- #1

- 235

- 0

**Motion of a Uniform Chain falling....I desperatly need help**

## Homework Statement

A uniform heavy chain of length "a" hangs initially with a part of length, "b", hanging over the edge of a table. The remaining part of length a-b is coiled up at the edge of the table. If the chain is released show that the speed of the chain when the last link leaves the end of the table is [2g(a^3-b^3)/(3a^2)]^1/2

## Homework Equations

mg=m*dv/dt+dm/dt*v

## The Attempt at a Solution

I have tried many times to do this problem, following the above equation, which my professor claims is correct for this problem; however, I get stuck everytime in the following situation (which my professor also claims is correct):

From the begining

Let m= z*dm/dx, than note that dm/dx=y. m=z*y. Than dm/dt=dz/dt*y.

z*y*g=z*y*dv/dt + v*dz/dt*y

Note that dv/dt= (d^2Z/dt^2)

(note from now on that z(dot)= dz/dt, as is common in this notation)

So dv/dt= (dz(dot)/dz)*z(dot)

Again note that d/dz{(1/2)*(z(dot))^2} = dz(dot)/dz * z(dot)

And then I get stuck, as I will end up in a loop of doing the same things over and over again. Worse yet, my professor told me explictly that this is what must be done for doing this problem using the equation given, and that I will (after I figure out the next step), need to use the method of frobenius to solve the problem.

The alternative method of solving this, which he himself did mention, is to use largrange or hamaltionan mechanics (but he said we would have to figure those out on our own, as he hasn't refreshed himself on the topic).

------

If I could get a hand setting this up with largrange, that would be awesome, I know that I need to some how split the problem into two to use the largrange, additionally I believe that I will need to take into account that it only has two degrees of freedom (motion on a plane); however, I can't figure out how to set up the largragian from that (I am a pretty inexperanced with the method; however, I would much rather learn to do this problem in largrange, than trying to muscule my way through the rest of the problem, when I don't quite understand how to exactly go about it).

Thank you in advanced for anyhelp.

Last edited: