Catenary

A catenary is the curve formed by a perfectly flexible, uniform chain hanging under its own weight. Its shape is described by the hyperbolic cosine function.

Hyperbolic cosine: The function cosh(x) = (eˣ + e⁻ˣ)/2, whose graph gives the shape of a catenary. The catenary’s equation is a scaled, shifted hyperbolic cosine.
Hyperbolic functions: Functions related to exponential functions that describe hyperbolas and appear in many differential equations. The catenary belongs to this family through its cosh equation.
Suspension bridge: A bridge whose deck hangs from cables supported by towers and anchored at both ends. Its main cables approximate a catenary when their own weight dominates.
Parabola: A U-shaped or inverted curve defined by a quadratic equation. A cable under uniform horizontal loading forms a parabola, unlike a chain under its own weight.
Static equilibrium: A state in which forces and torques balance, leaving a body at rest or moving uniformly. A hanging chain’s shape is set by force balance at every point.
Differential equation: An equation relating a function to one or more of its derivatives. Force balance can be expressed as a differential equation for the chain’s profile.
Overhead power line: A conductor suspended between supports to transmit electrical power. The conductor’s sag follows a catenary when weight per unit length is uniform.
Elastic curve: The deflected shape of a bent elastic beam under applied loads. A stiff beam resists bending, whereas an ideal chain has no bending stiffness.
Tension: A pulling force transmitted through a stretched rope, cable, string, or similar object. Changing tension along the chain counteracts the weight of each segment.
Linear mass density: The mass of a one-dimensional object per unit length. For a uniform chain, this quantity sets the load that shapes the catenary.

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