Catenary Realism: Calculating Weight of Electric Wire

[Stang]

Homework Statement

when a cable with non-zero mass is connected to a pole at both ends, the shape it assumes is called a catenary.
it can be shown that for an electrical wire whose linear mass density is .9 kg/m strung between poles 30m apart(and making a 22 degree angle at each end) the mathematical equation is
y=(38.1m)[cosh(x/38.1m)-1]
a) an electircal wire of linear mass density .9 kg/m is strung, between poles 30m apart, from west to east on earth. initially the current flowing is zero and therefore the wires shape is that of a catenary. what is the weight of the wire?HINT: the length of the wire(which is not 30m) is found by integrating dl over the catenary. Use the fact the
dl=dx[sqrt(1+(dy/dx)^2) in order to have an integration over the x-axis.

It shows a picture with an equation for the wire. y=(38.1m)[cosh(x/38.1m)-1]

Homework Equations

equation for the wire ...y=(38.1m)[cosh(x/38.1m)-1]
length of the wire... dl=dx[sqrt(1+(dy/dx)^2)
He also gave us all of the equations and proofs of hyperbolic functions.

The Attempt at a Solution

I was not sure what to do since i have never done a problem like this. I was neither shown in class how to do anything close to this.
I started with integrating the length of the wire function. I am having problems with the integration though. I am not sure what to do after this or if its even right what I am doing.

 

[SteamKing]

Well, what sort of problems are you having with the integration of dl?

 

[Stang]

Should i plug the equation of the wire into the length of the wire equation? Then i would derive the equation for the wire and then integrate the dl...

 

[Stang]

I can change the sqrt[1+(dy/dx)^2] to cosh(dy/dx), correct?

 

[paranormal]

I would first commend you for your efforts in attempting to solve this problem. It is always a good thing to push ourselves and try new things, even if we are not familiar with them. However, I understand your frustration and confusion with this problem, as it does involve some advanced mathematics and concepts.

In order to calculate the weight of the wire, we need to first understand the physical properties of a catenary. A catenary is the shape that a hanging cable or wire takes on when it is supported at both ends. This shape is determined by the forces acting on the cable, namely its own weight and the tension from the support points. The weight of the wire is directly related to its length and linear mass density, which is given as 0.9 kg/m in this problem.

To calculate the weight of the wire, we first need to find its length. This can be done by integrating the length function dl=dx[sqrt(1+(dy/dx)^2) over the x-axis. However, this integration can be quite complex and may require specialized techniques. It is possible that your teacher has not taught you these techniques yet, which is why you are having trouble with the integration. In this case, it would be best to consult with your teacher or classmates for help.

Once we have the length of the wire, we can then calculate its weight by multiplying it by the linear mass density. This will give us the total mass of the wire. However, we also need to take into account the angle at which the wire is hanging, as this will affect the tension and thus the shape of the catenary. In this problem, the angle is given as 22 degrees at each end. This information can be used to further refine our calculations.

In summary, calculating the weight of the wire in this problem involves understanding the physical properties of a catenary, integrating the length function over the x-axis, and taking into account the angle at which the wire is hanging. I hope this response has helped clarify the problem for you. If you are still having trouble, I would recommend seeking help from your teacher or classmates, as they may be able to provide more specific guidance and assistance.