(notes edited by Roberto Bigoni)

A thin, inextensible, perfectly uniform and flexible rope, suspended between the points
A and B, in condition of equilibrium in the constant Earth field ** g**, assumes the
shape done by a flat curve said

In order to deduce analytically the equation of such curve, we can choose a cartesian orthogonal reference system having origin in the middle point of AB, horizontal axis of abscissas and vertical axis of ordinates.

Let P be a point on arc VP. There are two forces wich act on the punctiform rope element
in position P, whose length be *dl* and mass be *dm*:

- the force
**t**, wich acts leftward, due to the tie A; - the force
**p**, due to the weight of segment VP, which acts downward.

The condition of equilibrium of the point P is due to the fact that the sum of the
tangential components of **t** and **p** is balanced by the reaction due to
inextensibility of the rope, while the
components of **p** and **t** perpendicular to the tangent balance themselves:
therefore their modules are equal.

If we name *θ* the angle the geometrical tangent to the point P forms with
the axis of the abscissas, we shall have

Let be *y(x)* the equation of the curve that describes the disposition of the rope
in the chosen system of reference and *(x ; y)* the cartesian coordinates of the generic
P point in such system: *tan θ* is given by the derivative of
the *y(x)*:

The weight **p** is due to the weight of the segment VP of the rope.

Let *v* be the abscissa of the vertex, *λ* the constant linear density
of the rope
and *g* the module of the gravitational acceleration: the module of the weight **p**
is given by

and therefore

Deriving both the terms of (5)

and therefore

If we introduce the constant *k*:

the differential equation admits the solution

The symbol *cosh* denotes the
hyperbolic cosine.

If we know the coordinates *a* and *b* of B
(and therefore the coordinates of A, symmetrical of B with respect to the origin O)
and the length *L* of the rope (wich obviously must be greater of AB), we have

From (10) we get

and from (11)

Dividing term by term the (12) and (13) we have

and we can express the function (9) in the following form

From (14) we can also deduce

Replacing (16) in (12) we obtain

Given the values of *a*, *b* and *L* , the equation (17) allows
to approximate the value of *k* (different from the banal solution 0).

Since *y(a)=b*, from (15) we have

Annulling the derivative of *y(x)* in (15)

we can get the abscissa v of vertex V

The ordinate of V turns out therefore

The modulus T of the tension, done by the sum of **p** and **t**,
acting on the point P(x,y) of the rope can be calculated
using the Pythagoras's theorem. From (8), (3) e (9) we have

therefore

Since the value of *g* is expressed as 9.81 m/s^{2}, the following Javascript application
requires measures in metric units.

last revision: May 2015