Determining the Shape of a Hanging Cable Using Basic Calculus
By: Marco Tavora Ph.D.
Tags: Calculus, Physics, Science, Math

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.”

Figure 1: From left to right, Jacob Bernoulli (source), Gottfried Leibniz (source), Christiaan Huygens (source) and Johann Bernoulli (source).

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.”

Figure 1: The figures submitted by Leibniz (left) and Huygens (right) to Jacob Bernoulli for publication in the Acta Eruditorum (source).

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 w₀).

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 T₀ and T, and its weight w₀s (see figure below). The first two forces are tangent to the chain.

Figure 2: This figure contains the parameters and variables used in the calculation.

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

Equation 1: Equilibrium conditions for the segment with length s.

The differential equation we need to solve is:

Equation 2: Differential equation we need to solve.

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

Equation 3: The derivative of Eq. 2

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

Equation 4: The derivative ds/dx written in terms of dy/dx.

Figure 3: The infinitesimal triangle used in Eq. 4

Eq. 3 then becomes:

Equation 5: Differential equation of the catenary.

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

Equation 6: Definition of u used to solve Eq. 5

Using Eq. 6, Eq. 5 becomes:

Equation 7: Eq. 5 expressed in terms of the variable u.

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

Equation 8: Eq. 7 after integration.

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

Equation 9: The variable u is zero at the minimum of the curve.

Substituting Eq. 9 in Eq. 8 we obtain:

Equation 10: Using Eq. 9 to determine c in Eq. 8.

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