Question

A telephone line hangs between two poles 14 m apart in the shape of the catenary , where and are measured in meters.

Answer 1

Answer:

a) At x=14 the slope will be given by:

$\frac{dy}{dx}(14)=a\sinh \left({\frac {14-C_{1}}{a}}\right)$.

b) Then, the angle between the line and the pole will be:

$\phi=\pi - \theta$

where $\theta$ is the angle between the tangent to the catenary and the x-axis.

Explanation:

The catenary has the following general form:

$y(x)==a\cosh \left({\frac {x-C_{1}}{a}}\right)+C_{2}$

a) The slope at any point will be given by the derivative of y.

$\frac{dy}{dx}(x)=a\sinh \left({\frac {x-C_{1}}{a}}\right)$

At x=14:

$\frac{dy}{dx}(14)=a\sinh \left({\frac {14-C_{1}}{a}}\right)$.

b) The angle between the tangent to the catenary and the x-axis at a given point will be given by:

$\frac{dy}{dx}(x)=tan(\theta)$ ⇒ $\theta=tan^{-1} (\frac{dy}{dx}(x))$

Then, the angle between the line and the pole will be:

$\phi=\pi - \theta$.