Project II, MATH 105

Click to download the TeX file for this project.

The Problem:

Suluclac Construction Company is building a suspension bridge over the Platte River. They need to know how much material will be required to construct the main support cables and what sort of cable they need to buy. The support cables will be attached at either end to the top of 100 meter tall concrete pillars. The two concrete pillars are 200 meters apart. The cable should hang down 50 meters at its lowest point. Gottfried Leibniz and Christian Huygens in 1691 determined that any cable hanging under the force of gravity must have the shape of the graph

y(x) = a cosh (x/a) + b

This shape is known as a catenary. The parameter a is the ratio of cable tension to cable density and cosh(x) = (ex + e-x)/2}. The only use of the parameter b is to provide a vertical shift, if necessary. SCC would like to hire your group to find two things for them. First, what values must a and b have in order for the catenary to fit the constraints imposed by the placement of the concrete pillars and the low point of the cable? They are especially interested in the parameter a since this tells them what tension the cable will be under. Second, what length cable do they need? You should try to give a formula for the cable length in terms of the cable function y(x). That way, SCC can use your result for other cable shapes as well.

Following are several hints for solving this problem. When you write your report for SCC, you must explain to them, step by step, how you solved the problems. You will need to use a combination of graphs, equations and text. You must try to convince them that your results are correct. It's no use to them if they have to solve the problem themselves in order to verify your results.

Hints for First Problem:

Graph the catenary for a=1 and b=0. Based on your result, you should construct a coordinate system for the bridge problem. Where will you put the y-axis? the x-axis?

To find both a and b you will have to solve two simultaneous equations. When x=0 what must y be? This information should give you one equation. What will x be at the left pillar? At the right pillar? What will y be at the left pillar? At the right pillar? What equation do you get from this information? Can you make a substitution to get one equation just involving a? How could you solve this equation using a graph or using the bisection method in Appendix A?

Hints for Second Problem:

First get several different approximations of the length. For the first approximation, go halfway across the bridge (that is, go half the length between the two pillars) and mark the point on the cable that is directly above you. Find the straight line distance from the top of the left pillar to that point and then the straight line distance from that point to the top of the right pillar. Using these values, find an approximation for the length of the cable. How does your answer compare to the true length? (Remember, the distance between two points (x_1,y_1) and (x_2,y_2) is given by the formula {(x_2 - x_1)2 + (y_2 - y_1)2}1/2

Next, go one quarter of the way across the bridge and mark the point on the catenary above you. Similarly, mark the points half way across and three quarters of the way across. Now find the straight line distances between each two consecutive points (so you should be finding 4 lengths in all). Is your total length greater than or less than the true length?

How would you make this approximation better? Write a formula (consider using summation notation) for your approximation when you use n points. What do you have to do to get the exact answer? Write a formula to express this. Is this the formula for a definite integral? If not, can you make it into one? (Hint: you might try pulling out a factor of delta-x).

What formula do you finally get for the length of the cable? Test this formula on cable shapes whose length you can determine using other methods. You might think about including these tests in your report, since they may help convince SCC that you have done the right thing.