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\(.........\)

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Answer & Solution

Correct Answer

(D) \(2\)

Detailed explanation

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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