## 2020

#### Mass Measurements

Consumers don't need to know about buoyancy

by Philip Stein

In my last column ("Masses and Weights," September 2002, p. 72), I wrote about weight and mass, and reviewed some of the influence quantities that can affect your measurement results. I discussed local gravity, temperature, air drafts, off-center loads and the need to calibrate a scale in the location where it is to be used.

As is often the case with precision metrology, the measurement itself--in this case a weighing--is simple both conceptually and in practice. The difference between routine measurements and high precision comes from the degree of care taken to be sensitive to all the little details and to make sure they have been taken into account.

There's also a need to be realistic. Weighing fruits and vegetables in the grocery store must be able to be done casually. If there are metrology details, they must be hidden from the general public; otherwise the system cannot succeed as an everyday activity.

The engineering of mass metrology and weighing has succeeded at this goal, perhaps better than any other parameter. Daily commerce can and does take place in the presence of some pretty daunting underlying complexities, and we neither notice nor do we need to take special care.

**Archimedes **

The most significant influence on weighing
was first articulated by Archimedes in his famous principle of buoyancy.
When an object is immersed in any medium (I think most easily of immersion
in water), a lifting force (buoyancy) is exerted equal to the weight of
the medium displaced. If a 1,000 cubic centimeter (cm^{3}) block
of aluminum, normally weighing about 2.7 kilograms (kg), is immersed in
water, it will displace 1,000 cm^{3} of water, which weighs about
1.0 kg. Under water, the aluminum will weigh 1.7 kg.

Anything with a density less than that of water will displace enough water so its net weight appears to be zero, and it will float. In a sense, the aluminum in the example is floating as well, but it's dense enough so it's heavier than its floating force and will sink.

**Weighing in air **

We weigh objects on a scale or balance, but
we don't usually weigh them under water, so why does buoyancy matter?
It turns out air also has weight and actually weighs quite a lot even
though we don't notice it--1,000 cm^{3} of air under some fairly
standard conditions weighs about 1.2 g. We must somehow take the buoyant
force of this air into account. When weighing is done at high precision,
the buoyancy must be similarly calculated to high precision.

What do we do when weighing fruits and vegetables, or for that matter when weighing gold and platinum for retail sale? Is buoyancy important here? Yes, it is quite important, but it is also necessary to be able to compensate for buoyancy without much fuss. For this reason, the common unit of weight is redefined.

**True mass and conventional mass **

The true mass of an object is a measure of its inherent inertia, and this is independent of buoyancy. We could determine true mass by weighing in a vacuum, without buoyant forces, but this is an enormous hassle. It's better just to weigh in air and make corrections.

If a mass standard made from stainless steel
at about 8 g/cm^{3} is compared with another mass of the same
material in a two-pan balance, the buoyancy is the same and no correction
needs to be made. When two masses are being compared, say one of stainless
and one of brass at 8.4 g/cm^{3}, the brass will appear to weigh
more because it is denser, displaces less air and has less buoyant force
on it. For the highest precision weighings, the exact density of the masses
and the current density of the surrounding air need to be known and calculated.

Air buoyancy is too important to be ignored for any weighing, though. The difference between the weight in air and the weight in a vacuum for stainless steel is about 150 parts per million (ppm) or 0.15%. To simplify these calculations so they are not necessary for routine commercial transactions, a new weight unit is used.

This is officially known as "the conventional value of the result of weighing in air," but is almost always (incorrectly) called conventional mass. It's the weight measured by balancing against stainless steel of standard density 8.0 in air of standard density 0.0012. The actual buoyant mass of stainless weights almost always varies less than 15 ppm from the conventional mass due to local variations in air density. A difference this small is negligible for virtually all commercial work, although laboratory comparisons at higher precision must still take actual buoyancy into account.

When a stainless weight is used as part of a calibration process for some other force, such as pressure or torque, the weight is calibrated to conventional mass rather than true mass, and in most cases buoyancy can be neglected as well.

Grocery scales, jeweler's scales and laboratory balances are all calibrated in conventional mass. Here, though, we're not weighing stainless steel, so there's still a small problem. The density of water--and therefore most groceries--is near 1, so there's still about a 100 ppm correction in standard air against conventional mass. This is small enough so it's not important for buying your staples, but it might be important for a pharmaceutical manufacturer mixing aqueous components by weight.

At the jewelry store, the density of 24k gold is about 19.3, silver is 10.5 and platinum is about 21. This means gold will have a buoyancy of 62 ppm but will be compared with stainless at 150 ppm, so it will appear about 88 ppm heavier than it should. If gold costs $320/oz, the buyer will lose about $0.03 per ounce. This is probably an acceptable amount--the uncertainty in the weighing equipment is probably larger.

**Other commercial practices**

Weighing procedures have also been simplified for commerce by the definition of grades, or classes of weights. There are many classes and many systems of classification, but they all have the same goal: to allow the description of weighing errors without requiring calculation on the part of the consumer. The most common weights in the United States are grade F (for field).

There are many requirements for these masses
(see *NIST Handbook 105-1, Specifications and Tolerances for Reference
Standards and Field Standard Weights and Measures* for details), but
the most important one is that their conventional mass is within 0.01%
(100 ppm) of the marked value. This is a small enough uncertainty that
they can be used without any further compensation (for buoyancy or other
effects) for virtually all daily transactions involving commercial use
of weights.

For more information about commercial weights and measures, including some fun exhibits at a virtual museum, go to www.nist.gov/owm.

**PHILIP STEIN** is a metrology and quality
consultant in private practice in Pennington, NJ. He holds a master's
degree in measurement science from the George Washington University in
Washington, DC, and is an ASQ Fellow.

IF YOU WOULD LIKE to comment on this article, please post your remarks on the Quality Progress Discussion Board at www.asqnet.org, or e-mail them to editor@asq.org.

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