Catenary

Definition

A catenary is the curve formed by a perfectly flexible chain suspended by its ends and acted on by gravity. Mathematically, it is described by the hyperbolic cosine function.

5 Must Know Facts For Your Next Test

1. The equation of a catenary in Cartesian coordinates is $y = a \cosh\left(\frac{x}{a}\right)$, where $a$ is a constant.
2. The shape of the catenary minimizes potential energy, making it an example of a variational problem.
3. Catenaries have applications in architecture and engineering, such as in the design of suspension bridges and arches.
4. In calculus, the arc length of a catenary can be determined using integration techniques involving hyperbolic functions.
5. The derivative of $\cosh(x)$ is $\sinh(x)$, which is useful in deriving properties related to the catenary.

Review Questions

• What is the general equation for a catenary curve?
• How does the derivative of $\cosh(x)$ relate to properties of the catenary?
• What are some practical applications of catenaries in engineering or architecture?

Related terms

Hyperbolic Cosine: $\cosh(x)$ is defined as $\frac{e^x + e^{-x}}{2}$ and describes the shape of a catenary.

Hyperbolic Sine: $\sinh(x)$ is defined as $\frac{e^x - e^{-x}}{2}$ and is the derivative of $\cosh(x)$.

Arc Length: The length along a curve between two points, often calculated using integrals.