*Question:* What is the shape of a flexible rope hanging from nails at each end and sagging under the gravity?

*Answer*:

First, observe that no matter how the rope hangs, it will have a lowest point (see Fig. 1)

Fig. 1

It follows that the hanging rope can be placed in a coordinate system whose origin coincides with the lowest point and the tangent to the rope at is horizontal:

Fig. 2

At , the rope to its left exerts a horizontal force. This force (or tension), denoted by , is a constant:

Fig. 3

Shown in Fig. 3 also is an arbitrary point with coordinates on the rope. The tension at , denoted by , is along the tangent to the rope curve. is the angle makes with the horizontal.

Since the section of the rope from to is stationary, the net force acting on it must be zero. Namely, the sum of the horizontal force, and the sum of the vertical force, must each be zero:

where is the hanging rope’s mass density and its length from to .

Dividing (2) by (1), we have

Since

, the slope of the curve at ,

and

,

we rewrite (3) as

and so,

i.e.,

To solve (4), let

.

We have

Integrate (5) with respect to gives

i.e.,

Solving (6) for yields

Integrate (7) with respect to ,

.

Hence,

.

Essentially, it is the hyperbolic cosine function that describes the shape of a hanging rope.

*Exercise-1* Show that .

*Exercise-2* Solve for .