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(A) The time period of a spring-mass system is given by $T = 2\pi \sqrt{\frac{m}{k_{eq}}}$,which implies $T \propto \frac{1}{\sqrt{k_{eq}}}$.For confi

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$1:\sqrt{2}:\frac{1}{\sqrt{2}}$

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(A) The time period of a spring-mass system is given by $T = 2\pi \sqrt{\frac{m}{k_{eq}}}$,which implies $T \propto \frac{1}{\sqrt{k_{eq}}}$.For configuration $(i)$: The equivalent spring constant is $k_{eq1} = k$. Thus,$T_1 \propto \frac{1}{\sqrt{k}}$.For configuration $(ii)$: The two springs are in series,so $\frac{1}{k_{eq2}} = \frac{1}{k} + \frac{1}{k} = \frac{2}{k}$,which gives $k_{eq2} = \frac{k}{2}$. Thus,$T_2 \propto \frac{1}{\sqrt{k/2}} = \sqrt{\frac{2}{k}} = \sqrt{2} \cdot \frac{1}{\sqrt{k}}$.For configuration $(iii)$: The two springs are in parallel,so $k_{eq3} = k + k = 2k$. Thus,$T_3 \propto \frac{1}{\sqrt{2k}} = \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{k}}$.Comparing the ratios: $T_1 : T_2 : T_3 = 1 : \sqrt{2} : \frac{1}{\sqrt{2}}$.

Five identical springs are used in the following three configurations. The time periods of vertical oscillations in configurations $(i)$,$(ii)$ and $(iii)$ are in the ratio:

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