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(B) The time period of a spring-mass system is given by $T = 2\pi \sqrt{\frac{m}{k}}$,where $m$ is the mass and $k$ is the spring constant.The spring

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$\frac{T}{\sqrt{2}}$

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(B) The time period of a spring-mass system is given by $T = 2\pi \sqrt{\frac{m}{k}}$,where $m$ is the mass and $k$ is the spring constant.The spring constant $k$ is inversely proportional to the length $L$ of the spring $(k \propto \frac{1}{L})$.When the spring is cut into two equal halves,the length of each half becomes $\frac{L}{2}$. Therefore,the new spring constant $k'$ becomes $2k$.The new time period $T'$ is given by $T' = 2\pi \sqrt{\frac{m}{k'}} = 2\pi \sqrt{\frac{m}{2k}}$.Dividing $T'$ by $T$,we get $\frac{T'}{T} = \frac{2\pi \sqrt{\frac{m}{2k}}}{2\pi \sqrt{\frac{m}{k}}} = \frac{1}{\sqrt{2}}$.Thus,the new time period is $T' = \frac{T}{\sqrt{2}}$.

$A$ spring has a certain mass suspended from it and its period for vertical oscillation is $T$. The spring is now cut into two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillation is now

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