A chain or a homogeneous flexible thin wire takes a form resembling an arc of a parabola when suspended at its ends. The arc is not from a parabola but from the graph of the hyperbolic cosine function in a suitable coordinate system.
Let's derive the equation
of this curve, called the catenary, in its plane with
-axis horizontal and
-axis vertical. We denote the line density of the weight of the wire by
.
In any point
of the wire, the tangent line of the curve forms an angle
with the positive direction of
-axis. Then,
In the point, a certain tension
whence the vertical component of
and its differential
But this differential is the amount of the supporting force acting on an infinitesimal portion of the wire having the projection
![]() |
(1) |
This may be solved by using the substitution
giving
i.e.
This leads to the final solution
of the equation (1). We have denoted the constants of integration by
![]() |
(2) |
Some properties of catenary
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