My recent post on tightropes reminded me of a great "hack." Here's the situation. Your car is stuck in the mud, so you grab a rope and tie it to the front of the car and then the other end to a very sturdy tree. Now for the trick—grab the rope in the middle of the length and pull perpendicular to the rope. Here is a diagram.

It's a cool and useful trick, but how does it work? In short, it's the same as standing on a tightrope. The forces at the point of contact have to add up to the zero vector if it's in equilibrium.

Rope Physics

Let's take a closer look at the point that you would pull on the rope. At this point, there are essentially three forces.

With the contact point in equilibrium, these forces have to add to zero. The only component of force that is interesting is that perpendicular to the rope. Assuming the magnitude of the two tensions is the same, then I get the following expression.

If the distance from the car to the tree has a value of L, then pulling perpendicular a distance of x would give the following for sinθ:

Why am I calling the perpendicular distance "x"? I have no idea, but I'm sticking with it. Now if I substitute this expression for sinθ, I get the following relationship between tension in the rope and the force I pull with.

With the term in front of F as the "force multiplier" (I made that term up). How about an example with actual real numbers? Suppose I have a rope that is 4 meters long and I pull to the side with a force of 20 Newtons such that it is displaced 10 cm. Putting these values into the above expression, I get a tension force of 200 Newtons—or a force multiplier of 10! Not too bad, right?

What if I pull with a force of 20 Newtons but only displace the rope 0.1 cm? Ah ha! I caught you trying to cheat. Yes, if you put these values into the expression above, you would indeed get a huge force multiplier. However, you don't get to pick how far the rope deviates from the straight line. This deviation is actually a factor of the initial rope tension (and of the "springiness" of the rope). Either way, start with a rope at a high tension and then pull to the side to get an even higher tension.

Oh, you aren't getting something for nothing—that's not how physics works. This is essentially a simple machine. You pull with a small force over some distance and get a much larger force out, but that larger force would only move the car a little bit (small distance). If you need to keep the car moving, you would have to re-tie the rope to get the tension back up.

Real Data

But does this really work? I'm not going to go and get my car stuck just to test this out (but you know, that's not a bad idea). Instead, I will do this on a small scale using some force sensors. I will take one force sensor and tie a string to it and a stationary object. Next, I will take a second force sensor and pull to the side of the string. The perpendicular displacement of the string will be measured with cart on track that measures position (this is pretty cool). Here is a picture.

Now I can collect the values of the two forces (the tension in the string and the side pulling force) along with the displacement distance. Here is a plot.

That's pretty cool. The tension force is indeed greater than the sideways force. I will assign a homework question that goes with this data (down below).

Rope Pulling Model

One of the things I have trouble thinking about is the initial tension in the rope. How does this effect the system? Honestly, I'm not too sure—so I'm just going to build a model to play with. My model will be in Python and you can play with it too.

Here is the plan. I'm going to replace the string with two springs. The two springs will have an unstretched length of a little under half the original rope length. This way the two springs will create a tension equivalent to the rope. Then when the middle point (between the two springs) is moved down, the two springs will stretch and increase the force.

Without too much detail (because you can look at the code), here is my two-spring model of a rope pull. Click the "play" button to run and the "pencil" to see (and change) the code.

Notice that the graph does indeed look similar to the actual data above. That's called winning.

Homework

Here are some questions for you to consider.