Discussion
A spiral mode has a real root of \( +0.020 \ \mathrm{s^{-1}} \). Its time constant and time to double amplitude are
Answer: D
The two measures are different and both are used:
\[ \begin{aligned} \tau &= \frac{1}{0.020} = 50 \ \mathrm{s} \\ t_{2} &= \frac{\ln 2}{0.020} = 34.7 \ \mathrm{s} \end{aligned} \]
so 50 s and 35 s. The time constant is the time to grow by a factor of \( e \), the doubling time to grow by a factor of two, and since \( \ln 2 < 1 \) the doubling time is always the shorter. Reporting both as 50 s ignores the logarithm, and the third option swaps them. Requirements on the spiral are written in doubling time, so the distinction decides compliance.
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