Mechanics Falling Chain Problem

In summary, the force required to keep the top end of the chain motionless as it unravels will become infinitely large as the chain falls, due to the increasing speed and tension. However, in a theoretical scenario, you can calculate the rate at which the amount of chain you are supporting increases.
  • #1
harrietstowe
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0

Homework Statement



A chain of length L and mass density σ kg/m is held in a heap. I grab an end of the chain that protrudes a bit out of the top. The heap is then released so that the chain can unravel with time. Assuming that the chain has no friction with itself, so that the remaining part of the heap is always in free fall, as a function of time what force must my hand apply to keep the top end of the chain motionless?

Homework Equations





The Attempt at a Solution


Mentally I'm trying to picture the problem as if at t=0 the heap of chain were on a table and I hold that last link of the chain up. Then let's say at some Δt later this hypothetical table disappears so that the heap falls and unravels and so as time continues the force I apply to that top link obviously increases until the chain is completely unraveled, call that t-end and at t-end I'll have to apply σ*L*g, the weight of the whole chain. I guess I'm stuck a bit determining how to mathematically express the force between these two extreme times.
Thanks
 
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  • #2
harrietstowe said:

Homework Statement



A chain of length L and mass density σ kg/m is held in a heap. I grab an end of the chain that protrudes a bit out of the top. The heap is then released so that the chain can unravel with time. Assuming that the chain has no friction with itself, so that the remaining part of the heap is always in free fall, as a function of time what force must my hand apply to keep the top end of the chain motionless?

Homework Equations





The Attempt at a Solution


Mentally I'm trying to picture the problem as if at t=0 the heap of chain were on a table and I hold that last link of the chain up. Then let's say at some Δt later this hypothetical table disappears so that the heap falls and unravels and so as time continues the force I apply to that top link obviously increases until the chain is completely unraveled, call that t-end and at t-end I'll have to apply σ*L*g, the weight of the whole chain. I guess I'm stuck a bit determining how to mathematically express the force between these two extreme times.
Thanks

The force you apply will very quickly become extremely large!

After all but 1 m of the chain has fallen 1 metre, it will be traveling at the speed that things travel at after they have fallen 1 metre.
At that point, you have to stop the first link of the rest of the chain as it travels zero distance. That requires an infinite force.
Never mind the "all but 2m of chain" which will be traveling even faster when the first 2 m go tight.
If this chain was tied to a strong beam instead of being held in your hand, the beam will flex slightly so the top of the chain is not motionless.

I suspect that the problem is theoretical and you are to ignore the fact the moving chain has to be stopped, and you just effectively calculate the rate at which the amount of chain you are supporting, increases.
 

1. What is the Mechanics Falling Chain Problem?

The Mechanics Falling Chain Problem is a classic physics problem that involves a chain hanging from a table or similar structure. The question is how much of the chain will hang off the edge when it is released from rest.

2. What are the key factors that affect the outcome of the Mechanics Falling Chain Problem?

The key factors that affect the outcome are the length and mass of the chain, the height at which it is released, and the force of gravity.

3. How is the Mechanics Falling Chain Problem typically solved?

The problem is typically solved using the principle of conservation of energy. This involves setting the potential energy of the chain at its initial height equal to the kinetic energy of the chain at the point where it is fully hanging.

4. Can the Mechanics Falling Chain Problem be solved using other principles or equations?

Yes, the problem can also be solved using the principle of conservation of momentum, as well as equations of motion such as Newton's second law.

5. What real-world applications does the Mechanics Falling Chain Problem have?

The problem has applications in various fields such as engineering and physics, particularly in understanding the behavior of flexible structures. It can also be used to study the dynamics of a swinging pendulum or a falling rope.

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