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(D) For figure $(a)$,the springs are in series. The equivalent spring constant $K_{eq}$ is given by:$K_{eq} = \frac{K 	imes 2K}{K + 2K} = \frac{2K}{3

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$2$

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(D) For figure $(a)$,the springs are in series. The equivalent spring constant $K_{eq}$ is given by:$K_{eq} = \frac{K 	imes 2K}{K + 2K} = \frac{2K}{3}$The period of oscillation $T$ is:$T = 2\pi \sqrt{\frac{m}{K_{eq}}} = 2\pi \sqrt{\frac{m}{2K/3}} = 2\pi \sqrt{\frac{3m}{2K}} = 3 	ext{ s}$For figure $(b)$,the springs are in parallel. The equivalent spring constant $K'_{eq}$ is:$K'_{eq} = K + 2K = 3K$The period of oscillation $T'$ is:$T' = 2\pi \sqrt{\frac{m}{3K}}$Dividing $T'$ by $T$:$\frac{T'}{T} = \frac{2\pi \sqrt{\frac{m}{3K}}}{2\pi \sqrt{\frac{3m}{2K}}} = \sqrt{\frac{m}{3K} 	imes \frac{2K}{3m}} = \sqrt{\frac{2}{9}} = \frac{\sqrt{2}}{3}$Given $T = 3 	ext{ s}$,we have:$T' = 3 	imes \frac{\sqrt{2}}{3} = \sqrt{2} 	ext{ s}$Comparing $T' = \sqrt{x} 	ext{ s}$ with $T' = \sqrt{2} 	ext{ s}$,we get $x = 2$.

As per the given figures,two springs of spring constants $K$ and $2K$ are connected to a mass $m$. If the period of oscillation in figure $(a)$ is $3 \text{ s}$,then the period of oscillation in figure $(b)$ will be $\sqrt{x} \text{ s}$. The value of $x$ is $.........$

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