Chain falling out from a tube

jeudi 29 août 2013

1. The problem statement, all variables and given/known data

A chain AB of length ##l## is located in a smooth horizontal tube so that its fraction of length ##h## hangs freely and touches the surface of the table with its end B. At a certain moment, the end A of the chain is set free. With what velocity will this end of the chain slip out of the tube.

(Ans: ##v=\sqrt{2gh\ln(l/h)}## )

2. Relevant equations

3. The attempt at a solution

I don't see how should I start with this problem. I tried to see the forces acting on the hanging part of the chain but that didn't help me.

The forces acting on the hanging part of the chain are weight, tension (T) due to the part of chain in the tube and the normal reaction from ground(N). Let the length of chain remaining in tube be x and mass per unit of chain be ##\lambda##. Applying Newton's second law on hanging part of chain,

$$\lambda hg-T-N=\lambda ha$$

Newton's second law for part in the tube,

$$T=\lambda xa$$

I am not sure if this helps and I don't know how to calculate N here.

Any help is appreciated. Thanks!

Attached Images