As per given figures, two springs of spring constants $K$ and $2\,K$ are connected to mass $m$. If the period of oscillation in figure $(a)$ is $3 s$, then the period of oscillation in figure $(b)$ will be $\sqrt{ x }$ s. The value of $x$ is$.........$
JEE MAIN 2022, Medium
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For figure $(a):$
$K _{\text {oq }}=\frac{ K \times 2 K }{ K +2 K }=\frac{2 K }{3}$
$T =2 \pi \sqrt{\frac{ m }{ K _{ eq }}}=2 \pi \sqrt{\frac{ m }{2 K / 3}}=2 \pi \sqrt{\frac{3 m }{2 K }}$
For figure $(b):$
$K _{ eq }=3 K , T ^{\prime}=2 \pi \sqrt{\frac{ m }{3 K }}$
$\frac{ T ^{\prime}}{ T }=\sqrt{\frac{ m \times 2 K }{3 K \times 3 m }}=\frac{\sqrt{2}}{3}$
$T ^{\prime}=\sqrt{2}$
$x =2$
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