On a frictionless horizontal plane, a bob of mass $m=0.1 kg$ is attached to a spring with natural length $l_0=0.1 m$. The spring constant is $k_1=0.009 Nm ^{-1}$ when the length of the spring $I > l_0$ and is $k_2=0.016 Nm ^{-1}$ when $ I < l_0$. Initially the bob is released from $l=0.15 m$. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T=(n \pi) s$, then the integer closest to $n$ is. . . . .
IIT 2022, Advanced
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$\ell > \ell_0 \rightarrow k=k_1$
$\ell < \ell_0 \rightarrow k=k_2$
Time period of oscillation,
$T=\pi \sqrt{\frac{ m }{ k _1}}+\pi \sqrt{\frac{ m }{ k _2}}$
$T=\pi \sqrt{\frac{0.1}{0.009}}+\pi \sqrt{\frac{0.1}{0.016}}$
$T =\frac{\pi}{0.3}+\frac{\pi}{0.4} \Rightarrow T=\frac{0.7}{0.12} \pi \Rightarrow T=5.83 \pi$
$T\approx 6 \pi$
So, $n=6$
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