Wikipedia:Reference desk/Archives/Science/2008 December 4

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December 4

Kinetic energy weapons or using "flying crowbars" to kill tanks

I was recently rereading Larry Niven and Jerry Pournelle's novel Footfall. In the book it mentions the alien invaders using several different KE weapons, where they just dropped something from orbit to smash the human opposition (the "Foot" from the title). But it also mentions using a smaller "flying crowbar" to destroy tanks that are advancing on their positions. Now these "crowbars" weight about as much as a crowbar (a kg or 2) and a heat shield, some sort of guidance and a rocket motor to drop them out of orbit. Anyway, this got me wondering about a few things. First, how fast would these "crowbars" be moving? Second how much KE would they have on impact? And, third, what would that impact be like. Would it be like a grenade, a couple pounds of TNT, a 155mm Artillery shell, a small "suitcase" nuke? Tobyc75 (talk) 00:45, 4 December 2008 (UTC)

Air resistance would limit the speed by the time it struck the surface to at most a few hundred miles per hour, which probably isn't enough for an object that size to destroy a tank. If it struck end-on, that would do the most damage, but probably still wouldn't be enough to breach the armor. Dropping it from higher and higher heights wouldn't make much difference due to the maximum terminal velocity possible in the Earth's atmosphere. However, if it was dropped from orbit, there would be the risk that it would pick up so much speed that it would burn up when it hit the atmosphere (I guess that's what the heat shield is all about). Incidentally, rockets aren't needed to control the trajectory to target, only adjustable fins, see JDAM. StuRat (talk) 02:53, 4 December 2008 (UTC)

(Edit conflict; this somewhat duplicates Stu's answer, but I'll post it anyway. One note: you do need rockets to get out of orbit in a controlled manner.)

The answer depends a great deal on how the object interacts with the atmosphere. Any satellite in low Earth orbit is moving at about 18,000 mph (8,000 m/s). Manned satellites are designed to lose most of speed using friction, so that they can land with ordinary parachutes. These weapons would be designed to retain as much speed as possible, but how much speed is that? It depends on the aerodynamics of the thing.

If we imagine that the object could be made so dense it would be practically friction-free in the atmosphere, then it could be "fired" by deflecting it into an orbit that would graze the Earth's surface (1/4 revolution or about 22 minutes later) and so impact at just about the same speed I mentioned abnove. At that speed a 1 kg object has a kinetic energy of mv²/2 = 1·8,000²/2 = 32,000,000 joules -- so the TNT equivalent would be about 32,000,000/4,164,000 = 7.7 kg or 17 pounds. For a 2 kg bar, twice that.

However, there's an important tradeoff that means those numbers are unrealistically large. If the object is so friction-free that it can punch through the atmosphere like that, it will also tend to punch through its target as well -- rather than delivering all that kinetic energy as heat in one place, it will make the same sort of hole that an ordinary bullet does. To deliver an explosive-like effect on the target, you probably need a significantly less aerodynamic object -- which will reduce its speed on arrival, and of course the kinetic energy varies as the square of the speed.

--Anonymous, 03:10 UTC, December 4, 2008.

If it punches into an armored vehicle, the kinetic energy would be released inside the vehicle; if it punches in and out, there would be significant damage from the vacuum effect. Unsurprisingly, we have an article on kinetic bombardment (I searched for Thor— it has been a while, but I think Project Thor is mentioned in the book). Mass drivers were used as weapons in Heinlein's The Moon Is a Harsh Mistress and in Babylon 5. --—— Gadget850 (Ed) ^talk - 03:24, 4 December 2008 (UTC)

Suppose they used spherical cannonballs - the terminal velocity is the square root of 2mg/dAC_d - m is the mass (2kg), g is 9.8ms^-2, d is the air density (around 1.2 kg m^-3), A is the cross-sectional area (for a 2kg iron sphere - volume is around 900 cc - which has a radius of around 4cm - so a cross-section of about 55 cm² - 0.0055 m²) and C (the drag coefficient) is 0.1 for a sphere...so we're looking at around 240 meters per second as the terminal velocity...close to 540 mph! But they could do much better than that - by streamlining, they could easily cut the drag by a factor of 10 and a longer, thinner cylindrical projectile could have maybe four times the mass for the same cross-sectional area. that's a six time speedup in the terminal velocity - so you're now up to about Mach 5. Much MUCH faster than an antitank round.

But it's even more than that - terminal velocity is what you get if you drop something in dense air - through dense air. In this case, the object picks up speed in the vacuum of orbit - by the time it starts to get into dense atmosphere, it's going vastly faster than terminal velocity and it comes down to a question of whether it'll slow down to terminal velocity before it hits the ground...and the answer is certainly "NO!"...but it's tough to calculate. It's nothing like the energy of a nuke...not even close...but compared to a kinetic energy needed to take out a tank - it's pretty good.

Dropping slightly-guided rocks from orbit would be a pretty effective way of taking out slow moving things like tanks. I think Niven & Pournelle got it exactly right. That's no accident though - Jerry Pournelle worked on 'Project Thor' for Boeing in the 1950s - where he developed the theory for...guess what?...kinetic energy weapons dropped from orbit! Check out kinetic bombardment.

SteveBaker (talk) 03:33, 4 December 2008 (UTC)

Losing weight by cold weather?

Hypothetically, if someone was comfortable in 10 degree Celsius weather wearing the same clothes as someone wearing the same clothes in 20 C weather, how many more calories would the colder person burn per hour? Vultur (talk) 08:08, 4 December 2008 (UTC)

I've had a quick look for studies. One paper found almost no difference between 20 and 30 Celsius[1]. Another study looking at the change from 22 C to 16 C found a slight increase of 0.74 MJ/day (around 180 kCal, or 6% total)[2]. A paper on rats suggests 210-240 kJ/day at 20 C and 260-310 kJ/day at 10 C [3]; this represents an increase of around 25%.

Extrapolating the result from the second paper suggests something like 300 kCal per day for a 10 C fall (from 22 C to 12 C; which will be similar to that in the original question). But there are a lot of variables involved. In cold weather, the body acts to minimise heat loss (see Thermoregulation) so calorie consumption may be less than expected. --Maltelauridsbrigge (talk) 13:47, 4 December 2008 (UTC)

A problem with your question is that the person is supposed to be comfortable in 10 degrees Celsius, and the question is what has caused this condition. Is it a different metabolism? Lova Falk (talk) 14:12, 4 December 2008 (UTC)

By the way, if you want to experiment with temperature as a help in losing weight, I would suggest the exact opposite. Warm up your environment to 40 degree Celsius, drink lots of water and eat only small servings. The warmth will help you to keep your appetite down. Lova Falk (talk) 14:21, 4 December 2008 (UTC)

By the way, how is it possible that some people can wear a t-shirt outside in sub-freezing temperatures?! ~AH1^(TCU) 00:13, 5 December 2008 (UTC)

Acclimatization. I'll do that sometimes in March for the 1-minute walk from my house to the corner store, but never in November when I'm still more accustomed to warmer temperatures. --Anonymous, 04:27 UTC, December 5, 2008.

I haven't worn any clothing heavier than short sleeved tee-shirt, short underwear and jeans for over 12 months, haven't switched on the heating in my house and have been quite comfortable. That's in Canberra, Australia, during the period in question temperatures got as low as -5C. As Anon said, Acclimatisation. Down side was it didn't help me loose weight - it just made me hungrier! --Psud (talk) 12:30, 6 December 2008 (UTC)

My point exactly. Heat up instead! Lova Falk (talk) 14:20, 7 December 2008 (UTC)

Knee arthroscopy

how many total knee arthroscopy has been done till date , when did first computer asisted surgery was done and who performed it where and when —Preceding unsigned comment added by 221.134.9.245 (talk) 08:26, 4 December 2008 (UTC)

I'm not sure on the total number but there have been a lot performed, it's a common surgical procedure. As for computer assisted surgery, have you read our article on the subject? —Cyclonenim (talk · contribs · email) 08:40, 4 December 2008 (UTC)

Drug making

How are presctiption or OTC drugs made? Do they begin somewhere as a plant or petroluem product? Most illegal drugs come from plants. But what about: Prozac, Hydrocodone, Zestril, etc... —Preceding unsigned comment added by JelloTube (talk • contribs) 09:41, 4 December 2008 (UTC)

They are mixed in factories from raw materials, pressed into the size/shape, packaged up and sent out to distributors/retailers. There was a great epside of 'How Stuff Works' (tv show on Discovery channel) that showed the making of them - a chemist was pouring huge vats of different chemicals into what was essentially a giant mixing-bowl, mixing it all very thoroughly, doing some tests on samples and then it was sent off where it was 'formed' into tablets. Hopefully someone will add more/links but hope that's a good start. 194.221.133.226 (talk) 10:12, 4 December 2008 (UTC)

They run the whole range from drugs that are made from pure chemicals (Benadryl), those extracted from plants (Quinine), and those extracted from microbes (penicillin), etc. Some drugs are even combinations of these where a natural product is collected and then chemically modified in vat to make something useful. Basically, pharmaceutical companies are willing to do whatever they can to get the next big drug, whether that means experimenting with synthetic chemistry or sorting through a wide variety of naturally occurring compounds. Dragons flight (talk) 10:37, 4 December 2008 (UTC)

For a little more detail on the kinds of academic fields dedicated to drug manufacturing (as opposed to drug research and development), process chemistry and chemical engineering are two related fields that deal with industrial manufacturing of all synthetic compounds, including drugs. A process chemist is one who works towards taking laboratory scale chemical reactions, and scaling them up to production scale. Just because a chemical process works in a little flask in a lab does not mean it necessarily works well in a giant tank in a factory. Chemical engineers perform similar jobs, but are more focused on the physical design of production-scale reaction vessels and processes. --Jayron32.talk.contribs 20:20, 4 December 2008 (UTC)

As Jayron32 alludes, process chemistry ("production-scale reaction vessels and processes") is more than just "works well in a giant tank in a factory" (emphasis mine). There also become practical concerns like safety, cost, and ultimate use of the product. Some things that work well and cheaply involve chemicals (or waste/byproducts) that are too expensive or too dangerous to handle to be a commercially practical large-scale reaction. Sometimes an intermediate might be hard to handle (process chemists love crystalline solids and chromatography is rarely viable for large-scale purification, whereas a med-chemist is happy tossing a few milligrams of a liquid into HPLC. Or else there might be certain contaminants that aren't permitted to exist for a certain end use. For example, a drug ingredient might be allowed to be only 98% pure, but a reaction that gives 99.5% pure with 0.5% as arsenic wouldn't be useful. Lots of process-chemistry routes have to redesign whole reactions and reaction sequences (and even add steps!) to avoid these types of problems. DMacks (talk) 07:07, 5 December 2008 (UTC)

Well, I wasn't being intentionally flippant about the complex nature of process chemistry; I fully understand all of the issues involved in taking laboratory-scale chemistry and scaling it up for industrial scale. I was only trying to give a general overview! But yes, the complex problems you list do indeed create quite an issue for drug manufacture. Many drugs which show very promising results end up not making it to market because of inherant difficulties in producing them for mass market. Since were at it explaining problems, from an engineering side there are also major issues. For one case, take thermodynamics. A reaction step which is slightly exothermic on the small scale may turn out to be unmanagably exothermic on a large scale. Heat dissipation and management is a huge part of industrial scale production of chemicals, including drugs. Where as a small 20 mL flask will tend to cool off rather quickly, a large 5000 liter tank may tend to get hot enough to boil its contents! Also of major concern is transport of intermediates. If a process needs several steps, and you are trying to keep all steps as part of a single sequence in a single factory, you often need to pipe stuff around, and that leads to a whole new set of problems. What I am trying to say here and above, and what DMacks is saying, is that its not a simple matter to take a viable drug from the R&D phase to the market phase... --Jayron32.talk.contribs 14:56, 5 December 2008 (UTC)

Creating a new page

I was searching for some information on a company called Einstein's Entertainers and there's nothing on wikipedia. How do I create a new page for them? —Preceding unsigned comment added by Eescience (talk • contribs) 09:52, 4 December 2008 (UTC)

Please see Wikipedia:Your first article.--Shantavira|^{feed me} 10:45, 4 December 2008 (UTC)

And be sure to take note of the "Things to Avoid" section; I'm guessing from the user id that Eescience might be personally involved in Einstein's Entertainers. --LarryMac | Talk 13:18, 4 December 2008 (UTC)

Decline in LED light output after several weeks of use

I have several LED night lights (two models, from different manufacturers). Initially they were all very bright. But after several weeks of use, their light output all dropped significantly (although still somewhat adequate for night lights). Is this normal behavior of LEDs, or is this caused by the LEDs being driven with too much current? (I suspect the latter, but I'm not an expert in LEDs.) Does any one know? --173.49.9.141 (talk) 13:00, 4 December 2008 (UTC)

The LED article contains a list of "failure modes" (many of which are well-beyond my knowledge). It looks like too much current can cause dimming, but there are a lot of possibilities there. --LarryMac | Talk 13:16, 4 December 2008 (UTC)

If they are "theatre-style" lights sold by Home Depot, return them for a refund; there's something wrong with the design of those night-lights that causes this exact failure on some samples. I didn't investigate closely but I believe the failure is actually in the automatic control circuit (the photocell and the associated control circuitry) rather than the LEDs themselves. We had a 50% fall-out rate and switched to using much cheaper LED night lights from IKEA.

Atlant (talk) 23:11, 4 December 2008 (UTC)

If they're battery-powered, have you tried changing the batteries? --Anonymous, 04:28 UTC, December 5, 2008.

Another LED question

A few years ago there was a sudden change in the design of LEDs that made them much brighter. They began to be used in traffic lights and other uses where formerly they would not have been bright enough. What was the change in LED design? —Preceding unsigned comment added by 98.17.46.132 (talk) 14:55, 4 December 2008 (UTC)

Our article on Light-emitting diode contains lots of good information to answer this sprinkled around the article. Its not all collected in one place, but you can deduce many of the more recent advancements in LED technology by reading the whole article. --Jayron32.talk.contribs 20:13, 4 December 2008 (UTC)

The Universal Expansion Paradox

Modern Astronomers say that the universe is expanding at an accelerating rate. From Hubble's law v=Hd where v is velocity, H is Hubble's constant, and d is distance if you calculate d in light years then v=Hy where y is the universe's present age minus the age of the universe you're looking at in years. From that, you may deduce that the velocity of expansion of the universe is decelerating. But this exactly opposes the statement that the universe is expanding at an accelerating rate! So which conclusion is correct? ----The Successor of Physics 14:20, 4 December 2008 (UTC) —Preceding unsigned comment added by Superwj5 (talk • contribs)

I think the key detail is that Hubble's constant isn't actually constant (it's constant in space, but not time). The expansion accelerating corresponds to Hubble's constant increasing. --Tango (talk) 15:00, 4 December 2008 (UTC)

In ΛCDM the scale factor goes roughly as (sinh t)^2/3. The Hubble "constant" is a'/a, i.e. coth t (I'm ignoring constant factors here). So the Hubble constant never actually increases, it just settles down to a fixed positive value. I think what's officially "accelerating" is the scale factor.

To the OP, in addition to treating H as constant, I think you've assumed that if an object emits some light while at a metric distance d from Earth, that we will receive that light after a time d/c. That's not true at cosmological scales. See this old ref desk thread and metric expansion of space. In the rather nice diagram on the right, which I cribbed from distance measures (cosmology), your d is the "angular diameter" and your y is the "lookback time". They already differ significantly at z = 0.5, and for the CMBR (z ~ 1000) they differ by a factor of more than 300. -- BenRG (talk) 19:22, 4 December 2008 (UTC)

The acceleration ought to mean a''/a>0, but if I can differentiate (which is far from certain), your scale factor comes out with a negative 2nd derivative, which ought to correspond to deceleration. Am I missing something? --Tango (talk) 12:26, 5 December 2008 (UTC)

I think you made a math error. I get $a''/a={\tfrac {2}{3}}(1-{\tfrac {1}{3}}\coth ^{2}t)$, which starts out negative and becomes positive. The expansion wasn't always accelerating (according to ΛCDM), that only started once the ordinary matter thinned out enough that its density dropped below the vacuum density. -- BenRG (talk) 13:37, 5 December 2008 (UTC)

Ah, I can differentiate, I just can't carry a factor of 1/3 from one line to the next! Thanks! --Tango (talk) 15:26, 5 December 2008 (UTC)

Fox killing 13 penguins

I was really saddened by this story [4] of a fox killing penguins in Germany (according to the article, they could tell it was a single fox by the footprints), little bastard! I can understand him going for one or two for food, but why kill so many (13 out of 32) but also not killing all and not eating any? Was it sadistic pleasure as in some human murderers or just wanting to practise killing or something else, and if so, do humans kill for this same motive as the fox did here too? Thank you in advance for information, --AlexSuricata (talk) 15:57, 4 December 2008 (UTC)

Animals love to play, we only think it wrong because we've trained ourselves to believe that massacres are wrong. Animals lack such morals, it's a game to them and it helps keep them fit, helps them to survive in the real world. If you have a cat, you'll see they follow similar behavior when chasing birds: they rarely eat them but almost always kill them. Humans may once of acted in the same way when we hunted just to stay alive, but now days we are far too adapted to being catered for, having our every need serviced without too much strain. Some people still kill for fun though, see hunting. —Cyclonenim (talk · contribs · email) 16:20, 4 December 2008 (UTC)

It's also possible the penguins mobbed the fox. They are social creatures, many were bred in captivity, and in the wild they have often have no natural predators while on land. It's quite possible their reaction after the fox grabbed the first was to get upset and "come to the aid" of the their distressed brethren. Not that they could actually help, but they might not know that. Dragons flight (talk) 17:33, 4 December 2008 (UTC)

Some predators kill more prey than they can immediately use. In the article on Fox hunting it says (of Surplus killings of chickens) that "Some anti-hunt campaigners maintain that provided it is not disturbed, the fox will remove all of the chickens it kills and conceal them in a safer place.." Presumably to eat them all sometime later. Yours, Daicaregos (talk) 18:07, 4 December 2008 (UTC)

I imagine that chasing penguins is incredibly fun for a fox. My pointer/whippet chases birds all day long and never gets bored of it. --Sean 19:40, 4 December 2008 (UTC)

Let's look at this instinct to "kill more than you can eat" from an evolutionary perspective. The first thought is that it would be advantageous for any species to only kill what it can eat, to allow the others to survive and breed to ensure a future food supply for itself. So, why kill more ? Well, if they can store them for later, that's a reason, but 13 penguins would surely rot before a fox could eat them all. There's also the angle of practicing catching them, but the obvious solution there is to catch and release. Cats will do that with a mouse several times for practice before they finally kill it. Another possible reason is to deprive the fox's competitors of potential food sources, so that they die out and the more successful fox and it's offspring are able to replace those other predators. That seems the only reason that might have applied in this case (well, if the weren't in a zoo, that is). Why not kill them all ? I imagine the fox got tired, was wounded, or was scared by approaching zookeepers. StuRat (talk) 20:44, 4 December 2008 (UTC)

You appear to be making the common mistaken assumption (even amongst biologists) of presuming the foxes behaviour must be optimal. Evolution doesn't work like that. Although it may make sense for the fox to simply practice on the same animal over and over or only kill the number it can resonably eat, there's no guarantee the fox evolved like that. If there is an advantage to practicing to kill or of catching extra and storing them, then the fox may have evolved like that (actually it could have even if it wasn't advantageous but that's a different issue altogether) and unless the evolutionary advantage is great enough it may have never evolved the 'sense' to practice on the same animal or limit kills to that which it can eat before they rot. IMHO, most likely the advantage in terms of greater availability of food simply is not going to be big. If there is a great evolutionary disadvantage to the foxes behaviour it would be the wasted energy and risk of injury due to it's practice Nil Einne (talk) 10:12, 5 December 2008 (UTC)

I had considered mentioning the possibility of sub-optimal evolution, specifically as it relates to variation from individual to individual. The "wasted energy" argument is a good one when food is scarce and energy must be conserved, but that obviously isn't the case here. StuRat (talk) 11:24, 5 December 2008 (UTC)

I strongly suspect that this behavior is completely optimal. Keeping body parts and brain parts and biochemical pathways that you don't need is inefficient. So if some tiny fraction of fox biochemistry/brain is required to shut off the killer instinct when enough food has been obtained - then there is a decision to make:

• Evolve that feature of the fox in order that in times of plenty you don't wipe out future food stocks.
• Evolve away from having that feature in order not to pay the 'price' of it during times of famine.

Since foxes do indeed lack the ability to shut off the killer instinct, the likelyhood is that foxes have evolved mostly in times of borderline nutrition - so they are better off not having to maintain a (mostly useless) feature. So just as fish that live in dark caves evolve to lose their eyesight - so foxes lost the ability to shut off the killer instinct because they didn't ever NEED to shut it off. The fox has evidently been "optimised" under pressure of borderline starvation and having these unnecessary behavioral controls would require them to eat a little more food in order to feed that extra tiny part of the brain or those sets of glands that produce inhibitory hormones. Foxes that lack that behavioral control would have a slight advantage over the ones that do...so evolution does the thing it does and foxes lose that skill.

SteveBaker (talk) 12:55, 5 December 2008 (UTC)

I don't buy that argument. The "price" of maybe a millionth of an ounce in extra brain mass would easily be worth even a rare and modest increase in the food supply. StuRat (talk) 14:24, 5 December 2008 (UTC)

But that's why evolution is so powerful. It may be a millionth of the weight of the foxes brain - but there are millions of foxes in each generation and hundreds of thousands of generations. So in a time when food is so scarce that some fraction of the population are dying (or at least, failing to reproduce) as a result of starvation - then for a VERY few of those millions of animals, that millionth of an ounce of extra brain will actually tip the balance. That produces a small skew into the population towards the foxes who are missing that element of self-restraint. But that small skew has a snowball effect so that each generation of starving fox pushes up the percentage who do not have self-restraint. When you crunch the numbers, that kind of evolutionary bias can come to dominate the entire population within just a few hundred generations. The smallness of the benefit is easily outweighed by the passage of time and the size of the population. SteveBaker (talk) 21:37, 5 December 2008 (UTC)

I don't think a fox's noggin is wired very delicately. The fox saw potential prey in front of it and went after it; I don't think you need to posit much more than that. Making a brain more complex is very expensive. Further, many carnivores "over-kill" and stow the meat for the future. They don't care about freshness to the extent you or I would. The phrase "make hay while the sun shines" seems to apply here; if there's easy prey about today, kill what you can so you don't have to try tackling a rabid moose the next time you get hungry. Matt Deres (talk) 20:19, 5 December 2008 (UTC)

Indeed I forgot to mention something. Remember we are talking about a completely human-made situation which will never occur without human involvment. Foxes are never going to naturally come across a bunch of chikens in a coop, let alone penguins, without human involvement. While humans have been involved in domesticated animals for long enough that we would expect some effect on foxes, there's no guarantee they will have adapted to deal with the situation (particularly as the situation was likely rare until perhaps the past few hundred years). Just because foxes kill all the prey they see in front of them doesn't mean they go out and seek new prey if they already have stored food. (I have no idea if they do, I expect they don't although they may still kill new prey if they happen to come across them which may make sense since fresh is better then stored.) In all likelihood, the most common situation a fox would encounter of multiple prey is if it comes across a nest. In that case, leaving some alive may not greatly increase the number of available prey since if you kill one of the parents (or the parent as the case may be) most of the offspring will probably die (whether of predation or environment). So there is probably no great advantage for the fox to only kill a few animals when it comes across multiple ones in most of the situations the fox has evolved in. (In other words, there's no reason why it would have evolved regardless of whether food was plentiful or not.) Indeed when it comes to the human situation, I'm not convinced keeping them alive is your best bet anyway. While humans tend to get pissed off when you kill all their chickens, they also tend to get pissed off when you kill one, just not so much. They may still set traps, put dogs, guard the coops better, make the coops more difficult to break in to or move the coops. So when you do happen get into the chicken house, it's probably best to kill them all and store them rather then hope you'll be able to kill some more without injuring yourself next time. P.S. I'm surprised no one has mentioned it but remember we have the issue here of selfish behaviour. While it's true not killing more then you need has the advantage of increasing the availability of food simply, it's likely to be more of a communal effect. If every other fox is killing everything it sees, you not killing everything you see is probably not going to effect the availability of food much at all. (In certain situations it could even be a disadvantage.) So what we have here has some similarities to altrusic behaviour (it isn't since your not directly helping others and regardless of what others do, what your doing could in some situations have a net benefit to you) and analysing something like that is a whole different kettle of fish Nil Einne (talk) 11:45, 6 December 2008 (UTC)

Double cross product of angular velocity

Hello,

I am unable to determine why relation 4 from http://scienceworld.wolfram.com/physics/Vorticity.html is true (in words, this relation is the del operator cross the cross product of the angular velocity and r = 2 times the angular velocity). I've been attempting to prove this relation in cylindrical coordinates as part of a larger derivation, but I can't seem to get it to work (I'm aware of the identity that breaks this form of the double cross product into four dot product terms). Any assistance regarding this would be much appreciated (even if it's as simple as how r and the angular velocity should be represented in cylindrical coordinates - I've been using [r1, r2, r3] and [0, omega, 0] respectively, although those don't seem to work too well). I suspect I'm over-complicating matters.

Thanks to all who assist. —Preceding unsigned comment added by 128.100.36.228 (talk) 18:29, 4 December 2008 (UTC)

that cross product of angular velocity and radius gives the linear velocity. So you then have to consider the curl of the velocity field, which by thinking about you can see it rotates in the same direction as the angular velocity and is proportional, but I guess you will have to calculate it to see the constant of proportionality. OK I have done if for you - if you look at the curl formula in Del in cylindrical and spherical coordinates you can see only the z component has a value. This leaves ${1 \over \rho }\left({\partial \left(\rho A_{\phi }\right) \over \partial \rho }-{\partial A_{\rho } \over \partial \phi }\right){\boldsymbol {\hat {z}}}$ The part subtracted does not vary as the φ component is constant at constant ρ giving ${1 \over \rho }\left({\partial \left(\rho A_{\phi }\right) \over \partial \rho }\right){\boldsymbol {\hat {z}}}$. The A φ is equal to the linear velocity magnitude, ie ρω. Omega is not a function of ρ, so can rearrange to ${1 \over \rho }\left({\omega \partial \left(\rho \rho \right) \over \partial \rho }\right){\boldsymbol {\hat {z}}}$ which amazingly is 2ω. Graeme Bartlett (talk) 21:11, 5 December 2008 (UTC)