Solving the Rope Falling off a Table Problem: A Contradiction?

[Mazerakham]

I am studying differential equations from this book by Edwards and Penney, and I seem to have stumbled on this rather bizarre contradiction which I can't seem to get myself out of.

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

 

[Fightfish]

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.

The mathematical input system used on these forums is LaTeX, not Math Type.

Perhaps the authors were incorrect to assume that

F = d/dt [mv]​

applies to this problem?

Er...that equation is the very definition of force? That's the mathematical statement of Newton's Second Law. So, there is no question of 'assumption' here; it is a fact.

Well, we'll just have to wait for your work before any further comments can be made.

 

[Studiot]

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[Fightfish]

In Mathtype
go to preferences > cut and copy preferences
Select the bottom radio box equation or application for website
Choose Physics Forum from the drop down box
Click OK

You will then be able to copy and paste directly from Mathtype into the forum

That's new to me ! Never knew Mathtype had such functions.

 

[PJC01]

As a scientist, it is important to recognize that contradictions and discrepancies in scientific problems and solutions are not uncommon. In fact, they can often lead to further investigation and discovery. In this particular case, it is possible that the authors' approach using differential equations and the one using conservation of energy may have different assumptions or simplifications that could lead to slight variations in the final answer.

It would be helpful to carefully examine the assumptions and equations used in both approaches to understand the potential sources of discrepancy. Additionally, it may be worth considering other factors that could affect the rope's falling time, such as air resistance or variations in the rope's weight or flexibility.

Ultimately, the most important aspect is to critically analyze and evaluate the solutions, rather than getting caught up in the contradiction itself. This could potentially lead to a deeper understanding of the problem and its underlying principles.