Question

On a frictionless horizontal plane, a bob of mass $m=0.1kg$ is attached to a spring with natural length $l_0=0.1m$. The spring constant is $k_1=0.009N{m}^{-1}$ when the length of the spring $l>l_0$ and is $k_2=0.016N{m}^{-1}$ when $l<l_0$. Initially the bob is released from $l=0.15m$. 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

Answer: 6

$\ell >{\ell }_0\to k=k_1$
$\ell <{\ell }_0\to k=k_2$
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, $n=6$