The formula to calculate the sag of a catenary curve (S) is:
\[ S = \frac{w \cdot d^2}{8 \cdot H} \]
Where:
A catenary curve describes the shape of a flexible chain or cable that hangs between two points under its own weight and is acted upon by gravity. The curve has a specific mathematical form and is relevant in various engineering and architectural applications, such as the design of suspension bridges and overhead power lines. Understanding the sag of a catenary curve is crucial for ensuring the structural integrity and functionality of such systems.
Let's consider an example:
Using the formula to calculate the sag of the catenary curve:
\[ S = \frac{10 \cdot 20^2}{8 \cdot 5} = \frac{10 \cdot 400}{40} = 100 \, \text{length} \]
This demonstrates that with a weight per unit length of 10, a horizontal distance between supports of 20, and a horizontal force of 5, the sag of the catenary curve would be 100 units of length.
