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  4. Consider two identical springs each of spring constant k and negligible mass compared to the mass M as shown. Fig. 1 shows one of them and Fig. 2 shows their series combination. The ratios of time period of oscillation of the two SHM is ( T b / T a )=√ x , where value of x is . (Round off to the Nearest Integer) <img class=img-fluid question-image alt=image src=https://cdn.tardigrade.in/img/question/physics/163a13ea60ff0a903ab528f534f5add9-.png />

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Q. Consider two identical springs each of spring constant $k$ and negligible mass compared to the mass $M$ as shown. Fig. 1 shows one of them and Fig. 2 shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac{ T _{ b }}{ T _{ a }}=\sqrt{ x }$, where value of $x$ is ________. (Round off to the Nearest Integer)
image

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Solution:

$T _{ a }=2 \pi \sqrt{\frac{ M }{ K }}$
$T _{ b }=2 \pi \sqrt{\frac{ M }{ K / 2}}$
$\frac{ T _{ b }}{ T _{ a }}=\sqrt{2}=\sqrt{ x }$
$\Rightarrow x =2$