How Does Earth's Slowing Rotation Affect Day Length Over Centuries?

[ePLUS]

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 61 centuries, what is the total 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 honestly stuck, I'm not sure how to approach this problem, i converted the 61 centuries and added 61 ms to it because the time increases every 100 years by 1.0 ms.

thank you to all that help

 

[andrewkirk]

This replaces my previous answer, which was based on my misreading the question.
The answer will be a number of milliseconds that is the integral, from 0 to $L=61\times 100\times 365.25$, of a linear function that is 0 at 0 and 61 at $L$.
That gives the total time inaccuracy.
My previous answer gave the total clock-speed inaccuracy.

 

[tnich]

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 61 centuries, what is the total 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 honestly stuck, I'm not sure how to approach this problem, i converted the 61 centuries and added 61 ms to it because the time increases every 100 years by 1.0 ms.

thank you to all that help

The way I read the problem, the gain is about 1/(365*100) ms the first day, 2/(365*100) ms the second day, 3/(365*100) ms the third, etc. What does that add up to over 61 centuries? You could correct for leap years, but I think that would be overkill given the (low) precision of the inputs.

 

[Cutter Ketch]

I suppose they could mean 61 ms as you and other responders have replied, but that is not how I read this at all. I believe this a “compound interest” problem. What they seem to be asking is: if you had a clock which was running at the correct speed on day one, how wrong would your clock be in 61 centuries?

Let me try to illustrate how this is different from your solution. Suppose the Earth slowed down just one time right at the beginning and never slowed down again. The Earth is turning 1ms per century slower than it was. From that single event your clock will be off by 1 ms at the end of the first century. However, the Earth is still slower than your clock. Your clock will be off by 2 ms after 2 centuries etc and after 61 centuries it is off by 61 ms. However, that was just one event. What if the Earth slowed down again at the beginning of the second century? Your clock will still have the 61ms error from the first slow down, but it will have AN ADDITIONAL 60 ms error from the second event. Keep doing this and the error at the end of 61 centuries is not 61 ms but 61+60+59 ... +3+2+1 = about 1.8 s which is very different from 61 ms. In analogy to interest, the error compounds. Now that is discrete compounding on the century basis. Your job is to determine what the banker’s call the continuous compounding. The Earth slows down continuously. How wrong is your clock?

 

[Merlin3189]

We're not told that the increase is a linear function of time. Maybe the rate of slowing increases or decreases over time. If it is due to some viscous drag effect, maybe the slowing is proportional to the speed of rotation, so that slowing decreases exponentially and never actually stops.

But we are not told, so perhaps we have to assume linear change. Then averages and summing linear series makes it easy.

 

[jbriggs444]

We're not told that the increase is a linear function of time.

The given information is that it is a linear function of calendar century. It is reasonable to infer that it is a linear function of calendar date. [To a reasonable accuracy over the relevant time frame]

 

[Cutter Ketch]

The given information is that it is a linear function of calendar century. It is reasonable to infer that it is a linear function of calendar date. [To a reasonable accuracy over the relevant time frame]

Hmmm ... you’right. It isn’t explicitly specified. However I think we had better assume a constant rate of angular deceleration. Much more knowledge would be required to justify anything else. I also think that it wouldn’t change the answer much. The velocity hardly changes in 61 centuries. If the torque depends on the velocity (which it does) the change would be small and the result would not differ significantly from the assumption of constant deceleration.

The compounding problem I described was in the assumption of a constant rate of deceleration.