Equation of motion for a chain sliding down an edge

[smayorgat]

I studied physics a long time ago and somebody just asked me this question. After trying for a while I couldn't work it out.

The situation is this: there's a chain of length $l$ on a table, of which a portion, of length $x_0$, is hanging out (enough so that when you stop holding it down, the chain starts sliding). Like in this drawing:

[PLAIN]http://b.imagehost.org/0596/chain.jpg

Friction is neglected.

How do you set up the differential equation of motion for the chain?

Thank you very much.

 

[arildno]

1. Remember that neither internal forces nor normal forces do any work on the chain.
That is: mechanical energy is conserved.

2. Assume that the vertical part of the chain never swings

3. Remember that the total length of of the chain remains the same.

4. Assume constant density.

5. For simplification, set the 0-level for the gravitational potential at the table surface.

6. Remember that kinetic&potential energies for different parts of the chain add together linearly.

7. Set up the equation of conservation of mechanical energy, differentiate that equation.

8. Hint: The vertical line segment should increase exponentially in length as a function of time, as long as there is still some horizontal segment moving on the table.

 

[Cleonis]

The situation is this: there's a chain of length $l$ on a table, of which a portion, of length $x_0$, is hanging out (enough so that when you stop holding it down, the chain starts sliding).

Friction is neglected.

A Google search with the terms "chain", "table" and "sliding" had as 3rd entry the following PDF-document:
https://wiki.brown.edu/confluence/download/attachments/2752884/Chain.pdf?version=1&modificationDate=1198253619000"