Question:
If two similar springs each of spring constant $\mathrm{K}_{1}$ are joined in series, the new spring constant and time period would be changed by a factor:
Correct Option: 1
Solution:
$\frac{1}{\mathrm{k}_{e q}}=\frac{1}{\mathrm{k}_{1}}+\frac{1}{\mathrm{k}_{2}}$
$\frac{1}{k_{e q}}=\frac{1}{k}+\frac{1}{k} \Rightarrow k_{e q}=\frac{k}{2}$
$k^{\prime}=\frac{k}{2}$
$\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{M}}{\mathrm{k}}}$
$T^{\prime}=2 \pi \sqrt{\frac{M}{k^{\prime}}}$
$\Rightarrow T^{\prime}=2 \pi \sqrt{\frac{M}{k}} \times \sqrt{2}$
$\mathrm{T}^{\prime}=\sqrt{2} \mathrm{~T}$
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