Pound (force)

Template:Two other uses

The pound or pound-force (symbol: lb, lb_f, or lbf) is a unit of force used in some systems of measurement including English Engineering units and the British Gravitational System.^[1] Pounds of force are not to be confused with foot-pounds or pounds-feet, which are a unit of torque, and may be written as "lb·ft" or "lb_f·ft".

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.^[2]

The 20th century, however, brought the need for a more precise definition. A standardized value for acceleration due to gravity was therefore needed. Today, in accordance with the General Conference on Weights and Measures, standard gravity is usually taken to be 9.80665 m/s² (about 32.174 049 ft/s²).^[3]^[4]

The acceleration of the standard gravitational field (g_n) and the international avoirdupois pound (lbm) define the pound-force as:^[5]

{\begin{aligned}1\,{\text{lbf}}&=1\,{\text{lbm}}\times g_{\text{n}}\\&=1\,{\text{lbm}}\times 9.80665\,{\tfrac {\text{m}}{{\text{s}}^{2}}}/0.3048\,{\tfrac {\text{m}}{\text{ft}}}\\&\approx 1\,{\text{lbm}}\times 32.174049\,\mathrm {\tfrac {ft}{s^{2}}} \\&=32.174049\,\mathrm {\tfrac {ft{\cdot }lbm}{s^{2}}} \\1\,{\text{lbf}}&=1\,{\text{lbm}}\times 0.45359237\,{\tfrac {\text{kg}}{\text{lbm}}}\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 lbm. A pound-force is the amount of force required to accelerate a slug at a rate of 1 ft/s², so:

{\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
v t e	newton	dyne	kilogram-force, kilopond	pound-force	poundal
1 N	≡ 1 kg⋅m/s²	= 10⁵ dyn	≈ 0.10197 kp	≈ 0.22481 lb_F	≈ 7.2330 pdl
1 dyn	= 10⁻⁵ N	≡ 1 g⋅cm/s²	≈ 1.0197×10⁻⁶ kp	≈ 2.2481×10⁻⁶ lb_F	≈ 7.2330×10⁻⁵ pdl
1 kp	= 9.80665 N	= 980665 dyn	≡ g_n × 1 kg	≈ 2.2046 lb_F	≈ 70.932 pdl
1 lb_F	≈ 4.448222 N	≈ 444822 dyn	≈ 0.45359 kp	≡ g_n × 1 lb	≈ 32.174 pdl
1 pdl	≈ 0.138255 N	≈ 13825 dyn	≈ 0.014098 kp	≈ 0.031081 lb_F	≡ 1 lb⋅ft/s²
The value of g_n (9.80665 m/s²) 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. lb_f·s²/ft. In other contexts, the unit "pound" refers to a unit of mass. In circumstances where there may otherwise be ambiguity, the symbols "lb_f" and "lb_m" and the terms "pounds-force" and "pounds-mass" can be used to distinguish.

Three approaches to units of mass and force or weight^[6]^[7]
v t e Base	Force		Weight		Mass
2nd law of motion	$m = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠F/a⁠$		$F = ⁠ W \cdot a / g ⁠$		$F = m \cdot a$
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	standard atmosphere	poundal per square foot	barye	pieze	pascal

In the gravitational systems, 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^[8]—the use of Newton's Second Law, F = m·a, requires another factor, g_c, usually taken to be 32.174049 lb_m·ft/lb_f·s² = 32.174049 lb_m/slug. "Absolute" systems are coherent systems of units: by using the slug as the unit of mass, the "engineering" FPS system avoids the need for such a constant. The SI is an "absolute" metric system with kilogram and meter as base units.

Notes and references

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

^ The Engineering ToolBox
^ 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.
^ 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.
^ Barry N. Taylor, Guide for the Use of the International System of Units (SI), 1995, NIST Special Publication 811, Appendix B note 24
^ The international avoirdupois pound is defined to be exactly 0.45359237 kg.
^ Comings, E. W. (1940). "English Engineering Units and Their Dimensions". Industrial & Engineering Chemistry. 32 (7): 984–987. doi:10.1021/ie50367a028.
^ Klinkenberg, Adrian (1969). "The American Engineering System of Units and Its Dimensional Constant g_c". Industrial & Engineering Chemistry. 61 (4): 53–59. doi:10.1021/ie50712a010.
^ The acceleration unit is the distance unit divided by the time unit squared.

[1] The Engineering ToolBox

[2] 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.

[3] 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.

[4] Barry N. Taylor, Guide for the Use of the International System of Units (SI), 1995, NIST Special Publication 811, Appendix B note 24

[5] The international avoirdupois pound is defined to be exactly 0.45359237 kg.

[6] Comings, E. W. (1940). "English Engineering Units and Their Dimensions". Industrial & Engineering Chemistry. 32 (7): 984–987. doi:10.1021/ie50367a028.

[7] Klinkenberg, Adrian (1969). "The American Engineering System of Units and Its Dimensional Constant g_c". Industrial & Engineering Chemistry. 61 (4): 53–59. doi:10.1021/ie50712a010.

[8] The acceleration unit is the distance unit divided by the time unit squared.

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Definitions

Conversion to other units

Foot-Pound-Second (FPS) systems of units

See also

Notes and references