# Standardized measurement for trade between world with different gravity?

So I was building my world with underwater civilizations and someone raised a question (see here) that I thought was interesting related to trade aspect: how would we have a standardized measurement when things weight differently?

In the case of the world I'm building, since there are kingdoms underwater and the aquatic races do trade with humans, how would they have a measurement so that there would be no misunderstanding about how much goods is in the contract?

Furthermore, this also applies to interplanetary trading (which doesn't really concern my story, but I'm curious about nonetheless). How would we have a measurement when we're trading between planets with different gravity?

• Mass is the same everywhere. The fun part is measuring it. – John Dvorak Dec 7 '16 at 23:07
• @JanDvorak yeah that's exactly my problem. I guess since my world has magic I can always handwave it with magical mass displaying device. – Juliette Evans Dec 7 '16 at 23:18
• No handwaving necessary. In nearly any environment that isn't in free-fall, a simple, age-old balance scale will do the trick. Volume displacement is another very easy option that can work for many substances in situations where balance scales won't work. You already have two good answers that espouse the virtues of mass, so I won't add another. – type_outcast Dec 7 '16 at 23:26
• The problem with mass (suggested by most answers) is that it's not easy to use a balance scale underwater, because of buoyancy. Buoyancy in air only matters for very precise measures, but it might be a concern in very dense atmospheres like those of Venus and Jupiter. – Pere Dec 8 '16 at 14:04
• I just edited my answer to account for buoyancy, thanks @Pere – Nobody Dec 8 '16 at 16:04

As said in the comments section, you would still use mass.

You just need different weighing scales for different environments, depending on the technology you use. This would maybe cause some confusion during first contact with a society which lives in a different environment, but no worse than the usual inter cultural units mess (could you Americans please stop using your inferior units system? ;-) ).

• Old fashioned balance scales work everywhere, although at high pressures you should roughly match the volume of your standard weighs to the volume of whatever you are weighing, so you might need more than one set of standard weighs (you might have a set of "volume" weighs with low density and a set of "heavy" weighs with high density or something like that). This is so that the effects of buoyancy cancel out (as was pointed out in a comment, should have really thought of this, thanks), otherwise they will limit the accuracy of the scale.
• Your usual kitchen/bathroom scale (which measures force) would have a label saying under which conditions it's accurate, it would do all of the math for the user so the output is just the usual mass unit. It would either require the user to dial in the volume of what is weighed or would take a guess. For example:

• "Only accurate at sea level or above."
• "Only accurate for weighing merfolk at 2000m below the surface." because you can probably guess the density of merfolk pretty well. If it's the fancy body fat measuring kind it could even calculate the density of the user.
• "Only accurate for weighing the cooking ingredients pictured above at 3500m below the surface." with a set of buttons to choose from ingredients

Scales in general would

• measure gravity (if they are intended to be used at wildly different depths/planets)
• and/or measure pressure (if they are intended to be used at varying pressures)
• measure volume/density or have a method to guess densities, for example by using a list of the densities of the things you are weighing (if they are intended to be used at high pressure)

Of course this would make possible some new methods for trying to cheat with weighing, which would be used until everyone knew about it, at which point the methods would stop working.

Use the volume of goods.

While using mass would be the correct way to go about measuring and comparing (ask any engineer1), it can be cumbersome and is likely to become difficult due to it involving calculus as soon as you switch altitudes, gravities, densities, etc..

Now volume on the other hand will always be the same. No matter how complex a container, you can always measure its volume by fully submerging it in a fluid and then measuring the displacement.
This can be done anywhere without issues and even the fluid does not really matter.
Even standartization is much easier, because the ur-volume will always have the same displacement as long as it's a good old solid.

1No, social-engineering is not an engineering discipline

• There are times for volume, and times for mass. Mass is not in any way more correct, it's just often more practical in our everyday life to buy groceries by mass, say. For some things even with the complications of buoyancy, mass is still more practical. It's the only real substitute for weigh. But depending on their style of life and stuff, it might be practical to use volume more often, good thought. – Nobody Dec 8 '16 at 16:12

This is why mass is a preferably metric for trading physical commodities, as opposed to weight, which depends on local gravity and atmospheric (assuming an atmosphere) or whatever the local medium's density happens to be. Go metric! The Kg is a measure of mass; the pound is a measure of weight.

As others have said, you could use mass rather than weight. Which in practice simply means that you have a conversion factor: 10 kilos weight on Earth = 6 kilos weight on Trantor, or whatever the conversion factor is.

If you're talking about trading with aliens, or a culture different enough that they don't use Earth units, it would be a no-brainer. There would be a conversion factor when you weigh things anyway. 10 kilos = 14 centons or whatever. It would be easy to determine the conversion factor: you get an artifact which has been carefully weighed to whatever precision is necessary on Earth, then you take it to this other planet and weigh it there. Like if your standard weight is 1 kilo, and on Trantor it weighs 2.7 centons, well, now you know the conversion factor. Presumably you get respected people from both cultures to observe the weighing and verify that no one switched weights or anything trickery, and the problem is solved.

Alternatively, you could use volume as the standard rather than weight. It's easy enough to measure volume: Fill a suitable size container with liquid, measure the fluid level, then immerse the object in the container and measure the change in fluid level. i.e. all you need is a big enough measuring cup. (Granted, this only works for things that are not damaged by being submerged in liquid. You probably don't want to measure the volume of paper or kittens this way.)

• 10 kiograms is 10 kilograms everywhere. It may weigh 60 Newtons on Trantor (and 98 on Earth) but it's still 10 kilograms. – JDługosz Dec 8 '16 at 7:20
• @JDługosz To physicists, kilograms are a unit of mass and newtons are a unit of weight. But to everyday people, kilograms are weight. I have never seen a bathroom scale or a scale at a butcher counter marked in Newtons. Most sources from non-physicists refer to the kg as a unit of "weight". metric-conversions.org/weight-conversion.htm, infoplease.com/ipa/A0001659.html, thefreedictionary.com/metric+weight+unit, etc. If I asked you how much you weighed, I bet you would give an answer in kg and not N. Note I said "kg weight" to distinguish from "kg mass". – Jay Dec 8 '16 at 14:37
• Common people conflate mass and weight because the conversion is a constant. Once we do commerce with the moon and mars; or in your case underwater, it will be important to merchants. If you asked how much I weigh (in some nonstandard context) then it would need careful explaining. – JDługosz Dec 9 '16 at 5:42
• Note that making this clear is the proper answer to the Question asked. Your misuse and insistance that it be allowed as such is part of the problem. «how would we have a standardized measurement when things weight differently?» so it may be pedantic with our Earth-bound culture, but is important specifically in the context he was asking. – JDługosz Dec 9 '16 at 5:46

Mass is mass and weight is not mass, it is a force which varies from place to place. On our very own Earth the weight of the same mass can differ by about 0.5% depending on where you measure it. For example, a quantity of gold which weighs 98.066 newtons (10 kg-force) in Copenhangen will weigh only 97.617 newtons (9.954 kg-force) in Mexico City; a difference of about 45 grams-force, about 1.5 oz-force [source]. I hear that it could be risky to come to Mexico City with 1.5 oz of gold missing.

(As an aside: The unit of mass is the last fundamental unit of measurement which is defined with reference to a specific object. One kilogram is by definition the mass of the object called the International Prototype of the Kilogram, a cilindrical piece of platinum-iridium alloy kept in a vault in a Parisian suburb. All other fundamental units have definitions which are independent of any man-made object.)

As others have pointed, mass is invariant and easy to measure in any planet, with a balance scale and a set of weights or even with a kitchen scale set to be used in a given planet gravity.

However, measuring mass underwater (or under other dense fluids) is difficult because objects experiment buoyancy, and buoyancy is the product of density of water and volume of object.

For objects of known density, that problem can easily be overcame with a little maths. Let $measured\ weight$ be the weight measured by a scale underwater, given in units of mass ($scale\ result$), calibrated to local gravity (in Earth a waterproof kitchen scale would be fine):

$$scale\ result=g·measured\ weight=weight-buoyancy=$$ $$=g·mass-g·density_{water}·volume=$$ $$=g·density_{object}·volume-g·density_{water}·volume=$$ $$=g·volume·(density_{object}-density_{water})=$$ $$=g·volume·density_{object}·(1-\frac{density_{water}}{density_{object}})=$$ $$=g·mass·(1-\frac{density_{water}}{density_{object}})$$

Therefore:

$$mass=scale\ result·\frac{1}{(1-\frac{density_{water}}{density_{object}})}$$

That is, to determine mass underwater you just to multiply for a factor depending of density of object and water. For example for steel (density=7,85 g/cm3) under fresh water (density=1 g/cm3), we only need to multiply what the scale says by 1.146.

Mass of objects with unknown density are very difficult to determine underwater. For heavy materials a quite rough approximation of density could be enough but for materials of density close to that of water (like wood or kelp) an small uncertainty in density would cause a larger error in mass. Therefore, a lot of commodities that in our land dwelling world are measured by mass are likely to be measured by volume or by units for underwater trade.