What is the solution to the classical string problem?

[Loren Booda]

Classical "string" problem

My freshman physics class was given the following problem. I can't remember if it can be solved by classical physics alone, or else needs a quantum mechanical start:

Half of a perfectly flexible string of length L and negligible width lies straight and motionless on an exactly horizontal (to gravity) frictionless table, while the other half hangs freely from its edge. How much time transpires until the string slips completely over the edge?

 

[jamesrc]

There's probably a better way to do this, but I'd try this:

$$F_{\rm net} = m\ddot{x} = \frac{\frac{L}{2}+x}{L}mg$$

where x is the distance between the end of the rope on the table and its starting point. The net force on the rope is just the weight of the fraction of the rope that is hanging off the table. If you solve that differential equation and find the time when x = L/2, that should be the answer.

 

[Loren Booda]

jamesrc,

Sorry, I think I misstated the problem. The weight of the hanging string is initially counterbalanced exactly by the friction of its other half lying on the table. How much time transpires until the string slips completely over the edge?

The system is in classical (albeit singular) equilibrium, but needs an infinitesimal impulse, perhaps quantal, to get started. Taking Q. M. into account, is there a standard answer to this problem or a distribution of possible times, given the minimum information needed?

 

[jamesrc]

Oh, I thought the table was frictionless and the rope was held until t = 0, when it was released. As it is now stated, I'm not sure how to go about solving the problem. It seems to me that the initial static equilibrium conditions will not help you solve the dynamics, since you know nothing about the kinetic friction characteristics (you would expect, for a Coulomb model of friction, that &mu;_k < &mu;_s). And since the rope is so idealized, I don't see how/why you could/would employ a more sophisticated friction model (well, maybe viscous, but I don't see a compelling reason to).

Anyway, once it starts (and it wouldn't matter how it started as long as it wasn't given an initial velocity), it should keep accelerating and should be solvable using differential equation similar to the one from my other post (with a friction term in there).

I guess in short, I don't know, so I'll defer to those who do and check into see how this develops.