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Ed Hartouni
Trad climber
Livermore, CA
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for the purposes of this thread we can assume that alpha is constant...
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Mighty Hiker
climber
Vancouver, B.C.
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Topic Author's Reply - Sep 4, 2010 - 02:38am PT
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Bruce may have been alluding to dark energy and dark matter - or is that Dark Energy and Dark Matter?
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Dr.Sprock
Boulder climber
I'm James Brown, Bi-atch!
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The high speed of the tip is explained by the law of the conservation of momentum. The initial motion is applied to the handle, and the initial wave along the whip's thong has a much larger mass than the end wave at the whip's tip or popper (a thin flexible piece of material tied at the end of the whip). Since the momentum is the product of the mass and speed of the moving object, the smaller the mass, the higher the speed, hence the light popper moves extremely fast. Also, the more flexible the popper, the shorter (and lighter) the last moving wave, hence an even higher speed. Many popular science explanations published capitalize on the fact that the general shape of a whip is tapered: thick at the handle and very narrow at the tip, hence the decrease of the mass. While tapering does contribute to the quality of the crack, it is not a deciding factor. Even "flat" whips can crack: the actual decrease of the mass of the moving part occurs simply because the whip ends: the closer the moving loop to the tip, the shorter the moving part. In this respect the whip crack resembles the "shoaling" action of a tsunami: a deep-water ocean wave piles up tremendously when entering into shallow waters. Recently, an additional, purely geometrical factor was recognized: the tip of the whip moves twice as fast at the loop of the whip, just like the top of a car's wheel moves twice as fast as the car itself.
quiz after lunch,
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Ed Hartouni
Trad climber
Livermore, CA
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Dr. Sprock, you haven't done the reading, look above, check out the references (they're not "popular science") and report back
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Dr.Sprock
Boulder climber
I'm James Brown, Bi-atch!
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i need diff equations,
i'm just not satisfied,
he ed, how is your math?
do you use it in heavy amounts at work everyday,
or is it sporadic calculus, used once a year?
i got a problem i need help with,
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Ed Hartouni
Trad climber
Livermore, CA
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everyday....
unless I'm catching up on my training...
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Dr.Sprock
Boulder climber
I'm James Brown, Bi-atch!
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ok Ed, i have been working on this seeming simple but yet, difficult problem.
if i want to wind a transformer, i usually use a square or rect bobbin.
if i wind on a circular bobbin, the wire velocity at the wire feeder will be constant.
but with a squate bobbin, there are accelerations from a dead stop to whatever,
4 times per revolution of the bobbin.
i want to program the stepper motor driving the bobbin to take out the x axis accelerations out of the wire, so that i can get a better wound coil.
the problem is to develop a set of functions that will describe the x velocity for given distance and set bobbin size.
i have tried setting it up in a number of ways, here are a few ideas,
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Ed Hartouni
Trad climber
Livermore, CA
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if you want a "constant tension" wind,
try this:
you set the feed motor velocity up so that it increases with an increased angle (as shown) and decreases with negative angle
This will keep the wire at a constant tension as it winds around your square bobbin
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Banquo
Trad climber
Morgan Hill, CA (Mo' Hill)
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Search google scholar for: folded free falling chain
Seems to be an old and difficult problem.
http://arxiv.org/PS_cache/physics/pdf/0510/0510060v1.pdf
I haven't been following this thread and apologize if I missed something. My thought problem goes something like this...
A tightly coiled rope sits on a ledge and one end is dropped. Ignore friction and air resistance. After some time, the rope reaches a point where it has just about reached straight in free fall. The CG of the rope has fallen half it's length (solve potential energy for 60m rope and I think this is about 24 m/s). The last tip of the rope has to accelerate from rest very suddenly putting the rope in tension and stretching it. As the tension pulls the tip of the rope off, it over accelerates as the stretch releases. If the last foot of the rope has a fold in it, it seems reasonable that it might snap as it pulls off.
Another model is holdong the two ends with the rope folded in half and hanging free, then drop one end.
Also:
"Falling Chains" Chun Wa Wong and Kosuke Yasui, American Journal of Physics -- June 2006 Volume 74, Issue 6, pp. 490-496
These guys actually did experiemnts:
"The motion of a freely falling chain tip" Tomaszewskia, Pieranskib & GeminardAmerican Journal of Physics -- September 2006 -- Volume 74, Issue 9, pp. 776-783
In the folded chain model, you can see the end whip depends on how close the ends are when it is dropped:
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Banquo
Trad climber
Morgan Hill, CA (Mo' Hill)
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With a tightly folded 1 meter chain they got peak velocity at the tip of almost 20 m/s and peak acceleration of 50g!
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Dr.Sprock
Boulder climber
I'm James Brown, Bi-atch!
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he Ed, i figured it out, you just find an equation for position, then integrate to get the total velocity equation.
but the integration is a pain, see here>
so i just need your help now in integrating:
Z= Root (D^2 -2D*cos t + 1)
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Ed Hartouni
Trad climber
Livermore, CA
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I'll check it out Dr. Sprock, you can always integrate it numerically....
Banquo - good reference, the rope is a bit different than a chain, but they share behavior in some regimes... I think that in the "whip crack" regime they are very different...
Look at my strobe picture up thread and see similarities to the figure you posted....
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Ed Hartouni
Trad climber
Livermore, CA
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to the paper by Tomaszewski, Pieranski and Geminard is a good one, and anticipates some problems with the rope.
Their Equation 10 has an analytic expression for the velocity of the chain tip as a function of how far the tip has fallen:
v(h) = √[gh(2L-h)/(L-h)]
as h → L, v(h) → ∞
this is good news for the OP as it indicates that the chain tip considered in the paper gets going very fast.
The analytic approximation does not correctly predict the maximum velocity of the tip...
their final sentence is my excuse for not getting to the solution yet:
Dynamics of the falling chain hides certainly a few more interesting details. The same, even to a larger extent, concerns the dynamics of the falling rope. In the latter case the dissipation plays a much more important role and elasticity becomes a crucial factor. Laboratory and numerical experiments are waiting to be carried out.
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Ed Hartouni
Trad climber
Livermore, CA
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looking at that paper, let's scale to a 100 m chain.
dropping the chain with the ends at the same height, and separated by 25m, would give a peak velocity of something like 320 m/s
This assumes that the peak velocity is well approximated by the analytic result but at the observed height of the maximum velocity in the experiments and numerical experiments, roughly 0.99 of the length of the rope...
This is close to the 343 m/s speed of sound... so that's quite good!
But the details of the constituent model of the rope will matter...
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Ed Hartouni
Trad climber
Livermore, CA
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here's another good article:
The weight of a falling chain, revisited by Hamm and Géminard
which concludes that the minimum radius of curvature (which is a part of the constitutive model of the real chain) is an important factor in the analysis of the dynamics...
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Mighty Hiker
climber
Vancouver, B.C.
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Topic Author's Reply - Sep 4, 2010 - 11:30pm PT
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Another aspect of this, raised by Ed's recent remarks, is that the science may show that it's possible for the tip to exceed the speed of sound. However, possible doesn't necessarily equal likely. It may be a rare event. If the math shows that it's just possible, then it may be time for some empirical tests, to see what actually happens. If you get ten different 50 - 60 m ropes, and drop each ten times from different positions (always dropping it the maximum twice the length of the rope), then it should become evident fairly quickly what actually happens.
If someone can borrow a radar gun, we could spend an afternoon at the FaceLift doing this, maybe at the base of the crazy wall.
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bmacd
climber
Relic Hominid
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The sound is just the tip of the rope flopping the full 360 degrees and hitting the other side of the rope.
The sound is not caused by breaking the sound barrier but by the rope tip smacking against the rope itself.
Problem Solved, Next question please ....
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Peter Haan
Trad climber
San Francisco, CA
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This is when MIghty Anders has to mooch up to Fattrad as Fatty can get his gun.
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Mighty Hiker
climber
Vancouver, B.C.
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Topic Author's Reply - Sep 5, 2010 - 01:27am PT
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Maybe I can outrun FatTrad, or grab his radar gun when he's busy during the BBQ. Hopefully he won't tase me.
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Chinchen
climber
Way out there....
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