Question

If two similar springs each of spring constant $K _{1}$ are joined in series, the new spring constant and time period would be changed by a factor:

• A. $\frac{1}{2}, \sqrt{2}$
• B. $\frac{1}{4}, \sqrt{2}$
• C. $\frac{1}{4}, 2 \sqrt{2}$
• D. $\frac{1}{2}, 2 \sqrt{2}$

Answer 1

Answer: (A)

$\frac{1}{k_{e q}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}$
$\frac{1}{k_{e q}}=\frac{1}{k}+\frac{1}{k} \Rightarrow k_{e q}=\frac{k}{2}$
$k^{\prime}=\frac{k}{2}$
$ T =2 \pi \sqrt{\frac{ M }{ k }} $
$ T ^{\prime}=2 \pi \sqrt{\frac{ M }{ k ^{\prime}}}$
$ \Rightarrow T ^{\prime}=2 \pi \sqrt{\frac{ M }{ k }} \times \sqrt{2}$
$ T ^{\prime}=\sqrt{2} T $