catenary

catenary

A 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

(

http://planetmath.org/HyperbolicFunctions

)

function

in a suitable

coordinate system

.

Let’s derive the equation

y

=

y

⁢

(

x

)

of this curve, called the

catenary

, in its plane with

x

-axis horizontal and

y

-axis vertical.  We denote the of the wire by

σ

.

In any point

(

x

,

y

)

of the wire, the

tangent line

of the curve forms an angle

φ

with the positive direction of

x

-axis.  Then,

tan

⁡

φ

=

d

⁢

y

d

⁢

x

=

y

′

.

In the point, a certain tension

T

of the wire acts in the direction of the value

a

.  Hence we may write

T

=

a

cos

⁡

φ

,

whence the vertical of

T

is

T

⁢

sin

⁡

φ

=

a

⁢

tan

⁡

φ

and its

differential

(

http://planetmath.org/Differential

)

d

⁢

(

T

⁢

sin

⁡

φ

)

=

a

⁢

d

⁢

tan

⁡

φ

=

a

⁢

d

⁢

y

′

.

But this differential is the amount of the supporting

σ

⁢

1

+

(

y

′

⁢

(

x

)

)

2

⁢

d

⁢

x

(see the

arc length

).  Thus we obtain the

differential equation

σ

⁢

1

+

y

′

⁣

2

⁢

d

⁢

x

=

a

⁢

d

⁢

y

′

,

(1)

which allows the

separation of variables

:

∫

𝑑

x

=

a

σ

⁢

∫

d

⁢

y

′

1

+

y

′

⁣

2

This may be solved by using the substitution (

http://planetmath.org/SubstitutionForIntegration

)

y

′

:=

sinh

⁡

t

,

d

⁢

y

′

=

cosh

⁡

t

⁢

d

⁢

t

,

1

+

y

′

⁣

2

=

cosh

⁡

t

giving

x

=

a

σ

⁢

t

+

x

0

,

i.e.

y

′

=

d

⁢

y

d

⁢

x

=

sinh

⁡

σ

⁢

(

x

-

x

0

)

a

.

This leads to the final solution

y

=

a

σ

⁢

cosh

⁡

σ

⁢

(

x

-

x

0

)

a

+

y

0

of the equation (1).  We have denoted the

constants of integration

by

x

0

and

y

0

.  They determine the position of the catenary in regard to the

coordinate

axes.  By a suitable choice of the axes and the equation

y

=

a

⁢

cosh

⁡

x

a

(2)

of the catenary.

Some of catenary

•

tan

⁡

φ

=

sinh

⁡

x

a

,

sin

⁡

φ

=

tanh

⁡

x

a

(cf. the

Gudermannian

)

•

The arc length of the catenary (2) from the apex

(

0

,

a

)

to the point

(

x

,

y

)

is

a

⁢

sinh

⁡

x

a

=

y

2

-

a

2

.

•

The

radius of curvature

of the catenary (2) is

a

⁢

cosh

2

⁡

x

a

, which is the same as length of the

normal line

of the catenary between the curve and the

x

-axis.

•

The catenary is the

catacaustic

of the

exponential

curve (

http://planetmath.org/ExponentialFunction

) reflecting the vertical rays.

•

If a parabola rolls on a

straight line

, the focus draws a catenary.

•

The

involute

(a.k.a. the evolvent) of the catenary is the

tractrix

.

Title

catenary

Canonical name

Catenary

Date of creation

2014-10-26 21:25:30

Last modified on

2014-10-26 21:25:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

29

Author

pahio (2872)

Entry type

Derivation

Classification

msc 53B25

Classification

msc 51N05

Synonym

chain curve

Related topic

EquationOfCatenaryViaCalculusOfVariations

Related topic

LeastSurfaceOfRevolution

Related topic

HyperbolicFunctions

Related topic

Tractrix

Related topic

EqualArcLengthAndArea

Defines

catenary

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