Two identical springs of constant K are conne | vedclass

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Question

(A) In the series combination,the effective spring constant $k_s$ is given by:$k_s = \frac{K \cdot K}{K + K} = \frac{K}{2}$In the parallel combination

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$1$:$2$

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(A) In the series combination,the effective spring constant $k_s$ is given by:$k_s = \frac{K \cdot K}{K + K} = \frac{K}{2}$In the parallel combination,the effective spring constant $k_p$ is given by:$k_p = K + K = 2K$The frequency of oscillation for a mass-spring system is $f = \frac{1}{2\pi} \sqrt{\frac{k}{M}}$.For the series combination,the frequency $f_s$ is:$f_s = \frac{1}{2\pi} \sqrt{\frac{K/2}{M}} = \frac{1}{2\pi} \sqrt{\frac{K}{2M}}$For the parallel combination,the frequency $f_p$ is:$f_p = \frac{1}{2\pi} \sqrt{\frac{2K}{M}}$The ratio of the frequencies in series to parallel combination is:$\frac{f_s}{f_p} = \frac{\frac{1}{2\pi} \sqrt{\frac{K}{2M}}}{\frac{1}{2\pi} \sqrt{\frac{2K}{M}}} = \sqrt{\frac{K}{2M} \cdot \frac{M}{2K}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$Thus,the ratio is $1:2$.

Two identical springs of constant $K$ are connected in series and parallel as shown in the figure. $A$ mass $M$ is suspended from them. The ratio of their frequencies in series to parallel combination will be

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