Infer damping from peak energies

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Problem: Infer damping from peak energies
For a freely decaying linear oscillator, two turning points on the same side are separated by four complete damped cycles and 8.0 s. Their spring energies are 2.0 mJ and 0.50 mJ. The damping force is proportional to velocity.
Question: What amplitude decay rate γ is consistent with these data?
Givens:
- Same-side peak energies: 2.0 mJ and 0.50 mJ
- Elapsed time: 8.0 s
- Amplitude envelope proportional to exp(−γt)
Assumptions: Constant mass, spring stiffness and positive viscous damping. Compare equal-phase turning points, where all mechanical energy is spring energy; do not assume instantaneous total energy is a pure exponential at every phase.
Choices (LaTeX):
A. \frac{\ln4}{8}\ \mathrm{s^{-1}}
B. \frac{\ln2}{2}\ \mathrm{s^{-1}}
C. \frac14\ \mathrm{s^{-1}}
D. \frac{\ln2}{8}\ \mathrm{s^{-1}}

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