Hello. Im stuck on this physics problem that i have for homework. Its the wording of the question thats odd to me. Im not sure exactly where to begin.

So after one century, the length of each day will be 24 hours and .001 s? And after the second century the length of each day will be 24 hours and .002 s? Is there a quicker way to do this than doing a 20-step summation!!?!? I started off by multiplying .001 x 365, because that will give you the total amount of added time after a year ... however after that it would be long summation, and Im sure that there must be a quicker way.

(FYI, the answer is 2 hours if you want to check your work :P)

The proper way to do this is by integration since the change is continuous.
Let the length of a day in seconds D be a continuous function of time t. D(t) is linear. The number of days in a century is about 365.25*100 = dc. So if the length of the day at t=0 is 24*3600, after dc days D = 24*3600+.001. The slope of the graph of D(t) vs. t is:

[tex]\frac{D(dc)- D(0)}{dc}[/tex] so:

[tex]D(t) = D(0) + \frac{D(dc)- D(0)}{dc}t [/tex]

Integrating D(t) from 0 to 20 centuries gives us the length (L(t)) in days:

all we are doing here is going from the discrete sum to the continuous integration (I know, I know this is not mathematically rigorous, I left out the limiting procedure, but hey, I am a physicist)