \frac{k \cdot k}{k + k} = \frac{k}{2}$.The frequency is $n_

Vedclass · Accepted Answer

$1:2$

Vedclass · Answer

(C) For the first system (series combination),the effective spring constant is $k_{eff1} = \frac{k \cdot k}{k + k} = \frac{k}{2}$.The frequency is $n_1 = \frac{1}{2\pi} \sqrt{\frac{k_{eff1}}{m}} = \frac{1}{2\pi} \sqrt{\frac{k}{2m}}$.For the second system (parallel combination),the effective spring constant is $k_{eff2} = k + k = 2k$.The frequency is $n_2 = \frac{1}{2\pi} \sqrt{\frac{k_{eff2}}{m}} = \frac{1}{2\pi} \sqrt{\frac{2k}{m}}$.The ratio of the frequencies is $\frac{n_1}{n_2} = \frac{\sqrt{k/2m}}{\sqrt{2k/m}} = \sqrt{\frac{k}{2m} \cdot \frac{m}{2k}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$.Thus,the ratio is $1:2$.

What is the ratio of the frequencies for the two given systems?

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