Question

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 $l>l_0$ and is $k_2=0.016\, Nm ^{-1}$ when $l < l_0$. 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 ______

Answer 1

$ \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$