---
title: "Catenary & Suspension Cable Interactive Lab | Sandbox Physics"
description: "Is every hanging cable a parabola? No. A cable that carries only its own weight hangs in a catenary."
canonical_url: "https://sandboxphysics.com/labs/catenary-cable/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/catenary-cable/"
last_modified: "2026-10-27"
---

M024 · Cable shape and friction

# Catenary & Suspension Cable

**Is every hanging cable a parabola?** No. A cable that carries only its own weight hangs in a catenary. It becomes a parabola when the load is spread evenly across the horizontal span, as when a heavy deck hangs from it.

Hang a cable between two supports at the same height and load it with its own weight, with a bridge deck, or with both. Change the span and sag, compare the shape with a separately computed parabola, check the forces at the anchors and measure the tension along the cable.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The cable is flexible and static. It hangs in one plane from rigid supports at the same height, under vertical loads spread continuously along it. The chosen sag sets how long the cable must be. Elastic stretch, wind, bending stiffness, structural strength and vibration are left out. Hanger spacing and rope thickness are drawn only for illustration; the deck carries the chosen effective line load.

Physics tutorial

## Weight chooses the cable shape

**Background** A hanging rope carries its own weight, spread along its curved length. A suspended bridge deck spreads its weight across the horizontal span instead.

**Why it matters** Each of these two loadings is worked out separately in the literature. Compare them, then solve the balance when both act at once.

Start with the essentials

- **Focus question**: When is the parabola a good model?
- **One-sentence intuition**: Whether the weight is spread along the cable or along the horizontal decides the equation and the shape of the cable.

### Core mathematical model

#### Two ways to distribute weight

$$
H y^{\prime\prime}=\lambda g\sqrt{1+(y^\prime)^2}+q
$$

Rope weight is spread per metre of cable; deck load is spread per horizontal metre. The horizontal part of the tension is the same everywhere.

#### Self weight gives a catenary

$$
y=a[\cosh(x/a)-1],\qquad a=\frac H{\lambda g}
$$

The origin is at the lowest point. The two anchors sit half a span to either side, both raised by the chosen sag above the origin.

#### Horizontal load gives a parabola

$$
y=\frac{q x^2}{2H},\qquad H=\frac{qL^2}{8d}
$$

This exact limit assumes zero rope weight. A light rope under a large deck load approaches it.

#### Tension along the cable and the anchor forces

$$
T=H\sqrt{1+(y^\prime)^2},\qquad 2V=\lambda g\ell+qL
$$

Together, the vertical forces at the two anchors hold up the rope weight and the deck load. The model integrates the cable length numerically, so the drawing does not affect it.

#### Fitting the tension with the model's slopes

$$
T_i^{\mathrm{obs}}\approx\widehat H\sqrt{1+p_i^2}
$$

The model supplies the cable slopes. The fit estimates the horizontal force from simulated noisy tension readings; it does not measure the shape.

### Common difficulties

#### The shape depends on how the load is spread

**Typical misconception** Every hanging cable is exactly a parabola.

**Better mental model** Weight spread per metre of cable gives a catenary; weight spread per horizontal metre gives a parabola.

#### Changing the sag means a new cable

**Typical misconception** When I change the sag, the same inextensible cable stretches.

**Better mental model** Each span and sag needs its own cable length, and the model recomputes it.

#### The dashed parabola is a reference, not a measurement

**Typical misconception** The dashed parabola was reconstructed from sensor readings.

**Better mental model** It is calculated separately, as a parabola with the same span and sag.

### Run the experiment

1. 01

   #### Start with self weight

   Choose Self weight and move the probe from the centre to an anchor.

   **What to observe:** The horizontal part of the tension stays the same, and the total tension grows toward the anchor.

2. 02

   #### Add a bridge deck

   Choose Deck dominated and compare the cable with the dashed parabola.

   **What to observe:** With a light rope, the cable comes close to the parabola.

3. 03

   #### Change the sag

   Drag the orange anchor handle or use its keyboard slider.

   **What to observe:** A shallower cable needs a larger horizontal force and a different length.

4. 04

   #### Measure the tension along the cable

   Add bounded noise, press Scan 25 stations and export the observations.

   **What to observe:** The fitted horizontal force uses the model's slopes; the residual shows how far the readings sit from the fit.

Physical conclusion

Ask how the weight is spread, include every force on the cable, and remember that the tension fit uses geometry the model supplies.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)