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 The New Kilogram By alby in ScienceSat May 15, 2004 at 12:21:08 AM EST Tags: Science (all tags) Somewhere in a heavily-protected vault in the suburbs of Paris is an unremarkable-looking cylinder of platinum and iridium with a unique property. No matter what happens to this object, its mass remains the same. If it were to get "heavier" or "lighter" through contact with the environment (being cleaned, accumulating dust, oxidation) it would still have the same mass: exactly one kilogram. Here it's important to explain what I mean by exactly: unlike most situations in physics I don't mean "to within the accuracy of our measurements"; I mean a one followed by a never-ending string of zeros. So, what is this object? And how can this be?

The International Prototype Kilogram

The international prototype kilogram is kept by the Bureau International Poids et Measures (BIPM) in Sèvres. The kilogram is the only SI unit to still be tied to a physical artifact and is defined as "the mass equal to that of the international prototype kilogram", which is why the mass of this object never changes.

Redefining The Kilogram

There are currently two different approaches being taken in the search for a new definition of the kilogram:

• The Watt Balance - Measures the weight (the force produced by the mass due to gravity, not the mass directly) of an object in terms of it's opposition to a force produced by a current-carrying wire in a strong magnetic field. This research is being carried out by the BIPM.
• The Avogadro Method - Uses the large-scale mass of a known number of silicon atoms to produce an extremely accurate value for Avogadro's constant. Avogadro's constant (≈6.022×1023) is defined as the number of atoms of an element that have the same mass as the element's atomic mass. That is to say that 6.022×1023 atoms of hydrogen weighs ≈0.001kg and 6.022×1023 atoms of helium weighs ≈0.004kg and so on. An accurate value for Avogadro's constant can thereby be used to create a definition of the kilogram. This research is being carried out by the Physikalisch-Technischen Bundesanstalt (PTB) in Braunschweig, Germany.
The Watt Balance

It's difficult to explain the principle behind a Watt balance without using diagrams; the BIPM do it much better than I could so I suggest you visit this page.

This basically involves creating a sphere of extrememly pure silicon and then measuring both its diameter and lattice spacing. With these measurements you can calculate the number of silicon atoms in the crystal and then by weighing it obtain a highly accurate value for Avogadros constant.

But wait! "Surely," I hear you cry, "this method still depends on a physical artifact!". Well you're right, it does. But what is important is that it doesn't depend on any one physical artifact. After determining the value for Avogadro's constant you could throw the sphere away and make another one next time you needed one. However, I wouldn't recommend you do this very often as each sphere has to be polished to within an accuracy of 50nm. To give you some idea of just how perfect this is; if you expanded one of the crystals used to the size of the Earth, the difference between the highest and lowest points would be no more than seven or eight metres.

Current Results

And the moment, current results from the two experiments for the mass of the kilogram differ by ≈ 1mg, clearly an unacceptable level of error. I wouldn't worry though: the platinum-iridium cylinder has done us good for 125 years so far.

• Unfortunately I subscribe to journals through the Institute of Physics and as such I can't link to the relevant papers. However, this story was inspired by an article in Physics World magazine: Redefining the kilogram.

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 The New Kilogram | 142 comments (120 topical, 22 editorial, 0 hidden)
 well it's clear to me (2.20 / 25) (#2) by circletimessquare on Fri May 14, 2004 at 10:34:27 AM EST

 we must liberate the kilogram from french oppression we must have a "freedom kilogram" installed in washington dc asap to prevent the ever encroaching arrogance and ambition of the gauls The tigers of wrath are wiser than the horses of instruction.
 You've already got one! (3.00 / 8) (#6) by alby on Fri May 14, 2004 at 11:13:35 AM EST

 As a member of The Metre Convention you are entitled to a copy of the original prototype kilogram. It's stored by NIST, but as to where it's kept, I don't know. I do know the UK's copy is kept at the National Physical Laboratory (we have copy number 18). -- Alby[ Parent ]
 your continentalist propaganda has no effect on me (2.16 / 6) (#7) by circletimessquare on Fri May 14, 2004 at 11:24:57 AM EST

 only scorched earth and rotting francogerman flesh will save us from the godless hordes of paneuropean hegemony The tigers of wrath are wiser than the horses of instruction. [ Parent ]
 feet, inches and gallons (2.00 / 8) (#10) by Wah on Fri May 14, 2004 at 11:46:20 AM EST

 it's the only way to be safe from fascists and the sons and daughters of former fascists. -- Help us cross the digital divide, yo. [ Parent ]
 Gallons? (none / 3) (#68) by squigly on Sat May 15, 2004 at 01:59:25 PM EST

 Is that the English gallon or the US gallon? [ Parent ]
 and is it related (2.80 / 5) (#85) by emmons on Sat May 15, 2004 at 10:15:40 PM EST

 To the Airspeed Velocity of an Unladen Swallow? --- In the beginning the universe was created. This has made a lot of people angry and been widely regarded as a bad move. -Douglas Adams[ Parent ]
 Australian or Arabian? (nt) (none / 0) (#101) by Wah on Sun May 16, 2004 at 01:23:53 PM EST

 -- Help us cross the digital divide, yo. [ Parent ]
 Arabian, I think (none / 0) (#128) by emmons on Tue May 18, 2004 at 12:53:37 AM EST

 But how the devil do they get all the way here? --- In the beginning the universe was created. This has made a lot of people angry and been widely regarded as a bad move. -Douglas Adams[ Parent ]
 well, i heard it was being prepared (1.70 / 10) (#12) by vivelame on Fri May 14, 2004 at 12:01:47 PM EST

 in the Pentagon. Some US soldiers infortunately couldn't hold their tongue, and besides, the US Army has enough on it's plate for now.. -- Jonathan Simon: "When the autopsy of our democracy is performed, it is my belief that media silence will be given as the primary cause of death."[ Parent ]
 Spheres ready to go, when they get back to Earth (2.40 / 5) (#11) by Wah on Fri May 14, 2004 at 11:50:38 AM EST

 However, I wouldn't recommend you do this very often as each sphere has to be polished to within an accuracy of 50nm. To give you some idea of just how perfect this is; if you expanded one of the crystals used to the size of the Earth, the difference between the highest and lowest points would be no more than seven or eight metres. Nice, they can use the gyros from the Gravity Probe B when it gets back. Of course, we must remember to compensate for the differences wraught by the relativistic curiosities left over from generations of bloodshed. The gyroscopes were painstakingly chosen and molded specially for this mission, and the spheres were polished to near-perfect roundness. If we were to enlarge these spheres to the size of the Earth, the highest peak or deepest trough would measure only 8 feet from sea level. They are the most spherical objects in the universe, aside from neutron stars. -- Help us cross the digital divide, yo.
 They may be perfect spheres ... (none / 1) (#28) by alby on Fri May 14, 2004 at 05:59:54 PM EST

 ... but unfortunately they're not pure enough. -- Alby[ Parent ]
 Please remove 0.0046 pounds from it (1.38 / 13) (#22) by bobpence on Fri May 14, 2004 at 05:37:52 PM EST

 So that 1 kg exactly equals 2.2 lbs, not 2.2046 as is currently the case. It would just be better, 'kay? Thanks. "Interesting. No wait, the other thing: tedious." - Bender
 Hrm (2.90 / 10) (#30) by awgsilyari on Fri May 14, 2004 at 06:01:43 PM EST

 That would cause almost every physical constant to change value. Pretty much any calculator or computer program with those values hard-coded would become out of date. (And don't try suggesting that they should have made the constants configurable; they're called constants for a reason). Imagine the chaos when people forget which programs use the "new" constants and which ones the "old." You'd have to remember to correct your inputs to be in "new" kg or "old" kg depending on the program you were using. It would probably lead to at least a few deaths, eventually. It would probably be better to redefine the pound such that 2.2 pounds is exactly 1 kg. At least then the recalibration wouldn't affect people doing science, only a few weird American engineers :-) -------- Please direct SPAM to john@neuralnw.com [ Parent ]
 Hey. (none / 3) (#45) by Ta bu shi da yu on Fri May 14, 2004 at 11:56:00 PM EST

 This comment reminds of when the state of Indian redefined the value of pi to become 4. This ensured that all mathematical and engineering calculations in the state were wrong. ---AdTIה"the think tank that didn't".ה[ Parent ]
 State of Indian? (none / 2) (#47) by Cruel Elevator on Sat May 15, 2004 at 12:33:30 AM EST

 As in, that country in South Asia? [ Parent ]
 Er. (none / 3) (#48) by Ta bu shi da yu on Sat May 15, 2004 at 12:35:08 AM EST

 Indiana. Oops. ---AdTIה"the think tank that didn't".ה[ Parent ]
 It never actually went through. (none / 2) (#49) by mold on Sat May 15, 2004 at 02:10:06 AM EST

 And I believe it was 3.2. A quick google found this. It is fun to joke about though. --- Beware of peanuts! There's a 0.00001% peanut fatality rate in the USA alone! You could be next![ Parent ]
 Touché (none / 1) (#60) by bobpence on Sat May 15, 2004 at 10:34:24 AM EST

 But pi is immutable; as the article alledges, the kilogram as it now exists is not. I would assert that a lot of engineering calculations are done with the 2.2 lb/kg value or its inverse. Therefore my painfully down-modded proposal would actually make these many engineering calculations more valid. I mean, is anyone else as anal as me so as to use 2.2046 rather than 2.2? "Interesting. No wait, the other thing: tedious." - Bender [ Parent ]
 Snopes calls bullshit on that one, too. (none / 2) (#116) by Zerotime on Mon May 17, 2004 at 12:36:05 AM EST

 http://www.snopes.com/religion/pi.htm --- "I live by the river With my mother, in a house She washes, I cook And we never go out."[ Parent ]
 Ummm... (none / 1) (#136) by Ta bu shi da yu on Wed May 19, 2004 at 01:55:16 AM EST

 ... well, I got it from "The Book of Heroic Failures" which was published in the 1980s. Also, if you actually bothered to read my comment, you'd notice I talked about Indiana and not Alabama. ---AdTIה"the think tank that didn't".ה[ Parent ]
 "Ummm..." yourself. (none / 0) (#138) by Zerotime on Fri May 21, 2004 at 12:20:46 AM EST

 If you'd actually bothered reading the article, you'd have come across the bit where it said that Indiana had tried to do the same thing. Hell, there's even other sources much the same thing, also involving Indiana. --- "I live by the river With my mother, in a house She washes, I cook And we never go out."[ Parent ]
 Interesting (none / 0) (#140) by Ta bu shi da yu on Sat May 22, 2004 at 08:09:09 PM EST

 However, if you look at the scopes text it never says that Indianapolous didn't submit this bill. I have no way of verifying the info, however, so I'll let this rest. ---AdTIה"the think tank that didn't".ה[ Parent ]
 1 in = 2.54 cm (none / 0) (#104) by J'raxis on Sun May 16, 2004 at 04:17:25 PM EST

 This would also be in keeping with the redefinition of the inch to be exactly 2.54 cm. Earlier, the equivalence was apparently 2.54000508 cm. — J’raxis [ J’raxis·Com | Liberty in your lifetime ][ Parent ]
 Nah (none / 0) (#132) by synaesthesia on Tue May 18, 2004 at 11:25:41 AM EST

 Just keep using use the term 'kilogram' to refer to 'Classic kilograms'. 'New kilograms' will quickly die (although for some reason God only knows, 'Cherry kilograms' will continue to exist). Sausages or cheese?[ Parent ]
 a bad, bad idea (none / 3) (#98) by anonymous cowerd on Sun May 16, 2004 at 09:52:54 AM EST

 If you do that then you won't have one conversion, you'll have two: from old kilograms to pounds and from new kilograms to pounds. And also, whenever you see something marked, say, "100 kilograms," you won't know what kind of kilograms they're talking about. Which may not matter if you're weighing a kilogram of navy beans, but which certainly would matter when making the manifest for a satellite launch. "But, but, that was a joke," you say. "No one could possibly be stupid enough to define two close-yet-significantly-different units of measurement which, to make it worse, share the same name." Let me introduce you, then, to the U.S. Survey Foot, which was created by precisely the kind or making-arithmetic-easy reason that you unseriously propose. A U.S. Survey Foot is close enough to the International Foot that you couldn't possibly tell them apart just by looking at the data (like you can, for instance, between feet and meters), but it is enough different - one part in a half-million - that geodetic calculations (e.g. involving state plane coordinates, or WGS coordinates derived from GPS measurements) done using the one unit automatically generate wonderfully confusing and potentially quite costly errors when accidentally compared with the other. Yours WDK - WKiernan@ij.net Minnamin, Gut mag alkan, Pern dirstan.[ Parent ]
 Question (2.42 / 7) (#24) by metalfan on Fri May 14, 2004 at 05:52:20 PM EST

 Who decided how massive the prototype kilogram would be, and how did he, she, or they decide?
 Before the standard prototype ... (3.00 / 7) (#31) by alby on Fri May 14, 2004 at 06:04:42 PM EST

 ... the kilogram was defined as the mass of a volume of pure water at standard atmospheric pressure. How they decided on that, I don't know! If you're really that interested, Measure for Measure by Alex Hebra is apparently an excellent text on the subject. -- Alby[ Parent ]
 Obviously... (2.80 / 5) (#33) by squigly on Fri May 14, 2004 at 06:19:32 PM EST

 The mass of 1 litre of water. A litre is 1000 cubic centimetres. A centimetre is 0.01 metres.  This was defined based on an estimate of the distance from the north pole to the equator through Paris.  This distance was taken to be 10 000 000 metres (I think). All seems fairly sensible powers of ten, but the actual powers used seem rather arbitrary.  Surely it would have made sense for the standard unit of volume to be the cube of the unit of length, and the unit of mass to be  the mass of one unit volume of water. [ Parent ]
 Oh, right! (none / 3) (#38) by metalfan on Fri May 14, 2004 at 08:14:55 PM EST

 Of course.  In elementary school and such a kilogram is still defined as a litre of water, i think.  Try telling an 8 year old that there is only 1 kilogram in the entire world. I'm not that interested. [ Parent ]
 Right... (none / 1) (#93) by Vesperto on Sun May 16, 2004 at 03:43:48 AM EST

 ...and what's a liter of water? How do you define that? _____________________________ If you disagree post, don't moderate. Not a Premium User.[ Parent ]
 Easy. (3.00 / 9) (#97) by henrik on Sun May 16, 2004 at 06:12:13 AM EST

 A liter is the volume of a cube with the sides 0.1 meter. A meter is defined as the distance light goes in 1/299792458 second. A second is defined as the time interval equal to 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium‑133 atom. But try telling that to 8-year olds. Akademiska Intresseklubben antecknar![ Parent ]
 And what is water? (none / 0) (#135) by Rich0 on Tue May 18, 2004 at 08:09:41 PM EST

 That was the easy part - the hard part is defining what water is... What isotopic composition, what temperature (well, that is easy at least - probably the triple-point), how do you make pure water? [ Parent ]
 How is the artifact used? (3.00 / 7) (#29) by coderlemming on Fri May 14, 2004 at 06:00:55 PM EST

 I enjoyed the picture of the current kilogram artifact sitting under 3 (or 4?) bell-jars.  But I have to wonder, what exactly is this artifact used for?  How often does it come out, who gets to use it, and what do they use it for?  God help the guy who accidentally drops it... --Go be impersonally used as an organic semen collector!  (porkchop_d_clown)
 Calibration. (3.00 / 11) (#36) by alby on Fri May 14, 2004 at 06:39:21 PM EST

 It might sound odd but the scales in your kitchen, the scales at the supermarket and the scales your dealer uses are all linked inextricably through what are called "calibration centres" to the original prototype kilogram. For example: The BIPM calibrates a copy and sends it to NIST (this actually happened years and years ago). NIST calibrate a number of copies and send them to regional centres. These regional centres calibrate equipment for commerical calibration centres. Your scales manafacturer pays a commercial calibration centre a fee to calibrate and "authenticate" their new scales. You use scales to weight carrots. -- Alby[ Parent ]
 Skipped a couple of steps (3.00 / 8) (#56) by localroger on Sat May 15, 2004 at 07:48:16 AM EST

 You forgot... (1.40 / 10) (#63) by rmn on Sat May 15, 2004 at 11:06:35 AM EST

 ??? Profit! No, sorry, wrong site. RMN ~~~ [ Parent ]
 Moved to voting. (none / 2) (#32) by alby on Fri May 14, 2004 at 06:05:56 PM EST

 Nice little text (none / 2) (#35) by nkyad on Fri May 14, 2004 at 06:31:24 PM EST

 I had almost forgot the might kilogram, hidden away from prying eyes and evil minds willing to disrupt the scales, in the dark Parisian catacombs. I was late to it, but you should have also mentioned that, before we could measure subatomic distances, the same catacombs once held "the" meter too. Don't believe in anything you can't see, smell, touch or at the very least infer from a good particle accelerator run[ Parent ]
 Not quite correct ... (none / 1) (#54) by alby on Sat May 15, 2004 at 05:13:24 AM EST

 The metre is defined in terms of the distance light travels in a fraction of a second, not in terms of subatomic distances. -- Alby[ Parent ]
 Yeah, old age leads to bad memory (none / 2) (#65) by nkyad on Sat May 15, 2004 at 01:30:52 PM EST

 I was commenting from vague memories of childhood. A meter today is the distance traveled by light in a vacuum in 1/299,792,458 of a second. In my defense, I was probalby thnking about the 1960 definition: "1,650,763.73 vacuum wavelengths of light resulting from unperturbed atomic energy level transition 2p10 ­ 5d5 of the krypton isotope having an atomic weight of 86." Don't believe in anything you can't see, smell, touch or at the very least infer from a good particle accelerator run[ Parent ]
 a 3rd standard (2.27 / 18) (#39) by horny smurf on Fri May 14, 2004 at 08:21:37 PM EST

 I propose the kilo be redefined as the amount of coke required to have a kick-ass friday night.
 How much... (2.71 / 7) (#46) by Pseudonym on Sat May 15, 2004 at 12:30:41 AM EST

 ...is that litres? And is it different if it's diet coke? sub f{(\$f)=@_;print"\$f(q{\$f});";}f(q{sub f{(\$f)=@_;print"\$f(q{\$f});";}f});[ Parent ]
 err (none / 3) (#51) by crazney on Sat May 15, 2004 at 04:49:01 AM EST

 he means the drug. (i think..) [ Parent ]
 or (none / 2) (#52) by crazney on Sat May 15, 2004 at 04:49:55 AM EST

 come to think of it, that would probably kill you. [ Parent ]
 wrong (none / 2) (#66) by GhostfacedFiddlah on Sat May 15, 2004 at 01:35:12 PM EST

 Unless you consider death part of a kick-ass weekend. [ Parent ]
 not necesarily (none / 1) (#83) by emmons on Sat May 15, 2004 at 09:49:59 PM EST

 He didn't say for how many people. --- In the beginning the universe was created. This has made a lot of people angry and been widely regarded as a bad move. -Douglas Adams[ Parent ]
 touche (n/t) (none / 0) (#122) by GhostfacedFiddlah on Mon May 17, 2004 at 10:46:18 AM EST

 [ Parent ]
 I wonder (none / 1) (#127) by emmons on Tue May 18, 2004 at 12:37:15 AM EST

 Heh, I wonder how many people read that and think "He spelled 'touch' wrong! Wait, that doesn't even make sense." =) --- In the beginning the universe was created. This has made a lot of people angry and been widely regarded as a bad move. -Douglas Adams[ Parent ]
 Perhaps the real question is (1.71 / 14) (#40) by omghax on Fri May 14, 2004 at 09:15:41 PM EST

 If the unit kilogram was traveling near the speed of light, would its mass change? OH THE PARADOX!!!1111 LO ... LO ... LOOOOOOOOOOOOOOOOOOL! I put the "LOL" in phiLOLigcal leadership - vote for OMGHAX for CMF president!
 No (2.87 / 8) (#42) by Elendur on Fri May 14, 2004 at 09:40:07 PM EST

 Not from the perspective of an observer to whom the kilgram is stationary, which is the only important case. [ Parent ]
 Exactly (2.83 / 6) (#62) by rmn on Sat May 15, 2004 at 11:04:16 AM EST

 Exactly. In fact, to some observers, the current kilogram might very well be travelling at almost the speed of light. RMN ~~~ [ Parent ]
 It doesn't need to. (none / 3) (#94) by Vesperto on Sun May 16, 2004 at 03:50:33 AM EST

 You see, it has already eroded with time. Also in time some freak will blow up the center of investigation and kill the technitians or, more likely, funds will be cut so that there is no alternative to the one kilo in France. Then, as decades go by, the kilo will erode and eventually you'll see an inflation of the Kilo, worldwide financial chaos, then anarchy, then most of the population dies and finaly the birdies will be eaten by the cats in peace. _____________________________ If you disagree post, don't moderate. Not a Premium User.[ Parent ]
 pondering this, I am (2.87 / 8) (#41) by khallow on Fri May 14, 2004 at 09:38:10 PM EST

 I wonder why the Avogadro method isn't a clear winner? Ie, set Avogadro's constant to be a specific, exact value just like the speed of light. Ah, here's the answer: The revolutions in science that took place during the 20th century had a profound influence on technology. One need only think of how the atomic clock and the laser have influenced our daily lives. However, these new technologies have had relatively little impact on mass measurement. For the last 50 years or so we have been certain that for all practical purposes the mass of a stable atom, say carbon-12, is constant in time and that all carbon-12 atoms have identical mass. Therefore, defining the kilogram in terms of the mass of a single atom of carbon-12 would be similar in spirit to the present definition of the second in terms of a physical property of caesium atoms. We could today redefine the kilogram in terms of a certain number of carbon-12 atoms but "weighing" the prototype in terms of an atom of carbon-12 using present technology is considered too imprecise by about a factor of 10. There are several research efforts around the world aiming to improve this situation and so we can be optimistic about replacing the present artefact definition of the kilogram with a definition based on physical constants. Stating the obvious since 1969.
 It is worth pointing out how accurately (2.37 / 8) (#44) by lukme on Fri May 14, 2004 at 11:31:12 PM EST

 you can weigh something on a good analytical balance to 6 significant figures, ie, 0.100000 g plus or minus .000001 (if I remember correctly). You could even do this with the old balances if you had enough patience. The accuracy of these two methods pales in comparison ----------------------------------- It's awfully hard to fly with eagles when you're a turkey.
 yeah ok (2.75 / 4) (#72) by mutualaid on Sat May 15, 2004 at 03:03:52 PM EST

 And then some guy three floors down on the other side of the building clears his throat and your balance reading jumps by a couple dozen nanograms. Coaxing accurate, reproducible data out of a balance to that precision is extraordinarily difficult. It's a fucking art. [ Parent ]
 Have you actually used one? (none / 0) (#113) by lukme on Sun May 16, 2004 at 11:24:01 PM EST

 I used an older mettler analytical balance, which weighted stuff to 1 microgram, and no, the reading didn't change when someone walked across the floor. You should look at the mettler website, http://www.mt.com/, and check out their latest balances which claim to meausre +- 0.1 microgram, and to be easy to use. Now, with a high performance STM it is possible to measure footsteps of people walking by the building. A good NMR, may be able to pick up the cars driving by (as well as the local radio stations :). ----------------------------------- It's awfully hard to fly with eagles when you're a turkey.[ Parent ]
 that doesn't matter (none / 2) (#82) by Bossk on Sat May 15, 2004 at 06:42:28 PM EST

 The point of the 2 described methods for obtaining a standard mass unit is to create a mass reference that is either permanent or easily reproducible, and not prone to decay/alteration like our current standard. An analytical balance is great once the standard is set, but the wieghts within need to be calibrated to the standard mass unit. The analytical balance should not be used to measure your standardized object because its precision is calibrated to a different value of weights(i.e., it is precise but inaccurate). [ Parent ]
 but, the std need to be at least that accurate(NT) (none / 0) (#111) by lukme on Sun May 16, 2004 at 09:17:17 PM EST

 ----------------------------------- It's awfully hard to fly with eagles when you're a turkey.[ Parent ]
 Actually it does matter. (none / 1) (#114) by lukme on Sun May 16, 2004 at 11:37:05 PM EST

 You need a standard that will be at lease as accurate as the measuring device you are trying to standardize. Yes I agree with the purpose of trying to tie the kilogram down to a standard that is not held in a vault somewhere. It would be great if you didn't have to go to paris to calibrate the weights. However, until a new standard is developed to the point of measuring fractions of micrograms, we are stuck with the way things are. ----------------------------------- It's awfully hard to fly with eagles when you're a turkey.[ Parent ]
 The Watt Balance? (2.20 / 5) (#50) by cpghost on Sat May 15, 2004 at 03:28:16 AM EST

 The Watt Balance - Measures the weight (the force produced by the mass due to gravity, not the mass directly) of an object in terms of it's opposition to a force produced by a current-carrying wire in a strong magnetic field. How do you get a pure, never-changing field of gravity? Is the Earth's field of gravity constant enough for this? What happens when dust and asteroids hit the Earth? Doesn't the gravity field increase due to a higher overall mass? cpghost at Cordula's Web
 Balance. (none / 2) (#53) by alby on Sat May 15, 2004 at 05:06:30 AM EST

 Because you're dealing with a balance (i.e. one weight against another) the effect is cancelled out. -- Alby[ Parent ]
 This doesn't balance out (none / 0) (#70) by bdoserror on Sat May 15, 2004 at 02:52:02 PM EST

 This does not cancel out, because the force on one side of the balance it not due to gravity but due to the force on the current in the magnetic field. That's the point. Thus, if the force of gravity changes, a different mass will be able to balance the force from the current, which won't have changed. -- "Complexity is easy, simplicity is hard."[ Parent ]
 A pity (1.27 / 18) (#55) by holdfast on Sat May 15, 2004 at 07:44:53 AM EST

 It's a shame you insist on spelling kilogramme incorrectly. As you note, this is an SI unit. SI stands for Systeme Internationale which is french. The word originated in France along with metre and all the associated units. They invented(?) them and that is how they are spelled. "Holy war is an oxymoron." Lazarus Long
 No. (3.00 / 12) (#57) by bafungu on Sat May 15, 2004 at 08:12:02 AM EST

 "Kilogramme" is not the "right" spelling. It is simply the French spelling. German, Italian, English, Spanish and many other languages choose to spell it "kilogram". You're in for a long and uphill battle if you're going to try to change that. [ Parent ]
 Yes (2.00 / 5) (#59) by holdfast on Sat May 15, 2004 at 09:11:27 AM EST

 I can't answer for those other languages but in English it is spelt "kilogram" unless you are emulating/mimicking the USA. Here is a list of other words you may be spelling wrongly then... Concorde theatre centre flavour honour defence aluminium and many others I am under the impression that to use US (non) English the way is to swop letters round, sometimes drop vowels you don't like replace S with Z. We all make spelling mistakes & typos but doing it delibarately as a matter of national identity is silly. Noah Webster has a lot to answer for. Get a proper dictionary like the OED! "Holy war is an oxymoron." Lazarus Long[ Parent ]
 Concorde (none / 3) (#67) by it certainly is on Sat May 15, 2004 at 01:38:53 PM EST

 Concorde is a plane, spelt Concorde in the US and Concorde outside the US. Concord is the name of several localities, in the US and France. They are all spelt Concord. Other than that, I am in complete agreement with you. Paediatrician, foetus, caesium, phosphorus, sulphur... kur0shin.org -- it certainly is
 Phosphorus (none / 1) (#81) by astatine on Sat May 15, 2004 at 06:21:21 PM EST

 doesn't change spelling when you cross the pond. I admit, though, that American English's disdain for æ and œ appears to know no bounds. Society, they say, exists to safeguard the rights of the individual. If this is so, the primary right of a human being is evidently to live unrealistically. — Celia Green[ Parent ]
 fosforus (none / 1) (#84) by it certainly is on Sat May 15, 2004 at 10:05:36 PM EST

 If they're going to demand -- and get! -- an international change of sulphur to sulfer because they're fuckwits and can only spell phonetically (sorry, fonetically), then they might as well apply the same change to all other words, starting with fosforus. kur0shin.org -- it certainly is
 I believe sulfur is spelled with an f now. (none / 1) (#92) by finality on Sun May 16, 2004 at 03:27:00 AM EST

 Although no doubt the Americans have failed to follow the rest of the world yet again. This account has been anonymised. If you can give a good reason why, email rusty@kuro5hin.org, as he is obviously lacking one.[ Parent ]
 wrong (none / 0) (#105) by holdfast on Sun May 16, 2004 at 04:19:56 PM EST

 sulphur also sulphate, sulphide and sulphurous "Holy war is an oxymoron." Lazarus Long[ Parent ]
 Why write at all? (none / 1) (#108) by KrispyKringle on Sun May 16, 2004 at 04:46:45 PM EST

 That's stupid. The reasons for changing spellings two hundred years ago (in order to foster a seperate national identity, as I was taught) were certainly petty, but now that those languages have diverged, which makes more sense; to insist upon a particular spelling that my audience may not agree with, or to use the spelling that my audience is most likely to be familiar with? Therefore, if one is writing for the British, he ought to use the British spellings, but if writing for the Americans, I see no reason to insist on the British (unless out of simple bullheadedness). Care to enlighten me? [ Parent ]
 Write for the version you learned. (none / 0) (#115) by Zerotime on Mon May 17, 2004 at 12:29:12 AM EST

 And damn the Americans torpedoes! --- "I live by the river With my mother, in a house She washes, I cook And we never go out."[ Parent ]
 Not quite (3.00 / 6) (#61) by rmn on Sat May 15, 2004 at 11:02:00 AM EST

 Spanish: "kilogramo" Portuguese: "quilograma" / "kilograma" German: "kilogramm" Italian: "kilogrammo" There is no "correct" spelling for the full word, since it's not based on anyone's name (like the newton, watt, etc., and even those vary a bit). You just have to abbreviate it as "Kg", regardless of how you spell or pronounce it. The BIMP, however, is spelled incorrectly in the article. It's not "Bureau International Poids et Measures", it's "Bureau International des Poids et Mesures". RMN ~~~ [ Parent ]
 umm (none / 1) (#89) by ffalcon on Sun May 16, 2004 at 02:05:40 AM EST

 There is no "correct" spelling for the full word, since it's not based on anyone's name (like the newton, watt, etc., and even those vary a bit). You just have to abbreviate it as "Kg", regardless of how you spell or pronounce it. Actually, you have to abbreviate it as "kg", to be precise. [ Parent ]
 Sort of. (none / 1) (#100) by alby on Sun May 16, 2004 at 12:14:18 PM EST

 Because the unit is called the kilogram, rather than representing one thousand grams, the 'k' should be lower case as it is only representative of a part of the larger word, rather than indicative like the (first) m in mm and the c in cm. Newton is capitalised when you are referring to the person and in lower case when writing the unit out in full. However, it's unlikely you'd actually do this and because the newton (see?) is named after a person the unit is a capital letter, N (see also Watt (W), Pascal (Pa) etc.) -- Alby[ Parent ]
 In Italian (none / 3) (#64) by l3nz on Sat May 15, 2004 at 11:08:51 AM EST

 it's "chilogrammo", and that's the correct spelling (though being written 'kg' in the short form). Popk ToDo lists - yet another web-based ToDo list manager. 100% AJAX free :-) [ Parent ]
 No. (none / 0) (#110) by chbm on Sun May 16, 2004 at 07:07:42 PM EST

 Kilograma escreve-se Kilograma e metro escreve-se metro independentemente do que os franceses ou polinesios digam. Yes, this post was mostly in portuguese. -- if you don't agree reply don't moderate --[ Parent ]
 Ha! (none / 0) (#119) by WWWWolf on Mon May 17, 2004 at 06:02:35 AM EST

 They invented(?) them and that is how they are spelled. Long ago, there was a joke involving the French trying to pronounce Finnish, with some comments along the lines of "if my grandfather would have been around, he would have tried to choke the guy and said 'Now pronounce it right or I'll kill you, you stupid Frenchman!'" I think a variation of this would be to make the French try to call the units "metri" and "kilogramma"... ...but personally, I think everyone should be entitled to call them whatever the hell they please, as long as we agree on the definitions of the units =) -- Weyfour WWWWolf, a lupine technomancer from the cold north...[ Parent ]
 Nifty neet-o science article (2.00 / 7) (#58) by The Terrorists on Sat May 15, 2004 at 08:19:41 AM EST

 Need more of these at k5! :) I have a set of nanites that may be of assistance in the ball-polishing department, as well. Watch your mouth, pigfucker. -- Rusty Foster
 Why wouldn't this method work? (none / 2) (#69) by clambake on Sat May 15, 2004 at 02:13:37 PM EST

 The mass of an sphere of pure platinum is 1 kg when it exerts a gravitational force of X (whatever this number happens to be) against an identically sized sphere of pure platinium when placed at a distance of exactly 1 m (or 1 cm, or 1 mm, depending on how accurate our gravitational force measuring machines are) apart.
 Low accuracy (none / 3) (#73) by dn on Sat May 15, 2004 at 04:04:53 PM EST

 Firstly, the attractive force can be measured, but it is so small that getting enough accuracy would be insanely difficult. Secondly, there are many sources of error. A tiny effect from magnetism or static electricity would cause a major error. Gravity gradients from the movement of nearby masses would also be a major problem; the morning dew drying more slowly on the shaded side of the building could probably be detected.     I ♥ TOXICWASTE [ Parent ]
 Whoa there, silver. (none / 2) (#91) by Empedocles on Sun May 16, 2004 at 02:38:07 AM EST

 Measuring G (the gravitational constant) is both insanely difficult and insanely inaccurate (compared to our measurements of other constants, anyway). There's so much variation in our measurements of G that several people (not the tinfoil hat crowd, either) have proposed a variable G based on other factors. An interesting subject if you want to look it up sometime, if nothing else. --- And I think it's gonna be a long long time 'Till touch down brings me 'round again to find I'm not the man they think I am at home[ Parent ]
 Try measuring it. (none / 0) (#99) by alby on Sun May 16, 2004 at 12:13:09 PM EST

 Our "gravitational force measuring machines" are not that good at all! You see there's this pesky force of gravity not only acting between the two spheres, but between them and every other massive object in the universe. -- Alby[ Parent ]
 It's also a mathamatical problem (none / 1) (#102) by simonfish on Sun May 16, 2004 at 03:03:22 PM EST

 In addition to all the inaccuracies, mass cancels out in the formulas, assuming you're willing to define gravitational and inertial mass as the same(which they should be, or should they? ;), and so any obvservations hafto be in reference to another force(non-gravitational), so you're defining in terms of the electromagnetic in all likely hood which just vastly confuses things even further. [ Parent ]
 Based on the meter (none / 2) (#71) by Gregoyle on Sat May 15, 2004 at 03:01:08 PM EST

 All metric units are based on the meter. The kilogram is 1000 grams; the gram is 1 cubic centimeter of water. The centimeter is .01 meters; the meter was supposed to be based on the size of the Earth, but the measurement was flawed and we stuck with the old one. So basically all the metric units are based on a semi-arbitrary measurement. If the French want to say that their kilogram is the kilogram, that's nice, but it really depends on how many people listen to them. I think the idea of a kilogram is much more powerful (and useful). ------- He's more machine now than man, twisted and evil.
 nope (none / 3) (#74) by opusman on Sat May 15, 2004 at 04:31:11 PM EST

 Wrong way around. A litre is defined as a kilogram of water. 1000 millilitres in a litre. 1 cubic cm = 1 millilitre. And so on. It all flows from the kg. What would americans know about metric measurements anyway? :) [ Parent ]
 Wrong wrong wrong! (2.50 / 6) (#96) by elgardo on Sun May 16, 2004 at 04:58:22 AM EST

 A litre is defined as one sq.dm. - i.e. a block that is 0.1m x 0.1m x 0.1m. Litre is a measurement of space, not a measurement of mass. There is no direct relation between amount of mass and amount of space the mass takes up, and I'll tell you why; Any material changes the amount of space it takes up, depending on temperature and pressure. 1 kg of water at 4 C does not take up the same amount of space as 1 kg of water at 25 C; the latter takes up more space and thus has less density. Because a litre is defined as one sq.dm., this means that one litre of water at 4 C is heavier than one litre of water at 25 C, within the same atmosphere. This can be proven by the fact that, if you mix the two, the lightest material will go to the top; the 25 C water will rise, while the 4 C water will fall. And if you still don't get it, try to boil a kettle of water, and think about why the hottest water at the bottom of the kettle starts rising to the surface - the process we actually call "boiling". [ Parent ]
 Liter of Water... (none / 0) (#118) by jameth on Mon May 17, 2004 at 03:06:10 AM EST

 at the prime meridian and the equator at sea level at boiling point, or something along those lines. It is quite well defined, I just don't care enough to check how. [ Parent ]
 So you're saying... (none / 1) (#124) by Gregoyle on Mon May 17, 2004 at 01:58:43 PM EST

 That the meter wasn't developed based on a faulty measurement of the earth's surface in a project commissioned by Napoleon? If such is the case it would be an EXTREME coincidence that exactly 1 cm cubed of water weighs exactly 1 gram. Read up on your history, man, the whole reason the metric system is named such is because of the meter. ------- He's more machine now than man, twisted and evil.[ Parent ]
 nope again (none / 1) (#131) by opusman on Tue May 18, 2004 at 07:30:46 AM EST

 I didn't say that. Just because the metre was originally defined one way does not mean it is defined so today. Today, the only physical artifact around which the metric measurement system is based is the kilogram. Therefore, it all flows from that. Or do you think the French went out and found a lump of platinum that just happened to weigh the same as 1000 cubic cm of water? [ Parent ]
 I am a chemist... Here is how it really works... (none / 2) (#134) by Rich0 on Tue May 18, 2004 at 07:59:38 PM EST

 FYI - I have an MS in chemistry...  (Not that it means much - but I do have some understanding of SI units.  And chemists have some insights into the properties of water that might not be obvious.) The meter is a fundamental unit derived from the second and the speed of light. The kilogram is a fundamental unit derived from a lump of metal in a vault. The kilogram was originally based on the mass of a liter of water, obviously, but it is no longer based on this - because it is difficult to prepare two 1 liter flasks of water that have the same mass. Keep in mind that the mass of a liter of water varies based on temperature, probably pressure, and certainly the purity of the water - both chemically and isotopically. It is VERY difficult to make pure water - at least pure enough for the most demanding mass-based measurements.  If you prepare pure water in two labs an ocean apart you'll get two slightly different masses for the kg - due mainly to isotopic variation of the water around the globe. The silicon method described above is far superior to water - since due to the huge market for the computer you are currently using there has been a lot of demand for highly pure silicon.  Consequently we know how to make pure silicon. The problem is that none of the methods above are as accurate as the lump of metal right now - and so the lump of metal remains... [ Parent ]
 Yegads! A Masters in Chemistry Yet... (none / 1) (#137) by dasunt on Thu May 20, 2004 at 12:56:22 AM EST

 FYI - I have an MS in chemistry... [ SNIP ] Keep in mind that the mass of a liter of water varies based on temperature, probably pressure A masters of science in Chemistry and you can't definately state that pressure and density of a volume of liquid are linked? What did you learn at the U? [ Parent ]
 Very nice. (2.40 / 5) (#75) by gilrain on Sat May 15, 2004 at 04:40:29 PM EST

 But I found your introduction misleading. Just because the mass of that object will always be called one kilogram does not mean that its mass never changes, as you conclude. It's a nice, dramatic way to start your article, but it is simply not true. Otherwise, I very much enjoyed it. :)
 yes it does (none / 3) (#76) by modmans2ndcoming on Sat May 15, 2004 at 04:47:14 PM EST

 mass is defined in Kilos. if the mass of the reference e object changes, then the definition of a kilo changes, as such the mass of the reference object is still 1 kilo and has not changes. what has changed is the WEIGHT of the reference object. I like the idea of the Watt method better than the mole method. like the meter being defined as a measurement of one of the forces over time, (electromagnetic), the watt method is similar. [ Parent ]
 Things are not ruled by definitions. (1.75 / 4) (#77) by gilrain on Sat May 15, 2004 at 05:02:40 PM EST

 Suppose, for some reason, the mass of the reference kilogram increases dramatically. Some people insist that the mass has not changed, since it is still one kilogram -- but the guy sitting under the thing is thinking, "Damn, I wish I had the old kilogram back!" And don't tell me that it's just the weight changing. That's bull. Suppose the mass increases enough so that gravity is effected dramatically. Are you going to be shouting, in the midst of natural disasters, that this can't possibly be happening, because it is still only one kilogram? You can't stop something from changing just by keeping its name constant. You can call a horse alive, and say that the state of that horse defines "alive" -- but as soon as it dies, people are going to start looking at you funny when you tell them that they are no longer alive just because alive is the state described by the dead horse. [ Parent ]
 Mass in kilograms is ruled by definitions, though (none / 2) (#78) by deadcow on Sat May 15, 2004 at 05:08:30 PM EST

 If you're talking "mass" by definition, yup, it changes. But technically and colloquially speaking, the mass of the physical kilogram doesn't change, so long as it's measured in kilograms. [ Parent ]
 I agree. (none / 3) (#79) by gilrain on Sat May 15, 2004 at 05:12:29 PM EST

 Yes, I understand that. I was just taking exception with the article's hyperbole. It stated that the mass never changes, which is just silly. All you can say is that we will always call its mass one kilogram, which is a very different statement. [ Parent ]
 again, you are mistaken (2.50 / 4) (#86) by modmans2ndcoming on Sun May 16, 2004 at 12:15:22 AM EST

 weight, which is the force a mass has on a surface due to gravity, and mass itself are different. the mass of the cylinder will not change because when the weight changes, the definition of 1 KG changes. [ Parent ]
 Uhhh, no (2.50 / 4) (#95) by esrever on Sun May 16, 2004 at 04:37:37 AM EST

 You're confused, the grandparent was correct.  Mass is not something that relies upon kilograms.  A unit of mass is a unit of mass.  We conveniently picked a certain amount of mass, called that amount '1 kilogram' and set about defining all other amounts of mass based upon that arbitrary measurement.  The rod does actually lose mass, but 'conveniently' the measure that we use to define it also reduces, so we still say that it is 'one kilogram' but that 'one kilogram' defines less mass than it used to previously. It's like if the measuring stick that you are using gets shorter every day, and you're still using it to define 'metre' then I guess you could claim you're getting taller?  After all, you'd have more metres in your height, wouldn't you? Audit NTFS permissions on Windows[ Parent ]
 Not quite right (2.87 / 8) (#80) by sjmurdoch on Sat May 15, 2004 at 05:45:01 PM EST

 The standard kilogram is losing a slight amount of mass each time it is cleaned, but mass defined by a kilogram is not changing. They do this by weighing the kilogram before and after cleaning, then taking the difference into account. Therefore the mass of the standard kilogram is a little less than 1kg, but they know by how much less. -- Steven Murdoch. web: My Home Page
 reference? (2.25 / 4) (#90) by ffalcon on Sun May 16, 2004 at 02:07:38 AM EST

 Do you have a reference for this? Interesting. [ Parent ]
 How often (none / 0) (#103) by metalfan on Sun May 16, 2004 at 03:41:06 PM EST

 do you need to clean a cylinder that is kept under three bell jars? [ Parent ]
 Often enough (none / 1) (#123) by Miniwheat on Mon May 17, 2004 at 01:49:57 PM EST

 It wouldn't be much good as a standard if all you did was keep it under three bell jars. It has to be used from time to time to weigh things. Major laboratories around the world have their own copies of the standard kilogram and these have to be compared with the original from time to time if they are to have any validity. This article may clear up some questions about the kiolgram standard which weren't answered by the original poster, or you can skim this usenet post from Henry Spencer if you have a short attention span or dislike small print. [ Parent ]
 Nice. (none / 0) (#126) by metalfan on Mon May 17, 2004 at 09:07:22 PM EST

 The small print isn't a problem for me (Ctrl-Plus, Ctrl-Plus; Mozilla, how i love thee!), but unfortunately I don't even have a long enough attention span to read the shorter link. [ Parent ]
 why not define mass by (none / 3) (#87) by modmans2ndcoming on Sun May 16, 2004 at 12:17:45 AM EST

 why not define mass by the force needed to over come the masses inertia from rest?
 Good sir... (none / 3) (#88) by jch on Sun May 16, 2004 at 01:12:11 AM EST

 ...that's what "mass" is--it's a measurement of an object's intertia. [ Parent ]
 umm, (none / 1) (#106) by modmans2ndcoming on Sun May 16, 2004 at 04:32:07 PM EST

 yeah that is great except that we do not define it in terms of inertia do we. [ Parent ]
 We don't? (none / 2) (#107) by KrispyKringle on Sun May 16, 2004 at 04:39:39 PM EST

 Sure we do. Abbr. m Physics. A property of matter equal to the measure of an object's resistance to changes in either the speed or direction of its motion. The mass of an object is not dependent on gravity and therefore is different from but proportional to its weight. --http://dictionary.reference.com/search?q=mass How do you define it? [ Parent ]
 Definition of newton (none / 1) (#112) by Idran1701 on Sun May 16, 2004 at 09:55:51 PM EST

 Also, if you mean why isn't mass determined based on a certain number of newtons required to move it from rest in a vacuum, or some such, it's because the newton is itself derived from mass, as well as distance and time. Specifically, 1 newton is defined as the force required to accelerate a 1 kilogram object at 1 meter/second/second. [ Parent ]
 It's not that easy (none / 1) (#125) by Miniwheat on Mon May 17, 2004 at 02:02:40 PM EST

 That's a nice idea, but first you would have to define both "force" and "inertia" in terms of only the six other basic SI units (Length, Time, Electric current, Temperature, Luminous intensity and Amount of substance) and reproducable physical constants such as the speed of light in a vacuum or the wavelength of a specific transition between two energy levels of a specific atom. Your definition would have to meet very strict standards for reproducability in order to be considered as a replacement for the current SI standard. If you know how to do this, then I know more than a few scientists who would be quite eager to see your work. [ Parent ]
 Inaccurate (2.16 / 6) (#109) by gidds on Sun May 16, 2004 at 05:56:25 PM EST

 It's hard to take seriously an story with such an important error so early on. No matter what happens to this object, its mass remains the same. Er, no, its mass changes. The measurement of its mass on one particular scale doesn't change. Not the same thing at all. Andy/
 I highly suspect that to be intentional (none / 2) (#120) by Milo Minderbender on Mon May 17, 2004 at 07:28:13 AM EST

 I know that I, at least, clicked on the link to view the entire story because I read the intro and thought, "But that's impossible!". It's not an error; it's a hook to get you to read the article...and a clever one at that! -------------------- This comment is for the good of the syndicate.[ Parent ]
 Fiction story (2.57 / 7) (#117) by Shubin on Mon May 17, 2004 at 02:46:04 AM EST

 Some time ago I invented a plot for a sci-fi story. I'm not sure I'll ever have time to write it, so here is the idea : The whole story is something in the Tom Clansy style. A group of international terrorists steals the The international prototype kilogram right before the copies are to be made. This causes a great crisis in measurements, influencing gravimetry, astronomy, geology etc etc... The best detectives are trying to find the prototype, but without success. Finally, terrorists being followed closely, decides to hide the item in the territiory of Russia. The plane where it travels is accidentaly shot down by Ukrainian missile and crash lands in Ukraine. The final scene : someone sells the odd-looking cylinder made of platinum to the black market dealer. The dealer weights thing on his scales and says : "950 grams"...
 scale probably wrong then. (none / 3) (#121) by xmnemonic on Mon May 17, 2004 at 09:05:46 AM EST

 scale probably wrong then. [ Parent ]
 boring (none / 0) (#129) by Liet on Tue May 18, 2004 at 04:25:31 AM EST

 that sounds like the most boring thriller ever written or not written. please let it be not written. think of the trees [ Parent ]
 Trees ? (none / 0) (#130) by Shubin on Tue May 18, 2004 at 06:57:26 AM EST

 You say trees ? Does your Internet connection work on steam power ? Anyway if you think it's the most boring thriller ever made I feel I must write it. A chance to become an author of the most borind thriller in the world is irresistible. [ Parent ]
 you got it wrong (none / 0) (#139) by FreeLinuxCD.org on Fri May 21, 2004 at 05:07:11 PM EST

 You mean the dealer's scale shows 1050 grams right?? They are usually hacked to show more than the actual weight. Otherwise they lose money :) oktay [ Parent ]
 my dad's idea (none / 0) (#141) by maw on Sun May 23, 2004 at 08:12:33 PM EST

 My dad once had a similar idea. In his story, a scientist discovers that the principles behind a common technology that many people daily depend on, such as refrigeration, are invalid. Hence, all the refrigerators, air conditioners, etc stop working. As far as I know, he's never sketched out the plot, though. -- I have no idea what you're talking about, but that's ok, since you don't either.[ Parent ]
 How do balances handle differences in density? (none / 0) (#133) by expro on Tue May 18, 2004 at 12:22:58 PM EST

 I saw no mention that these weighings are carried out in a vacuum, but perhaps I missed it? It seems to me that when dealing with extremely-accurate balances that weighings would need to be carried out in a near vacuum? Otherwise, you have to account for the volume of air displaced which will be buoying up the item being weighed, which would only be equal between items of identical density, and how do you establish that the density is identical...
 this is where... (none / 0) (#142) by Morphine007 on Wed Nov 17, 2004 at 02:10:24 PM EST

 ...the triple bell jars come into play.... [ Parent ]
 The New Kilogram | 142 comments (120 topical, 22 editorial, 0 hidden)
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