Evaluate the following series.
The first swing of the pendulum is 25cm. if each swing is 75% of the preceding swing how far does the pendulum travel before coming to a rest?
This is a geometric series problem.
Recall that 1+x+x2+x3+... = 1/(1-x) whenever |x| < 1
In your problem we have that the total distance travelled in centimeters is:
25(1 + 0.75 + 0.752 + 0.753 + ...) = 25/(1 - 0.75) = 25/0.25 = 100
This is a limit problem, similar to how many steps does it take to get to one if your first step you take half the distance and the next you take half of the previous distance. You will never get to the destination, but with each step you will get infinitesimally close.
Now to the problem. You must determine how close to zero you want to get. I let the distance be 0.0001cm, arbitrarily. You could let it be smaller and solve for that n, or larger and solve for than n. It depends on how close you want to get
The formula to determine how many swings it takes to get to 0.0001cm is:
25*(0.75)^n = 0.00010, divide both sides by 25
(0.75)^n = 4.0*(10)^-6 take the natural logarithm of each side
n*ln (0.75) = ln 4.0*(10)^-6 divide by ln (0.75)
n = ln 4.0*(10)^-6/ln (0.75)
n= 43 This is the number of swings to reach the .0001 cm swing.
The total distance travelled is, (I'll give you the first ten), 25 + 18.83 +14.06 + 10.54 + 7.91 + 5.93 +4.45 +3.33 +2.50 + 1.8.......
Note the significant figures in your answer. You will approach a number that changes very little with each successive addition. It depends upon how far you want to push the limit.