# Consider a damped oscillator with p < No. There is a

Chapter 5, Problem 5.25

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QUESTION:

Consider a damped oscillator with $$\beta<\omega_{0}$$. There is a little difficulty defining the "period" $$\tau_{1}$$ since the motion (5.38) is not periodic. However, a definition that makes sense is that $$\tau_{1}$$ is the time between successive maxima of $$x (t)$$.

(a) Make a sketch of $$x (t)$$ against $$t$$ and indicate this definition of $$\tau$$ on your graph. Show that $$\tau_{1}=2 \pi / \omega_{1}$$.

(b) Show that an equivalent definition is that $$\tau_{1}$$ is twice the time between successive zeros of $$x(t)$$. Show this one on your sketch.

(c) If $$\beta=\omega_{\mathrm{o}} / 2$$, by what factor does the amplitude shrink in one period?

### Questions & Answers (2 Reviews)

QUESTION:

Consider a damped oscillator with $$\beta<\omega_{0}$$. There is a little difficulty defining the "period" $$\tau_{1}$$ since the motion (5.38) is not periodic. However, a definition that makes sense is that $$\tau_{1}$$ is the time between successive maxima of $$x (t)$$.

(a) Make a sketch of $$x (t)$$ against $$t$$ and indicate this definition of $$\tau$$ on your graph. Show that $$\tau_{1}=2 \pi / \omega_{1}$$.

(b) Show that an equivalent definition is that $$\tau_{1}$$ is twice the time between successive zeros of $$x(t)$$. Show this one on your sketch.

(c) If $$\beta=\omega_{\mathrm{o}} / 2$$, by what factor does the amplitude shrink in one period?

Step 1 of 6

(a) The condition for the displacement of the damped oscillator when $$\beta<\omega_{\circ}$$ is given as,

$$x(t)=A e^{-\beta t} \cos \left(\omega_{1} t-\delta\right)$$ ………. (1.1)

Here, $$A$$ is the amplitude, $$\beta$$ is the decay parameter, $$t$$ is the time, $$\omega_{1}$$ is the angular frequency, and $$\delta$$ is the phase.

The graph between displacement of the damped oscillator and the time:

Here, $$\tau_{1}$$ is the time between the two successive maxima.

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### Review this written solution for 101862) viewed: 778 isbn: 9781891389221 | Classical Mechanics - 0 Edition - Chapter 5 - Problem 5.25

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Textbook: Classical Mechanics

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