Question 39: Oscillations

Question

On a frictionless horizontal plane, a bob of mass $m=0.1 \mathrm{~kg}$ is attached to a spring with natural length $l_{0}=0.1 \mathrm{~m}$. The spring constant is $k_{1}=0.009 \,\mathrm{Nm}^{-1}$ when the length of the spring $l>l_{0}$ and is $k_{2}=0.016 \,\mathrm{Nm}^{-1}$ when $l < l_{0}$. Initially the bob is released from $l=$ $0.15 \mathrm{~m}$. 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 __________.

Accepted Answer

Time period of oscillation
$$
\begin{aligned}
& =\mathrm{T}_1+\mathrm{T}_2 \\
& =\pi \sqrt{\frac{m}{K_1}}+\pi \sqrt{\frac{m}{K_2}} \\
& =\pi \sqrt{\frac{0.1}{0.009}}+\pi \sqrt{\frac{0.1}{0.016}} \\
& =\frac{\pi}{0.3}+\frac{\pi}{0.4} \\
& =\frac{\pi(0.4+0.3)}{0.12} \\
& =\frac{70 \pi}{12} \\
& =5.83 \pi 	ext { seconds } \\
& \simeq 6 \pi 	ext { seconds }
\end{aligned}
$$

Oscillations

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