Правка разделов «Statement», «Solution», «Answer»
en/3.5.39.md
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| ### Statement | |||
| − | $3.5.39.$ [Insert the problem statement] | ||
| + | $3.5.39.$ A sapphire monocrystal with quality factor $Q=10^9$ and natural frequency $\omega_0=10^4$ s$^{-1}$ can be used for resonance detection of small forcing forces. How long (in order of magnitude) should we wait for the monocrystal to establish oscillations? | ||
| ### Solution | |||
| − | From $Q=10^9$ and $\omega_0=10^4$ s$^{-1}$, we have $\gamma=\omega_0 | ||
| + | From $Q=10^9$ and $\omega_0=10^4$ s$^{-1}$, we have $1/\gamma=2Q/\omega_0=2\times10^5$ s. The amplitude of the transient response decays according to the exponential $e^{-\gamma t}$, which becomes less than $5$% for $t\approx3/\gamma$. Thus, the steady-state response is firmly established after roughly $6\times10^5$ s. An order-of-magnitude estimate is $10^5-10^6$ s. | ||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | $10^5-10^6$ s | ||
| @@ -1,11 +1,11 @@ | |||
| ### Statement | ### Statement | ||
| $3.5.39.$ [Insert the problem statement] | $3.5.39.$ A sapphire monocrystal with quality factor $Q=10^9$ and natural frequency $\omega_0=10^4$ s$^{-1}$ can be used for resonance detection of small forcing forces. How long (in order of magnitude) should we wait for the monocrystal to establish oscillations? | ||
| ### Solution | ### Solution | ||
| From $Q=10^9$ and $\omega_0=10^4$ s$^{-1}$, we have $\gamma=\omega_0 |
From $Q=10^9$ and $\omega_0=10^4$ s$^{-1}$, we have $1/\gamma=2Q/\omega_0=2\times10^5$ s. The amplitude of the transient response decays according to the exponential $e^{-\gamma t}$, which becomes less than $5$% for $t\approx3/\gamma$. Thus, the steady-state response is firmly established after roughly $6\times10^5$ s. An order-of-magnitude estimate is $10^5-10^6$ s. | ||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | $10^5-10^6$ s | ||