Rope Race: Newsletter Challenge (July 2016)

>0...

I'm giving you a GA for identifying the problem (though I think you will find that "slightly" is a bit of an understatement):
The (idealised) rope will only have a defined speed for as long as the hanging section falls vertically from the edge of the table. So the correct answer is that different parts of the rope will be travelling at different speeds when the rope loses contact with the top of the table.

In fact, I'm not even certain there is an analytic solution to the equations of motion once the falling section of the rope is no longer entirely vertical (I'd be happy to be corrected on this...); however, what is clear is that the centre of gravity will not be as low as L/2 when the rope leaves the top of the table, so even the kinetic energy will be lower than stated in your GA that you acknowledge is not-really-an-answer. BTW, I don't know this to be the case, but it is conceivable that the end of the rope could be travelling extemely fast at the point when it reaches the edge of the table (like the end of a whip).

Speculation over, I would like to convert this to a solvable challenge:
Subject to all the usual idealising assumptions, how much rope is hanging over the edge at the instant when the falling part ceases to be vertical, and what is the speed of the rope at this moment?

P.S. If we make every possible idealising assumption, just one word of the original challenge needs to be changed to allow the sensible numerical answer:

"You have a rope of length L and a frictionless table with height 1.5L. Put the rope on the top of the table and pull one end to the border of the table so that a small piece hangs down. Release the rope, so that the rope starts to fall down. What is the speed of the rope at the very instant when it loses contact with the edge of the table?"

BTW, the problem is quite straightforward once you have properly identified the constraints. To my mind this makes it as close to an ideal challenge (of its type) as you will find.

Had this problem been posed to scientists they would make all sorts of assumptions about frictionless tables, non-stretching rope with no thickness and infinite flexibility, bits that hang over the end but have no length and the lack of air resistance. But the question was posed to engineers, who live in the real world and solve problems in the real world, so is it surprising that instead of providing an idealized answer they come up with more questions to clarify a poorly defined task. And are then skeptical about the additional information provided. The setter of the challenge should have specified at the outset that this is not a problem that is solved by common sense but one that needs to be solved by the application of pure maths rather than applied maths. If the setter had done that, how many of us would have ignored the challenge as irrelevant and moved on the the next topic?

Far too many, I'd say.

If you won't (can't?) quickly address a simplified problem that needs simplified maths there is no way I would trust you with anything real. You would need this maths and more to design a high speed pulley system or a chain-driven production line. Wthout it you would end up grossly overspecifying or with an unsafe system (or, more likely, both, as the ropes or chains would both be too heavy and fly uncontrollably around the place)

Fortunately for the last 25 years my clients have taken a different view and trusted me enough that I have been able to make a good living as a Consultant Design Engineer, designing and building their plant and process equipment, including high speed pulley and chain driven conveyor systems. None of it was under designed, and some of it was over designed at the request of the client, but it all has adequate safety measures incorporated to render it safe to operate. Rather than working on simplified theoretical models I try to take in all the complexities to arrive at an optimum solution. You have obviously never seen the damage inflicted by a failed trawl net tow line on a trawler if you believe ropes and chains are too heavy to fly uncontrollably when severed. I have never witnessed such a failure but I have repaired a steel 6x6 RHS stanchion that was sliced through by a whipping steel rope.

Evidently you know more than your previous indicated.

Unfortunately we are either divided by a common language or you share my inability to read carefully on some occasions.

I wrote the ropes (chains) would be too heavy and fly allover the place. The specific mistake I was referring to was making them too heavy as a precaution against their breaking under intended loads, under wich condition the ropes/chains become more likely to whip all over the place (so over-specified and more dangerous).

To minimise further misreading: under designed in my parlance means indaequate design work (quality or quantity) and (usually) over-specified to compensate. So (in my parlance) you are saying that you don't overspecify, except when clients err on the side of (possibly ill-advised) apparent caution.

To be a bit more specific: too light a rope/chain and it won't do its job (probably break extremely dangerously), or it will need to be replaced too early; on the other hand if you make it too heavy you will not only need stronger constrainers to keep it safe, but you will probably need additional ones as well (and even if you have infinite power to drive the heavier rope/chain the limiting speed is probably lower due to the rope's movement not being as well controlled by the load and so having more degrees of freedom).

I would guess that you mainly simulate rather than calculate?

This is a reply to Fyz's supplementary question. When does the rope leave the edge of the table.

I'm struggling a bit here:

Looks to me as though if you consider just a tiny bit of the rope then as alpha tends to zero, l tends to L

Even if you look at the whole edge ie. alpha=45° (π/4) then l = 2√2L/π = 0.9L

This is not what my intuition tells me, and, I don't believe it's the result Fyz has calculated, so, I've clearly made a mistake.

"...the correct answer is that different parts of the rope will be travelling at different speeds when the rope loses contact with the top of the table."

Different parts of the rope may have different velocities, but every point on the rope wll have the same speed. --Unless you're suggesting that the rope stretches as it moves. (It might, but the stretching will be minimal. Physicists 'idea' ropes never stretch.)

This assumes that one section of the rope follows the same path as the section that preceded it.

Try holding a rope at one end and whirling it around you. Is the speed of the end you are holding the same as the end furthest away?

P.S. In the absence of air resistance that could be a straight rope; our rope has more degrees of freedom...

I suspect you're right, but I see it as not having any contact with the surface if there's no friction...for it to have a speed component it would have to be moving, so anything over 0 would be the speed...

Aw, come on... Contact would be defined as providing the table providing an upward forces. And while you're about it, see the answer I'm about it, see the note I'm about to write to jkwarner

Could this problem be visualized as a liquid of unknown viscosity flowing around an elbow in an open ended, friction-less tubing?

The stiffness of the rope,as the viscosity of the liquid, will affect the speed of the liquid or rope as it turns the corner.

If we consider the flexibility of the rope as a group of infinitely flexible strings,then the same result would be achieved if we rolled a series of spheres off of the table.

The speed should be the same at both ends.

Ideal rope running through the corner in an ideal tube would work, as would an ideal liquid in an ideal (no-friction, zero-dimension limit) frictionless tube (liquid redistribution at the corner vanishes as the dimension approaches zero, as therefore do viscous losses).

Rolling balls (on a smooth table??) would need some sort of coupling, and would suffer the same problem discovered by Randall. If rolling instead of having a smooth surface they would have lower speed (due to the rotational kinetic energy). The uniform speed assumption would still fail, but at a later point in the motion than for sliding arrangements.

That's how I would have done it. Potential energy = kinetic energy. You shouldn't over analyze Newsletter Challanges!

GA

Better still that Randall saw the flaw in this answer and demonstrated it with his simulation in post #25. I would therefore supplement this energy equation with a simple momentum analysis to show where it first breaks down.

Hint:
The rate of change of momentum at the corner is d/dt(mv). Of course the important term in the expansion is the one we rarely consider (the one that involves the change in the amount of mass that is moving)

Have fun

Some problems are much easier to analyze from an energy standpoint. A classic example is calculating the average force of a bullet of a given velocity from the depth of penetration in body armor. Average force x depth = kinetic energy. Much easier than calculating time rate of change of momentum.

This is a case where we need more detail than the global energy equation can supply.

It is the method that Randall used to calculate the velocity under the assumption that the rope followed itself around the corner. Energy balance still holds, of course, but we are looking for the point where the speed of the rope ceases to be uniform, so the energy solution is no longer sufficient.

Energy balance is a useful part of the solution for the point where the assumption breaks down, but insufficient for the whole. After that point, the falling part of the rope no longer travels vertically, so even the potential energy side isn't straghtforward (in fact the whole thing becomes analytically too difficult, at least for me)

I believe we have to integrate the acceleration as each infinitesimal portion of the rope crosses the edge of the table.

As each slice of rope falls it feels the retarding force of the remainder of the rope on the table which starts out at Velocity = 0. We assume that no energy is lost as the rope bends through any angle, and that the horizontal velocity causes the rope to assume a generally parabolic path as it streams over the edge (this deviates from the diagram which implies a 90 degree downward bend without explanation of how it occurs).

So the acceleration 'A' would necessarily increase as: g* L`/(L-L`) where: L` is the length over the edge. Acceleration 'A' increases finally achieving the value of 'g' (9.8m/s² or 32 feet per second per second) as the final bit of the rope crosses the edge.

Expanded Definitions (& assumptions):

The OP doesn't specify the position where the speed of the rope is to be measured (or calculated). Assuming the desired point is the tail end, as in the last part of the rope to go over the edge, then 'speed' is composed of two vectors. The vertical component will be constrained to zero by the table, and the magnitude of the horizontal component can be considered numerically equivalent to the speed being sought.

The problem of calculating the 'speed' of the head end of the rope is more complex. The free end would begin with Vx=0, Vy=0, but as the rope slides, each portion has a steadily increasing Vx, with Vy increasing from 0 as noted. We assume a rope does not stretch, so as each infinitesimal slice of the rope passes the edge is pulls on both the rope still on the table and the rope that has passed the edge and is now falling.

Since the height of the rope that has previously passed the edge is less than the height of the table, it will now feel a pull at an angle with both horizontal and vertical components.

That force imparts a horizontal acceleration on the fallen rope, pulling it through some small angle A which increases as the rope accelerates. Since the rope does not stretch the hypotenuse of the triangle from the head, the side of the table, and the distance Vx the rope traveled is reduced, The head of the rope feels this retarding force at some angle B which begins at 90^o (vertical) and rotates towards the horizontal. Therefore the vertical acceleration of the head of the rope is slowed as more of the rope acquires a horizontal component, at the expense of the previously acquired vertical velocity component, because only the energy imparted by the fall remains constant.

Therefore, as the rope falls, it assumes a roughly parabolic shape. The rope is not in free fall until the tail leaves the table. At that moment the horizontal force drops to zero and the horizontal velocity remains constant, and the entire rope begins falling freely until the head hits the floor some small distance from the edge of the table, carried by the net horizontal velocity.

That's the physics, I don't have the tools to show the notation here.

I don't believe a single specific speed is possible to calculate without knowing how much of the rope is "a small amount". Is that small amount 0.01, or 0.1, or 0.25 of L? Those numbers are all well under half, so could all be considered small.

Secondly, I haven't really attempted a solution, but I strongly suspect that the answer will also be dependent on the stiffness of the rope, and I have no idea how to specify or make use of a stiffness value. The Challenge did not specify an infinitely flexible rope, but I suspect that was the intention.

Finally, I'd think it would also depend on the thickness of the rope, since the center of gravity starts "a small amount" below half the rope thickness above the table. I can imagine an infinitely thin string, but not an infinitely thin rope.

"The Challenge did not specify an infinitely flexible rope, but I suspect that was the intention."

Clearly, simplifying assumptions have to be made if the problem is to be soluble, so any unstated parameter has to be assumed "ideal". To be honest, definition of idealisation is a common issue with C4 challenges. This is also true of University examination questions at the higher level, and the examinee is expected to either to assume ideality or state additional assumptions and then settle down to a best attempt to answer the question. I believe that there is little point in these challenges if participants do not take the same approach.
Thus, the rope has to be placed perpendicular to the edge of the table, be infinitely thin, inextensible, with zero bending rigidity, "small" means you are expected to calculate the limit as the overhang tends to zero, and you should include any other idealising assumptions you need to make (reductio ad absurdum: do we expect every relevant challenge to state that you can ignore rotation of the Earth, Brownian motion...).

However, the question as posed is unfortuately unasnwerable, because (as observed by Randall) at some point the rope will no longer fall in a vertical line, so different parts of the rope will have different speeds. In order to have anything sensible to solve we need a different question.

The closest in terms of the question would be to substitute "the kinetic energy of the rope", but that is an exceedingly difficult question, and probably not even capable of an analytic solution, so contribution #4 proposed the alternative of discovering the situation at the time when the rope stops falling vertically.
Have fun! And when that is solved I have a different (but related) case that allows the "leaving the top of the table" condition, but does not have the same value answer as Randall assumed was expected for this challenge.

Answer: Indeterminate given the information provided.

How much is "a small amount" If it is 10% then the rope has 0.9L to accelerate from rest. If it is 1% the rope has 0.99L to accelerate from rest.

The table is frictionless but as no statement has been made about the flexibility of the rope, the rope cannot be assumed to be infinitely flexible, so there is a minimum radius as the rope curves round the edge of the table. When the rope reaches near the end the lack of flexibility in the rope will lift the trailing end off the table prior to the trailing end reaching the corner. the horizontal inertia of the trailing end when it breaks contact will flip the rope outwards so it will not free fall vertically. How much it flips out is determined by the mass of the rope as well a the bend radius. Thus you lose rope length at the trailing end as well as the leading end.

If the rope is infinitely flexible and has no mass it does not move, it would not leave the table, so still no answer can be determined.

No friction on table,even on the corner? Then it is the same as dropping a crooked rope from the same height- .5L.

When the end strikes the floor,of course,it stops,but the remainder falls at the same speed as the first because it is not in free-fall until it clears the table.

As I stated before,a series of ball bearings would be the same,without connections in between,because all objects fall at the same speed.

Unless the rope stretches,all parts must fall at the same speed while in free fall.And if it stretches,it is not in free fall.

As Randall astutely observed, the rope will eventally not fall vertically, which means that its end will have to leave the table before the CofG has fallen 0.5L below the table top. So even the kinetic energy will not be as great as assumed. (The assumed situation would however apply if you ran the ideal rope through an ideally bent and ideally-fitting tube...)

If you try a reasonably close practical experiment (string not rope for flexibility and safety), you will find that the string does not land immediately below the edge of the table; this shows that eventually there is insufficient tension in the rope to arrest its forward motion it as it goes over the edge of the table; and this is in spite of the delayed starting condition (so slower speed) and frictional losses helping to hold the string back horizontally.
(Your wife's fine gold chain running over a surface you have prepared by rubbing with PTFE plumbers' tape would be a closer approximation - but we probably wouldn't be fiddling on CR4 if we had that degree of autonomy)

P.S. The curator at the museum where you are located may be able to provide even more appropriate materials

What frictional losses?????? Wasn't this a frictionless table?????? No friction....no tension!!!!!

Two significant problems with your posting:
1) The comment on friction clearly related to the "reasonably close practical experiment"
2) There is still tension in the rope under the ideal situation of the OP. It is needed to accelerate the horizontal section, and it serves to reduce the acceleration of the vertical section so that they are equal. The tension is zero at the ends and increases as you approach the corner of the table (from whichever end).

I hope this helps

I forgot to say: the centre of gravity of a crooked rope will not be 0.5L below the top of the table when the end of the rope leaves the top.

Also, you wrote: "Unless the rope stretches,all parts must fall at the same speed while in free fall.And if it stretches,it is not in free fall."
Usbport made a similarly invalid comment, answered in post #14. And of course a crooked rope (which you acknowledge) can whip all over the place.

The rope can whip around all it wants,it cannot increase or decrease it's free fall speed(ignoring air resistance).

The OP did not ask for velocity,he asked for speed.

He also did not ask for the speed of the rope when the first end touches the floor.

The end may land further away than the rest of the rope,but while in free fall,it is moving at the same vertical speed.

To get a little more technical, the OP did not specify if the table was horizontal,or tilted,and if tilted at what degree.If not specified,then any value can be taken.So how about a fraction of a degree from vertical?

Is the edge of the table square,or smoothly curved?Again,open to speculation.

The OP did specify at the instant that it clears the table,not later,when it touches the floor.

Overall, a very poorly written challenge.

Aren't the challenges there for you to make the best of? It is clear what the challenge should be. One of your carps isn't even relevant: the height of the floor is given so that you know the rope does not hit the ground before leaving the edge of the table.
The only real defect in the challenge is that it did not apparently recognise that the rope would not continue to follow the same path around the corner of the table, and that the solution becomes impractically difficult once the movement of the hanging end is no longer vertical.

"The end may land further away than the rest of the rope,but while in free fall,it is moving at the same vertical speed."

Spin a wheel with its axle horizontal and stationary. The top moves horizontally. the sides vertically at the same speed. But the speed reduces as you approach the centre of the wheel. The wheel does not have a single speed in any direction. It has an average (in this case zero). Ropes being flexible are harder to analyse, but the principles are unchanged. As the rope is twisting around we can be certain that:
. a) its CofG will not be as low as L/2 when the end leaves the table, and
. b) its average vertical speed will be less than its average speed, and (because there is additional energy in the twisting) its average speed will be less than sqrt(g.h) where h is the height of the CofG below the table edge.
This makes the problem as posed too difficult for me. It's reasonably straightforward to solve the motion up to the point when the behaviour stops being simple, and I have done this, including finding the point at which the analysis fails. I'll propose this as a future challenge - if no-one else cares to solve this on the basis that it wasn't the challenge as presented.

The speed of the rope before clearing the table is not the question..the speed at the "instant" of clearing the table of the rope,and I take this to mean the entire rope,not the end or middle of the rope.

I interpret "Instant" to be a very short interval of time,before the rope has a chance to rebound from the bend.

The smaller the interval,the less certain the speed or position of the rope will become.

The OP did not ask for Velocity,as I previously stated,he asked for Speed.

Drop your spinning wheel from a height,and regardless of the velocity of the spokes,the ENTIRE wheel is in free fall.

The rope could be on the table in a serpentine configuration instead of a straight line.

Again, a matter of semantics, or semiotics.

Every good engineer is very concerned about details,as is a good physicist.

As stated, I understand the timing perfectly. The problem is that the situation is only readily analysed up to the point at which the assumption fails. At the time that the end of the rope leaves the table, all I'm able to say is that the kinetic energy E is less than M.g.L/2, that the average speed S of the rope is less than √(2.E), and that the average vertical component of the velocity is less than -S.

Tell me, what is the speed of the wheel when the centre is stationary?
If you have two wheels attached to a car that is spinning out of control having fallen off a cliff, do the two wheels still have one speed?

If you tell someone that the wheel has zero speed they will normally interpret this as saying the wheel is stationary, which is isn't if the wheel is rotating
These ambiguities are the reason you (andI) should state "average speed" or "average velocity" in these cases.
The exception would be when you are referring to an object where the meaning is well defined by the context, such as a car moving down a road.

Another question: what is the speed of the rope when it is half-way, and one part is moving vertically and the other horizontally? Is it the speed of every section of the rope √(g.L/2), is it the (undirected) average velocity of the rope (√(g.L/4), or? I would say √(g.L/2). This becomes very difficult when differnet parts of the rope are moving at different speeds in different directions, and is of little use in predicting outcomes; average components of velocity become far more useful at this stage.

This matter of interpretation of the OP's intent could go on forever,without a solution.

Do not confuse Speed with Velocity.

The OP simply asked for speed"at the instant..."

To get an average,you must combine measurements from the past states of the rope,which does not apply here,therefor I will not use the average speed..

It will only get deeper and deeper as we discuss it,therefor I will withdraw from any further waste of time.

I will await the "Answer" from the OP,the only one that really knows his intention.

I thought you were saying that speed of an object whose parts are moving with different velocities was well-defined. As I've heard multiple incomptinle definitons for such a thing I was asking what you meant. I made some suggestions, couched in terms of velocity, which is better defined, and so can be used to define speed. This is not confusing the two, but using one as a tool to help clarify the other.

I hope I'm wrong, but your accusatory response suggests to me that did you not have a clear idea what you meant by speed in this context, and worse yet that you'd rather belittle others than work things through.

In support of the above hope, I'll give two different results for average speed of a uniform ring with its perimeter rotating in-plane at 3-m/s and central velocity (out-of-plane) of 4-m/sec:

Speed1 = average velocity without stating direction: 3-m/sec
Speed2 = average speed of all its elements: 5-m/sec (I deliberately made the numbers trivial)

Which if either do you mean?

BTW the second might not uniquely define the result in all cases

Without direction?

Velocity is a vector measurement of the rate and direction of motion or, in other terms, the rate and direction of the change in the position of an object. The scalar (absolute value) magnitude of the velocity vector is the speed of the motion.

There was no intention to impune your intelligence,education,training,intuition,attitude,perceptional acuity or experience, I was merely stating my opinion of a truly ambiguous challenge.

Of course, it isn't what you see as much as it is how you look at it.

"No one can insult me unless I let them."---Eleanor Roosevelt

I wasn't actually interested in the challenger's analysis at ths point, as I have every expectation that "the answer" will be the one Randall gave (and said was incorrect). That incorrect case has all the rope moving at equal speed, so there's no definition to required.

However, my recollection is that you said that the speed of a rope with complex motion was well defined - so I'm still trying to understand what you meant.

Using your terminoalogy for velocity/speed (which I believe to be exactly equivalent in meaning to what I writted:
Do you take the average speed to be the scalar value of the average velocity? If so, the rope's average speed is less the speed of any part of the rope until at least 75% has left the top of the table.
Or do you take it to be the mass-averaged speed of the individual secions of the rope?

Note that when applied to a car the term speed almost universally means the first of these. When applied to a rope going well-controlled through pullies it almost universally means the second (all speeds are pretty-much identical, of course).

I know other definitions that are appropriat in other contexts (and generally clear from the context). However, I have no idea what definiton one might use for the speed of a rope when the different parts are travelling at different speeds - except if one part happens to cause damage, in which case the relevant speed is usually the speed of the bit that caused the damage.

Let's take the wheel, give it a spin,and drop it so that the rotation is perfectly horizontal to the ground.The wheel,as a whole will fall at a uniform speed,will it not?

Presumably you mean "with a spatially uniform vertical component of velocity"?

Now suppose the axle is horizontal?

But either way I don't think it is the same definition as the speed of a rope passing well-controlled through a pulley system.

Vector misuse... (L and v should equally be marked as vectors)

And once corrected it fails if the rope is not moving uniformly, which (as Randall observed) it won't be

For what it's worth this is what Algodoo makes of it:-

https://youtu.be/-sRxBtt09R0

This rope is fairly long about 25 foot.

I can't find out how to get rid of all the stiffness of the rope, and, I had to add the rounded edge of the table because the rope is modeled as a series of short straight sections.

As often seems the case with Algodoo the result is probably closer to reality than the ideal simulation.

Thank you. I doubt that it is that far off the ideal. You can see the separation at the top begins at about the right time, and slso indications of whipping of the top end of the rope as it leaves the corner. I hadn't expected it to take quite as long for the bottom end to be pulled out noticeably, but it's probably right. If it looks essentially the same when you double or halve the length of the rope (everything else except the available drop unchanged) it's probably extremely close.

IMO the bigest visible departure is likely to be the (minor) continued rotation of the link as it goes (beyond vertical) around the curve. Moment of inerta of course. This would reduce if you could concentrate the mass at the centre of each link. (Call it a chain and see if that allows you to do this?)

If you idealize the problem enough, the rope will fall, but possibly not during our lifetimes. The time is going to depend on the finite mass per unit length being nonzero, the initial length hanging over the edge and gravity (earth normal or black hole event horizon).

F=MA

Acceleration of rope instantaneously for length of rope hanging H and mass per unit length D under influence of gravity G is:

Acc=(H/L)*G

Integrating from H(t=0) to L will yield the velocity when the end of the rope leaves the edge.

AT time =0, if H approaches 0, Accel approaches 0 and time to fall approaches infinity.

If density is 0, then there is no acceleration and time=infinity.

For H>0 and D>0, V will have finite solutions for a designated G. This experiment would not work well on the International Space Station.

At the black hole event horizon, the braking effect of radiation emission from the rope converting to gamma rays and the relativistic velocity of the rope will not be insignificant.

While you're correct that the time would be infinite for a rope of zero density, the only fanciful assumption in this problem is the frictionless tabletop. So while I agree that for a purely mathematical analysis based on an infinitesimal length of rope being pulled over the edge, the time would seem interminably long for the rope to get started, the time would in fact be quite finite.
Of course that depends on # additional assumptions: 1) The rope actually slides when dL is extended over the edge. If the rope's material deforms and the flexure absorbs the energy released by the 'fall' then without motion the rope will not continue.

However, the OP explicitly stated that the initial displacement/tug be sufficient to cause the head of the rope to begin falling, and therefore the body of the rope to begin sliding, the possibility of an indefinitely protracted beginning of the observation of motion is precluded.

"...the only fanciful assumption in this problem is the frictionless tabletop."

As I see it, we must also assume a sharp edged table; otherwise a radius of curvature would need to be specified. ...and we must assume infinite flexibility; zero energy loss to bending. The sharp edge may not be fanciful, but the zero bending loss certainly is.

All I did was identify boundary limits for the idealized case. I see the discussion going two directions, idealized where the only terms are mass per unit length and gravity and semi-idealized where flexibility and edge radius become relevant terms and ballistics comes into play. So far I haven't seen any treatment of center of mass radial distances and electromagnetic radiation emission ala James Clerk Maxwell.

I mean, if we're going down the rabbit hole, the rabbit should be big enough to make it worthwhile.

this depends a bit on the elasticity of the rope, it's more clearly if we say it's a chain!

the gravitation pulls only at the end x (0<x<1) of the rope about the tborder of the table, the end on the top of the table (1>(1-x)>=0) act's as a break, the part x accelerates the rope! have an integral about the Length L.

Have you seen my hometriner? or did you get it?

I do believe that after reading the question that the small piece hanging off the end of the table would not be enough weight or have enough force to drag the rest of the rope off the table, so the rope would never loose contact with the table.

In the OP, it states: "Release the rope, so that the rope starts to fall down."

Thus, it MUST start to fall, and if it starts, it will drag the rest of the rope off the table, since there is no friction (or other force) to stop it.

I still believe there can be no single correct answer without knowing what fraction of the rope is left hanging over the edge, and I agree with those who say that different parts of the rope will have different speeds at the instant of interest.

You've hit on another boundary condition. At zero rope diameter and zero edge radius, a freebody diagram would indicate that the vectors are wrong to impart lateral force on the rope on the tabletop from the vertical force on the section hanging over the edge.

The problem requires either elastic deformation of the edge to convert vertical to lateral force or a radius at the edge or the rope centerline to provide the vertical to lateral force conversion.

The type of the problem is indicated by the word frictionless, therefore no need to consider non-ideal issues.

Since the time of Galileo it has been known that the velocity of objects falling in a gravitational field do so independently of mass. It seems simplest then to use a standard equation of motion, which can be derived using basic calculus:

v² - u² = 2as where v = final velocity, u = initial velocity, a = acceleration and s=distance

The tip of the rope is given an instantaneous impulse and moves downwards by the force of gravity. I assume u = 0. The bottom tip of the rope descends and what the rest of the rope does is irrelevant. When the tip of the rope has moved downwards by L, the instantaneous velocity may be computed by substituting into the previous equation we get:

v = √(2gL) where g is acceleration due to gravity

No need to discuss how many angels may dance on the head of a pin. Next problem please!

Even a frictionless table top exerts vertical forces on the portion of the rope still above the table, preventing the tip of the rope from falling with an acceleration of g.

The acceleration must start out at a low value, and approach g asymptotically at the time the tip has fallen a distance somewhat less than l, since the rope is no longer straight at the time it looses contact with the table.