cable.html

Sagging Cable

Let m and n be the number of letters in your first and last names, respectively. A cable is stretched from one side of a creek to the other. On the west bank of the gorge the cable is attached to a pole at a height of 10*m+25 feet above the creek. On the east bank of the gorge the cable is attached to a pole at the same height as the west end. The two poles are feet apart. The cable sags so that the middle of the cable is 2*m+n feet lower than either end. The cable hangs in the shape of a catenary; its height is given by the function

where c and k are constants and the center of the cable is at x=0.

Your report: Conduct investigations of this cable and write a report on your investigations. Organize your report, making sure that you investigate each of the following three situations and report on the questions posed in each section. Be sure to state your questions clearly, show your reasoning as well as your calculations, and answer each question.

Problem 1: Investigate this catenary

1. Determine the values of the constants c , and k for your cable. Use the heights of the poles to solve for c . Then use the lowest point of the cable to solve numerically for k . You will need to use the fsolve command to solve for k .
2. Sketch the graph of your cable.
3. Find the length of the cable.

Problem 2: Now suppose you move the poles 1 foot farther apart, but leave them the same height and use the same length of cable. Investigate the cable.

What is the sag in the cable now?

Problem 3: Now replace the poles in their original positions and suspend a horizontal footbridge from this same cable. The cable will assume the shape of a parabola . Investigate this parabola.

1. Find a and c, then determine the sag in the cable. Is this more or less sag than without the bridge?
2. Plot on the same coordinate system, for values of x between and , the catenary of the original cable and the parabola created by suspending a horizontal bridge from the original cable. Compare these graphs over this domain.
3. Plot the catenary and the parabola over a small domain near the origin, such as between x = -10 and x = 10. How do these graphs compare over this domain? Does it seem possible that these graphs have the same length?
4. Plot the catenary and the parabola over a larger domain, such as between x = -1000 and x = 1000. How do these graphs compare over this domain? Discuss the similarities and differences of these graphs.
5. Find all points at which the parabola and the catenary intersect. Can you explain how the cable with greater sag has the same length as the other cable?

The Most Common Maple Commands

Academic Honesty Statement:

Place the following statement (by copying and pasting) at the end of your report and sign it in ink. Your instructor will not grade your report unless this signed statement appears at the end of your report.

I understand that I may work with others if I give them credit in this statement. I also understand that I am required to write my report--that to copy all or part of someone else's report or to allow someone else to copy all or part of my report constitutes plagiarism, which is a serious violation of academic honesty.

I worked with (replace this parenthetical remark with first and last names of those with whom you worked) on this project. I wrote my own report. I did not copy any of this report from anyone else and I did not allow anyone else to copy any of this report.

Signed: