Club Obi Wan

This month's "American Scientist" magazine happens to have an excellent article on why whips crack.

The url is :http://www.sigmaxi.org/amsci/Issues/Sci ... smath.html

Here is the content of the article:



Science Observer
September-October, 2002

Crackin’ Good Mathematics

A couple of years ago, Alain Goriely of the University of Arizona visited
Hungary for a mathematical conference. He took a break to listen to some local bands and watch entertainers crack bullwhips. Watching a whip snake rapidly through the air and then outplay the bands with a thunderous snap, Goriely wondered what created the crack.



Image by Tom Dunne.

The fictional Zorro may have defended himself with a long whip, but a
bullwhip was not designed as a weapon. The bullwhip—really just a long whip—was developed as a way to control a herd of cattle or to signal someone out of yelling distance. It consists of a handle followed by a long, braided section called a thong that tapers down to a fine end, or “cracker.” Experts crack bullwhips in a variety of styles. Most pull the whip over the head and back, cracking it far in front of them. Others swing it sidearm. Some swing it their own way.

In the early 1900s, some scientists wondered whether a whip’s crack came from a sonic boom. That is, perhaps part of the whip moves faster than the speed of sound, around 750 miles an hour, and the clap of noise comes as the sound barrier is broken. Presumably the cracker creates the crack. By the 1920s, high-speed photography revealed that a whip’s cracker can indeed break the sound barrier. The question for Goriely, however, was: How can the relatively slow speed of a whip pulled back and forth generate such high speeds at the tip?

Goriely said, “The first thing I did—after looking at the old papers—was buy a whip.” He bought one through the online marketplace eBay for $15. “It was a really crappy whip,” Goriely said. Although he bought a book and videotape on bullwhip cracking, he couldn’t get a squeak out of his. He moved up to a better whip, one that cost $70, and started cracking it right away. “You realize that you need a good whip to make it work,” he said. In fact, someone fancying a decent whip can easily pay a few hundred dollars or more.

More than a good whip, Goriely needed a good teammate. He had one in graduate student Tyler McMillen. McMillen said, “We had been doing problems with elastic rods, so we thought a bullwhip would be an interesting problem.” Goriely and McMillen started by examining past approaches. In general, previous investigators selected a law of conservation, made an assumption about the shape of the whip and combined that information to calculate the velocity of the tip. Different conservation models—say, conservation of energy vs. conservation of linear momentum—can give different results: The tip approaches infinite speed in some models and maintains a constant speed in others. McMillen said, “The problem is: You can’t assume the shape of the rod, because the rod obeys a physical law, and you must solve for that shape.”

This solution demanded cracking and computation. Goriely said, “I was in my backyard cracking, and Tyler did all the hard work.” McMillen started with what he called a pretty simple model of an elastic rod, basically a rod that can bend as a whip does. He let the radius of his model vary, so that its shape mimicked a whip’s taper, and he modeled the unfurling of the whip as a traveling loop. McMillen said, “We derived the equation for the rod with the varying cross section and looked at how this loop would change as you allowed the cross section to vary. That required some advanced mathematical techniques.” Goriely and McMillen found that in fact a loop can stay the same size as it goes down the rod and still accelerate.

The crack happens when the loop reaches the end of the whip and opens. When Goriely and McMillen’s model used a tapering that made a virtual whip’s tip just one-tenth the diameter of the handle, the tip reached speeds 32 times faster than the original speed of the loop. As a result, this model bullwhip’s tip can break the sound barrier rather easily. (In real whips, the increase in speed could be much higher, because McMillen said that real whips often taper even more.) Getting to this answer required conservation of energy, as well as conservation of linear and angular momentum. In other words, Goriely and McMillen created a much more realistic picture of the dynamics of cracking.

This work, however, goes beyond curiosity about the sound of a whip. For one thing, whip cracking provided Goriely with a new teaching tool. He said, “I do take it to show the students, because people are very impressed by whip cracking.” Moreover, Goriely and McMillen’s model provides added insight for the general problem of the motion of waves in complex materials. One day, investigators might use this work to understand the motility of bacteria and sperm or the waves along DNA as it unfolds to make RNA. The role of a bullwhip might eventually stretch far beyond herding dogies—all because cracking caught Goriely’s attention.—Mike May


-Sergei

“You realize that you need a good whip to make it work,” he said. In fact, someone fancying a decent whip can easily pay a few hundred dollars or more.

I can see it now, as I try to convince the misses... "But dear it's scientific fact!"

Though I don't think I'll let her read on to the part comparing it to sperm.

Great article Sergei, thanks for posting.

Mike

Great posting Sergei,

There is an interesting foot note to that scientific paper Goriely published. At first he was making the arguement that it was the loop traveling the length of the bullwhip that was generating the sound. I had a couple of email discussions with him and illustrated that a whip can make a crack with out linear movement. (trying to bolster my non-mathmatical expierence) At the time Goriely was resolute that his math wasn't wrong.
I haven't re-read his newly puplished paper but form the gist of the article it looks like he has agreed (somewhat) that the sound is generated by the cracker. I know I wasn't the only practical whip cracker debating his mathmatical findings. Some of that discussion was covered on the WE group email list on Yahoo some months back.
I'm interested to see how his observations have changed and of course I'm glad tha whips are still news to the scientific community.

Smiles,
Seb~

Oh, yeah, I remember Goriely now. He was the guy that I saw published an article in May about the loop breaking the sound barrier and not the end of the cracker. Interesting if he really backed off this claim since this American Scientific article. Or whether the article is out of synch with Goriely.

I found the reprint of that article (Scientific American):

t takes a dexterous hand to coax a whip to crack. Now researchers report that they have discovered the mechanism responsible for the startling sound. It has long been thought that the crack results from the tip of the whip traveling fast enough to break the sound barrier and create a sonic boom. But the new findings suggest otherwise. Apparently, it's the loop in a whip that is the real noisemaker.

Though by no means a master whip cracker, Alain Goriely of the University of Arizona was nonetheless intrigued by the phenomenon and set out to study it at a theoretical level. Together with Tyler McMillen, a graduate student in applied mathematics, he modeled the behavior of the leather strips in a paper to be published in Physical Review Letters. Previous whip work (one of just three papers on the subject in the past century) had resulted in the puzzling observation that the sonic boom occurs when the tip of the whip is traveling at about twice the speed of sound. But if the tip were truly the cause of the crack, why wasn't the sound heard earlier, when the tip first reached the speed of sound? Goriely and McMillen's calculations have revealed the answer. "The crack of a whip comes from a loop traveling along the whip, gaining speed until it reaches the speed of sound and creates a sonic boom," Goriely says. He notes that even though some parts of the whip travel at greater speeds, "it is the loop itself that generates the sonic boom."

Although the whip's tip has lost the distinction of being the source of the menacing crack, it is still a force to be reckoned with: according to Goriely's calculations, "the tip can reach speeds more than 30 times the initial speed [of the whip]." --Sarah Graham

I'm wondering the same thing. I emailed Alan a moment ago and it usually takes a day or two for him to reply. If he has backed off his earlier claim, I'm hoping my debate with him played some small part. The clincher to my side of the debate was the same spin crack I demonstrated at the park (TX- Indy meet) and it's on that video clip. Almost no linear movement to the whip, yet a crisp crack is developed. Alan couldn't refute the practical evidence and demonstration. His response was, that technique didn't fit the model he was using. My answer was his model was flawed. I don't think I have to tell you that a prof with a list of degrees as long as my arm doesn't like being told they are wrong, espcially by some whip cracking bumpkin. even if the bumpkin is right.

Keep 'em cracking,
Seb~

LOL, your demonstration didn't fit his model! So then he decides to ignore it? Let us know what his response is from your email. I will bet you that there will be some heavy CYA language in his response back to you. You have to expect it.

-Sergei

I can't imagine what his response will be. I don't think he'll be too excited to read my email and respond. Some times Academics are so stodgy and he wasn't too pleased to debate his work with me the first time (since I'm not a mathmatician nor in any of his classes). I can't say I'd really blame him. After all, he and his young assistant spent time devloping thier findings and publishing a scientific paper to his peers. Then here comes this nobody with no background in math beyond high school algebra, and tells him he is wrong.
Honestly, his only flaw (besides publishing too soon) was he didn't have much expierence in actually throwing a whip and only knew how to toss the forward loop (cattleman's, targeting, circus, etc..) so his intial testing model didn't have a full dynamic of how a whip can crack.

,
S~

I think it is very telling that these highly educated scientists are just now "discovering" what whip crackers have known for decades. These are the same folks that have proved the big bang is fact and that God doesn't exist.

Thanks for sharing the article, Sergei.

-Kyle

Interesting. I studied this a while back, too. That article takes 2 paragraphs to illustrate Newton's Second Law of Motion. Acceleration is equal to Force over Mass. Acceleration increases as the force travels over a continuous decreasing mass. Similar to what Sebastian said, whether there is linear motion or not, I think in this one application, is unimportant.

I do think there is a principle that is being overlooked. After watching the video, I noticed that cracks like the Overhead crack were much louder. I started thinking what might cause the louder crack. More acceleration? Where would that come from.

Then I remembered that NASA accelerates spacecraft in planetary flybys using the "slingshot effect". I was wondering if the loop or extra curl in the whip helps with the acceleration of the popper. Researching this theory I found a quote by R. C. Johnson:

R. C. Johnson, Department of Mathematical Sciences, University of Durham, Durham, England wrote:At first sight the underlying slingshot mechanism seems puzzling, for there is an uneasy feeling that something is conjured from nothing. It is presumed to be rather sophisticated when, despite its obvious importance, NASA's public information is either highly technical or utterly confusing.

Indeed, a clear elementary mathematical treatment is less accessible than it might be - given that by its usage the slingshot effect is a modern triumph of Newtonian Mechanics.

Triumph? Yes. Modern? Mmmmm..maybe not. He then goes on to try to explain the effect with some really big formulas. I wonder if that's what Goriely calls "...some advanced mathematical techniques.."

I then came to fork in the road. I could research this further using advanced mathematical tools or I could apply Pecards to my Aldens. Guess which one I chose?

Cheers,
Pyr.

Pyroxene wrote: I then came to fork in the road. I could research this further using advanced mathematical tools or I could apply Pecards to my Aldens. Guess which one I chose?

Cheers,
Pyr.

Well you are a good researcher Pyroxene. I think I'll take the Yogi Berra philosophical route at the fork in the road. In fact that was his title to his last book, "When you get to the fork in the road, take it."

Anyway, I will end this post with another Yogi Berra quote:
"You can observe a lot just by watchin'."


-Sergei

p.s. And here's one for Goriely. ""It ain't the heat; it's the humility."