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 sin?:

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 sin?, 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.

- Make a new plot. On this plot show the theoretical force multiplier as a function of perpendicular displacement. Use the actual data above to also calculate this experimentally. Finally, add data to this plot from the python model.
- See if you can adjust the python model to give results that are very close to the experimental data.
- What happens to the force-multiplier as you increase the initial tension in the string? Use the python model to explore this.
- What if you have a slack string? What happens then? You can build this into the python code by having the unstretched length of the two springs greater than the distance between fixed points. Be careful not to make the springs push, but rather just pull.

This process has been used by sailors for centuries. It’s called “sweating” a line, and is used, for example, to raise a sail. The halyard (the line that raises the sail), is attached to the head of the sail, and to a hard point on the mast, then is pulled horizontally from the mast, taking out the slack at the hard point as the sail goes up.

Does anyone ever do these homework questions?

maybe, you want to add twist to the center point and let the rope do the work for you

I prefer a Spanish Windlass. They used it to raise masts on old sailing ships. Use two ropes, put a tree branch between the two ropes near the center, and wind it up. The rope knots up and pulls. Until somebody comes along with another vehicle or a Mule (2 HP) to pull you all the way out.

I shot a mule buck deer in 2012 that was so big, that while strung up, his nose drug on the butcher’s floor like an Elk. When I got to the butcher where many animals are unloaded, I said, “I weigh 185 pounds and I got that animal in the back of my vehicle by myself.” No one believed it. But I had used a rope and the rope trick of this article. I pulled the rope in the middle sideways like plucking a guitar string. I used a second rope for syncing the position after each pull.

Very good idea, but again the ropes you use would need to be made with steel wire or the elasticity encountered would mean you would need an incredibly large balloon structure before mitigating the elastic effect.

If this were a problem solving question, with rope, my answer would be to thread one end of the rope through the attachment loop on the car, the other end to the tree, and cut off 3 small sections of the rope from the unused end at the tree.

Bring the end of the threaded rope out of the car tie in loop, 3/4 way back towards the tree, here use one of the 3 short lengths to wrap around the tree to loop section making a sliding gripping knot (called a prussic knot), by repeating this, a 2:1 or 3:1 or as many steps are included, you can create an effective levered pulley system by rope alone.

A real world example of this is a simple lifting operation with a crane where you have a load that is held by two rigging straps. The hoist line of the crane is at the midpoint where that would be your third rope pulling sideways to the two straps that are holding the load. Every rigger knows that as the straps are shorter and the angle goes down closer to the load the tension on each strap is logarithmically higher. At a 30 degree angle of the strap to the load, the tension on each strap is as much as the tension on the Hoist line. Check it out.

I just happened to have gotten my car stuck in the mud yesterday and just happened to try this exact trick. It didn’t work at all and the reason is the difference between theoretical science and real world engineering. After getting stuck I went to the store and bought some long tow straps and a winch. I tied one end to a huge tree and one end to my stuck car and the winch sat nicely between them. Before using the winch I tried the sideways pull trick, out of curiosity, and found that in the real world the tow straps stretch and as they stretch the force angles change such that I lost much of force gain I was hoping for. With my 30 foot run I was able to stretch about 3 feet sideways by pulling with pretty good force (maybe 100lbs). Your formula shows the force multiplier of about 2.5 which generates a nice stretching force but not really pulling a car out of the mud force. So I did use the winch. It took quite a while to drag it six feet uphill over a slippery slope, but overall the winch worked out. A different real world situation, for example, with a steel cable or with less mud and side force trick might have worked just fine.

I’d just call AAA.

Pretty sure you want two trees, with one of them being the end point for the vehicle, otherwise the car would travel roughly towards the origin of the force (the center of the rope in this case).

I did the math on this when planning a zip line for a dog run.

I ran aircraft cable between two trees. The dog is attached to the cable via a pulley and some chain. I went with 3/8 cable after doing the math. Every fall before the snow flies I have to tighten up the cable because she has stretched it. That’s 3 times so far. I intend to replace it after 5 years if it lasts that long. I haven’t got a good way to estimate how much it is weakened when she stretches it out.

#huskyownerproblems

That’s a much better idea. Pull, don’t push. Use a rope instead of a contraption. But I think you still might need another rope depending oh how far you need to pull the tree. Like what if you pull the rope so far from strait that your leverage advantage starts to disappear. Then you need to tie new rope between the anchor and the lead tree to keep it from pulling back when you release the perpendicular rope and start the cycle over again to move it as far as needed.

Similar to the Gotham tightrope equation from yesterday! ðŸ˜‰

Likewise, rather than use a winch puller to lead a tree I am trying to fell toward an anchor tree in a specific direction, I could use this technique to do this in a much more safe manner. I could use a ratcheting puller to get a fair amount of tension on the first cable (between the felling tree and an anchor tree in the desired fall path) and then have a second cable with a winch or ratcheting cable pulling the first cable to a third tree in a perpendicular direction to the angle of fall. Sweet! (Yes, of course I will be wearing all of my safety gear!)

Interesting idea. What you need is 3 ropes between the anchor and the car. The main rope is just a rope with a cinch and anchor that stops the car from slideing backwards, but lets it pull forwards only. Then you use the two other ropes connecting the anchor to the car as your simple machine with any sort of expanding force between the two to separate the two ropes. In my idea, you’d use a simple inflatable ballon in a bag to convert air pressure into an expanding balloon between the two pull ropes.

So once you build this 3 rope system, you can inflate and deflate the ballon and each inflation or deflation of the ballon spreads the two pull ropes apart and pulls the car forward. Then you pull the middle third rope tight so the car can’t slip backwards, deflate the ballon or if you’re not using a ballon to separate the ropes (maybe you’re using a car jack to separate the ropes), then you release the car jack and let the two pull ropes relax, and you retighten the pull ropes when there’s nothing separating them and then you repeat.