To find the distance the pendulum travels in the fifth oscillation, we start with the initial distance for the first oscillation, which is 1/2 inches. The distance decreases by a factor of \( \frac{2}{3} \) for each subsequent oscillation.
Let's calculate the distances for each oscillation:
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First Oscillation (O1):
- Distance: \( 1/2 = 0.5 \) inches
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Second Oscillation (O2):
- Distance: \( O1 \times \frac{2}{3} = 0.5 \times \frac{2}{3} = \frac{1}{3} \approx 0.33 \) inches
-
Third Oscillation (O3):
- Distance: \( O2 \times \frac{2}{3} = \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} \approx 0.22 \) inches
-
Fourth Oscillation (O4):
- Distance: \( O3 \times \frac{2}{3} = \frac{2}{9} \times \frac{2}{3} = \frac{4}{27} \approx 0.15 \) inches
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Fifth Oscillation (O5):
- Distance: \( O4 \times \frac{2}{3} = \frac{4}{27} \times \frac{2}{3} = \frac{8}{81} \approx 0.10 \) inches
To determine the distance of the fifth oscillation to the nearest whole inch, we calculate:
\[ \frac{8}{81} \approx 0.0988 \]
When rounded to the nearest whole inch, the distance is 0 inches.
Thus, in the fifth oscillation, the pendulum will travel approximately 0 inches.