Do you have an idea for two easy experiments to measure the coefficients of static and sliding friction of a rope wrapped around a steel rod?

Question

I'm researching a phenomenon dependent on the Capstan/Euler-Eytelwein-Equation. To model a simulation, I need the friction coefficients. I came up with the idea of hanging one mass $m_1$ on one side of the rope, and another mass $m_2=4m_1$ on the other one, then wrapping the rope around the rod (with an angle of $\frac {\pi}{2}$ and letting the system slide down. By measuring the time it took $m_2$ to traverse a certain distance I can compute the acceleration acting on the system. By comparing it to the frictionless case I get the force of friction. Does this sound sensible? But what could I do for static friction?

Answer 1

The experimentation seems perfect. Remember, all it requires is just one wrap, the rope going up and over just that once for simplicity. For the static part, you'll only be able to get a real close approximate value, depending on how much precision you put into the experiment. You'll have to place one of the masses on a table so it doesn't pull tension on the rope by its own weight, then the other "mass" mass should be quite close but not equal to the sitting "mass" mass, and that should be suspended by the rope. As you can see the setup isn't at equilibrium and it would have been falling if not for the table. Now you'll have to acquire really small masses (as small as you can make it for precision) and gradually add those to the suspended mass. You'll reach a point where the sitting mass probably start lifting up a tiny bit then falling back down (you're almost there). The point of target is when the sitting mass lifts off the table by a small amount and doesn't fall back, that's the point when the weight of the sitting mass equals the weight of the varying suspended mass plus max static friction. Remember: when doing the math, the normal force is as a result of both eventual masses weight on the rod. I think you should be fine from this point on. I hope this helps, leave any question in the comments below.

Answer 2

I've tried some experiments of this type, and I found it a lot more difficult to get reasonable results than I'd expected. Experiments with friction are infamously hard to reproduce.

For static friction, just use a spring scale to pull a weight over a rod.

I found some helpful information in Blau, Friction science and technology. In experiments on friction, it's important to prepare and clean the surfaces. There are phenomena of breaking in, running in, and wearing in. There can be a complicated variation as a function of time. You may get long periods of constant force and then a transition.

I did experiments using 6 mm nylon cord with a kernmantle construction, running over cylindrical metal rods with a diameter of roughly 5-10 mm (similar to some materials used in rock climbing). With these materials, I did not get good quantitative agreement with the rule that tension varies as $e^{\mu\theta}$.