Catenary

Calculated Worked Example

Let's verify the calculations for a specific setup. We will use the exact parameters and formulas implemented in this module to find the properties of the catenary.

1. Set Catenary scale a = 2, Half-span L = 2, and Probe fraction u = 1.
2. First, calculate the actual x-coordinate of the probe. Since the probe fraction u = 1, the x-coordinate is x = u * L = 1 * 2 = 2.
3. The selected point height is given by a * (cosh(x / a) - 1). Substituting the values:
   2 * (cosh(2 / 2) - 1) = 2 * (cosh(1) - 1).
   Since cosh(1) ≈ 1.543080634815, the height is 2 * 0.543080634815 = 1.08616126963.
4. The selected slope is the derivative sinh(x / a). At x = 2:
   sinh(2 / 2) = sinh(1) ≈ 1.17520119364.
5. The total cable length is given by the integral of the arc length from -L to L, which simplifies to 2 * a * sinh(L / a).
   Substituting the values: 2 * 2 * sinh(2 / 2) = 4 * sinh(1) ≈ 4.70080477458.

Interactive Experiment

Use the Catenary interactive module to explore how changes in tension, weight, and span affect the shape and length of the hanging chain.

1. Set Catenary scale a to 1, Half-span L to 1, and Probe fraction u to 0. Observe the selected point at the lowest point (0,0). The support sag is cosh(1) - 1 ≈ 0.543081 and the total length is 2 * sinh(1) ≈ 2.350402.
2. Change the Probe fraction u to -1. The selected point height now equals the right support sag, and the slope is -sinh(1), showing the reflection symmetry of the curve.
3. Set Catenary scale a to 0. The module rejects the nonpositive scale, demonstrating that tension must be strictly positive to maintain the shape; division by zero is avoided.
4. Restore Catenary scale a to 1, then set Half-span L to 0 with Probe fraction u at 1. The supports coincide at (0,0), and the span, sag, length, and slope all become 0. This is labeled the collapsed-span limit.
5. Set Catenary scale a to 0.5, Half-span L to 3 and Probe fraction u to 1. Read finite sag approximately 100.357818061 m and length 201.713157370 m; both supports and the low point fit the independent labelled axes.
6. Set Catenary scale a to 3, Half-span L to 0.00000001 and Probe fraction u to 1. The small positive sag is about 1.66666666667e-17 m, displayed in scientific notation rather than replaced with zero.
7. Compare the solid cyan catenary with the dashed parabola. While they look similar near the origin, they diverge significantly further out.

Model Limitations

This module calculates the exact theoretical geometric properties of a static, uniform, flexible, and inextensible chain under uniform gravity. It assumes equal-height supports.

It is not a dynamic rope solver; it does not simulate swinging, elasticity, or non-uniform mass distribution. The displayed visual is scaled automatically to fit the span and sag within the viewing area, with independently scaled horizontal and vertical axes. Thus visual angles and pixel aspect ratios are not physical measurements; distances do not correspond directly to meters unless referenced against the labeled axes. Both a and L are in meters, while u and slope are dimensionless. The collapsed L = 0 state is a geometric limit, not a finite loaded chain. Results use standard IEEE 754 floating-point arithmetic, retaining at least six significant digits, but may exhibit standard rounding differences.

References

For further derivation of the differential equation and the properties of hanging chains:

• LibreTexts: The Catenary — Provides the statics derivation showing how balancing tension and weight leads to the hyperbolic cosine shape.