Правка разделов «Statement», «Solution», «Answer»
en/3.5.38.md
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| @@ -1,10 +1,10 @@ | |||
| ### Statement | |||
| − | $3.5.38.$ [Insert the problem statement] | ||
| + | $3.5.38.$ The figure shows the dependence of the square of the amplitude of the velocity of forced oscillations on the frequency of the forcing force, the amplitude of which is constant. Determine the natural frequency of the oscillator, its damping coefficient and quality factor. | ||
| ### Solution | |||
| − |  | ||
| The energy (and hence the square of the amplitude of the velocity) of a weakly-damped forced oscillator, where $\gamma\ll\omega_0$, near resonance is proportional to | |||
| \[\frac{1}{(\omega-\omega_0)^2+\gamma^2}.\] | |||
| This means that the resonance curve peaks at $\omega=\omega_0$ and the width of the curve at half of the maximum height is $2\gamma$. Reading from the provided graph, we have $\omega_0\approx540$ s$^{-1}$ and $\gamma\approx60$ s$^{-1}$, and the quality factor is $Q=\omega_0/(2\gamma)\approx4.5$. | |||
| @@ -14,4 +14,4 @@Solution | |||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | $\omega_0\approx540$ s$^{-1}$, $\gamma\approx60$ s$^{-1}$, $Q\approx4.5$ | ||
| @@ -1,10 +1,10 @@ | |||
| ### Statement | ### Statement | ||
| $3.5.38.$ [Insert the problem statement] | $3.5.38.$ The figure shows the dependence of the square of the amplitude of the velocity of forced oscillations on the frequency of the forcing force, the amplitude of which is constant. Determine the natural frequency of the oscillator, its damping coefficient and quality factor. | ||
| ### Solution | ### Solution | ||
|  | ||
| The energy (and hence the square of the amplitude of the velocity) of a weakly-damped forced oscillator, where $\gamma\ll\omega_0$, near resonance is proportional to | The energy (and hence the square of the amplitude of the velocity) of a weakly-damped forced oscillator, where $\gamma\ll\omega_0$, near resonance is proportional to | ||
| \[\frac{1}{(\omega-\omega_0)^2+\gamma^2}.\] | \[\frac{1}{(\omega-\omega_0)^2+\gamma^2}.\] | ||
| This means that the resonance curve peaks at $\omega=\omega_0$ and the width of the curve at half of the maximum height is $2\gamma$. Reading from the provided graph, we have $\omega_0\approx540$ s$^{-1}$ and $\gamma\approx60$ s$^{-1}$, and the quality factor is $Q=\omega_0/(2\gamma)\approx4.5$. | This means that the resonance curve peaks at $\omega=\omega_0$ and the width of the curve at half of the maximum height is $2\gamma$. Reading from the provided graph, we have $\omega_0\approx540$ s$^{-1}$ and $\gamma\approx60$ s$^{-1}$, and the quality factor is $Q=\omega_0/(2\gamma)\approx4.5$. | ||
| @@ -14,4 +14,4 @@Solution | |||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | $\omega_0\approx540$ s$^{-1}$, $\gamma\approx60$ s$^{-1}$, $Q\approx4.5$ | ||