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 54 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.)?
2. Relevant equations
Sum of an arithmetic sequence (see below)
The Attempt at a Solution
I've seen this question "answered" a number of times via a google search (for example, https://www.physicsforums.com/showthread.php?t=208510"), but nobody seems to ask the more fundamental question: What are they even asking? Most attempts at answering the question use an arithmetic sequence. For example, if the day is 1ms longer at the end of each century, then at the first day it was 1/36525 ms (365.25 days/year, 100 years/century) longer than at the beginning of the first century, and at the second day it was 2/36525 ms longer than at the beginning of the first century, and more generally at the n'th day it was n/36525 ms longer than at the beginning of the first century. So the sum of all these gains is (1 + 2 + 3 + ... + k) / 36525, where k is the total number of days in 54 centuries.
I have three problems with all of these solutions.
First of all, it makes no sense. The answer you get is meaningless in the context of physics and has no physical interpretation that I can come up with. This is normal in a math book to have problems with no physical interpretation, but I'm suspicious of seeing a physics problem like this.
Second, I'm not sure what this formula is that people are using for the sum of an arithmetic sequence. In the link above, for example, someone claims that the sum is (n/2) · [2·a1 + (n-1)·d]. What are a1 and d? And what happened to n*(n+1) / 2?
Third, the wording of the question says "what is the total (in hours) of the daily increases in time". The daily increases. The increases between each day. The increases between each day are constant, 1/36525 ms. So the sum of the daily increases for a century should be 1ms, and the sum of the daily increases for 54 centuries should be 54 ms, or 1.5 × 10^-5 hours.
Anyone have any insight on what the heck this question is talking about?
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