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When a spring with natural frequency $\lambda / 2 \pi$ is driven with a sinusoidal force $\sin (\omega t)$ with $\omega \neq \lambda,$ it oscillates according to $$y(t)=\frac{1}{\lambda^{2}-\omega^{2}}(\lambda \sin (\omega t)-\omega \sin (\lambda t))$$ $$\text{Let}y_{0}(t)=\lim _{\omega \rightarrow \lambda} y(t)$$ (a) Use L'Hopital's Rule to determine $y_{0}(t)$ . (b) Show that $y_{0}(t)$ ceases to be periodic and that its amplitude $\left|y_{0}(t)\right|$ tends to $\infty$ as $t \rightarrow \infty($ the system is said to be in resonance; eventually, the spring is stretched beyond its structural tolerance). (c) $CAS$ Plot $y$ for $\lambda=1$ and $\omega=0.8,0.9,0.99,$ and $0.999 .$ Do the graphs confirm your conclusion in $(b) ?$

          When a spring with natural frequency $\lambda / 2 \pi$ is driven with a sinusoidal force $\sin (\omega t)$ with $\omega \neq \lambda,$ it oscillates according to
$$y(t)=\frac{1}{\lambda^{2}-\omega^{2}}(\lambda \sin (\omega t)-\omega \sin (\lambda t))$$
$$\text{Let}y_{0}(t)=\lim _{\omega \rightarrow \lambda} y(t)$$
(a) Use L'Hopital's Rule to determine $y_{0}(t)$ .
(b) Show that $y_{0}(t)$ ceases to be periodic and that its amplitude $\left|y_{0}(t)\right|$ tends to $\infty$ as $t \rightarrow \infty($ the system is said to be in resonance; eventually, the spring is stretched beyond its structural tolerance).
(c) $CAS$ Plot $y$ for $\lambda=1$ and $\omega=0.8,0.9,0.99,$ and $0.999 .$ Do the graphs confirm your conclusion in $(b) ?$

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