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
- The equation of a catenary curve is given by $y = a \cosh\left(\frac{x}{a}\right)$, where $a$ is a constant.
- The hyperbolic cosine function, $\cosh(x)$, resembles the shape of a parabola but has different properties.
- Catenaries are used in engineering to design arches and bridges because they naturally form to minimize bending moments.
- The length of a catenary between two points can be found using integration involving hyperbolic functions.
- 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?
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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.
