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 of this curve, called the catenary, in its plane with -axis horizontal and -axis vertical. We denote the 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 of the wire acts in the direction of the value . Hence we may write
whence the vertical of is
and its differential (http://planetmath.org/Differential)
which allows the separation of variables:
This may be solved by using the substitution (http://planetmath.org/SubstitutionForIntegration)
This leads to the final solution
of the equation (1). We have denoted the constants of integration by and . They determine the position of the catenary in regard to the coordinate axes. By a suitable choice of the axes and the equation
of the catenary.
Some of catenary
(cf. the Gudermannian)
The arc length of the catenary (2) from the apex to the point is .
If a parabola rolls on a straight line, the focus draws a catenary.
|Date of creation||2014-10-26 21:25:30|
|Last modified on||2014-10-26 21:25:30|
|Last modified by||pahio (2872)|