# Falling Chain Problem

#### PhysicsKid99

1. The problem statement, all variables and given/known data
A uniform chain of length l and mass M contains many links. It is held above a table so that one end is just touching the table top. The chain is released freely. What is the force between the links? What is the time for the topmost link to fall to the table?

2. Relevant equations

v=u+at, v^2 = u^2 + 2as
3. The attempt at a solution
I think that the time to reach the table is sqrt(2s/a), because v=at, so (at)^2=2as, so rearranging that would give me that answer. I'm trying to visualise the part about the force though; is it that the only force exerted on each chain the weight of the chain below it? Or is it tension, in which case Mg-T=Ma, so T=Mg-Ma?

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#### OldEngr63

Gold Member
Falling chain problems are still somewhat of an open research area because the correct model is not entirely clear. Here is a paper on the topic:

"Falling chains as variable mass systems: theoretical model and experimental analysis," by C.A. de Sousa, P.M. Gordo, and P. Costa, Physics Education, 28 May, 2012.

The particular problem stated by the OP is not addressed in this paper, but some of the relevant modeling questions are.

#### haruspex

Homework Helper
Gold Member
2018 Award
1. The problem statement, all variables and given/known data
A uniform chain of length l and mass M contains many links. It is held above a table so that one end is just touching the table top. The chain is released freely. What is the force between the links? What is the time for the topmost link to fall to the table?

2. Relevant equations

v=u+at, v^2 = u^2 + 2as
3. The attempt at a solution
I think that the time to reach the table is sqrt(2s/a), because v=at, so (at)^2=2as, so rearranging that would give me that answer. I'm trying to visualise the part about the force though; is it that the only force exerted on each chain the weight of the chain below it? Or is it tension, in which case Mg-T=Ma, so T=Mg-Ma?
Suppose there is some tension in the chain as it falls. What does that tell you about the vertical acceleration of the top link? What does it tell you about the vertical acceleration of the bottom link that's not yet in contact with the table?

"Falling Chain Problem"

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