Experimental Rope Logarithms

Sue from Math Mama Writes... sent me an email about wrapping a rope around a pole. In that post, Sue thinks about rope looped around a post. When you wrap a rope around a post, the friction between the rope and the post can help you hold something (like a horse) that is much stronger than you.

The first case she thinks about is using several posts. What if you wrap a rope around one post and pull? What if you use 2, or three posts? The idea is that if one posts 'multiplies' the force by 10, two posts would have an effect of multiplying by 100 and so forth. That seems reasonable.

The next thing, what about one post with just half a turn? What about multiple turns around the same post? This is what I looked at first since it is the easiest (you don't need multiples of the same post).

Setup

Here is a basic diagram of my experimental setup.

i-9b4757d92a7f9457afa0e8d57ca313c3-2010-06-01_untitled_1.jpg

Where these force probes are Vernier probes using Logger Pro. One of the probes is clamped to the table and the other held by me. First, let me test one probe connected to another by a string (no wrapping). Here is a plot of force 1 vs. force 2:

i-9e6a0b0fd36bd3ebc3cc088188bdd77d-2010-06-01_0turnscmbl.jpg

Fitting a linear function, I should get a slope of 1, but I get 1.152. This means that the two probes are not perfectly calibrated (I guess I should have checked this before hand). Now what about half of a turn around the post? (I calibrated the probes - they match a little better now)

i-38f1b8fe8b4508309d1fe801be1897e0-2010-06-01_untitled_2.jpg

So, for 1/2 turn, there is a force difference with the relationship:

i-554f1b4dbcfd552f806f8a2bb6e7b0b3-2010-06-01_la_te_xi_t_1_1.jpg

Now, what about one complete turn?

i-7ee84eb1834ed07bbd7cdc81a1132c35-2010-06-01_untitled_3.jpg

This gives a relationship of:

i-5d54beb4b3366594e80ab2450a4e324d-2010-06-01_la_te_xi_t_1_2.jpg

I tried two loops, but I just could get good data. How about multiple posts? I already have data for one loop around one post. What about two posts?

i-19326291661bfddf37b8e315dfca5e6a-2010-06-01_untitled_4.jpg

Can I get three posts? Yes, I can.

i-5293cbd09034938fcf6aa44b2953b309-2010-06-01_untitled_5.jpg

I am not going to try 4 posts. I know my force probes would be unhappy.

Back to the theory

For multiple posts, the idea is that if 1 posts multiplies the force by a factor of a, then two posts should have a total multiplicative factor of a2 and three would have a factor of a3. In general, if one posts produces a force multiplication of a and the number of posts is n, then:

i-d70ad6272ca8978d4643c82b74115573-2010-06-01_la_te_xi_t_1_3.jpg

Let me take my three whole turn data points. I already have the ratio of F1 to F2 (the slope of the graphs). So, re-writing the above expression, I get:

i-92a8f0e518b37bfb08d66587af99e82f-2010-06-01_la_te_xi_t_1_4.jpg

If I plot the natural log of the ratio of forces vs. the number of wraps, I should get a constant. Here is that plot.

i-d3098d6fd60fc3129e36c13cd1050cb3-2010-06-01_untitled_6.jpg

So, the natural log of a is the slope of this line - .8825. This would give a value for a as 2.41. This should be the same as the slope for one turn, but it is not. Oh well, it isn't too far off.

What about n = 1/2? Does that work? The natural log of the slope for half a turn is 0.385 which is fairly close to half the value of ln(a).

The physics

Physically, how does this work? The normal model for friction says that the frictional force is proportional only to the force the two surfaces are pushing against each other. Not sure if that works here.

Energy? This seems like it is cheating. Do you get more energy than you put in? No, because you don't even put in any energy if the rope doesn't move. This is not a simple machine. What about forces? This seems like it makes more force and thus violates Newton's third law. Well, it doesn't. If I pull with 1 Newton and a horse on the other end pulls with 10 Newtons, these are not "equal and opposite forces". You could think of it this way. The horse pulls on the rope with a magnitude of 10 Newtons. So, looking at the post, the horse pulls one way with 10 N, me the other way with 1 N. This leaves 9 Newtons. The ground exerts 9 Newton force on the post. If the horse were strong enough, it could pull the post out of the ground.

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I suspect the friction law holds. The wrap works only so long as the force between load and belay translate into a force between the line and the post. If whoever is tailing the line let's loose, there is much less friction and the load flies.

For a minute there, it looked additive. Two posts (6.2) looked like about double the effect of one (3.1), and a half post looked like about half the effect (1.5). But then three posts would have been near 9.3, and it wasn't. It was near 18. (I'm glad you able to do it on three posts.)

I'd love to understand this better. But it is way cool to know Sawyer was right.

I'm the author of the post Jlu linked to. The capstan, or rope friction, problem is actually pretty easy to work out from Coulomb's friction law and the equations of staticsâI show the derivation in that post Rope friction is a standard topic in every engineering mechanics book I've seen and taught to all civil and mechanical engineering students.

An important thing to remember is that friction formulas are inequalities, not equalities. They only lead to equations when the system is on the verge of slipping. This makes friction testing tricky, because it's hard to set up a test that puts things constantly in that state. If you're continuously acquiring data, you're not likely to be on the verge of slipping all the time. I suspect that's why the slopes of your graphs don't have the relationship to each other that they should. Also, the fact that your intercepts are jumping around and not always close to zero is a clue that the test setup is a little off.

@Dr. Drang,

Your post was very nice. But really, I think our roles are sort of reversed. It seems you looked at the theoretical aspect of the friction on the rope and I just did the experiment. Honestly, I would like to have done the theoretical approach, but the experiment seemed quicker at the time.