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G_Gnome
Boulder climber
Sick Midget Land
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Ho man, I am feeling saucy today. If you only knew what I almost posted in response to Juan there.
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Ed Hartouni
Trad climber
Livermore, CA
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Mar 10, 2007 - 12:37am PT
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A progress report on the paper...
...it probably does have all the ingredients for determining the velocity of the rope, and whether or not it is possible for the rope to exceed the speed of sound and emit a shock wave we hear as a crack.
The ingredient is a loop. In a wip this is a distinctive feature of making the crack. It turns out that in the case of the wip, the tapering of the wip from the handle to the tip is an important part of the explanation. Of course, ropes don't taper.
Another difference is that the loop "falls," it's accelerated by the earth's gravity, and that same gravity puts a tension gradient in the rope, the tension is higher near the top of the rope than at the bottom. However, it still seems somewhat difficult to understand how the rope loops get to such high velocities, even in the wip scenarios.
More later
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Mighty Hiker
Social climber
Vancouver, B.C.
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Topic Author's Reply - Mar 14, 2007 - 10:12pm PT
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Bump. In case any of the resident physicists, and would-be physicists, haven't seen it already.
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Ed Hartouni
Trad climber
Livermore, CA
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Mar 15, 2007 - 12:44am PT
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I'm still working on it...
...here's how it might work.
Tie the rope end to an anchor and through the coiled loops over the edge. Let's not get complicated, the rope free falls, never in contact with the wall.
The loops accelerate as they fall, uncoiling along the way. My first estimate of the speed at which the rope falls is just the free fall velocity at the end of a rope of length L:
v = sqrt(2gL)
where g is the acceleration of gravity, roughly 10 m/s/s.
For a L = 60 m of rope, this is 35 m/s (about 78 mph).
The top of the loop actually travels at twice the speed of the loop, or 70 m/s... but the speed of sound at sea level is something like 330 m/s, the velocity is off by an order of magnitude.
Based on the cheesy kinematic estimate I'd say a rope can't crack like a wip.
However, the wip is more complicated and a proper dynamical analysis has to be done. Basically, the acceleration of the loops has to be taken into account properly. The tip of the rope as it wips through the air can be accelerated by factors of 10 or more then the loop velocity. If this were so, then the rope could crack, it's tip exceeding the speed of sound.
But the analysis has to be done.
So far, I'm looking at the following references:
A. Goriely and T. McMillen, Phys. Rev. Lett. 88 2443011 (2002)
R.S. Falk and J.-M. Xu, SIAM, J. Numer. Anal. 32 1185 (1995)
B.D. Coleman and J.-M. Xu, Acta Mechanica 110 173 (1995)
and I haven't quit my day jobs yet.
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Ed Hartouni
Trad climber
Livermore, CA
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Mar 15, 2007 - 12:50am PT
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the speed of sound changes with altitude:
340 m/s at sea level
334 m/s at 5,000 ft
328 m/s at 10,000 ft
322 m/s at 15,000 ft
it looses 6 m/s for every 5,000 ft elevation gain, or 1.2 m/s every 1,000 ft... so at altitude in the mountains it might be more likely to happen than at sea level...
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TradIsGood
Happy and Healthy climber
the Gunks end of the country
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Mar 15, 2007 - 11:01am PT
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My memory of this is that the swing of the whip imparts a certain kinetic energy to essentially the entire rope. So the energy is approximately 1/2 mvv where m is mass of rope and v is the velocity of the hand (rope) while the rope is essentially in a straight line behind the whipper.
Then the hand stops. So the rope held in the hand stops, but most of the kinetic energy has not yet been converted to heat. As a result some of the rope must be accelerated (conservation of energy).
As the mass of the moving part of the rope declines, the velocity (squared) must increase. So the key is to have a small tip, giving low mass and smaller turning radius. It is the turning radius that allows for the mass to reduce more than linearly with the amount of "uncracked" rope.
Of course, this unquantitative analysis does not address the question of whether it would be easy to reproduce in a climbing rope. But it does seem, at least qualitatively, to me that it would be more likely in a static rope than a dynamic one.
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G_Gnome
Boulder climber
Sick Midget Land
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Mar 15, 2007 - 01:06pm PT
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Are you sure you are going down the right path here? Seems to me the most relevant piece of data to look at is the unrolling of the end loop once the rest of the rope ceases to fall. And I just don't see how a static rope is going to be substantially different than a dynamic rope given the small loads (ie. it's own weight) on the rope.
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TradIsGood
Happy and Healthy climber
the Gunks end of the country
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Mar 15, 2007 - 01:34pm PT
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g-gnome.
Pretty sure the analysis of the whip and falling (whipping) rope is the same.
As far as the static vs dynamic...
Static rope would exert higher force regardless of the load, so higher acceleration and resulting higher velocity.
Another way of looking at it is that the kinetic energy of the rope is reduced by the work done stretching the rope (Force times distance). Hmm, maybe that explanation is not so clear.
In fact, I guess I could find a way to question both of the above qualitative arguments. The acceleration is surely higher with the static, but it is accelerated for a shorter time (less stretch).
Kinetic energy argument is also less clear. If static will, in fact, retain more kinetic energy for the whipping action, then it stands to reason that it is heated less.
Ok. rambling over. Static rope stretches less (by definition). So the potential energy of the dynamic rope is lower at the end of the fall. Since it is lower, the dynamic rope must have heated more than the static, since they both have zero kinetic energy at the end.
So less heating means the final kinetic energy of the tip was larger. (I think - anybody disagree?) :-)
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TradIsGood
Happy and Healthy climber
the Gunks end of the country
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Mar 15, 2007 - 01:50pm PT
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hmm - are densities of static and dynamic rope the same? arg.
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G_Gnome
Boulder climber
Sick Midget Land
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Mar 15, 2007 - 02:31pm PT
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I just meant that I suspect that the actual velocity from falling is a small part of the whole, it is probably the whip (which I know gets it's momentum from the fall) that accounts for 90% of the total. Also, I can get a 'not very tapered' fly line to snap even without a tapered leader on the end by casting too fast. It is pretty hard to get a snap unless you actually change direction of the line before the line has straightened out that enables the tip to whip fast enough to actually crack, yet that change in direction is not going to happen with a falling climbing rope.
Good luck with the math on this one!
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TradIsGood
Happy and Healthy climber
the Gunks end of the country
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Mar 20, 2007 - 05:45pm PT
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Ed, here is another reference. I do not have easy access, but found this in "The Physics of Golf".
B. Bernstein, D. A. Hall, and H. M. Trent, "On the dynamics of a bull whip," J. Acoust. Soc. Am. 30, 1112 (1958).
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Mighty Hiker
Social climber
Vancouver, B.C.
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Topic Author's Reply - Mar 20, 2007 - 06:45pm PT
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I wonder if this is one of those deceptively simple seeming problems, that in fact is more complicated than it seems.
There are some variables - rope stiffness, elasticity, diameter, length, and wear, whether it is in contact with itself or the rock as it falls, what kinds of wave patterns it makes. (Presumably a fuzzier rope has more air resistance.) Leaving those aside, it would be interesting to do some empirical tests, on an overhanging wall, and use a radar gun to see what kinds of velocities are attained. There'd have to be lots of repeats, measuring from different angles and so on, but it might fairly quickly give an approximation of the possible velocity.
Looking forward to more on this, though I know Ed's quite busy right now.
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ChrisW
Trad climber
boulder, co
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Mar 20, 2007 - 07:10pm PT
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The article also said there's noway to advoid the crack. With a little practice there is a technique to advoid it. Can't really explain it very well. But i will try. I guess all u do is keep a few feet of rope in your hand and release right when the end of your rope is going to crack. Some how this works.
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Klimmer
Mountain climber
San Diego
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Mar 20, 2007 - 07:12pm PT
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Yes, it is very possible to make something go faster than the speed of sound which is appox. = 340m/s at normal room temperatures (speed of sound also varies with temperature).
When a Bull-whip is used properly, the tip goes faster than the speed of sound and the crack you hear is a mini "Sonic Boom."
Ever use a wet towel to slap your friend on the bum? Have you ever made the same towel crack sharply in sound? If you have, then my friend you made a small sonic-boom. The towel went faster than the speed of sound.
When a car backfires, the exhaust is expelled faster than the speed of sound and you get a large "Sonic Boom." Ever wonder why the tail-pipe doesn't explode when you hear that backfire? Because the gases travelling faster than the speed of sound through the tail pipe, don't make the "Boom" until the gases hit the atmosphere right at the opening. Some back-fire sonic booms can be unbelievably loud, but no damage.
Google, "Sonic Boom in a Bottle" for a great experiment to do. I do this demo for my Physics students, and we get gases to go faster than the speed of sound coming out of a 1 liter bottle and it makes a great big sonic boom, but the plastic bottle is left intact. I 've been using the same bottle for about 7 years, doing this demo for my students.
Bullets flying over head make a sharp crack sound. It's a mini sonic boom! Etc, etc.
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TradIsGood
Happy and Healthy climber
the Gunks end of the country
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Mar 20, 2007 - 08:48pm PT
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Can we assume that because the tip of the whip exceeds the speed of sound that any air does?
For example, the diameter of a Cessna 177 propeller is 78 inches. At 2500 rpm the tip travels at a speed of 1700 feet per second or over 1.5 times the speed of sound.
But what does the air do? Does any air move at the speed of the tip? The plane only cruises at that rpm at about 140 knots.
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scuffy b
climber
The town that Nature forgot to hate
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Mar 20, 2007 - 09:06pm PT
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The propellor is designed to minimize the speed of the air. It's
just supposed to part and rejoin at the trailing edge.
Maybe a chewed-up prop could make booms.
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TradIsGood
Happy and Healthy climber
the Gunks end of the country
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Mar 20, 2007 - 09:17pm PT
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That is my point. And it also rolls over at the tip in a vortex, which may be turbulent, but I do not expect air velocity would be supersonic. (But I am not aerodynamicist.)
But one could imagine that the air simply flows around the tip of a whip or rope and never reaches the speed of either.
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Kevster
Trad climber
Evergreen, CO
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Mar 21, 2007 - 01:12am PT
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Thought I would add on the static vs dynamic that I have had both types of ropes go "Boom" and blow apart their ends when dropped on an overhanging face.
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gunsmoke
Trad climber
Clackamas, Oregon
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Mar 21, 2007 - 10:55am PT
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I watched my rope end crack on a drop from an overhanging route. Blew out the last inch into a fray. The end didn't make contact with the wall at the point of the crack. ChrisW is right about being able to prevent it. When I chuck a rope I give a slight upward tug just before the tip reaches its bottom point.
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scuffy b
climber
The town that Nature forgot to hate
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Mar 21, 2007 - 11:09am PT
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Maybe the tip of the propellor on that Cessna is actually
creating sonic booms. The task is not to get air moving along
with the prop tip, rather to get the tip moving fast enough
(there's all that power) to "outrun" the rejoining air at the
trailing edge. That might create a tiny moving, ever-collapsing
vacuum continually booming upon collapse.
When accelerating a propellor, is there a speed where the noise
level increases suddenly?
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