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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?

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(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?

ANSWER:

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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