course mth174

A chain of uniform mass per unit length, with total length l, is loosely coiled on a table at height H above the floor. It starts to fall through a hole in the table.a. How long does it take before the chain first touches the floor?

Since the chain is coiled at this moment, will it have no potential energy, just as we saw in the rope and cylinder situation?

The mass is distributed according to M/L. When the chain begins to fall through, I guess we are assuming an ideal situation and it falls at the same rate through the hole until the whole entire chain reaches the floor. You can slice each segment into slices gM/L yi* d'y - I'm not sure if this would be applicable here given that we wouldn't necessarily have to consider the work being done on the chain, but if the chain is coiled up, its in a circle. Would what be the best approach for understanding this (and the one below)?

b. How long does it take before the whole chain has reached the floor? "

You get to decide where PE is 0. Could be tabletop, could be floor, could be the center of the Earth, could be at infinite separation from the Earth (the latter two probably aren't the simplest choices in this situation).

First assume that the length l is greater than the height h of the table.

When the position of the end of the chain is y < H relative to the floor, what is the KE of the moving part of the chain?

What therefore is the velocity of the moving part of the chain?

Having answered this question you know velocity as a function of y. Velocity is dy/dt, so setting dy/dt equal to your v(y) function you get a differential equation which can be solved for y as a function of t. Plug in initial conditions and you get y as a function of t. The rest should be easy.