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Catenary Hoodie

THE CHAIN FINDS THIS CURVE | Pixelated Physics

\[y = a\cosh(x/a)\]
$54.99 USD
$54.99 USD
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What shape does a hanging chain make? Not a parabola, though it looks close. A uniform flexible chain settles into the catenary, y = a cosh(x/a), the shape with the least potential energy for its length. Jakob Bernoulli posed the problem in 1690, and Huygens, Leibniz and Johann Bernoulli solved it in 1691. Black pixel-art hoodie; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Returns & replacements21 days

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The physics

A chain hanging freely under its own weight takes the shape of a hyperbolic cosine, the catenary. That shape minimizes the chain's potential energy, and the tension at each point is proportional to its height.

A uniform flexible chain hangs under gravity between two supports. What shape does it take?

Not a parabola. The shape that minimises potential energy for a given length is the catenary y = a cosh(x/a). Local tension satisfies T = μ g y.

A parabola through the same apex and endpoints is close — the largest gap is about 8.7 px, roughly 8.7 % of the 99.9 px sag — but the Euler–Lagrange identity y/√(1+y′²) = a holds only for the cosh.

On the front, the filled arch is the catenary; the thin magenta ghost is the matched parabola for contrast.

Jakob Bernoulli posed the problem in 1690; Huygens, Leibniz and Johann Bernoulli published the cosh solution in 1691.

Equations

\[y = a\cosh(x/a)\]
\[T = \mu g y,\qquad V = \mu g\int y\,ds\]
SymbolMeaningUnit
\(y\) height of the cable above the directrix \(\mathrm{m}\)
\(x\) horizontal distance from the vertex \(\mathrm{m}\)
\(a = T_0/(\mu g)\) shape parameter (horizontal tension over weight per length) \(\mathrm{m}\)
\(T\) local tension \(\mathrm{N}\)
\(\mu\) mass per unit length \(\mathrm{kg/m}\)
\(g\) gravitational acceleration \(\mathrm{m/s^2}\)
\(V\) gravitational potential energy of the cable \(\mathrm{J}\)

Sources

  1. Bernoulli, Leibniz, Huygens (1691) solutions to the hanging-chain problem posed by Jakob Bernoulli
  2. Halliday, Resnick & Walker, Fundamentals of Physics (catenary / hanging cable notes in continuum mechanics supplements)
  3. Taylor, Classical Mechanics (2005), variational problems and the catenary
  4. Weisstein, Catenary, MathWorld

Source Bernoulli, Johann (1691). Solutio problematis funicularii. Acta Eruditorum, June 1691 (with the solutions of Leibniz and Huygens in the same issue).

Field notes

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