1
$\begingroup$

So, I was solving this problem1 and I realized that the system given seems to be non-deterministic when analysed with classical laws.

enter image description here

The situation is thus: A rope is wrapped for an angle of $\theta_0$ (symmetrically) around a fixed pole (the circle in the diagram). One end is fixed to a wall, and the other end is pulled with a force/tension $T_0$. The string is massless, but there is friction between the string and the pole of coefficient $\mu$. There is no slipping (the system is static). If necessary, the length of the string and the radius/relative position of the pole can be considered as givens.

Now, if you follow my solution2, one gets that the tension in the string along the cylinder, where $\theta=0$ is the part on the right where the string just touches follows the following constraints: $T_0e^{-\mu\theta}\leq T(\theta)\leq T_0e^{\mu\theta}$. Extending this, you get that the tension $T_1$ in the part of the string connected to the wall follows the constraint $T_0e^{-\mu\theta_0}\leq T_1\leq T_0e^{\mu\theta_0}$.

However, I don't see any way to further analyse this system with Newtonian mechanics. Note that the inequality appears even if I consider a string with mass, though the calculation becomes more complicated.

So basically I'm stuck with an inequality. Note that the system, as given, should be completely defined — $\theta_0$, $T$,$\mu$, the radius of pole, and the relative positioning of pole are all that should be required to physically create this system. However, the macroscopic, measurable quantity $T_1$ somehow cannot be determined from Newton's laws.

How does one resolve this apparent paradox? Classical systems are supposed to be able to fully determine measurable quantities. This doesn't seem to be the case here.

1. This was a course on Classical (Lagrangian & Hamiltonian) mechanics, and our professor gave us some Newtonian mechanics problems as a first assignment. We're not yet sure why, but most were quite interesting. The assignment due date is over, this is just a residual point that has been bothering me.

2. This is lifted from the document I submitted. I have blurred out the end because it contains my own resolution of the apparent paradox here. I may post it as an answer later, but for now I'd prefer to give everyone a fresh look at this situation.

$\endgroup$
  • $\begingroup$ Just a quick point: Did you notice that, the question in your link was asking about the maximum tension on the wall. $\endgroup$ – Ali Jul 30 '13 at 16:41
  • 2
    $\begingroup$ Why do you think you only get inequalities? The total force acting between the pole and the string is surely determined, isn't it? $\endgroup$ – Luboš Motl Jul 30 '13 at 16:50
  • 2
    $\begingroup$ @Manishearth: Just want to check that you understand that a statics problem can be underdetermined. Like a book on a four-legged table. There is no issue with "determinism", just with the limitations of the rigid-body formalism. $\endgroup$ – BebopButUnsteady Jul 30 '13 at 17:03
  • 1
    $\begingroup$ @manishearth: What do you mean by "measurable"? The weight on each of leg of a four legged table is measurable if I put a scale under each foot. The reading of those scales is not determined by solely rigid body concerns. I mean everything in these problems is measurable. $\endgroup$ – BebopButUnsteady Jul 30 '13 at 17:07
  • 1
    $\begingroup$ Related: Capstan equation. $\endgroup$ – Qmechanic Jul 30 '13 at 17:34
2
$\begingroup$

The system is not completely determined. For example, if the pole had been rotating very slowly clockwise before being brought to rest by the friction of the string (and then welded to the wall, so as to be a "fixed pole", but with the friction force remaining) you would get one answer. If counterclockwise, you get the opposite.

$\endgroup$
  • $\begingroup$ Yeah, that's how I resolved the paradox myself. I reasoned that the distribution of microscopic elongations in the string (even if inelastic, a string must have some infinitesimal elongation to carry tension) is dependant on the way the system is formed, and the distribution of tension is dependant on the distribution of elongation. $\endgroup$ – Manishearth Jul 30 '13 at 17:22
  • 1
    $\begingroup$ @Manishearth: Yeah, I think this is related to those questions of the form "How can I have friction force if nothing's moving?" $\endgroup$ – Mike Dunlavey Jul 30 '13 at 17:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.