# Determining the Shape of a Hanging Cable Using Basic Calculus

How to Derive the Equation of the Catenary

A perfectly flexible chain in equilibrium suspended by its ends and subject to gravity has the shape of a curve called the catenary. The name was coined in 1690 by the Dutch physicist, mathematician, astronomer, and inventor, Christiaan Huygens in a letter to the prominent German polymath Gottfried Leibniz.

The catenary is similar to a parabola which led the great Italian astronomer, physicist, and engineer, Galileo Galilei, the first to study it, to mistakenly identify its shape as a parabola. The correct shape was obtained independently by Leibniz, Huygens, and the Swiss mathematician Johann Bernoulli in 1691. All of them were responding to a challenge proposed by the Swiss mathematician Jacob Bernoulli (Johann's older brother) to obtain the equation of the “chain-curve.”

The figures that Leibniz and Huygens sent to Jacob Bernoulli are shown below. They were published in the *Acta Eruditorum, *the* “*first scientific journal of the German-speaking lands of Europe.”

Johann Bernoulli was delighted that he had successfully solved the problem his older brother Jacob failed to solve. Twenty-seven years later, he wrote in a letter:

The efforts of my brother were without success. For my part, I was more fortunate, for I found the skill (I say it without boasting; why should I conceal the truth?) to solve it in full…. It is true that it cost me study that robbed me of rest for an entire night. It was a great achievement for those days and for the slight age and experience I then had. The next morning, filled with joy, I ran to my brother, who was struggling miserably with this Gordian knot without getting anywhere, always thinking like Galileo that the catenary was a parabola. Stop! Stop! I say to him, don’t torture yourself any more try ing to prove the identity of the catenary with the parabola, since it is entirely false.

— Johann Bernoulli

# Finding the Equation of the Catenary

To find the equation of the catenary the following assumptions are made:

- The chain (or cable) is suspended between two points and hangs under its own weight.
- The chain (or cable) is flexible and has a uniform linear weight density (equal to
*w*₀).

The treatment here follows closely the book by Simmons. To simplify the algebra, we will let the *y*-axis pass through the minimum of the curve. The length of the segment from the minimum to the point (*x*, *y*) is denoted by *s. *The* *three forces acting on the segment are the tensions *T*₀ and *T, *and its weight *w*₀s (see figure below). The first two forces are tangent to the chain.

For the segment to be in equilibrium horizontally and vertically, the following two conditions must be obeyed:

The differential equation we need to solve is:

We now have to re-write this equation in terms of *y* and *x *only. We first differentiate it to obtain:

The derivative *ds/dx* can be written in terms of *dy/dx* as follows:

Eq. 3 then becomes:

To quickly solve Eq. 5 we conveniently introduce the following variable:

Using Eq. 6, Eq. 5 becomes:

This equation can be integrated by variable separation and a simple trigonometric substitution *u *= tan *θ*:

Since the *y*-axis pass through the minimum of the curve we have:

Substituting Eq. 9 in Eq. 8 we obtain:

Substituting *c*=0 into Eq. 8 and solving for *u* we obtain:

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