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The problem, which is a variation on the classic rope falling off a table, goes as follows:

"Suppose that a flexible 4-ft rope starts with 3 ft of its length arranged in a heap right at the edge of a high horizontal table, with the remaining foot hanging (at rest) off the table. At time t = 0 the heap begins to unwind and the rope begins gradually to fall off the table, under the force of gravity pulling on the overhanging part. Under the assumption that frictional forces of all sorts are negligible, how long will it take for all the rope to fall off the table?"

The authors of the textbook chose to start with F = d/dt [mv]. From here, they set up a differential equation, multiply by an integrating factor, then solve the resulting exact differential equation. Once the function t(x) has been found, the authors conclude T = (approximately) 0.541 s.

I found that when I tried applying conservation of energy, the work was quite a bit simpler, and I reached a final answer which seemed (to me) just as realistic as the authors': 0.839 s.

As soon as I figure out how Math Type works in this forum, I will post the work which leads to these two different answers. I'm curious because, in the frictionless system, Conservation of Energy should hold. Perhaps the authors were incorrect to assume that

F = d/dt [mv]

applies to this problem? I'll appreciate any responses.

-Mazerakham