Mass, "Weight"
(C. B. of Philadelphia, PA.
2000-10-25)
Is there a [unit of] measurement smaller than a milligram?
Here's a list of the smaller official units of mass in "concrete" terms:
- gram (g): A paper clip.
- milligram (mg): Cubic millimeter of water. Mass of a typical ant.
- microgram or gamma: Dust mite (dermatophagoides pteronyssinus).
- nanogram (ng)
- picogram (pg): A typical bacterium (Escherichia coli).
- femtogram (fg)
- attogram (ag): A typical virus, or 20 prions.
- zeptogram (zg, 10-21g):
3 gold atoms, or 33 water molecules.
- yoctogram (yg, 10-24g):
60% of a hydrogen atom.
- unnamed
(10-27g):
110% of an electron.
- unnamed
(10-30g):
561 eV/c2
- unnamed
(10-33g):
40% of a neutrino.
The zeptogram and yoctogram have been officially recognized
by the CGPM since 1991.
An atom of hydrogen is about 1.66 yg.
An electron is about 0.00091 yg.
This is roughly equal to the next unit down
the list (namely, yg/1000 or 10-27g),
which doesn't yet have an official name.
(2012-11-07)
Solar mass :
1.98855(24) 1030 kg
[ CODATA 2010 ]
The unit of mass in the astronomical system of units.
The gravitational influence of a celestial body is measured by
its standard
gravitational parameter, equal to
the product of its mass into
Newton's universal constant of gravity.
Both factors may only be known at a
modest level of precision
in SI units,
but the product can be determined with astronomical
precision (literally). For example:
| Heliocentric gravitational constant |
1.32712440042(8) | 1020 |
m3/s2 |
|
Geocentric gravitational constant |
3.986004415(8) | 1014 |
m3/s2 |
Thus, the ratio of the mass of the Sun to that of the Earth
(atmosphere included) is known with excellent
precision, namely: 332946.0438(8).
Although the Sun loses
millions of tons per second,
it will take more than 2000 years
for this to affect the least significant digit of that last ratio.
This is good enough to use the changing mass of the Sun
as a very practical unit which allows the mass of
large celestial bodies in the solar system
to be expressed with much more precision than SI units (kilograms)
would allow, using the values of their relative
gravitational constants, as defined above.
| Body | Mass | Reciprocal |
| Sun | 1 | 1 |
| Jupiter |
9.5479194 (74) 10-4 |
1047.3486(8) |
| Earth + Moon |
3.040432685(9) 10-6 |
328900.5558(11) |
| Earth |
3.003489661(7) 10-6 |
332946.0438 (8) |
| Moon |
3.69430242(46) 10-8 |
27068710 (34) |
The Earth is 81.30059(1) times as massive as the Moon.
Astrodynamic Constants (NASA / JPL)
Basic Astronomical Data for the Sun
(BADS) by Eric Mamajek.
("Biker" of Jerome, ID.
2000-10-09)
What is a slug, in the [engineering] weight measurement system?
The slug is a unit of mass.
The word was coined in a 1902 textbook by the British physicist
A.M. Worthington
to designate the British engineer's unit of mass, which appeared
in engineering calculations late in the 19th century.
The slug is defined as the mass which would accelerate
at a rate of 1 ft/s2 under a force of one pound-force (lbf).
Since 1 lbf is the force exerted on a mass of one pound by a
standard gravitational field (of exactly 9.80665 meters per square second),
a slug is thus exactly equal to 196133/6096 pounds
(about 32.1740485564 lb or 14.593902937206 kg).
It's worth making a few technical points about this:
- The slug is the unit of mass in a coherent system called either
"British engineering system" or "English gravitational system".
On the other hand, the Imperial
(formerly "English") unit of mass is the pound (lb),
which is now defined in metric terms
(0.45359237 kg exactly).
- The "metric equivalent" of the slug
is the hyl of exactly 9.80665 kg
which is the unit of mass of the so-called "metric-technical system".
The hyl is also called "metric slug" or
designated by the German acronym TME
(Technische Mass Einheit ).
A mass of one hyl gets accelerated at a rate of one meter per square second
by a force of one kilogram-force (namely, 9.80665 N).
- The SI unit of mass is the kilogram,
not the gram or the hyl.
- Both the pound and the slug are units of mass.
The latter weighs about 32 times as much as the former,
even on the surface of the moon.
On the moon, however the weight of a pound-mass
(lb or lbm) is only about one sixth of a pound-force (lbf).
(2007-05-13)
Surviving customary units of mass, in the electronic age.
What are the units of mass available on modern electronic balances?
The customary units listed below are mostly kept alive by
gold traders.
A common feature of electronic analytical and/or precision balances
is the ability to use various customary units of mass.
Copying each other over the years
(often misspelling "baht" and/or "mesghal") manufacturers have picked from the
following limited catalog of units, which caters to all international traders.
In East Asia, the catty is to the tael
(TL)
what the pound (lb) is to the ounce (oz).
There are 16 taels to the catty...
The Taiwanese tael
(37½ g) thus corresponds to a catty
of 600 g, whereas the "tael of Singapore"
(defined as 1/12 lb or 4/3 oz)
corresponds to a catty of 4/3 lb
(about 604.79 g).
1000 grams (g) to the kilogram (kg).
7000 grains (gn) to the
avoidupois pound (lb).
Note that the abbreviation "gr" is best shunned
(as
it could stand for either grams or grains).
| Name | Symbol | Value in grams (g) | Definition |
Usage |
| (kan) | kw |
3750 g (since 1891) |
1000 mom | Japan |
| pound | lb |
453.59237 g (1959 treaty) |
7000 gn | avoirdupois |
| (troy pound) | lbt |
373.2417216 g (12 ozt) |
5760 gn | illegal 1879 |
| (newton-mass) | "N" |
101.97162129779282... |
1 N/G | dubious (*) |
tael (S)
tael (M)
tael (C)
tael (T)
cay, luong
tael (H) | TL |
37.79936416666666...
37.793
37.584
37.5
37.5
37.429 g (or 37.43 g)
|
1/12 lb
37½ g
10 chi
jewelry |
Singapore
(Malaysia)
(China)
Taiwan
Vietnam
Hong-Kong |
| troy ounce | ozt |
31.1034768 |
480 gn | precious metals |
| ounce | oz |
28.349523125 |
437½ gn | avoirdupois |
| baht, tical | |
15.244 g (or 15.2 g) |
(235¼ gn) | Thailand |
| tola | tol |
11.6638038 |
180 gn | India |
| mesghal | Ms |
4.6083 (24 nukhuds) |
1 g / 0.217 | Iran |
| mithgál | |
4.25 |
modern | gold dinar |
| troy dram (**) | dr. 3 |
3.8879346 (3 scruples) |
60 gn | troy (1/8 ozt) |
momme
chi
don | mom |
3.75 |
0.001 kw | Japan
/ pearls
Vietnam
South Korea |
| | |
2.975 |
0.7 mithgál | silver dirham |
| (avrdps) drachm | dm |
1.7718451953125 |
1/16 oz | US ammo |
| pennyweight | dwt |
1.55517384 |
24 gn | troy (1/20 ozt) |
| gram | g |
1 |
0.001 kg | SI |
| carat | ct |
0.2 g
(CGPM, 1907) |
200 mg | precious stones |
| grain | gn |
0.06479891 |
1 lb/7000 | troy & avdp. |
| milligram | mg |
0.001 |
0.001 g | SI |
(*)
A "newton-mass" unit was (improperly) introduced by some instrument
makers as the mass (about 102 g)
on which a standard gravitational field
of 9.80665 m/s2
would exert a force of exactly 1 N (1 newton).
This is yet another offspring of the ongoing confusion between mass and weight
(the latter being the force exerted by gravity on a given mass).
(**)
This is not the avoirdupois drachm (symbol dm.)
which is the smaller unit of only
1/16 oz (1.7718451953125 g) still used for loading ammunition in the US.
The unit found on electronic scales is the troy dram
(symbol: dr. or 3)
which belongs to the deprecated apothecaries' weight system,
(illegal for trade in the UK since 1985).
It's equal to 3.8879346 g :
60 grains (1/8 ozt) or 3 scruples.
The troy system and the apothecaries' weight system are
fully compatible (units with the same names have the same values)
but some units are unused in either system.
By contrast, the troy and avoirdupois systems are incompatible;
they only have one unit in common, the grain (gn).
Some popular
conversion tables make a clear distinction between the
"drachm" (dm) and the "dram" (dr):
16 dm to the ounce and 8 dr to the ounce...
Unfortunately, this misguided "clarification" is inconsistent with
ancient usage: Actually, the former unit (1/16 oz) belongs
exclusively to the avoirdupois system and the latter (1/8 ozt) to the troy system,
while each unit may be called by either name (drachm or dram)
within its own system.
-
Ohaus
|
Shimadzu
|
A&D
|
Mettler Toledo
|
Acculab
|
Sartorius
(2007-06-03)
Royal French weights:
18827.15 grains to the kg.
The ancient livre de Charlemagne and the
poids de marc system.
From the early definitions of the kilogram
survives only an exact equivalence between the old French units of mass
poids de marc and the metric system;
there are exactly
18827.15 French grains to the kilogram.
Once it had been realized that a definition of the kilogram as the mass of a
cubic decimeter of water was not satisfying
(by the metrological standards of the late 18th century)
the kilogram was evaluated using the best system of
weights then available.
In pre-revolutionary France, that was based on a
famous artifact known as the pile de Charlemagne,
which is still preserved in the
Musée National des Techniques in Paris, France.
In spite of its name, the "Pile de Charlemagne" was built in the 14th
century—half a millenium after the days of Emperor Charlemagne.
It consists of 13 copper weights in the form of
truncated cones (larger base on top).
Except for the smallest one, all of
those are hollow, so that the next smaller weight may fit in it snugly.
The largest weight has a handle and a lid and serves as a box for the whole
thing, which stands at a height of about 9cm, with a top diameter of about 15.5cm,
and a lower diameter around 14cm (for a volume of about
1.54L, and a mass of about 12.2376 kg).
The nominal mass of this revered standard was exactly 50 marcs
(of 4608 French grains each). The royal French
livre (better known as
livre poids de marc )
was once defined as 1/25 of the "pile de Charlemagne" mass.
When the kilogram (then called the grave)
was first defined (on August 1, 1793), it was equated to
18841 grains of
the above poids de marc system,
from a single measurement by
Lavoisier and
Haüy.
 
Early in 1799, an accurate equivalence of
18827.15 grains
to the kilogram was established...
That last measurement was due to the French academician
Louis
Lefèvre-Gineau (1751-1829)
and the Italian engineer
Jean-Valentin
Fabbroni (1752-1822)
[elected to France's Corps Législatif in
1809].
Defining the kilogram as the mass of a cubic decimeter of water at 4°C
(close to the densest point at 3.984°C)
they weighed a hollow brass cylinder of known dimensions,
first in the air, then in water at 4°C.
The new determination was enacted on May 30, 1799, and it became the
final legal
equivalence between the kilogram and the "old" French units.
Because of that, the metric equivalent of the French
grain no longer depends on
the actual mass of the "pile de Charlemagne".
However, we may remark that,
at 9216 grains to
the livre , the
Pile de Charlemagne has a nominal mass of
230400 / 18827.15 or about 12.237646165...kg
whereas its actual mass has been
measured to be 12.2376429 kg.
Interestingly, the 0.27 ppm difference translates into 0.005 grains/kg,
which goes to show that the above 18th century determination (which settled,
once and for all, the conversion factor between grains
and kilograms) was indeed
fully accurate.
Since the newer equivalence was quite different from Lavoisier's original one
(13.85 grains is about 3/4 of a gram)
the standard weights that had been sent to all departmental
chef-lieus
had to be recalled. New ones were made.
The same thing did not happen for prototypes of the meter because the
difference with revised standards of length was considered acceptable.
The old French system of 18 onces to the
livre had been introduced in the wake of Charlemagne's
monetary reform.
The once
was understood to be exactly
the same as the Roman uncia
but there were 18 of those to
Charlemagne's livre
(French pound, poids de marc )
as opposed to 12 unciae to the
libra (Roman pound).
So, a French pound was exactly
1½ Roman pounds.
The above thus provides a paper trail
to what may be construed as a "legal" value of the ancient Roman
pound in metric terms, namely:
1 Roman pound (libra) =
12 onces (of 512 grains)
= 0.326337231... kg
(J. W. of Tustin, CA.
2001-02-07) Biblical Units
How many pounds was a talent?
How many ounces was a shekel?
A talent was the mass of a cubic foot of water.
The exact value of the talent thus depended on what foot was in use
in a specific part of the world at a certain period in history.
If there was such a thing as a modern Imperial talent
(based on water at 62°F)
it would be about 62.288 lb (or 28.25 kg).
The Roman talent was also defined as 80 Roman pounds
("librae", plural of "libra").
The above value of the libra,
from the days of Charlemagne, makes the Roman talent equal to about
26.107 kg.
Incidentally, this would imply a value
of about 0.2969 m for the Roman foot
(water at 62°F has a density of 10 lb per Imperial gallon).
For some obscure reason,
a foot whose length is derived backwards
from a given value of the talent is called a geometric foot.
The ancient Sumerian talent is estimated at about
28.8 kg (about 63.5 lb)
from the mass of surviving standard weights (basalt statuettes in the form of
sleeping ducks).
Outside of Rome, the talent was normally divided into 60 minas;
a mina (or maneh)
was thus roughly equal to a modern avoirdupois pound.
The shekel was always some submultiple of this mina:
The Babylonian shekel was 1/60 mina, the Phoenician shekel was 1/25 mina,
the Egyptian shekel was 1/100 mina,
whereas the "modern" Palestinian or Syrian shekel is 1/50 of a mina.
Solomon's mina of gold (1 Kings 10:17) was divided into
100 units (unnamed in the Hebrew text of 2 Chr. 9:16) not necessarily
related to the Biblical shekel of the sanctuary
(bishekel hachodesh)
whose value ought to be determined by the last words of Ezekiel 45:12.
Unfortunately, Bible scholars have been advocating
at least
two contradictory renditions of that verse, namely:
- 50 shekels to a mina
(Septuagint, according to Walther Zimmerli):
"[...] 5 shekels are to be 5, and 10 shekels are to be 10,
and 50 shekels are to amount to a mina with you."
- 60 shekels to a mina (King James and other English versions, also
supported by Rabbi Nosson Scherman, in the Stone Edition Tanach):
"[...] 20 shekels, 25 shekels, and 15 shekels shall be your mina."
The latter may have exhorted traders to check their minas against smaller
standard weights... If you know for sure, please
tell me.
Beqa'
Weight (half a shekel? =
5.99 g)
7th - 6th Century BC
ginapa
(2001-06-11)
How many pounds in 1 ton?
There are many different kinds of tons.
In the US, you're most likely to encounter the short ton
(2000 lb, or about 907.185 kg) unless you're primarily concerned with ships,
for which the displacement ton and the gross ton are in fact units of
mass both equivalent to the British long ton of 160 stones
(2240 lb, or about 1016 kg).
The long ton is retained in this international context
because it's almost exactly equal to the mass of a cubic meter of seawater.
This is a prime example of crossbreeding between the metric and Imperial systems.
Another example of interbreeding between the metric system and the Imperial system
(and the troy system)
involves a much smaller "ton", the assay ton,
which is slighly more than an ounce.
It's defined to make 1 milligram
per assay ton equivalent to one troy ounce (ozt) per ton.
There are 2 or 3 kinds of assay tons, depending on which reference "ton" is used.
The most common one seems to be the short assay ton
of 29.1666... g, which corresponds to the ton of 2000 lb.
A troy ounce (ozt)
per ton is
a milligram (mg) per assay ton.
| Ton, in lb | Assay ton, in g |
|---|
| short ton | 2000 lb |
175 / 6 | 29.16666... g |
| long ton | 2240 lb |
98 / 3 | 32.66666... g |
| troy ton | 2016 lb |
147 / 5 | 29.4 g |
Other types of tons include the very important metric ton
(better spelled tonne, which corresponds to 1000 kg or about 2204.62 lb)
and the totally unimportant and unused troy ton of 144 stones
(2450 lbt = 2016 lb = 914.44221792 kg).
The pound is understood to be the common
avoirdupois pound ("lb" or "avdp lb") of exactly 0.45359237 kg
(a 1959 international statute now defines the pound in metric terms).
For the record, the troy pound (lbt) was officially abandoned on
January 6, 1879 (175 lbt = 144 lb).
However, the troy ounce (ozt) is still widely used for
precious metals.
As if this were not bad enough, a few units of volume
are also called tons:
This includes, most notably, the international register ton
of 100 cubic feet (2831.6846592 L).
Of lesser importance is the British water ton of (exactly) 224 Imperial gallons,
which originally corresponded to the volume occupied by a long ton (2240 lb)
of distilled water at 62°F, when the Imperial gallon was still defined in like
terms as a "10 pound gallon".
(Under the modern definition of the Imperial
gallon, in metric terms, the British water ton is exactly 1018.32416 L.)
On the other hand, the unit variously called shipping ton, freight ton
or marine ton is 40 cubic feet (1132.67386368 L),
which happens to be equal to the so-called
ton of timber (of 480 board feet).
There's also
a fluid ton of 32 cubic feet (906.139090944 L),
a corn ton of 32 bushels
(which means exactly 1127.65024534016 L in the US and 1163.79904 L in the UK),
and a British tun, spelled with a "U", of two pipes or 252 Imperial gallons
(1145.61468 L).
Whew!
On 2001-10-26, Darren Finck wrote:
I just wanted to drop you a line to tell you that I enjoyed your treatise on the "ton"(s).
[above]
For completeness, it would be interesting if you were to also mention and/or
describe the origination/relation of the "refrigeration ton" and/or the "explosion ton" units.
Regards, Darren Finck
|
Thanks for the kind words, Darren.
First a general remark:
The adjective "extensive" qualifies (loosely speaking) physical quantities for
which the measure of the whole is the sum of the measures of the parts.
Volume and mass are examples of extensive quantities
(pressure and temperature are not).
Choosing some "stuff" of reference, like water under normal conditions, establishes
a "conversion factor" (coefficient of proportionality)
between any pair of extensive quantities and/or
the units which measure them.
New "practical" units may thus be created ad nauseam,
including many flavors of tons which correspond to various
extensive properties of a ton of "stuff".
This is how some of the "tons" mentioned above as units
of mass gave rise to units of volume
(a volume of one ton being the volume occupied under standard
conditions by a mass of one ton of water).
This is also how a unit of mass may become a unit of force
(the corresponding weight in a standard gravitational field,
equal to 9.80665 m/s2 ).
In particular, the ton of thrust is a unit of force equal to the standard
weight of a metric ton/tonne, namely 9806.65 N.
[The newton (N) is the SI unit of force.
Applying for 1 second a force of 1 N
to a mass of 1 kg, initially at rest, will make it move at a speed of 1 m/s.]
 |
|
The ton unit pertaining to nuclear explosions
is a unit of energy equal to 1000 000 000
thermochemical calories (of exactly 4.184 J)
and is thus exactly equal to 4184 000 000 joules.
(The kiloton and megaton are a thousand and a million times as large.)
| |
Video :
How G.I. Taylor (1886-1975)
obtained the classified tonnage
of the Trinity bomb from
dimensional analysis.
|
Detonating
1000 kg of TNT
(227.134 g/mol) yields only 64% of such a ton:
C7H5N3O6
® 6 CO
+ 5/2 H2
+ 3/2 N2
+ C + 608.8 kJ
The carbon (C) produced appears as black smoke.
Some residues may subsequently burn in air to give
more energy (393.51 kJ per mole of carbon,
241.826 kJ per mole of hydrogen gas,
282.98 kJ per mole of CO).
The total heat
of combustion of TNT is thus about 3305 kJ/mol, which
translates into 3½ of the above tons of energy
for 1000 kg of TNT (227.13 g/mol)...
What's wrong?
Well, to optimize the energy of the initial blast, an oxidizer
(ammonium nitrate = AN) must be added to TNT
to form a balanced high explosive, called amatol.
The optimal proportion for a given total weight is 78.7% AN and 21.3% TNT,
matching the stoichiometry of the following reaction.
(A slight excess of AN seems better for dynamic reasons,
so the usual mix is 80/20.)
2 C7H5N3O6
+ 21 NH4NO3
®
47 H2O + 14 CO2
+ 24 N2 + 9088.6 kJ
This yields 1.0174 tons of energy when 1000 kg of the mix are detonated,
which justifies quantitatively the term "ton of TNT "
commonly used for the above ton of energy,
although "ton of amatol" would have been more proper...
Other types of "tons" are used to measure energy in a more peaceful context:
Burning a ton of crude oil releases
about 10 times as much energy as exploding a ton of TNT/amatol.
On the other hand, the best grade of coal (anthracite) is supposed
to be about 30% less efficient than oil.
Burning pure carbon completely into carbon dioxide would
release about 393.51 kJ/mol, which is more than 7800 cal/g
(a mole of carbon is 12 g).
However, actual coal can be much less efficient;
see below.
This gave rise to two other "ton" units for measuring energy,
the ton oil equivalent (toe) and the ton coal equivalent (tce):
1 tce = 0.7 toe.
Both refer to metric tons (1000 kg) but, unlike the ton of TNT,
they are usually defined as round multiples of the IT calorie
(International Steam Table calorie
of exactly 4.1868 J instead of 4.184 J):
1 tce = 29 307 600 000 J
1 toe = 41 868 000 000 J
Natural gas is an important source of energy as well, so that the toe has also
been given the following equivalences in term of gas quantities,
using the different units of measurement preferred in various regions of the Globe
(these values are, unfortunately, slightly incompatible with each other
and with the above):
- USA : 42900 cubic feet (about 1214.8 cubic meters).
- Europe : 1270 cubic meters.
- Japan : 0.855 metric tons of LNG ("Liquefied Natural Gas").
Standard Calorific Values :
Coal = 7000 cal/g. Oil = 10000 cal/g.
Actual Calorific Values (CV):
[ NB: 1 cal/g = 1.8 Btu/lb ]
Brown coal = 2250 cal/g.
Firewood = 4300 cal/g (= 7740 Btu/lb).
Bituminous coal = 6000 cal/g.
Crude oil = 10800 cal/g. |
Now, the ton of refrigeration
or ton of cooling is a unit of power
(which can't be compared with any of the above units of energy).
It was first defined as the power released by a ton (2000 lb) of water when it freezes
in one day (86400 seconds) or, conversely,
the power absorbed by a ton of ice which melts in a day.
This would be about 3502.6 W (watts),
but the ton of cooling is now conventionally defined as
exactly 12000 Btu/h (about 3516.852842 W),
based on the rounded value of 144 Btu/lb for the latent heat of fusion of water.
In the United Sates,
air conditioning units
are now rated using the Btu of cooling, which is a unit of power simply
equal to a Btu per hour
(about 0.293 W, more precisely 0.2930710701722222...W).
The labeling of A/C units is in terms of thousands of Btu [per hour]
(typically: 024, 030, 036, 042, 048, or 060), but
betrays its origin in terms of tons of cooling
(2, 2½, 3, 3½, 4, or 5 tons of refrigeration).
The electrical energy fed to the motor of an A/C unit may allow the transfer of a greater
energy "uphill", from cold to hot.
The ratio of these two energies is called the coefficient of performance (COP),
which is normally much more than 100%.
This would be clear if refrigeration and electrical powers were both expressed
in the same units (W/W), but this fact is obscured in the US,
where the so-called EER (Energy Efficiency Ratio) is used instead:
An EER of 10 means 10 Btu/h/W,
(a COP of about 2.93 W/W, or 293%).
An EER of 15 is 439.6%.
Last, and probably least, we're told that the "ton" is also an informal
British unit of speed equal to 100 mph (160.9344 km/h or 44.704 m/s).
[Colloquially, in the UK, a ton can be 100 times
as large as any commonly understood unit.]
|
robster
(
"
and the numeral "1". When typing, a cursive
"
