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?
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 lets 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.