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

[AnkhUNC]

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.)?

Homework Equations

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.

 

[Dick]

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.

 

[Moetasim]

It seems like the question is asking for the total amount of time gained over 97 centuries, not just the daily increase. To calculate this, we can use the formula for the length of a day: T = 24 hours + (1 ms/86,400 s) * t, where t is the number of days. We know that at the end of 1 century, the day is 1 ms longer, so we can plug in t = 36,525 days (since there are 100 years in a century and 365.25 days in a year). This gives us T = 24 hours + 1 ms, or 24 hours and 0.001 seconds.

Now, we can use this value for T and plug in t = 36,525 * 97 days to find the total amount of time gained over 97 centuries. This gives us T = 24 hours + 97 ms, or 24 hours and 0.097 seconds. This means that over 97 centuries, the day has increased by 97 milliseconds.

To convert this to hours, we can divide 97 ms by 1000 to get the change in seconds, and then divide by 60 to get the change in minutes, and finally divide by 60 again to get the change in hours. This gives us a total of 0.000027 hours, or about 0.098 seconds.

So, over 97 centuries, the total increase in the length of a day is approximately 0.098 seconds. This may seem like a small amount, but it is important to consider the long-term effects of this gradual change on Earth's rotation and the impact it may have on our daily lives.