Mechanics Falling Chain Problem

[harrietstowe]

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

 

[PeterO]

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.