# Pound (force)

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Pound-force
Unit system English Engineering units, British Gravitational System
Symbol lbf
Unit conversions
1 lbf in ... ... is equal to ...
SI units    4.448222 N
CGS units    444,822.2 dyn
Absolute English System    32.17405 pdl

The pound-force (symbol: lbf[1], sometimes lbf,[2]) is a unit of force used in some systems of measurement including English Engineering units and the British Gravitational System.[3] Pound force should not be confused with foot-pound, a unit of energy, or pound-foot, a unit of torque, and may be written as "lbf⋅ft". They should not be confused with pound-mass (symbol: lb), often simply called pound, which is a unit of mass.

## Definitions

The pound-force is equal to the gravitational force exerted on a mass of one avoirdupois pound on the surface of Earth. Since the 18th century, the unit has been used in low-precision measurements, for which small changes in Earth's gravity (which varies from place to place by up to half a percent) can safely be neglected.[4]

The 20th century, however, brought the need for a more precise definition. A standardized value for acceleration due to gravity was therefore needed.

### Product of avoirdupois pound and standard gravity

The pound-force is the product of one avoirdupois pound (exactly 0.45359237 kg) and the standard acceleration due to gravity, 9.80665 m/s2 (about 32.174049 ft/s2).[5][6][7]

The standard values of acceleration of the standard gravitational field (gn) and the international avoirdupois pound (lb) result in a pound-force equal to 4.4482216152605 N:[8]

{\displaystyle {\begin{aligned}1\,{\text{lbf}}&=1\,{\text{lb}}\times g_{\text{n}}\\&=1\,{\text{lb}}\times 9.80665\,{\tfrac {\text{m}}{{\text{s}}^{2}}}/0.3048\,{\tfrac {\text{m}}{\text{ft}}}\\&\approx 1\,{\text{lb}}\times 32.174049\,\mathrm {\tfrac {ft}{s^{2}}} \\&\approx 32.174049\,\mathrm {\tfrac {ft{\cdot }lb}{s^{2}}} \\1\,{\text{lbf}}&=1\,{\text{lb}}\times 0.45359237\,{\tfrac {\text{kg}}{\text{lb}}}\times g_{\text{n}}\\&=0.45359237\,{\text{kg}}\times 9.80665\,{\tfrac {\text{m}}{{\text{s}}^{2}}}\\&=4.4482216152605\,{\text{N}}\end{aligned}}}

This definition can be rephrased in terms of the slug. A slug has a mass of 32.174049 lb. A pound-force is the amount of force required to accelerate a slug at a rate of 1 ft/s2, so:

{\displaystyle {\begin{aligned}1\,{\text{lbf}}&=1\,{\text{slug}}\times 1\,{\tfrac {\text{ft}}{{\text{s}}^{2}}}\\&=1\,{\tfrac {{\text{slug}}\cdot {\text{ft}}}{{\text{s}}^{2}}}\end{aligned}}}

## Conversion to other units

Units of force
newton
(SI unit)
dyne kilogram-force,
kilopond
pound-force poundal
1 N ≡ 1 kg⋅m/s2 = 105 dyn ≈ 0.10197 kp ≈ 0.22481 lbf ≈ 7.2330 pdl
1 dyn = 10−5 N ≡ 1 g⋅cm/s2 ≈ 1.0197 × 10−6 kp ≈ 2.2481 × 10−6 lbf ≈ 7.2330 × 10−5 pdl
1 kp = 9.80665 N = 980665 dyn gn ⋅ (1 kg) ≈ 2.2046 lbf ≈ 70.932 pdl
1 lbf ≈ 4.448222 N ≈ 444822 dyn ≈ 0.45359 kp gn ⋅ (1 lb) ≈ 32.174 pdl
1 pdl ≈ 0.138255 N ≈ 13825 dyn ≈ 0.014098 kp ≈ 0.031081 lbf ≡ 1 lb⋅ft/s2
The value of gn as used in the official definition of the kilogram-force is used here for all gravitational units.

## Foot–pound–second (FPS) systems of units

In some contexts, the term "pound" is used almost exclusively to refer to the unit of force and not the unit of mass. In those applications, the preferred unit of mass is the slug, i.e. lbf⋅s2/ft. In other contexts, the unit "pound" refers to a unit of mass. The international standard symbol for the pound as a unit of mass is lb.[9]

Three approaches to units of mass and force or weight[10][11]
Base Force Weight Mass
2nd law of motion m = F/a F = Wa/g F = ma
System BG GM EE M AE CGS MTS SI
Acceleration (a) ft/s2 m/s2 ft/s2 m/s2 ft/s2 gal m/s2 m/s2
Mass (m) slug hyl pound-mass kilogram pound gram tonne kilogram
Force (F),
weight (W)
pound kilopond pound-force kilopond poundal dyne sthène newton
Pressure (p) pound per square inch technical atmosphere pound-force per square inch atmosphere poundal per square foot barye pieze pascal

In the "engineering" systems (middle column), the weight of the mass unit (pound-mass) on Earth's surface is approximately equal to the force unit (pound-force). This is convenient because one pound mass exerts one pound force due to gravity. Note, however, unlike the other systems the force unit is not equal to the mass unit multiplied by the acceleration unit[12]—the use of Newton's Second Law, F = ma, requires another factor, gc, usually taken to be 32.174049 (lb⋅ft)/(lbf⋅s2). "Absolute" systems are coherent systems of units: by using the slug as the unit of mass, the "gravitational" FPS system (left column) avoids the need for such a constant. The SI is an "absolute" metric system with kilogram and meter as base units.

## Notes

1. ^ IEEE Standard Letter Symbols for Units of Measurement (SI Units, Customary Inch-Pound Units, and Certain Other Units), IEEE Std 260.1™-2004 (Revision of IEEE Std 260.1-1993)
2. ^ Fletcher, Leroy S.; Shoup, Terry E. (1978), Introduction to Engineering, Prentice-Hall, ISBN 978-0135018583, LCCN 77024142.:257
3. ^ "Mass and Weight". engineeringtoolbox.com.
4. ^ Acceleration due to gravity varies over the surface of the Earth, generally increasing from about 9.78 m/s2 (32.1 ft/s2) at the equator to about 9.83 m/s2 (32.3 ft/s2) at the poles.
5. ^ BS 350 : Part 1: 1974 Conversion factors and tables, Part 1. Basis of tables. Conversion factors. British Standards Institution. 1974. p. 43.
6. ^ In 1901 the third CGPM declared (second resolution) that:

The value adopted in the International Service of Weights and Measures for the standard acceleration due to Earth's gravity is 980.665 cm/s2, value already stated in the laws of some countries.

This value was the conventional reference for calculating the kilogram-force, a unit of force whose use has been deprecated since the introduction of SI.

7. ^ Barry N. Taylor, Guide for the Use of the International System of Units (SI), 1995, NIST Special Publication 811, Appendix B note 24
8. ^ The international avoirdupois pound is defined to be exactly 0.45359237 kg.
9. ^ IEEE Std 260.1™-2004, IEEE Standard Letter Symbols for Units of Measurement (SI Units, Customary Inch-Pound Units, and Certain Other Units)
10. ^ Comings, E. W. (1940). "English Engineering Units and Their Dimensions". Industrial & Engineering Chemistry. 32 (7): 984–987. doi:10.1021/ie50367a028.
11. ^ Klinkenberg, Adrian (1969). "The American Engineering System of Units and Its Dimensional Constant gc". Industrial & Engineering Chemistry. 61 (4): 53–59. doi:10.1021/ie50712a010.
12. ^ The acceleration unit is the distance unit divided by the time unit squared.

## References

• Obert, Edward F., “THERMODYNAMICS”, D.J. Leggett Book Company Inc., New York 1948; Chapter I, Survey of Dimensions and Units, pages 1-24.

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