To calculate the sag (S) in a hanging cable, multiply the weight per unit length (**w**) by the square of the distance between the support points (**d²**). Then divide the result by 8 times the horizontal tension (**H**). This will give you the vertical sag of the cable.

The Catenary Equation Calculator is a tool used to determine the sag or curvature of a hanging chain or cable between two fixed points. This mathematical concept, known as a catenary curve, has been studied and applied in various fields, including architecture, engineering, and physics.

### Formula:

`$S = \frac{w \times d²}{8 \times H}$`

Variable | Meaning |
---|---|

S |
Sag (vertical distance between the highest and lowest points) |

w |
Weight per unit length of the cable |

d |
Horizontal distance between the two support points |

H |
Horizontal tension in the cable |

**Solved Calculations :**

**Example 1:**

**Given Values**:

**w**= 10 kg/m**d**= 20 meters**H**= 1000 N (Newtons)

Calculation | Instructions |
---|---|

S = (10 × 20²) / (8 × 1000) | Multiply the weight by the square of the distance, then divide by 8 times the tension. |

S = (10 × 400) / 8000 | Perform the multiplication and division. |

S = 4000 / 8000 | Perform the final division. |

S = 0.5 meters | The result gives the sag in meters. |

**Answer**: S = 0.5 meters

**Example 2:**

**Given Values**:

**w**= 5 kg/m**d**= 15 meters**H**= 500 N

Calculation | Instructions |
---|---|

S = (5 × 15²) / (8 × 500) | Multiply the weight by the square of the distance, then divide by 8 times the tension. |

S = (5 × 225) / 4000 | Perform the multiplication and division. |

S = 1125 / 4000 | Perform the final division. |

S = 0.28125 meters | The result gives the sag in meters. |

**Answer**: S = 0.28125 meters

**What is Catenary Equation Calculator ?**

The **catenary equation** is used to describe the curve of a cable or chain suspended under its own weight between two fixed points. This curve is important in applications such as power lines, suspension bridges, and any system that involves cables.

This calculator let you easily calculate the **catenary sag**, which is the vertical displacement between the highest point of the curve and the lowest. With the help of this you can easily find solutions to the **catenary equation between two points**, ensuring that your cable systems are safe and within load-bearing limits.

Moreover, it helps in determining other key measurements, such as the length and tension of the cable, by using formulas like the **catenary length calculator** or **catenary tension calculator**.

Whether you’re designing for large infrastructure projects or smaller mechanical systems, this tool provides accurate results, enabling you to calculate everything from **sag curves** to **catenary constants**. This ensures that your designs are optimized for stability and strength, with all measurements complying with standard engineering practices.

**Final Words:**

The **Catenary Equation Calculator** is a valuable tool for calculating the shape, sag, and length of a catenary curve, typically used in mechanics and engineering applications. With this calculator, engineers can easily find solutions to **catenary equations** between two points, ensuring safe and efficient designs for power lines, suspension bridges, and other structures that rely on precise tension calculations.