SINH

Application:

The Catenary Curve

Imagine a power line, a clothesline, or a chain hanging between two poles. The curve it forms is not a parabola, as is often mistakenly assumed. Instead, it's a catenary, and its shape can be described by a function that includes the hyperbolic cosine, cosh(x), which is closely related to sinh(x). The catenary's vertical position, y, at a horizontal distance, x, from its lowest point is given by the equation:

$y=a\cosh \left(\frac{x}{a}\right)$

Here, 'a' is a constant that depends on the tension in the cable and its weight per unit length.

The sinh(x) function comes into play when we consider the slope of the catenary curve at any given point. The slope of the catenary is given by the derivative of the position function with respect to x:

$\frac{dy}{dx}=\frac{d}{dx}\left[a\cosh \left(\frac{x}{a}\right)\right]=a\cdot \frac{1}{a}\sinh \left(\frac{x}{a}\right)=\sinh \left(\frac{x}{a}\right)$

So, the slope of the hanging cable at any point is directly proportional to the sinh of its horizontal position. This is crucial for engineers who need to calculate the tension in the cable at different points.

Example Table: Calculating the Slope of a Catenary

Let's assume we have a power line with a lowest point at (0,10) meters and a characteristic constant a=10 meters. We can use the sinh(x) function to find the slope of the cable at various horizontal distances from the lowest point.

The slope at a horizontal distance x from the center is given by:

$\text{Slope}=\sinh \left(\frac{x}{10}\right)$