Launch with Chains: Solving a Hard Physics Problem

[benf.stokes]

I've read the FAQ and this is not homework, it is just a hard problem I can't solve.

A mass M attached to an end of a very long chain of mass per unit length $$\lambda$$
, is thrown vertically up with velocity $$v_{0}$$.
Show that the maximum height that M can reach is:

$$h=\frac{M}{\lambda}\cdot \left [ \sqrt[3]{1+\frac{3\cdot \lambda\cdot v_{o}^{2}}{2\cdot M\cdot g}}-1 \right ]$$

and that the velocity of M when it returns to the ground is $$v=\sqrt{2\cdot g\cdot h}$$

Conservation of energy cannot be used because inelastic collisions occur in bringing parts of the rope from zero velocity to v

I start by setting up that the total mass at a position y is:
$$M_{total}=M+\lambda\cdot y$$ and thus the momentum at any position is given by:

$$p=(M+\lambda\cdot y)\cdot v$$ but I can't figure out an expression for v and using

$$F=\frac{dp}{dt}$$ I get an differential equation I can't solve.

Any help would be appreciated.

 

[tiny-tim]

hi benf.stokes!

Conservation of energy cannot be used because inelastic collisions occur in bringing parts of the rope from zero velocity to v

No, you can use conservation of energy …

the question asks for the maximum height, which would be in the limiting case of no energy loss.

As for the question itself, I don't understand what position the mass and chain are in at the start.

 

[benf.stokes]

Hi tiny-tim. I think that conservation of energy can't be used is a part of the problem,my bad, and if you used conservation of energy wouldn't you get at most a square root and not a cubic one?
At the start both the mass and the chain are at rest on the floor. Thanks for the reply

 

[benf.stokes]

i already arrived at the solution. Thanks anyway tiny-tim

 

[rdl]

I can understand the difficulty in solving this problem and the frustration that comes with it. It is important to remember that difficult problems often require time, effort, and collaboration to solve. I would suggest seeking out other scientists or experts in the field of physics to discuss and brainstorm potential solutions.

One approach to solving this problem could be to use the equations of motion for a particle in free fall, taking into account the changing mass of the system as the chain is released. This would involve setting up and solving a differential equation, as you have attempted, but with careful consideration of the changing mass and velocity.

Another approach could be to use the principle of virtual work, which states that the work done by a conservative force can be expressed as the difference in potential energy between two points. In this case, the conservative force would be gravity and the two points would be the starting point and the maximum height reached by the mass. This would involve setting up an integral and solving for the maximum height.

In both approaches, it may also be helpful to consider the energy dissipated due to inelastic collisions and how it affects the overall motion of the system.

Overall, it is important to approach this problem with patience and persistence, and to seek out help and collaboration when needed. Good luck with solving this challenging problem!