PAUL BIRCH

INCH POUND TRICE UNITS

The inch-pound-trice system of units is a modified and updated scientific version of traditional British or Imperial weights and measures. It was invented as an educational tool or as a possible successor to the metric Système International by Leonard Cottrell and Andrew Usher. I have added some units and symbols. Unlike the foot-pound-poundal system currently used in engineering, with its somewhat awkward relationship between its basic units, the inch-pound-trice system employs a new unit of time designed to make the value of the acceleration due to gravity at the Earth's surface come out numerically equal to one, thus allowing the pound weight to be directly related to the pound mass. The disadvantage is obviously the requirement for a new unit of time, called the trice, which turns out to be approximately one twentieth of a second.

There are a number of possible versions of the inch-pound-trice system, depending upon the exact values given to the units. The version described here has exactly 19.65 trice per second and an inch and pound essentially unchanged from their present values of 25.4 mm and 0.45359237 kg; the speed of light is then 600,654,080 inches per trice. Another possibility would be to have exactly twenty trice per second, a "metric" inch of a fortieth of a metre and a standard gravity of exactly ten metres per second; a further possibility would be to adjust the inch a little more to make the speed of light come out at exactly 600,000,000 inches per trice. Other versions would adjust the trice to second ratio to tweak the speed of light to the same round number. However, my tentative opinion is that this would probably be a mistake. It is debatable whether forcing the fundamental constants to exact round numbers (quite different from suppressing redundant constants by setting them equal to unity) is on balance the best solution. In any event I do not personally see the inch-pound-trice system as being a genuine contender to replace SI or other metric systems; I see its principal application as being in the field of education. For another take on the idea, and some sample problems, go to Leonard Cottrell's site http://www.planck.com/poundinch.htm.


Units:

Here's a set of possible unit names, with their abbreviations and metric equivalents. The seven basic units — inch, trice, pound, rankine, que, radian and beaver — are given in bold. Exact, or very nearly exact, metric equivalents are given in red bold:

Length: Inch, in, i (i) (25.4mm)
Time : Trice, tr, t (t) (1/19.65 second)
Acceleration: Ax, a (i/tt) (9.8075115 m/s/s )
Mass : Pound, lb, l (l) (0.45359234 kg)
Force: Poundax, lbf, la (li/tt) (4.4486121 N)
Temperature: Rankine, R (not °R) (5/9 C° or K)
Frequency: Pert, p (cycles per trice, c/t) (19.65 Hz)

Multiples: K,M,G,T,(P,E,Z,Y) (×1E3,6,9,12,15,18,21,24)
Submultiples: m,u,n,p,f,(a,z,y) (÷1E3,6,9,12,15,18,21,24)

Energy: En, E (inch poundax, ila, lii/tt) (0.11299475 J)
Power : Ent, E/t (inch poundax per trice, ila/t, lii/ttt) (2.2203468 W)
Specific energy: Enl, E/l (inch poundax per pound, ii/tt) (0.24911079 J/kg)
Specific power : Entl, E/lt (inch poundax per pound per trice, ii/ttt) (4.8950271 W/kg)
Pressure: Psi, Y (poundax per square inch, la/ii, l/itt) (6895.3625 N/m/m)

Charge: Que, q ((lii/t)1/2, q) (3.9069403 millicoulomb)
Current: Cute, U (lii/qtt, q/t) (76.771377 mA)
Potential (voltage): Pot, P (lii/qtt, q/t) (28.921545 V)
Resistance: Rex, X (lii/qqt, -) (376.723 ohm)
Inductance: Dux, D (lii/qq, t) (19.171655 H)
Capacitance: Cap, C (qqtt/lii, t) (0.13508754 mF)
Field strength (electric): Fel, F (li/qtt, q/it) (1138.6435 V/m)
Field strength (magnetic): Mag, M (li/qtt, q/it) (3.7981055 µT)

Angle: Radian, rad, r (1 radian)
Solid angle: (square radian, rr) (not steradian, sr) (1 sterad)
Angular velocity: (radian per trice, r/t) (19.65 rad/s)
Angular acceleration: (radian per square trice, r/tt) (386.1225 rad/s/s)

Amount of substance: Beaver, b (0.273161E27 particles) (453.59234 mole)
Concentration (1): (beaver/cubic inch, b/iii) (27,679.903 mole/litre)
Concentration (2): (beaver/pound, b/l) (1000 mole/kg)
Concentration (3): (pound/pound, l/l) ( 1 kg/kg)

Amount of radiation : (beaver, b, 0.273161E27 particles) (453.59234 moles of particles)
Amount of ionisation: (beaver, b, 0.273161E27 ion pairs) (453.59234 moles of ion pairs)
Ionisation rate (1): Bet, b/t (beaver per trice, b/t) (5.367614E27 ion-pairs/s)
Ionising dose : Bel, b/l (beaver per pound, b/l) (0.169665E18 roentgens)
Ionising dose rate: Belt, b/lt (beaver per pound per trice, b/lt) (0.374048E18 roentgens/kg)
Radioactivity (1): (bet, beaver per trice, b/t) (5.367614E27 Bq = 0.1450706E18 curie)
Specific radioactivity (1): (belt, beaver per pound per trice, b/lt) (11.833563E27 Bq/kg = 0.3198260E18 curie/kg)
Radiation flux (1): (beaver per square inch per trice, b/tii) (8.319818E30 particles/m/m/s)

Radioactivity (2): Dit, d/t (disintegrations per trice, d/t) (19.65 Bq)
Specific radioactivity (2): Dilt, d/lt (disintegrations per pound per trice, d/lt) (43.320837 Bq/kg)
Ionisation rate (2): (dit, disintegrations per trice, d/t) (19.65 ion-pairs/s)
Specific ionisation: (dil, disintegrations per pound, d/l) (2.2046228 ion-pairs/kg)
Specific ionisation rate: (dilt, disintegrations per pound per trice, d/lt) (43.320837 ion-pairs/kg/s)

Absorbed radiation: (En, E, lii/tt) (0.11299475 J)
Absorbed radiation rate: (Ent, E/t, lii/ttt) (2.2203468 W)
Absorbed radiation dose: (Enl, E/l, ii/tt) (0.2491108 J/kg = 0.2491108 Gy = 24.91108 Rad)
Absorbed radiation dose rate: (Entl, E/lt, ii/ttt) (4.8950271 W/kg)


Notes:

1) The conventional two or three letter abbreviations (in, tr, lb, lbf, rad) are deprecated, but may be employed for clarity when individual units are used in isolation
2) The symbol for micro becomes the english letter u; the greek µ may also be used.
3) The symbol for kilo becomes upper case K.
4) Use of the higher multiples and submultiples is generally deprecated (use full engineering notation).
5) Symbols may be freely concatenated and doubled or tripled to indicate squares or cubes, etcetera. However, it is preferable to use only a single multiple or submultiple at a time; otherwise care must be taken to avoid ambiguity, in particular with E (exa or en), p (pica or pert) and a (atto or ax). The symbol a for i/tt should not be placed at the beginning unless standing alone.
6) The pound-mass or libra is a basic unit here given the name pound and symbol l. The pound-force is a derived unit here given the name poundax and symbol la to avoid confusion with the mass unit. It is the product of the mass unit, the pound, with the unit of acceleration, which thus receives the name ax and symbol a.
7) The names and symbols for frequency, energy, specific energy, power, specific power and pressure are strictly redundant, but are included for convenience.
8) The unit of pressure should be pronounced like the greek letter psi, not spelled out.
9) The units for electromagnetism are symmetrical units, and, dimensionally, qq=lii/t. Thus the units of current and voltage are the same (q/t), as are those for inductance and capacitance (t), and for electric and magnetic fields (q/it). Resistance is dimensionless. However, distinct supplementary names and symbols have been provided for convenience.
10) The full names cap and fel may be used instead of the symbols C and F whenever confusion with coulombs and farads is believed likely.
11) No special units for light, corresponding to the candela, lumen or lux, have been provided, being judged unnecessary. Derived units like E/t, E/tii, E/trr, E/tiirr, E/tiirrp will do for any sort of electromagnetic radiation, including visible light.
12) One beaver is the amount of substance in which there are as many elementary entities as there are atoms in 12 pound of carbon-12, that is, 0.273161E27 particles. This makes one beaver per cubic inch 27,679.903 mole/litre, which is impossibly big. However, one beaver per pound is 1000 mole/litre (for aqueous solutions of density 1000g/litre); dividing by the molecular weight of the species in question gives us the fractional concentration by mass. A one millibeaver solution (1mb/l) is thus the same as a one molar solution (1 mole/litre).
13) The units of radioactivity and ionisation based on the beaver are very large. 1 bet = 0.145E18 curie, which is a very large unit. The attobet, ab/t (.145 curie) and attobelt, ab/lt (.320 curie/kg) may be more practicable and could be used if desired with concatenated multiples or submultiples - such as 1 Kab/t (145 curie) and 1 uab/lt (.320 microcurie/kg). The ionising dose of one beaver per pound is 0.170E18 roentgens, which is again huge, so the attobeaver per pound or attobel (.170 roentgen) would usually be more practical.
14) The name bel (beaver per pound, b/l) is deprecated except when used small submultiples, to avoid possible confusion with the logarithmic bel and decibel, dB.
15) The natural life of a pure radioactive substance (half-life/ln2) is given by the reciprocal of the product of its specific radioactivity (in belts) and its molecular weight.
16) If the beaver is an inconveniently large basis for units for radioactivity, the alternatives based upon the dit (disintegration per trice) are very small, and in practice the megadit, Md/t (19.65 Rutherford or 19.65 MBq), might be more useful. Note that ionisation is the disintegration of an atom into an ion pair, so the unit name remains appropriate.
17) No special units have been defined for absorbed radiation dose, since derived units like E, E/t, E/l and E/lt will suffice. Note that the SI unit is the Gray, corresponding to an energy absorption of 1 J/kg, which is equal to 100 Rad (or when multiplied by the "relative biological effectiveness" 100 rem) and approximately equal to 87 roentgens.


Definitions:

Time: One trice is the duration of 467,818,411 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.

Length: One inch is the distance travelled by light in vacuo in 1/600654080 of a trice.

Mass: One pound is the mass of a collection of photons with frequency summing to 3.13104E48 cycles per trice.

Charge: One que is that charge which placed one inch from an identical charge in vacuo experiences a force of 600654080/4pi pound inch per trice squared (47,798,533 la).

Amount: One beaver is the amount of substance in which there are as many elementary entities as there are atoms in 12 pound of carbon-12 (so 1 b = 0.273161E27 particles).

Temperature: One rankine is 1/491.688 of the thermodynamic temperature of the triple point of pure water (so 1 R = 1 Fº and 32 ºF = 491.706 R).

Angle: One radian is the angle subtended by an arc of a circle equal to its radius.


Electromagnetism and Maxwell's Equations:

First some fundamental constants:

Electron charge e = 41.008868E-18 q
Speed of light c = 600654080 i/t
Planck's constant h = 0.115228590E-30 Et (lii/t)
Fine structure constant alpha = 1/137.0360
Impedance of free space z0 = 2 alpha.h/e2 = 1
Permittivity of free space epsilon-nought = 1/z0c = 1/c = 1.66485180E-09 C/i (t/i)
Permeability of free space mu-nought = z0/c = 1/c = 1.66485180E-09 D/i (t/i)

In these symmetric rationalised units the equations of electromagnetism take the following forms:

The electric field from a charge is E = qcr/(4pi.r3)
The magnetic field from a current element is dB = di2/(4pi.r2)
The force between charges is F = q1q2cr/(4pi.r3)
The force between currents is d2F = di1×di2/(4pi.cr2)
The Lorentz equation is F = q(E + (v/c)×B)

Maxwell's equations in free space, where ro is the charge density and J the current density, are then:

div B = 0
div E = c.ro
curl B = d(E/c)/dt + J
curl E = -d(B/c)/dt

In a medium of permittivity epsilon and permeability mu, Maxwell's equations become:

div B = 0
div E = (c.ro)/epsilon
curl B = mu(epsilon.d(E/c)/dt + J)
curl E = -d(B/c)/dt

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© Paul Birch, 1st July 2002.

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