Hanging cables

Definition

Hanging cables, or catenaries, are curves formed by a cable suspended under its own weight and subject to hyperbolic functions. The shape of these cables can be described mathematically using the hyperbolic cosine function $\cosh(x)$.

5 Must Know Facts For Your Next Test

1. The equation for a hanging cable is $y = a \cosh(\frac{x}{a})$, where $a$ is a constant that depends on the physical properties of the cable.
2. The curve described by a hanging cable is called a catenary.
3. $\cosh(x)$ and $\sinh(x)$ are essential hyperbolic functions used to describe hanging cables.
4. Hanging cables provide real-world applications for integration techniques in calculus.
5. The hyperbolic cosine function $\cosh(x)$ has properties similar to those of the trigonometric cosine function but applies to different types of problems.

Review Questions

• What is the mathematical equation that describes the shape of a hanging cable?
• Which hyperbolic function is primarily used in describing hanging cables?
• How does the constant 'a' in the catenary equation affect the shape of the curve?

Related terms

Catenary: A curve formed by a perfectly flexible chain suspended between two points under its own weight, described mathematically by $y = a \cosh(\frac{x}{a})$.

$\cosh(x)$: $\cosh(x)$, or hyperbolic cosine, is defined as $\frac{e^x + e^{-x}}{2}$ and describes the shape of catenaries.

$\sinh(x)$: $\sinh(x)$, or hyperbolic sine, is defined as $\frac{e^x - e^{-x}}{2}$ and is often used alongside $\cosh(x)$ in problems involving hyperbolic functions.