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(A) The time period of a spring-mass system is given by $T = 2\pi \sqrt{\frac{m}{k_{eq}}}$,where $k_{eq}$ is the equivalent spring constant.For config

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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 $T = 2\pi \sqrt{\frac{m}{k_{eq}}}$,where $k_{eq}$ is the equivalent spring constant.For configuration $(i)$: $A$ single spring is used,so $k_{eq,1} = k$. The time period is $T_1 = 2\pi \sqrt{\frac{m}{k}}$.For configuration $(ii)$: Two springs are connected in series. The equivalent spring constant is $\frac{1}{k_{eq,2}} = \frac{1}{k} + \frac{1}{k} = \frac{2}{k}$,so $k_{eq,2} = \frac{k}{2}$. The time period is $T_2 = 2\pi \sqrt{\frac{m}{k/2}} = 2\pi \sqrt{\frac{2m}{k}} = \sqrt{2} T_1$.For configuration $(iii)$: Two springs are connected in parallel. The equivalent spring constant is $k_{eq,3} = k + k = 2k$. The time period is $T_3 = 2\pi \sqrt{\frac{m}{2k}} = \frac{1}{\sqrt{2}} T_1$.Therefore,the ratio of the time periods is $T_1 : T_2 : T_3 = 1 : \sqrt{2} : \frac{1}{\sqrt{2}}$.

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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