OBSERVATION CHAMBER / ACTIVE
Observe the static catenary of a uniform flexible chain and read how span, chain length, linear density, and gravity set its sag and support tension.
01 / OBSERVE THE CHAIN
A finite catenary will appear here when Canvas2D is available.
03 / READ THE FORCE BALANCE
04 / RECORD THE OBSERVATION
Compare the cyan catenary with the violet parabola. The parabola is close for a shallow chain, but it is not the exact equilibrium curve.
05 / CHECK THE NUMERICS
Self-test has not run.
No report yet.
MODEL NOTE
Assume a uniform, inextensible, flexible chain. With equal-height supports, symmetry gives the lowest point at midspan. Every small chain element balances its horizontal and vertical tension components.
EQUATION NOTE
The arc-length constraint is S = 2a sinh(D / 2a). This page solves it by bisection for the positive a. The residual is recomputed from the accepted value.
LOAD NOTE
For fixed D and S, changing μ or g changes w = μg and scales H, V, and T linearly. It does not change a, sag, or the catenary-versus-parabola comparison.
FORMULA / CAVEAT
y(x) = a cosh((x − D/2) / a) − a cosh(D / 2a)
S = 2a sinh(D / 2a), sag = 2a sinh²(D / 4a)
H = wa, V = wS/2, T = wa cosh(D / 2a) = √(H² + V²)
The catenary is the exact static shape for a uniform chain under gravity. A parabola matches the local shallow-sag approximation. It becomes less faithful as S/D grows.
HOW TO OBSERVE
This observatory runs a deterministic static catenary model in your browser. Span, chain length, sag, tension, calculations, reports, and images stay local. It does not measure visitors, materials, structures, or real loads.
This chamber belongs to Structures & Symmetries.
STRUCTURE Back to Structures & Symmetries Route.