Due: Wednesday, January 29
The Catenary*
If the ends of a chain are attached to the points (-1, 0) and (1, 0), the chain will take the shape of the curve (called a catenary) given by:
where the constant a depends upon the length of the chain.
(A) Show that for any value of a, the graph of y = f(x) passes through the two points (-1, 0) and (1, 0).
(B) Find the length of this chain as a function of a. (Hint: You will need to use the arc length formula for a curve. If you have not learned this yet in calculus, then look it up in your text. Alternatively, you may use the built-in ArcLength function in the Student[Calculus1] library.)
(C) Find the value of a for which the chain has length 4. Plot the graph of this curve. (Note: Use the fsolve command here.)
(D) For which value of a does the chain hang 1 unit below the x-axis at its lowest point?
(E) Find the length of the chain using the value of a that you obtained in part (D).
(F) Discuss briefly the history of the hanging chain (catenary)
problem. Describe the contributions of Jacob Bernoulli, Christiaan
Huygens, Galileo Galilei, Gottfried Leibniz, Joachim Jungius, David Gregory
and Johann Bernoulli.
* This problem is a modification of one in Discovering Calculus with Mathematica by Knoll, Shaw, Johnson, and Evans, Wiley (1995).
This catenary is part of a Vertical
Motion Simulator at the NASA Ames Research Center.