The Geometry of Catenary Curves

Author:: Justin Mears

Topic:: Calculus, Geometry

Introduction:

A flexible cable always hangs in the shape of a catenary given by

where

and

are constants and

. Below, you will find the graphs of several members of the family of functions

for various values of

in light blue and a slider that allows you to investigate specific members of the family, given in red.

Question 1

How does the graph change as varies? You must be very specific with how you describe the change and in what way is varying.

Question 2

What happens if we reintroduce the constant into the family? What type of transformation will occur to the graphs of the family for each value of ?

Question 3

Plot several members of the family on the graph below with

Question 4

Using principles from physics, it can be shown that when a cable is hung between two poles, it takes the shape of a curve that satisfies the differential equation (d.e)

where

is the linear density of the cable,

is the acceleration due to gravity,

is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function

is a solution of this differential equation.

Question 5

Plot the solution function on the graph below and play around with the shape of the graph as you vary both

and

Question 6

Setting and varying , how does the shape of the graph change?

Question 7

Setting and varying , how does the shape of the graph change?

Question 8

A cable with linear density kg/m is strung from the tops of two poles that are 200 m apart. (a) Use the previous questions to find the tension so that the cable is 60 m above the ground and its lowest point. How tall are the poles?

(b) If the tension is doubled, what is the new low point of the cable? How tall are the poles now?