Hanging cables

Definition

Hanging cables are curves formed by a flexible chain or cable suspended by its ends and acted on by gravity. The shape of a hanging cable is described by the hyperbolic cosine function, also known as a catenary.

5 Must Know Facts For Your Next Test

1. The equation of a hanging cable is $y = a \cosh\left(\frac{x}{a}\right)$, where $a$ is a constant determining the curvature.
2. The hyperbolic cosine function, $\cosh(x)$, is defined as $\cosh(x) = \frac{e^x + e^{-x}}{2}$.
3. A catenary curve minimizes the potential energy of the system, resulting in an equilibrium shape under uniform gravitational force.
4. Unlike parabolas, catenaries have exponential components due to their dependence on hyperbolic functions.
5. Applications of integration can be used to find the length of a hanging cable over an interval using $L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx$ with $f(x) = a \cosh\left(\frac{x}{a}\right)$.

Review Questions

• What is the general equation for the shape of a hanging cable?
• How do you define the hyperbolic cosine function?
• Explain how to use integration to find the length of a hanging cable.

Related terms

Hyperbolic Functions: Functions that include hyperbolic sine ($\sinh$), hyperbolic cosine ($\cosh$), and others which are analogs to trigonometric functions but for hyperbolas.

Catenary: A specific type of curve described by the equation $y = a \cosh\left(\frac{x}{a}\right)$, representing the idealized shape of a hanging flexible chain or cable.

$\cosh$ (Cosine Hyperbolicus): $\cosh(x)$ is defined as $(e^x + e^{-x})/2$, and it describes one component of solutions to certain differential equations modeling physical phenomena like hanging cables.