How Does Earth's Slowing Rotation Affect the Length of Our Days Over Centuries?

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SUMMARY

The Earth's rotation is slowing, resulting in an increase in the length of each day by 1.0 millisecond (ms) per century. Over 97 centuries, this results in a cumulative increase in daily time, which can be calculated using the formula for the sum of an arithmetic series. The gain per day, denoted as A, is derived from dividing the total increase by the number of days in a century. The total increase in hours over 97 centuries can be determined by applying the arithmetic series formula to the cumulative gains.

PREREQUISITES
  • Understanding of arithmetic series and their summation
  • Basic knowledge of time measurement (milliseconds, seconds, hours)
  • Familiarity with the concept of Earth's rotation and its effects
  • Ability to perform unit conversions (e.g., from milliseconds to hours)
NEXT STEPS
  • Study the formula for the sum of an arithmetic series
  • Learn about Earth's rotational dynamics and its historical changes
  • Explore time measurement conversions, particularly between milliseconds and hours
  • Investigate the implications of Earth's rotational changes on daily life and scientific measurements
USEFUL FOR

Students in physics or astronomy, educators teaching about Earth's dynamics, and anyone interested in the effects of planetary rotation on time measurement.

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Homework Statement



Because Earth's rotation is gradually slowing, the length of each day increases: The day at the end of 1.0 century is 1.0 ms longer than the day at the start of the century. In 97 centuries, what is the total (in hours) of the daily increases in time (that is, the sum of the gain on the first day, the gain on the second day, etc.)?

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The Attempt at a Solution


I'm having a little trouble getting a grasp on what the question is actually asking. I'm assuming I need to take the 1 sec it changes per century, divide that by 100, then divide that by 365 to find the change per day correct? But I'm not sure even that is correct. I'm feeling a little lost.
 
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Your thoughts on starting are correct. Find the gain per day corresponding to 1ms/century and call it A. Then the gain after the first day is A, the gain after the second day is A+2A, the gain after the third is A+2A+3A. Etc etc. So you finally have A*(1+2+3+4...+N) where N is the number of days in 97 centuries. There is a simple formula for (1+2+3+4...+N) in terms of N. It's an arithmetic series.
 

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