COSH

Application:

The Shape of a Hanging Power Line

Imagine two power poles, both 50 meters tall, spaced 100 meters apart. A power line is suspended between them. We want to model the shape of this power line, which will form a catenary curve. The equation for a catenary is given by:

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

where:

• y is the vertical position of the cable.
• x is the horizontal position from the center point between the two poles.
• a is a parameter that depends on the tension in the cable and its weight per unit length. It's related to the sag of the cable.

Let's assume the lowest point of the cable is 40 meters above the ground. The height of the poles is 50 meters. The sag, which is the vertical distance between the highest and lowest points, is 50−40=10 meters.

We can set up a coordinate system where the origin (0,0) is at the lowest point of the cable. In this case, the equation simplifies to:

$y=a\cosh (\frac{x}{a})-a+40$

Let's find the value of 'a'. The poles are at x=−50 and x=50. The height of the cable at these points is 50 meters. So, we can plug in one of these points to solve for a:

$50=a\cosh (\frac{50}{a})-a+40$

$10=a\cosh (\frac{50}{a})-a$

$10+a=a\cosh (\frac{50}{a})$

$\frac{10}{a}+1=\cosh (\frac{50}{a})$

Solving this equation for 'a' requires numerical methods. The approximate value of 'a' turns out to be around 124.9. Let's use this value to create a table showing the height of the power line at various horizontal positions.

Table: Height of the Power Line at Various Horizontal Positions