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Appendix 2UnitsA physical quantity is expressed as the product of apure number and a unit. Physical laws which existbetween physical quantities are conveniently expressedin systems of consistent units such that the form of theequation is independent of the system of units chosen.

Four systems will be listed here, namely the SystemeInternational dUnitCs (SI); the centimetre, gram,second system (c.g.s.); the British absolute systembased on the foot, pound and second (f.p.s.); and theBritish Engineering system based on the foot, slug andsecond (f .s.s.

In all these systems F = ma and weight W = m g,where g is the gravitational field strength. A standardvalue of the field strength at the surface of the earth isgiven as 9.80665 N/kg (m/s2) or approximately 31.174ft/s2or pdUlb or IbUslug.

By definition,1 slug = 32.174 lb1 Ibf = 32.174 pdl

also 1 kgf = 9.80665 N

Note that these are exact relationships and do noConversion of British units to SI units is achievevary with location as does g.

using the following exact conversion factors:1 ft = 0.3048 m1 Ib = 0.45359237 kg

Using these values,f t

l p d l= l I b - = l k gS2= 1 kg 0.4536 m 0.3048 s-*= 0.1383 kg m sK2= 0.1383 N

1 Ibf = 1(z)dl = l ( g ) ( $ ) N= lx32.174x0.1383N= 4.448 N

Table A2.1Quantity I i $ f a n d s y m b o l

C.Q.S. f.p.s.MassLengthTimeAngleForceEnergyWorkPowerPressureMoment of forceMoment of inertiaVelocityAcceleration

f.s.s.kilogram, kg gram, g pound, Ib slugmetre, m centimetre, cm foot, ft foot, ftsecond, s second, s second, s second, sradian, rad radian, rad radian, rad radian, radnewton (kgm s-), N dyne poundal, pdl pound force, Ibf

I joule (mN),J erg foot poundal, ft pdl foot pound forcewatt (J s- ) ,W erg s-l ft pdl s- ft Ibf s-lpascal (N m-), Pa dyne cmP2 pdl ft-* 16f ft-N m dyne cm pdl ft Ibf ftkg m2 g cm2m s-l crn s-l ft s-1 ft s-m sP2 crn sP2 ft s-2 ft 5-2

Ib ft2 slug f t2

[l micron = 10-6 m, 1 litre = 10-3 rn3,1 tonne = 1O3 kg, 1 bar = 1O5Pa]

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250 Appendix 2: Units

Similarly for work and energy:1 t pdl = 0.042 J1 f t Ibf = 1.356 J

also 1 h.p. = 550 f t lbf s- l = 745.700 WOther useful conversion factors are

DensityPressure

1 IbK3 16.0185 kg m P 31 Ibf inp2= 6894.76 N rn- (Pa)1atmosphere (atm) = 1.01325x I d N m-Note that in the above calculations the symbol for the

unit is treated as if it were an ordinary algebraicquantity.Table A2.2

Factor Prefix Symbol1021091o610 3Id10lo-10-210-310-610-910-1210-151 o-l8

teragigsmegakilohectodecadecicentimillimicronanopic0femtoatto

TGMk

mCLnPfa

The use of the prefixes is illustrated by the following:O.OOOOO1 m = 1pm (micron)1OOOOOO N = 1MN (meganewton)

0.1 m = 100mmlo4 N = 10 kNId kg = 1 Mg (not 1kkg)

Other systems of units are still seen in which the useof a mass unit is avoided by writing m = (W/g)so thatF = ( W / g ) a . Alternatively the acceleration may beexpressed as multiples of g to give F = W ( a / g ) .

The use of variable units of force such as the poundweight (Ibwt) and the gram weight (gmwt) is nowmoribund and must be discouraged. For practicalpurposes a force equal to the weight of the unit masswill often find favour in elementary applications, so thekilogram force (kgf), or its close equivalent 1da N ,may continue to be used in non-scientific applications.

Occasionally one sees the use of systems involvingIbf, in, s; or kgf, cm, s. In these cases the corresponding

unit masses are 386 Ib and 981 kg respectively.When labelling the axes of graphs or writing th

headings for tables of values, the following scheme iunambiguous.

The approximate value for the density of steel (p) i7850 kg/m3, sop = 7850 kg m-3= 7.850x I d kg m P3= 7.850 Mg m V 3

It follows that= 7850kg m-3= 7.850lo3kg m-3

PMg m-3 = 7.850The practice of heading a list of numbers in the mannep lo3 kg m3 leaves a doubt as to whether the I d referto the physical quantity or to its unit of measurementHowever

Plo3 kg m3

for example, has no such ambiguity.ReferencesFor further information, the following booklets shouldbe consulted:1. British Standards Institution, BS 5555:1993 (incor

porating I S 0 IOOO), spec8cution fo r SI units andrecommendations fo r use of their multiples and ocertuin other units.The Symbol Committee of the Royal SocietyQuantities, units, and symbols2.