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 $1/\gamma=2Q/\omega_0=2\times10^5$ 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.