If a rope is in free fall, does tension force act on it or not?

[MatinSAR]

Homework Statement

My question: If a rope is in free fall, does tension force acts on it or not?

Relevant Equations

Please see below pictures.

This is the question.

To this point everything is clear.
I have problem with following part:

The authors claim that each part of the remaining rope is under constant acceleration. So it is in free fall and only gravitional force acts on it.

If we release a rope like above, before it hits the ground I think only gravitional force acts on it. But when it reaches the ground and drops on it, I think there is a tension force acting on upper parts due to the lower parts. Why I am wrong?

 

[kuruman]

But when it reaches the ground and drops on it a there is a tension force acting on upper parts due to the lower parts. Why I am wrong?

A piece of rope can only pull but not push on something. How do you figure that the lower part of the rope already on the table can push up on the upper part that is still falling?

 

[MatinSAR]

How do you figure that the lower part of the rope already on the table can push up on the upper part that is still falling?

I meant that lower parts pull down the upper parts. Sorry for my bad English.

 

[erobz]

In the classical limit of reality is this is just a (stiff) stretched slinky falling? I think a sonic wave propagates down the rope, so for a very brief moment the top is moving and the bottom is still. Then after the wave reached the bottom, the entire rope is in free fall and under no tension.

 

[kuruman]

I meant that lower parts pull down the upper parts. Sorry for my bad English.

It has nothing to do with English. Just think about it. The lower parts cannot pull down on the upper parts while both are in free fall. If the lower parts are on the table at rest, they can only push up not pull down.

 

[pines-demon]

This was originally a homework exercise in uni, but the teacher told us that he had a disagreement with another teacher with respect to the final result, so he asked us to forget about it

 

[MatinSAR]

In the classical limit of reality is this is just a (stiff) stretched slinky falling? I think a sonic wave propagates down the rope, so for a very brief moment the top is moving and the bottom is still. Then after the wave reached the bottom, the entire rope is in free fall and under no tension.

You are making it harder . It is a rope that's in free fall. It doesn't act like a spring. So
It falls without deformation.

It has nothing to do with English. Just think about it. The lower parts cannot pull down on the upper parts while both are in free fall. If the lower parts are on the table at rest, they can only push up not pull down.

Thanks for your time @kuruman . So it's not pulling down when the lower parts at rest. That makes sense.
But does it push up the upper parts? In post #2 you've mentioned that it cannot. Can I say that because this rope doesn't act like a spring it cannot push up the upper parts?

A piece of rope can only pull but not push on something.

This was originally a homework exercise in uni, but the teacher told us that he had a disagreement with another teacher with respect to the final result, so he asked us to forget about it

Same thing happened in my class. The teacher told us he isn't sure about the final result for the reason that I've stated in post #1.

Sorry for the late answer. I wasn't online after 3 AM.

 

[kuruman]

Thanks for your time @kuruman . So it's not pulling down when the lower parts at rest. That makes sense.
But does it push up the upper parts? In post #2 you've mentioned that it cannot. Can I say that because this rope doesn't act like a spring it cannot push up the upper parts?

No it cannot push on the upper parts. Mass element $dm$ which is in free fall collides with the table and stops. The information that it stopped cannot be communicated to the part of the string that is above and is still falling. The energy is lost to heat and vibrations of the table as would be the case in a perfectly inelastic collision.

If you want to put some math in this, the relevant equation to use (derived here) is $$m~\frac{dv}{dt}=(u-v)\frac{dm}{dt}+F_{\text{ext}}.$$In this equation
$m=~$the mass of the rope still falling
$v=~$the velocity of the mass still falling relative to the lab frame
$u=~$the velocity of mass $dm$ that is stopped in time $dt$ relative to the lab frame
$F_{\text{ext}}=~$the external force in this case $-mg$.

Because all parts of the rope that are still in motion move as one at all times, $u=v$ and the equation becomes $$m~\frac{dv}{dt}=-mg.$$ If the rope is released from rest,the solution is $$v(t)=-gt.$$ This is what one would get if one modeled the continuous rope as consisting of separate mass elements $dm$ arrayed in a continuous vertical line and released from rest all at the same time.

You can clearly see now why a mass element starting at height $x$ above the table has speed given by $v^2=2gx$ when it hits the table as claimed by the textbook.

 

[MatinSAR]

If you want to put some math in this, the relevant equation to use (derived here) is

Interesting.

@kuruman Many thanks.

 

[hutchphd]

You are making it harder . It is a rope that's in free fall. It doesn't act like a spring.

In the real world any rope acts a bit like a spring (it will be slightly elastic....so the slinky-like behavior is probably true but very fast) I think this is probably easier to understand than idealized rope behavior. One needs to specify the problem much more diligently, as your prof discovered. In my ideal world a rope does not push. So I think it not worth too much worry. But I do love the ideal slinky problem.

 

[MatinSAR]

In the real world any rope acts a bit like a spring (it will be slightly elastic....so the slinky-like behavior is probably true but very fast) I think this is probably easier to understand than idealized rope behavior. One needs to specify the problem much more diligently, as your prof discovered. In my ideal world a rope does not push. So I think it not worth too much worry. But I do love the ideal slinky problem.

Thanks for your reply @hutchphd ...

 

[sophiecentaur]

I did think about it and I concluded that the situation, as stated in the problem, is an impossible one without some extra information.

The rope must have somewhere to go when it lands; it cannot be an ideal rope that all those statics problems assume. If we assume the rope has three dimensions and some modulus then, when an element lands, it will hit the element below it. Either (when there is no lateral force) it will stand on its end and a compression force will propagate back up or (in real conditions) the element will be deflected to the side. At this point there will be a lateral component of force as well as a vertical one and the rope will take on a curved shape. This is the start of a wave motion which will change as the lateral displacement increases.
At some stage, the downwards velocity of the (ever accelerating) falling rope will 'beat' the lateral velocity due to contact will reverse and the rope snake towards the middle and out the other side. Eventually you will get a spiral / figure of eight (or?) under the point where the rope was dropped.

No wonder the teacher backed out of the problem. "Too hard guv".

I think the situation will be along the same lines as the Chain Fountain experiment (a real chain and not an ideal one).

 

[MatinSAR]

No wonder the teacher backed out of the problem. "Too hard guv".

I agree. Thanks for your time @sophiecentaur .

 

[kuruman]

I agree. Thanks for your time @sophiecentaur .

We have no idea what the two unnamed teacher disagreed on or found "too hard" about this problem. Often problems requiring some idealization to reach the intended answer trigger objections in the form of a "yes, but in the real world $\dots$" argument. In this specific case, I believe that OP's question

If we release a rope like above, before it hits the ground I think only gravitional force acts on it. But when it reaches the ground and drops on it, I think there is a tension force acting on upper parts due to the lower parts. Why I am wrong?

has been adequately answered within the idealization parameters needed to clarify Example 9.10 in OP's textbook. I think that this thread has run its course and is now moving in the direction of the disagreeing teachers.

 

[sophiecentaur]

We have no idea what the two unnamed teacher disagreed on or found "too hard" about this problem. Often problems requiring some idealization to reach the intended answer trigger objections in the form of a "yes, but in the real world $\dots$" argument. In this specific case, I believe that OP's question

has been adequately answered within the idealization parameters needed to clarify Example 9.10 in OP's textbook. I think that this thread has run its course and is now moving in the direction of the disagreeing teachers.

Are you saying it’s a soluble and straightforward problem. It describes the rope just stopping when it lands. Does it have finite linear density and finite density?
Can you describe what happens when it lands? Is it not ‘too hard’ for you? Help me with this please.

 

[MatinSAR]

We have no idea what the two unnamed teacher disagreed on or found "too hard" about this problem. Often problems requiring some idealization to reach the intended answer trigger objections in the form of a "yes, but in the real world …" argument. In this specific case, I believe that OP's question

I thought that @sophiecentaur was talking about my prof not 2 teacher's mentioned by @pines-demon ...
My professor was not sure about final result for the reason I've mentioned in post #1.

If a rope is in free fall, does tension force acts on it or not?

But he mentioned that this book is verified so it cannot be wrong.

has been adequately answered within the idealization parameters needed to clarify Example 9.10 in OP's textbook. I think that this thread has run its course and is now moving in the direction of the disagreeing teachers.

Thanks again for sharing your time @kuruman ...

 

[kuruman]

Are you saying it’s a soluble and straightforward problem. It describes the rope just stopping when it lands. Does it have finite linear density and finite density?
Can you describe what happens when it lands? Is it not ‘too hard’ for you? Help me with this please.

I am saying that it is soluble and straightforward within the parameters of an idealization. The rope has finite linear mass density $\rho$ but zero volume. Using this idealized model, the solution is as presented in post #8 and explains Example 9-10 posted by OP. I do not disagree with you that the, so called, real life is more complicated. However, if you make it sufficiently complicated and the probem cannot be solved, it stops being a useful teaching tool.

I am arguing in favor of using simplifications to illustrate the gist of basic underlying principles so that they can be taught to novices. One cannot start with realistic models to explore the basic ideas of acceleration, force, momentum, energy etc. with people who have just begun to learn physics. Just imagine what it would be like to teach projectile motion taking air resistance into account right from the start. So I see no profit in "yes but $\dots$" objections to idealized models when the goal is pushing across basic physical principles.

 

[sophiecentaur]

within the parameters of an idealization. The rope has finite linear mass density ρ but zero volume.

I can't argue with that but i'd say that the scenario would need a big caveat before hitting students with it because it seriously goes against experience so it could generate a lot of questions from a class.

If the rope is already travelling horizontally by a finite amount them my objection would not apply initially but eventually the problem would still arise. (If the rope is long enough).
[edit: or the rope could fall onto a moving table]

 

[sophiecentaur]

So I see no profit in "yes but …" objections to idealized models when the goal is pushing across basic physical principles.

If you don't take care in simplifications you can end up with paradoxes, as in the Capacitors sharing Charge problem. You are forced to introduce a 'missing' mechanism to explain that one. The falling rope seems to fall into that category.

 

[pines-demon]

If you don't take care in simplifications you can end up with paradoxes, as in the Capacitors sharing Charge problem.

What is that again?

 

[sophiecentaur]

Here

Guaranteed to confuse students.

 

[pines-demon]

Here

Guaranteed to confuse students.

Definitely confusing. I never thought about this, what is the solution? is it radiated in terms of EM waves?

 

[sophiecentaur]

Definitely confusing. I never thought about this, what is the solution? is it radiated in terms of EM waves?

Two things: For early students, you can tell em that there's alswys some resistance in a real circuit and that will dissipate just the right amount of energy.

Another reason is radiation by EM waves but it's another can of worms for beginners.

You can tie those two together. The dissipation by EM waves as current changes in a circuit can be characterised as an extra 'Radiation Resistance' in the circuit. (Antenna theory uses this.)