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A simple pendulum with a small spherical bob oscillates in air; its amplitude falls from 10 cm to 8 cm in 40 s. Assuming Stokes' law governs the damping and the ratio of the viscosity of air to that of carbon dioxide is 1.3, estimate the time for the amplitude to decrease from 10 cm to 5 cm when the same pendulum oscillates in carbon dioxide. (Use ln 5 = 1.601, ln 2 = 0.693.)

1. 161 s
2. 231 s
3. 142 s
4. 208 s

Correct answer: 161 s

Solution

In air: ln(10/8) = k_air * 40, so k_air = ln(1.25)/40. Since damping is proportional to viscosity, k_CO2 = k_air/1.3. Time in CO2 for 10->5 cm: t = ln(2)/k_CO2 = ln(2)*1.3/k_air. This evaluates to about 161 s.

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