Catenary

Definition

A catenary is the curve formed by a perfectly flexible, uniform chain suspended under its own weight and acted upon by gravity. It is mathematically described by the hyperbolic cosine function.

5 Must Know Facts For Your Next Test

1. The equation of a catenary curve is given by $y = a \cosh\left(\frac{x}{a}\right)$, where $a$ is a constant.
2. The hyperbolic cosine function, $\cosh(x)$, resembles the shape of a parabola but has different properties.
3. Catenaries are used in engineering to design arches and bridges because they naturally form to minimize bending moments.
4. The length of a catenary between two points can be found using integration involving hyperbolic functions.
5. A catenary differs from a parabolic curve which represents the shape of a hanging chain under uniform horizontal force rather than gravitational force.

Review Questions

• What is the mathematical equation for a catenary curve?
• How does the hyperbolic cosine function describe the shape of a catenary?
• In what practical applications might one encounter catenaries?

Related terms

Hyperbolic Functions: Functions that include hyperbolic sine ($\sinh$), cosine ($\cosh$), tangent ($\tanh$), and their inverses. They are analogs of trigonometric functions but for a hyperbola.

$\cosh(x)$: $\cosh(x)$ or hyperbolic cosine is defined as $\cosh(x) = \frac{e^x + e^{-x}}{2}$. It describes the shape of a catenary.

$\sinh(x)$: $\sinh(x)$ or hyperbolic sine is defined as $\sinh(x) = \frac{e^x - e^{-x}}{2}$. It is related to $\cosh(x)$ through various identities.