What is the solution to the classical string problem?

In summary, classical mechanics doesn't seem to be able to solve this problem without a quantum mechanical approach.
  • #1
Loren Booda
3,125
4
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?
 
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  • #2
There's probably a better way to do this, but I'd try this:

[tex] F_{\rm net} = m\ddot{x} = \frac{\frac{L}{2}+x}{L}mg [/tex]

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.
 
  • #3
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?
 
  • #4
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.
 

What is the "Classical string problem"?

The "Classical string problem" is a mathematical and physical problem that involves studying the behavior and properties of a string under tension. It is a fundamental problem in classical mechanics and has applications in various fields such as engineering, physics, and mathematics.

What are the main assumptions made in the "Classical string problem"?

The main assumptions made in the "Classical string problem" include the string being inextensible, having a constant linear density, and being under uniform tension. The problem also assumes that the string is ideal, meaning it has no thickness and is perfectly flexible.

What are the key equations used in solving the "Classical string problem"?

The key equations used in solving the "Classical string problem" are the wave equation, the boundary conditions, and the initial conditions. These equations describe the motion of the string over time and allow for the determination of its shape and behavior at any given point.

What are some applications of the "Classical string problem" in real-world scenarios?

The "Classical string problem" has various applications in real-world scenarios, such as understanding the vibrations of musical instruments, analyzing the structural stability of bridges and buildings, and predicting the behavior of ropes and cables under tension in construction and engineering projects.

What are some limitations of the "Classical string problem"?

One limitation of the "Classical string problem" is that it assumes the string is ideal and neglects factors such as thickness and elasticity. It also does not take into account the effect of external forces or the possibility of the string breaking. Additionally, the problem becomes more complex when dealing with non-uniform strings or multiple strings interacting with each other.

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