You might look at an overhead power line and see an engineering problem. After all, those transmission towers are impressively huge. But if you've ever seen those cables, you probably noticed they seem to hang fairly low. Why they hang low is a great physics question that can be modeled with masses and springs.
Let's start by creating a model. Suppose I string a cable between two points so it is supported horizontally from the ends. Obviously the total force on this cable must be the zero vector because the cable is in equilibrium. This means the sum of the forces from the two end points must be equal in magnitude to the gravitational force. Yes, a gravitational force acts on the cable because the cable has mass. Massless cables exist only in physics homework problems.
But what about the forces acting on only a small portion of the cable? Imagine that I can isolate a section near the middle, where the cable hangs lowest. Since this piece also is in equilibrium, the net force must be zero (zero vector).
You can see that three forces act on this piece of cable. Gravitational force pulls down, of course, with a magnitude dependent upon the value of the gravitational field and the mass of the section of cable. Tension provides the other two forces, which should be tangent on the cable and the same magnitude at the bottom.
Seeing this, you understand why the cable must droop. Because the cable has mass (and therefore a gravitational force pulling down), there must be an upward component of the tension to keep it at rest. A truly horizontal cable would have no upward force to balance the downward gravitational force. Yes, you can reduce the sag by increasing the tension, but you'll never get the cable horizontal.
You can mathematically derive the sag in a cable, an exercise that gets you the famous catenary cable shape. You often find this in advanced mechanics courses. Of course, I like doing things a little differently, so I will derive the sag with a Python model using masses and springs, which looks something like this:
I thought about drawing a diagram, but found writing a program easier. But how does it work? Look at each mass (well, not the two on the ends) and you see four forces acting on them:
- The gravitational force pulling straight down.
- The spring force from the stretch or compression of the spring on the left.
- The spring force from the stretch or compression of the spring on the right.
- A drag force proportional to the mass velocity. This ensures the masses don't continue moving around forever.
The great thing about these forces is you can calculate them at any instant since they depend on the position and velocity of the mass. Once I know the total force on a particular mass, I can determine the change in momentum and the change in position. Yes, doing this for many masses quickly grows tedious, which is why I use a computer.
You could make a numerical model that looks cool, but unless you compare it to actual data, you've created a video game, not a useful model. I'm a physicist, not a game developer, so let's get some data. Instead of a cable, I am going to use a string of beads because you can find them all over the place in Louisiana.
No, a string of beads is not a cable. But it has a fair amount of mass that makes it easier to measure the tension using a force probe. So here's the plan: I will measure the tension on one side of the cable and the vertical drop in the middle (the sag). Then I can change the span distance and measure again.
For comparison, I will use a numerical model with the same mass and length as the string of beads. In this case, I broke it into 20 pieces. It looks like this:
If you want to play with the code, just click the pencil icon. I'll suggest a few changes in the homework questions below.
Now I can make the span size of my model the same as the beads and compare the tension and vertical drop.
It's not a perfect match, but close enough for me.
If you want to experiment a little, let me offer some things to consider.
- What happens if you increase or decrease the number of masses in the numerical model? Warning: Things can get crazy with a large number of masses. If this happens, try refining your code.
- The numerical model uses small springs between the masses. What happens if you change the spring constant?
- One problems I experienced was determining the length of the cable. In my method, all of the masses start off in a line between the two end points. Of course, the cable is longer than the distance between the endpoints, which leads to some weird behavior at the beginning. See if you can revise the code to prevent this.
- I used fixed points in my code. What happens if one of the end points is free? You shouldn't have much difficulty modifying the code to do this.
- Find the mathematical equation for a catenary cable, and see how well the mathematical function and the numerical model agree with each other.
- What happens to the magnitude of the tension as you move down the cable?
- Try modeling a cable with end points at different heights.
- Take a look at an actual overhead power line. Estimate the diameter and density of the cable and determine the tension needed to achieve the drop height you see.
