Trivial counting problem

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Homework Statement:

Assuming the length of the day uniformly increases by 0.001 second per century, Calculate the net effect on the measure of time over 20 centuries. ( in hours)

Relevant Equations:

Length of the day after 20 centuries is 24 hours and 0.21s
Since the length of day is greater by 0.21 seconds, thus the change in time = ##\frac{0.21*365*100}{60*60}## hours = 2.1 hours.

Here, I do not understand why the length of the day increases by 0.21 seconds instead of 0.2 seconds.
 

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  • #2
Doc Al
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Here, I do not understand why the length of the day increases by 0.21 seconds instead of 0.2 seconds.
I don't understand either of those numbers. If after one century the day is 0.001 second longer, how much longer is a day after 20 centuries?

And once you've got that straight, what's the average increase of a day over those 20 centuries?
 
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  • #4
WWGD
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Since the length of day is greater by 0.21 seconds, thus the change in time = ##\frac{0.21*365*100}{60*60}## hours = 2.1 hours.

Here, I do not understand why the length of the day increases by 0.21 seconds instead of 0.2 seconds.
I assume because the increase by 0.001 seconds can happen at any time in the century , since this has not been specified. So they may be assuming the second(ha-ha) the century starts, the 0.001 second delay is in effect.
 
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  • #5
SammyS
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Isn't it true that 20×(0.001) = 0.020 ?
 
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  • #6
BvU
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It's a series addition exercise. First year 0.001 s/day, second year 0.002 etc.
 
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  • #7
SammyS
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It's a series addition exercise. First year 0.001 s/day, second year 0.002 etc.
The Problem Statement is quite clear.
Problem Statement:​
Assuming the length of the day uniformly increases by 0.001 second per century, Calculate the net effect on the measure of time over 20 centuries. ( in hours)​

As one century elapses, the length of a day will have increased from 24 hours to 24 hours and 0.001 seconds.

It's not a yearly increase of 0.001 seconds per day.
 
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  • #8
BvU
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Length of the day after 20 centuries is 24 hours and 0.21s
Where did you find that 'relevant equation'?
@JC2000: can you do an integration ? or is your curriculum not that far yet ?
 
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I too found the wording of the question awkward, I have posted the solution as in my book. It seems that the question can be interpreted as a summation (each day of the first century increasing by 0.01s and each day of the second century being 24 hours + 0.02s and so on) or as responder SammyS puts it :

As one century elapses, the length of a day will have increased from 24 hours to 24 hours and 0.001 seconds.
Re #2 :

I don't understand either of those numbers. If after one century the day is 0.001 second longer, how much longer is a day after 20 centuries?

And once you've got that straight, what's the average increase of a day over those 20 centuries?
I interpreted this as responder SammyS did (#7). If I understand correctly, the day should be 0.020s longer after 20 centuries with an average increase of 0.001s per century(?) (If the question is interpreted as a summation then the average would be different...)

Re #3 : The same question has been asked previously as pointed out by responder archaic, yet the solution offered by OP differs (2hours as compared to 2.1 hours). OP in that case seems to have interpreted the increase to mean that each day of the century increases by 0.01(?). The solution going by that interpretation would be 2.1 hours (without factoring in leap years?). Now I see that the 'relevant equation' I mention is arrived at by integration in the solutions provided.

I assume because the increase by 0.001 seconds can happen at any time in the century , since this has not been specified. So they may be assuming the second(ha-ha) the century starts, the 0.001 second delay is in effect.
Re #4 : I think that is the interpretation going by the solution. (Just to clarify : If the first century had a day 24 hours long then would the summation be upto 0.019?)

Re #5, #6, #7 : My book interprets this question as a summation (I understand this after going through all of your responses) while I interpreted it as responder SammyS does (#7) if I understand correctly. I thought that the total increase should be 0.020s and confused 0.21s with 0.021s (though 0.21 and 0.020 are arrived at through entirely different routes).

Re #8 : The solution in my book offered no explanation regarding the summation interpretation (once again, I realised this only after reading all the responses). I have a rudimentary understanding of integration ( I could make sense of the solutions on the link provided by archaic).
 
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  • #10
Doc Al
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If I understand correctly, the day should be 0.020s longer after 20 centuries
Right.
with an average increase of 0.001s per century(?)
No, what is the average length of a day above the base of 24 hours? (Analogy: You have a strange job where your hourly rate starts at 0 and increases uniformly by $1 every hour. If you work for 1000 hours, what is your average hourly rate? How much money did you make?)
(If the question is interpreted as a summation then the average would be different...)
How can it be anything but a summation, since you have to add up all those time increments? But it's a simple summation. Sure, you can think of it as integration -- but you won't need calculus. Try to draw the plot of day length (above 24 hours) as a function of time: it's a straight line. Find the area under that line.
 
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I forget which book this is from but I’ve seen it and it’s a horribly worded question. They are not asking for the increase in the length of a day after 20 centuries, they are asking for the sum of the increases in day length per century for 20 years. So their answer is 0.001+0.002+...+0.019+0.02=0.21 seconds. But it’s a stupid question. The increase in length of a day after 20 centuries is 0.02 seconds.
 
  • #12
Doc Al
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I forget which book this is from but I’ve seen it and it’s a horribly worded question. They are not asking for the increase in the length of a day after 20 centuries, they are asking for the sum of the increases in day length per century for 20 years. So their answer is 0.001+0.002+...+0.019+0.02=0.21 seconds. But it’s a stupid question.
If that is the intended meaning of the question (and the given solution) then I must agree that it is a stupid question. o_O

More charitably, one could interpret the problem as follows. Imagine two clocks. One clock stays the same, keeping a day at 24 hours. The other clock slowly and uniformly speeds up so that each day it adds just enough so that at the end of each century the day has grown another 0.001 seconds longer. After 20 centuries, how many additional hours will the second clock show as having passed?
 
  • #13
BvU
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So
Length of the day after 20 centuries is 24 hours and 0.21s
wasn't a relevant equation; it was an excerpt from the problem + 'and 0.21 s', the 0.21 s being the answer to a fuzzy question (but nicely phrased by Al)
 

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